Paradigms in Mathematics, Physics, and Biology - their Philosophical Roots

Paradigms in Mathematics, Physics, and Biology: their Philosophical Roots

DFM Strauss (2001)

© TEKSKOR BK P.O. Box 28378 Bloemfontein 9300 2001 ISBN 0-620-31329-3

Preface The Renaissance optimism of the 14th and 15th centuries soon witnessed the rise and successes of the modern natural sciences. Galileo claimed that the language of nature is written in mathematical symbols and Alexander Pope assessed the significance of Newton's Principia (1686) with his well-known appreciation: “Nature and Nature's laws hid in night: God said: Let Newton be! and all was light.” The belief that the mathematical natural sciences will be able to understand and explain all of reality climaxed during the age of reason, the Enlightenment of the 18th century. This classical era of modernism – dominated by an unbridled trust in the capacities of human reason – eventually was challenged by the rise of historicism at the beginning of the 19th century and after the turn from “thought” to “language” by the end of the 19th century through the new emphasis on understanding (hermeneutics) also by the rise of contemporary postmodernism. However, recently Alan Sokal and Jean Bricmont exposed the unfounded way in which prominent postmodern thinkers abuse the mathematical natural sciences in their writings (Fashionable Nonsense: Postmodern Intellectuals’ Abuse of Science, Picador, New York, 1998; cf. the German edition: Eleganter Unsinn: Wie die Denker der Postmoderne die Wissenschaften mißbrauchen, C.H. Beck, München 1999). These authors did not develop the positive side of their critique by exploring the indubitable interaction between philosophy and the natural sciences. The aim of this book is precisely to look at the constructive role of decisive philosophical pre-suppositions in the natural sciences. Scientific reflection is embedded in theoretical thinking. The latter differentiates into identifying and distinguishing on the basis of similarities and differences – which, in the case of scholarly activities, are articulated in a unique language [see the sketch on page (iv)] . Yet the reality in which we live transcends the limits of logical thinking and lingual articulation even though we have to acknowledge that we do not have access to the world outside the domains of thought and language. Scholarly reflection is constantly involved in an analytical awareness of a given “more-than-logical” diversity which is always articulated by a language which is constantly subject to subtle alterations in meaning and interpretation. The complicated interaction of these and other even more central and direction-giving factors constitute the conflicting paradigms operative within the disciplines. Of course this recognition also at once explains the historicity of scientific endeavours – the fact that we have to recognize the inevitable historical underpinnings of human activities. It may therefore often be helpful also to pay (iii)

attention to the history of the natural sciences in order to come to a better understanding of the divergence between alternative theoretical paradigms within them. The claim that the natural sciences are free from philosophical pre-suppositions is itself indeed the effect of the influential philosophical orientation of positivism and neo-positivism. Against the background of developments within modern philosophy of science this work highlights diverging trends of thought within the disciplines of mathematics, physics and biology while focusing upon the inevitable philosophical distinctions entailed in these differences. Although many of the selected issues discussed delve into some of the frontiers of natural scientific reflection the reader with a secondary school acquaintance with these disciplines will be able to follow the argumentation expressed in the various chapters.

The Author (January 2001)

Identification = Synthesis i.e., the bringing together of the features united in a concept

Similarities

Analysis

Abstraction

Differences

Modal analogies - for example physical space/original space; social distance/spatial distance

Social distance shown in

Spatial distance

Entitary analogies - for example the elbow of my finger; the head of the mountain; the modal grid of reality

Ante- and retrocipations

Metaphors

(iv)

Similarity

= ANALOGY = Differences-evinced-in-the-similarities

Difference

Example Similarities-shown-in-the-differences

Contents

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (iii)

Chapter I Fundamental Questions in the Philosophy of Science An Initial Comment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 (Neo-)positivism and reactions to it. . . . . . . . . . . . . . . . . . . . . . . . 3 Background to neo-positivism . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Reaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Contemporary philosophy of science . . . . . . . . . . . . . . . . . . . . . . . 5 The relation between Popper, Kuhn and Sneed. . . . . . . . . . . . . . . . 6 The Dynamics of Theory Formation . . . . . . . . . . . . . . . . . . . . . 6 Kuhn's critics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Foundational problems and basic distinctions . . . . . . . . . . . . . . . . . . 8 The unique nature of science . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Non-distinctive characteristics . . . . . . . . . . . . . . . . . . . . . . . 11 The distinctive characteristic of scientific (theoretical) thought . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Is Philosophy a Science? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Philosophy and the Special Sciences . . . . . . . . . . . . . . . . . . . . . . 14 Philosophical Foundational Questions in the Special Sciences . . . . . . . . . 15

Chapter II Foundational Philosophical problems in Mathematics Introductory remarks . . . . . . . . . . . . . . . . . . . . . . . Definitions of mathematics . . . . . . . . . . . . . . . . . . . . The Infinite in Greek Thought. . . . . . . . . . . . . . . . . . . A few further contours from the history of the infinite . . . . . . Infinitesimals and the second foundational crisis of mathematics . Cantor and Aristotle . . . . . . . . . . . . . . . . . . . . . .

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(a) Aristotle's objections against the actual infinite . . . . . . . . . . . . . . . . 32 (b) Continuity in Aristotle and Cantor-Dedekind . . . . . . . . . . . . . . . . . 34 Non-denumerability: Cantor's Diagonal Proof. . . . . . . . . . . . . . . . . . . 36

Comment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The third foundational crisis in Mathematics . . . . . . . . . . . . . Divergence of opinion . . . . . . . . . . . . . . . . . . . . . . . . . Questioning completed infinitude . . . . . . . . . . . . . . . . . . . The influence of intuitionism on the approach of Dooyeweerd . . . . Brief systematic assessment of the relationship between the potential and the actual infinite . . . . . . . . . . . . . . . . . . . . . . . . . (v)

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Chapter III Basic questions in Physics The Prejudice against Prejudices . . . . . . . . . . . Discrepancy between philosophers of science and the practitioners of science . . . . . . . . . . . . . . . . Property terms – the Achilles’ heel of positivism . . . The measurement of time and modal time orders . . . Time in the aspects of number and space . . . . . The kinematical and the physical time order . . . The uniqueness of Constancy and Dynamics . . . . . Perpetual motion . . . . . . . . . . . . . . . . . Closer reflections on constancy and change. . . .

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The core of Einstein’s theory of relativity . . . . . . . . . . . . . . . . . . . . . 62 An alternative formulation of the first main law of thermodynamics . . . . . . . 63 The theory of relativity and relativism . . . . . . . . . . . . . . . . . . . . . . . 63

Determinism and indeterminism . . . . . . . . . . . . . . . . . Order and delimitation in physics . . . . . . . . . . . . . . . . . . . The finite and limited cosmos in Greek culture . . . . . . . . . . Are there inaccessible limits in the natural sciences? . . . . . . . The unlimited but finite universe in Eintein’s theory of relativity Complementarity – limits to experimentation . . . . . . . . . . . Entities with a physical qualification . . . . . . . . . . . . . . . The unity and identity of an entity. . . . . . . . . . . . . . . . . . . Physically qualified entities . . . . . . . . . . . . . . . . . . . . . . The wave particle duality and the idea of the typical totality structure of an entity. . . . . . . . . . . . . . . . . . . . . . . . Physically qualified structural interlacement . . . . . . . . . . .

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Paradigms in Mathematics, Physics, and Biology: Their Philosophical Roots Chapter IV The Mosaic of philosophical stances in modern biology Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Biotically qualified entities . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 What guarantees the identity of living things? . . . . . . . . . . . . . . . . . 85 The origin of living things – a biological boundary question . . . . . . . . . . 87 Are viruses a transitional form between material and living entities? . . . . . . 88 Nominalist structural understanding in modern biological literature . . . . . . 90 Structureless continuity versus structural discontinuity . . . . . . . . . . . . . 92 Continuity of descent? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 The structure of a nuclear living cell . . . . . . . . . . . . . . . . . . . . . . 97 Physical-chemical constituents in the living cell . . . . . . . . . . . . . . . . 98 Organelles – the different organs in the cell . . . . . . . . . . . . . . . . . . 100 The quest for a basic denominator . . . . . . . . . . . . . . . . . . . . . . . 101 Conflicting views despite “the same facts”! . . . . . . . . . . . . . . . . . . 103 (vi)

Neo-Darwinism . . . . . . . . . . . . . . . . . . . . . . . . . Vitalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Holism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Emergence evolutionism . . . . . . . . . . . . . . . . . . . . . Pan-psychism. . . . . . . . . . . . . . . . . . . . . . . . . . . Metabolism as first level of freedom . . . . . . . . . . . . . . . A new mechanistic approach . . . . . . . . . . . . . . . . . . . Structural diversity founds structureless fantasies . . . . . . . . Structural dimensions of the cell – an enkaptic structural whole

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Continuity or discontinuity between the various levels? . . . . . . . . Is the fossil-record conclusive? . . . . . . . . . . . . . . . . . . . . Is there anything distinctive to human tools? . . . . . . . . . . . . . Do animals share the dimension of (human) logicality? . . . . . . . . The human being as “Homo symbolicus”?. . . . . . . . . . . . . . . The anatomical conditions of human speech . . . . . . . . . . . . . . Do human beings have ‘speech-organs’? . . . . . . . . . . . . . . . Does human experience of the world differ from that of the animals? . The unspecialized traits of the human body . . . . . . . . . . . . . . Is the human being to be seen as a deficient creature? . . . . . . . . . The ontogenetic uniqueness of being human. . . . . . . . . . . . . . Human freedom – the predominantly negative approach of modern philosophy . . . . . . . . . . . . . . . . . . . . . . . . . Autonomous freedom versus natural causality . . . . . . . . . . . . . Bridging the abyss teleologically. . . . . . . . . . . . . . . . . . . . Entelechie negatively described: the influence of Hans Driesch . . . . Reinforced dialectics: Existentialism and Existential Phenomenology Freedom at the molecular level. . . . . . . . . . . . . . . . . . . . . The rejection of structural conditions: nominalism . . . . . . . . . . The common root of diverging trends in modern philosophy . . . . . Human freedom: a subjective response to normative conditions. . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter V Remarks about the mystery of being human

Consulted Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Index of Technical Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Index of Persons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

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Paradigms in Mathematics, Physics, and Biology: Their Philosophical Roots Chapter I Fundamental Questions in the Philosophy of Science An Initial Comment Since the concept natural science is usually identified with physics it is proper to begin our train of thought with the words of a well known physicist of this century. Carl Friedrich von Weizsäcker once commented on the presuppositions of modern natural scientific thought, saying that “it is an empirical fact that virtually all leading physicists of our time philosophize” (1972:42). This is no brand new revelation, although it is carefully hidden behind the mask which a particular philosophical approach forced upon physics at the beginning of the 20th century. Positivism – also later known as neo-positivism – convinced natural scientists that all good science must be practised without any prejudices or presuppositions whatsoever, and that good science could only be practiced by always referring to “empirical phenomena” when making scientific statements. We can also thank this philosophical trend for a particular elaboration of the opposition between thought and experience. Einstein, for instance, was strongly influenced by Ernst Mach – a leading natural scientist who supported positivism around the end of the 19th century – but later developed towards giving priority to theoretical thought. In his Preface to the English translation of Werner Heisenberg's work on the relationship between physics and philosophy, Northrop comments on Einstein's emphasis that “...the physical scientist only arrives at his theory by speculative means. The deduction in his method runs not from facts to the assumptions of the theory but from the assumed theory to the facts and the experimental data” (cf. Heisenberg, 1958:3-4). In his Herbert Spencer lecture at Oxford on 10 June 1933 Einstein himself categorically stated that no bridge could be erected between pure logical thought and our experience of reality: “Pure logical thinking can give us no knowledge whatsoever of the world of experience; all knowledge about reality begins with experience and terminates in it” (contained in Coley & Hall, 1980:144). This statement touches on the core of epistemology – i.e. the question how we obtain knowledge of reality. This question, however, is a typical 1

philosophical basic question for every conceivable particular scientific discipline. The answer we give to it can therefore not be divorced from particular philosophical presuppositions and orientations. The correctness of this statement coheres inter alia with the important influence exerted historically – and still today – by particular philosophical approaches. After all, the words quoted from Einstein truly gain depth when understood against the background of Immanuel Kant's “Kritik der reinen Vernunft” (Critique of Pure Reason) which was originally already written in 1781 (second impression 17872). Kant was convinced by his British predecessor, David Hume, that exact natural laws cannot be founded in “experience.” Kant could not, however, agree with Hume's conviction (strikingly similar to a facet of Einstein's statement above) that we obtain all knowledge from sensory perception or “experience.”1 Kant was impressed by the ability of human beings to mentally formulate the laws which the perceptible things in nature obey. He consequently focused on the (epistemological) question how such knowledge is possible. Kant was particularly impressed by the contribution of Galileo to the development of the modern natural sciences. Galileo's formulation of his law of inertia followed exactly the way of a pure thought experiment. In his famous 1638 treatise on “two new sciences” Galileo used the following thought experiment: if a body is put in motion on an indefinitely extended track, then this body would continue its motion infinitely, i.e. it would not discontinue its motion except if something exerts power on it (e.g. gravity or friction). From this Kant draws the following conclusion: if it is possible for Galileo to formulate a thought experiment out of the spontaneous subjectivity of his theoretical thought and to deduce a natural law from it – the kinematical law of inertia – this must mean that elements of knowledge are previously present in the human mind, which in the first place makes our knowledge of reality possible. Without going any deeper into this particular question, it is already clear that we can only really understand Einstein's statement if it is placed in the context of this foundational philosophical epistemological approach. Einstein's view of “experience” reaches back to the train of thought of Hume, while his view of the mind (thought) appears to be similar to that of Kant, while simultaneously differing from it since he assigns a more independent status to theoretical thought.2 Kant's so-called ‘Copernican’ revolution in epistemology, in ascribing the primacy no longer to the ‘object’ but to the (formal law-giving) subject, reinforced the notion of things within nature as ‘objects’. Someone inclined to de1 2

“To hate, to love, to think, to feel, to see, all this is nothing but to perceive.” Hume made this statement in his book: A Treatise of Human Nature, I,2,6. Although Einstein differs from Kant's view, he nevertheless did not want to be precise as far as his own relationship to Kant is concerned. In April 1922 the Société Française de Philosophie arranged a discussion about the significance of Einstein's theory of relativity for philosophy. Answering a question concerning his relationship with Kant Einstein merely stated that ‘each philosopher has his own Kant' and that he cannot infer from the question what Kant interpretation it presupposes.

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fend the neutrality of observation normally would be willing to accept as the most general observation-term the notion of an ‘object’: all the different things in nature are to be seen as ‘objects’. However, this observation-term in itself displays the tremendous subjectivistic assumption so deeply ingrained in our Western notion of science – as such causing the inability to appraise things in nature as genuine subjects, i.e. as being subjected to God's creational order (law-order) for their existence as material things, plants or animals. Although these things could be objectified by humans, this objectification pre-supposes their primary existence as subjects and not as objects! In this brief discussion of the philosophical foundations of the natural sciences we want to provide a brief survey of a few perspectives, using a few prominent approaches in particular natural sciences as examples. The intention is to simultaneously place the discussion in the context of recent developments in the newer philosophy of science which has been greatly stimulated in particular by Thomas Kuhn's The Structure of Scientific Revolutions (1962, 19702) – the focus of the rest of this introductory chapter. (Neo-)positivism and reactions to it (Neo-)positivism can be described as the philosophical idolization of the experimental method on the basis of sensory perception. The central concept of the Wiener Kreis, in the twenties and thirties of the 20th century, was the “verification principle.” As we briefly indicated above, the modern epistemological heritage draws a distinction between two original sources of knowledge: the mind and the senses (thought and experience – cf. Kant), on occasion amended with intuition as an ultra-sensory and super-rational organ of knowledge. Following Kant we could distinguish between transcendentalists and empiricists. The latter, particularly well-represented in the British tradition, would eventually develop into (neo-)positivism, which designed a scientific methodology which begins with particular sensory data/sense-impressions on the one hand and the logical construction of entities from these on the other. Thence the progression of empirical perception-hypothesizing-testing (verification)-theory formation (verified hypothesis), a methodical approach certainly not unknown to the “experimental” natural sciences. Background to neo-positivism Related to Kant we find the 19th century positivism of Ernst Mach which, on the grounds of empirical (i.e. sensory) perception, includes only mathematics and physics in the house of the sciences. This delimitation of science led to the position of Wittgenstein, the mathematician-engineer-philosopher, to the point of view that the limits of my language are the limits of my world (Tractatus 5.6.). According to Wittgenstein the task of philosophy is to delimit the controversial terrain of the natural sciences (= physics) (4.113) – and the totality of the natural sciences constitutes the totality of true propositions (4.11). That which transcends the propositions of physics (which is meaningful) and logic (the propositions of which are tautologies and therefore meaningless: 4.461) cannot be known or lingually expressed – it belongs to the 3

sphere of nonsense. The objection that the Tractatus itself would be a victim of such a delimitation of science (to logic and natural science) is obviated by Wittgenstein with his comment that his propositions serve an illuminating end: “anyone who understands me eventually recognizes them as nonsensical, when he has used them – as steps – to climb up beyond them. (He must, so to peak, throw away the ladder after he has climbed up it” (6.54)!

Reaction The famous philosopher of science of the 20th century, sir Karl Popper, reacted strongly against this attempt at delimitation by Wittgenstein. He investigates, for instance, the following sentence by Wittgenstein: “Philosophy is no theory, but an activity” (4.112). This sentence clearly does not belong to the totality of natural scientific propositions, and therefore also not to the totality of true propositions. On the other hand it is not a false proposition either, since if it was, then its negation would have to be true and therefore belong to the natural sciences. The only possibility would then be the mentioned conclusion of Wittgenstein (6.54): the sentence is nonsensical. Although Wittgenstein admits with this that the Tractatus is nonsensical, he declares in the final paragraph of the Preface that the truth of his notions appear to him unassailable and definitive. He is even of the opinion that he has on all cardinal points found the final solution to the problems. Popper reacts sharply to this: “This shows that we can communicate unassailably and definitely true thoughts by way of propositions which are admittedly nonsensical, and that we can solve problems finally by propounding nonsense.” The implication is that “[i]t means that all the metaphysical nonsense against which Bacon, Hume, Kant and Russell have fought for centuries may now comfortably settle down, and even frankly admit that it is nonsense. For we now have a new kind of nonsense at our disposal, nonsense that communicates thoughts whose truth is unassailable and definitive; in other words, deeply significant nonsense” (cf. 1968, as well as 1966:296ff)! Popper asks himself how one can oppose this position of Wittgenstein. Every possible objection against it is after all philosophical and therefore nonsense! According to Popper this is simply fortified dogmatism, since all that is required, is to delimit the concept sense (or: meaning) in an appropriately narrow way so as to rid oneself of all awkward questions by simply saying that you do not find them meaningful. Every reasoned objection to this conception of meaning is simply rejected as nonsensical: “Once enthroned, the dogma of meaning is for ever raised above the possibility of attack. It is unassailable and definitive.” This meaning-conception of Wittgenstein, with its included delimitation of science, is just as untenable as the “verification principle” of neo-positivism. The term “logical positivism” (or logical empiricism) was brought into being to refer to a group of philosophers, logicians and mathematicians who became known in Vienna as “der Wiener Kreis.” The movement originally centered around Moritz Schlick, with philosophically oriented members such as 4

Carnap, Neurath, Feigl, Waismann, Zilsel and Kraft, as well as natural scientific and mathematically oriented members like Frank, Menger, Gödel and Hahn. Carnap, Neurath and Hahn in 1929 published a manifesto entitled “Wissenschaftliche Weltauffassung, Der Wiener Kreis.” In this circle Wittgenstein's Tractatus was also discussed and from it (cf. 4.024) they borrowed their famous verification principle: the meaning of a statement lies in the manner by which it is verified. In his Language, Truth and Logic (1936) A.J. Ayer explains that factual assertions are subject to the following criterion of verification: a sentence is meaningful for any specific person if and only if that person which perceptions would lead him (under certain conditions) to accept the proposition as true or to reject it as false (1967:35). A closer analysis causes Ayer to distinguish between a strong and a weak sense of verification. A proposition is verifiable in the first sense, if and only if the truth thereof can be conclusively determined in experience. A proposition is verifiable in the latter sense if it is possible to render the experience probable (1967:36-38). Ayer fully realizes that general formulations of laws cannot be conclusively verified – in consequence he has to accept verification in the weak sense. In a later preface (1946) he is nonetheless of the opinion that there exists a class of empirical propositions which are conclusively verifiable. These are the basic propositions which refers exclusively to the content of a single experience and which can be identified as unique. Ayer is convinced that he has eliminated all metaphysics by means of this verification-criterion. Contemporary philosophy of science The newer theory of science of the past 40 years has realized, due to the influence of Popper, Toulmin, Polanyi (originally a chemist) and especially Thomas Kuhn (physicist) that even physics is inevitably gripped by a theoretical picture of reality (paradigm) and that it is possible to speak meaningfully of an ultimate commitment in every scientific activity – a central heart conviction out of which the scientist accounts for the deepest fundamental questions of his/her scientific practice. This realization came about partly because of the non-verifiability of the (neo-positivist) verification principle. As an alternative to (neo-)positivism Popper defends a critical rationalism, that is a rationalism which recognizes that faith in the rationality of reason is itself not rational: “we may describe it as an irrational faith in reason” (1966:321).1 This cre1

Stegmüller once remarked: “One should not delimit science in order to create room for faith. Much rather it is the case that one already has to believe something in order to be able to speak about knowledge and science as such” (“Man muss nicht das Wissen beseitigen, um den Glauben Platz zu machen. Vielmehr muss mann bereits etwas glauben, um überhaupt von Wissen und Wissenschaft reden zu können” New Introduction, Stegmüller, 1969:33); “somewhere an ultimate knowing must be given; without that we could not even start” (“Irgendein absolutes Wissen muß es geben; ohne dieses könnten wir überhaupt nicht beginnen” 1969:194); “We should already ‘dispose of’ an absolute evidence. i.e. we have to believe in it in advance, ...” (“Absolute Evidenz müssen wir schon ‘haben’, d.h. wir müssen an sie bereits glauben, ...” 1969:194); “in science believing is found, in religion one knows (or: pretends to

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ated room for the conviction that the range of science should not be narrowly reduced to the limits of (the methods of) mathematics, physics and logic (as the modern natural science-ideal claims) – it is as wide as all of creation. The relation between Popper, Kuhn and Sneed The current and dominant (even if implicit) positions of natural scientists and many philosophers with regard to the theory of science claims that the “empirical sciences” developed in a linear and continually progressive manner, that is that in these sciences we can speak of a linear accumulation of knowledge. Broadly seen this accumulative growth methodically gains access to more and better observatory and measuring instruments which give rise to the discovery of new facts. The depth dimension of this concerns the theoretical lines of connection which can be accessed when empirical regularities are replaced by regularities which can be mathematically formulated, and which are then imbedded in theoretical designs. This progression implies at the same time that one can get rid of the ‘unscientific elements’ in the scientific tradition. Stegmüller points out that it is exactly this image of the cumulative increase of knowledge which is false in principle according to Kuhn (cf. 1975, pp.484ff). The Dynamics of Theory Formation In his work on The Structure and Dynamics of Theories formation Stegmüller gives an elaboration of Kuhn's thought as reflected in the spirit of his critics (cf. 1976:135ff). His assumption, however, is that his sketch is an approximation of the impression which Kuhn's work would make in the mind of impartial critical readers. The fascinating historical backdrop which is relevant here, is provided elsewhere by Stegmüller when he compares Hume, Carnap, Popper and Kuhn. The classical rationalistic conviction that science can provide definitive, indubitable knowledge was already seriously doubted by David Hume, particularly with reference to “empirical knowledge.” It was already widely accepted that the experiential sciences would proceed inductively, without it being at all clear what exactly would characterise this method. Hume admits that the empirical sciences could only with aid from the inductive principle reach generalizations which bring into view the regularities which provide insight into future occurrences out of past experience. It is remarkable that according to Hume no rational foundation exists for this principle! Every attempt to provide such a foundation leads either to an infinite regression, or a logical circle. In consequence the empirical sciences follow the inductive method, while maintaining that this method is irrational, since the foundational inductive principle cannot be founded or legitimated. Using the concept of logical probability Carnap attempted to rescue the rational status of induction, which consequently implies for him that the experien-

know)” (“in der Wissenschaft wird geglaubt, in der Religion weiss man (oder: behauptet man, zu wissen)” 1969:212).

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tial sciences proceed both inductively and rationally.1 Popper, as is well known, reacted with sharp criticism to the supposedly inductive nature of natural scientific research. The discovery of scientific theories is and remains entirely speculative and their can be no foundation or legitimation by any means – at most a deductive method of testing. This means that theory formation proceeds rationally and non-inductively – against verification Popper's sets falsification.2 According to Stegmüller, what would appear to be unique and unprecedented in Kuhn's work, “is the fact that he appears to impute irrational behavior to the practitioners of the exact natural sciences (of all people!). And indeed he appears to impute it to both of the forms of the scientific practice distinguished by him. Anyone engaged in normal science is a narrow-minded dogmatist clinging uncritically to his theory. Those engaged in extraordinary research leading to scientific revolutions are religious fanatics under the spell of conversion, trying by all means of persuasion and propaganda to convert others to the new paradigm as revealed to themselves.” (1976, p.vii).

Not only do the natural scientists work in an irrational manner – according to the critics of Kuhn it would appear as if he is also a proponent of the non-inductive nature of the natural sciences. A comparison between the four mentioned figures provides the following picture: (1) Hume: the natural sciences proceed inductively and non-rationally; (2) Carnap: the natural sciences proceed inductively and rationally; (3) Popper: the natural sciences proceed non-inductively and rationally; (4) Kuhn: the natural sciences proceed non-inductively and non-rationally; (cf. Stegmüller, 1975:487-490). Kuhn's critics The first obvious matter emphasized by Kuhn is that no single process thus far exposed through historical research shows even the least similarity with Popper's doctrine of falsification. An experience standing in contradiction to a theory does not, according to Kuhn, indicate a shortcoming in the theory, since it only discredits the person who holds the particular theory! Of course Popper admits that his intention was solely to give an account of what Kuhn has called exceptional research and that he consequently neglected the phenomenon of normal science. The normative methodology designed by Popper, still means that he wants to overcome the nature of normal scientific practice (in Kuhn's sense) in permanent revolution. Stegmüller nonetheless points out that it would be mistaken to believe that the differences between Kuhn and his critics are limited to their evaluation of the 1

2

Keep in mind that the older inductivists were first of all directed towards what Hans Reichenbach called the context of discovery as distinct from the context of justification, whereas modern inductivists are focussed on the latter. Blaise Pascal (1623-1662) already defended a similar view. According to him a hypothesis could never be ‘verified’ by experience. Experience may however prove it false.

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nature of normal scientific practice. According to Kuhn it after all never occurs that a new theory emerges because an old theory could not give account of experiential data. The old theory is rather replaced immediately by a new theory, without the mediation of any experience (cf. Stegmüller, 1980/I:28). Reacting to these perspectives, and especially in reaction to the negative (relativistic and irrationalistic) picture which critics of Kuhn created of Kuhn's contributions, Stegmüller aimed rather to determine what Kuhn established as a competent theorist of science, and to process this material logically (1980/I:29). Although it would continue our train of thought beyond the purposes of the current context to go into the Stegmüller-Sneed modification of Kuhn (cf. Strauss, 1987), it is still meaningful to mention its central point. In the Stegmüller-Sneed modification of Kuhn attention is given to the problem of the immunity of scientific theories against falsification – with as a central concept the non-falsifiable structural core of a “theory.”1 Foundational problems and basic distinctions In distinction to Kuhn's emphasis on the nature of scientific revolutions, Holton gave emphasis to the persistent themes in science – with as central concept the role of a paradigm or disciplinary matrix linked to the idea of a Gestalt-switch. The foundational problem with regard to this difference of opinion has not only repeatedly played a role in the history of the natural sciences, but simultaneously has dominated the history of philosophy to a great extent and can still be found back today in practically all areas of scientific practice. At the beginning of this elaboration we briefly noted the thought experiment of Galileo with regard to the nature of a body in motion on a straight course. With this argumentation Galileo – albeit influenced by particular earlier thinkers from the transitional period from the middle ages to the modern age – abandoned the classical Aristotelian notion that a moving body must be kept in motion by some or other causal force. Galileo realized that motion must be acknowledged as an original given in its own right. The concept cause after all already presupposes the given nature of a continuously moving body – since we can only meaningfully consider changes in motion which might occur (whether decelerating or accelerating) on the grounds of this given. In both cases a cause is necessary – but not a cause of motion: rather a cause for the change in motion. In pure kinematics the concept change strictly speaking has no place. In a physical sense acceleration can never occur in a discontinuous way. A discontinuous acceleration would require an infinite physical force – a physical impossibility (cp. the remark by Janich, 1975:69). Plato was already confronted by the problem of continuity and change. In his famous theory of ideas he strove to gain knowledge of all things. Since he also 1

Newton's law of gravity is used to substantiate this argument – a physical law evincing modal universality. Galileo's law of inertia is another example of a universal modal law which could only be formulated on the basis of modal abstraction and not on the basis of ‘empirical experimentation’. Later on we shall explain that modal laws holds universally in an unspecified way, whereas typical (entitary) laws are only applicable to a limited class of entities.

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studied with a student of Heraclitus, namely Cratylus, he was confronted from the beginning with the doctrine that everything changes. Heraclitus famous maxim is that no-one can enter the same river twice. Plato asks if this approach does not suspend the possibility of knowledge: if everything constantly changes it would after all not be possible to know anything, since as soon as we attempt to approach knowledge the thing has already essentially changed and has therefore escaped the grasp of our knowledge. The only way out Plato could find was speculative. He assumes a static being – the so-called aujto; to ei\do" (essential being) – which is not subject to any change and which guarantees the knowability of every thing. In his dialogue Cratylus he states it in the following manner: if the aujto; to ei\do" of knowing changes into a different aujto; to ei\do" no knowledge (either subjectively or objectively) would be possible (440 a-b). Plato postulates a trans-sensory sphere in which these supposed eidè reside. This world of ideas can only be thought of mentally, it cannot be observed through sensory perception. Note that we find here an earlier expression of the opposition between thought and experience which we have already mentioned with regard to the manner in which Einstein gave priority to theoretical thought above (experimental) experiential data. Although we no longer find the speculative solution offered by Plato acceptable, we cannot deny that Plato built an extremely significant insight into his argument. He realized that it is not possible to determine any change except on the basis of durability or constancy. Stated differently: changes can only be determined on the basis of something relatively constant. Although we shall return to this insight with regard to Einstein's theory of relativity, we want at this stage to indicate a few further problems which have played a role in the history of philosophy. Time and again thinkers have been confronted with the question whether reality can be illuminated in terms of a single point of view, or whether a multiplicity of explanatory approaches should not be recognised. The earliest philosophical designs already attempted to find repose in a single explanatory perspective which is elevated to the foundational denominator in terms of which all of reality must be understood. The Pythagoreans, for instance, were of the opinion that everything is number. Since the mechanics of Galileo modern physics has been enamored by the (mechanistic) conviction that all physical phenomena can be described exhaustively in terms of mechanical movement (of mass points, whether charged or not). If the way of multiple explanatory approaches is chosen, the next problem would of course be to attempt to determine which various explanatory approaches can actually be distinguished and to trace what the mutual coherence and relation among these various possible approaches might be. The discipline of modern biology negatively illustrates the possibility of multiple explanatory approaches. We need only consider the following trends which have yielded prominent representatives: the mechanistic approach, physicalism (including neo-Darwinism), (neo-)vitalism, holism, organismic biology, emergence evolutionism and pan-psychism (whether monistic in the manner of Thailhard de Chardin or pluralistic in the manner of Rensch). 9

We could also refer to this as the problem of unity and diversity, or even unity in diversity. A third problem is closely linked to both the preceding questions, namely the question regarding the relationship between that which can be considered to be universal and that which can be considered to be individual. People who designate (“objective”) reality to universalia (universals) have historically been known as realists. Plato for instance confesses that the ideas of his world of ideas posses reality and universality independently of human knowing thought. Universality in this case functions in two senses: (i) it exists independently of the knowing human soul and (ii) it exerts an appeal on everything – wherever. Varieties of this realistic approach later received the label of platonism. In opposition to this platonic realism we traditionally encounter those who are of the opinion that no universality exists outside the knowing human soul. Only the human soul is able to abstract the immense multiplicity of things outside itself to the unity of particular concepts and/or words. The only form of universality acknowledged by this alternative is that of concepts or words in the human soul. The latter function as substitutes, mere nomina (names) for the things outside the soul. It is for this reason that this approach is known as nominalism. Nominalism not only denies the reality of universality outside the human soul, but also implicitly denies the order-side of the diversity within reality. At most it offers as an alternative the notion that the ordering instance in reality is human thought itself. Although these two isms – realism and nominalism – are virtually of the same age as philosophy itself, both are still alive and well in contemporary scientific thought. To bring their actuality to your attention it is enough merely to note that the dominant trend in 20th century mathematics is platonic while the dominant trend in contemporary biology is nominalistic! With this fleeting introduction of three fundamental problems of science – the problematic relationships between constancy and dynamics, unity and diversity, and universality and individuality, we have by no means exhausted the menu! We will however suffice with only pointing out that a further cardinal foundational problem is constituted by the relationship between laws and that which is factually subject to laws. Answers to the question of the status of laws commonly diverge in two directions: firstly the direction towards order in the variety of explanatory options which provide access to (theoretical) analysis and study of reality and secondly the direction towards the typical structure of particular groupings of entities. Behind this stands the quest for truth, which, according to the physicist Stafleu (1980) can differentiate in these two tracks when it is focused on the question of order or structure. One way to form a clearer understanding of this important distinction can be explored by beginning with a brief investigation into the nature of science – with the intention of determining what makes science unique, i.e. what distinguishes science from our everyday non-scientific thinking. 10

The unique nature of science To answer the question “what is science?” we need to identify the nature of science by noting those particular characteristics which distinguish science from everything which is not science. Identification and distinction (i.e. analysis) presupposes dealing with similarities and differences. Indicating what science is has two sides: the similarity-side and the difference-side. Science has characteristics which are not distinctive since in some regards science is similar to non-scientific activities. As an illustration of what is meant with identification and distinction, we can ask how we can distinguish between a material thing and a plant: (i) There are fundamental similarities between plants and physical things. A plant, for instance, has a characteristic mass – a feature shared with material things. Similarly both plants and material things have a certain spatial extension, a certain durability and a certain unity. (ii) Only when we pay attention to the fact that plants are alive do we come into touch with the difference-side in the comparison of matter and plants – with the distinctive characteristic of being-a-plant/“plantness.” A starting point would appear to be given in the observation that every scientific activity is a thought-activity. This characteristic still does not distinguish science from non-scientific activities, since someone who is not scientifically engaged can also think. When we focus this question more closely by asking: what sort of thought is scientific thought? we can begin with those characteristics shared by scientific and non-scientific thought. Non-distinctive characteristics (a) Is scientific thought systematic thought? Certainly, but this is by no means a distinctive characteristic of science. The judge who is preparing a verdict in court must similarly work systematically in his argumentation – but this does not mean that a legal verdict changes into a legal scientific dissertation. (b) Is it verifiable thought? While the answer to this must also be in the affirmative (taking into account the big controversy in contemporary theory of science over the meaning of this characteristic), it is still not a distinctive characteristic, since the judge mentioned above must also verify that every bit of evidence under consideration is trustworthy. (c) Is it methodical thought? Since there are also non-scientific methods of doing, it will always be necessary to distinguish between scientific and non- scientific methods. Indicating methodology as the distinctive characteristic of science only leads to a tautology (repetition): scientific method is scientific! (d) Particularly tired and disseminated is the notion that science centers in the relationship between a scientific researcher (the “knowing subject”) and that which is studied, the “study object.” In the first place we need to note that the subject-object relation is common to non-scientific human experience: we need but consider the human subject using social objects (like furniture), or technical objects (tools), economic objects (money), 11

semiotic objects (books), aesthetic objects (paintings), ethical objects (engagement or wedding rings), legal objects (property), and the like. All these objects indicate concrete entities which can equally much be studied by various special sciences (each from its own distinctive perspective). Let us briefly look at the many perspectives which provide access to the analysis of a social object like a lounge chair. The chair has four legs (numerical aspect: interest of mathematical number theory); it is big or small (spatial aspect: mathematical theory of space); it could be a rocking chair (kinematic aspect: kinematics); it is strong or weak (physico-chemical aspect); it could be useful in human life (as biotical object; since the chair itself is not alive – biology studies reality from the perspective of the biotical aspect); it is comfortable (sensitive/psychic aspect: psychology); someone conceived it (analytical aspect: logic); it is culturally shaped (historical aspect: historical science, which would be interested in the historical development of various chair styles); it has a name (a verbal sign – semiotic aspect: general semiotics and linguistics); it is used in the interaction among people (social aspect: sociology); it has a price (economic aspect: economics); it is beautiful or ugly (aesthetic aspect: aesthetics); it belongs to someone who has a subjective right (the comptence of enjoying it and disposing over it) to it (juridical aspect: legal science); it is someone's favourite chair (ethical aspect: ethics); and it is trustworthy – everybody trusts that the chair will bear the weight of a person sitting on it (faith aspect: perspective of theology as a science). From this example it is clear that the cardinal question is not: with what object (entity, event or societal relationship) does this or that science engage itself, but: from what perspective (aspect, way of being, mode, modality, function, facet) of reality are certain things, events and societal relationships studied by a special science. Only when we have brought the diversity of aspects of reality into sight can these scientifically distinguished aspects serve as gateways to the study of data. How do we bring these aspects into sight? Only once we have asked this question have we stepped beyond the similarities of science and non-science. The distinctive characteristic of scientific (theoretical) thought Since the aspects of reality indicate the framework within which all entities function concretely we can also refer to them as creational ways of being. The manner in which something is approached is also known as the modus operandi, and from this Latin term modus we derive the term modality (= way of being, aspect). When we therefore identify a particular modality (aspect) as such and distinguish it from other modalities, we must abstract the concerned aspect. This process is referred to as modal abstraction. Whoever is engaged in modal abstraction, relinquishes the non-relevant aspects and focuses theoretical-logical attention on one particular aspect. The distinctive characteristic of theoretical-logical (=scientific) thought, therefore, is modal abstraction. Exactly because all concrete entities function in all the various aspects of reality (cf. the lounge chair example above), the ab12

stracted modalities (aspects) provide access to the analysis of the structures of such entities. Although non-scientists certainly have an analytical awareness of the diverse aspects, this does not mean that in our non-scientific experience of reality we ever achieve modal abstraction. One would not, for instance, in one's non-scientific experience, reflect on the nature and structure of the numerical aspect when one notices six people walking past one's house, just as little as one would develop an economic price theory when one notices that a car costs R30 000. Without an analytical consciousness of the diversity of aspects in reality, one would not, however, have any conception of what was meant when someone comments that a certain car is so beautiful but so expensive. Beauty (aesthetic aspect) and price (economic aspect) are facets of one's total experience of cars, although we would still maintain that these aspects are generally noticed in a non-abstractive manner of and in entities. A particular kind of abstraction is part of our non-scientific experience, referred to with the apparently contradictory term of concrete abstraction (or: entitary abstraction). A little child who first notices a dove and learns its name can already abstract concretely, for instance when s/he shortly thereafter refers to a sparrow as a “dove.” The child actually indicates the concept bird with the name (verbal sign) dove. This is only possible because the child has lifted out certain bird-characteristics out of the concrete sensory perceived image of a dove (e.g. a beak, wings, feathers) and simultaneously relinquished the specific characteristics which distinguishes the dove from the sparrow.1 This kind of abstraction is part of our everyday life, since we are continually identifying all sorts of entities, placing them in certain categories. Otherwise how would one be able to identify a particular horse as a horse (=belonging to the category of horses), or a particular car as a car? Without general concepts such as cars and horses (in which the detail of particular cars and horses are relinquished), this would be impossible. This kind of abstraction does not provide us with theoretical insight into the nature of any aspect, since – as we shall still see – the aspects belong to a separate dimension which must be distinguished from the dimension of entity-structures. Is Philosophy a Science? If it is true that modal abstraction is the distinctive characteristic of science, then it should be obvious that we can distinguish within the category of sciences between the kind of science limited to the perspective of a particular aspect, and the kind of science which pays attention to the foundational coherent interlacement among all the aspects of reality – a coherence which also serves as the foundation of a theoretical analysis (=an analysis via the gateway of abstracted aspects) of the interlacement which exists between the wide diversity of concrete entities, events and societal relationships. The latter kind of sci1

Note the symmetry between analysis and abstraction: analysis rests on the two legs of identification and distinction which is equivalent to the nature of abstraction as it is revealed in lifting out and relinquishing (compare the sketch in the Preface).

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ence (focused on a coherent total view of the diversity of reality) we call philosophy. Every science limited to the perspective of a particular modality we call a special science. Philosophy and the Special Sciences The fact that theoretical thought entails modal abstraction has important implications for every (modally delimited) particular special science. Modal abstraction entails theoretical analysis and we have briefly pointed out that analysis on the basis of similarity and difference is aimed at the identification and distinction of data. Theoretical analysis would therefore be aimed at the identification of a particular aspect in distinction from other aspects. If reality contained only one aspect, analysis as such would be impossible, because we can only identify an aspect by simultaneously distinguishing it from all aspects which differ from the identified aspect. Theoretical analysis (modal abstraction) must therefore always simultaneously consider at least two differing aspects. If a special scientific discipline delimits its domain of study to the perspective of a single aspect, then it is obvious that the identification of the domain of study of a special science can never be seen as an activity of that special science or as an activity taking place within the perspective of a particular aspect – simply because more than one aspect is involved in the identification of any aspect! Since only philosophy can engage more than one particular aspect in its theoretical purview, this implies that no special science can indicate its own delimited domain of investigation without proceeding from some or other philosophical view of the coherence which can account for the similarities and differences among the diverse aspects of reality. In other words: the nature of modal abstraction as distinctive characteristic of science implies that all science has a philosophical base! This course of reasoning is closely linked to the argument which follows: The answer which a special scientist gives to the question: what is science? can never be a special-scientific answer, since every such a description discusses the special science and thus transcends its limits. A description such as: mathematics consists of subsidiary disciplines such as set theory, algebra, topology and the like, is no mathematical statement since the description is not in the least an axiom/proposition/proof/argument in set theory, algebra, topology and the like! This course of reasoning is valid for every particular modally perspectival science. Even theology does not escape this truth. While every student of theology becomes acquainted through the study of this discipline of the encyclopedia of theology (which is responsible for the identification and delimitation of the subsidiary disciplines of theology), this discipline is never itself classified as a subsidiary discipline of theology (next to e.g. the bibliological, dogmatological and ecclesiological groups). In this way theology admits that the question: what is theology? is not a theological question! This situation is remarkable furthermore since, although no definition of any special science can have a particular-scientific character, no such definition can be given without accounting for the scientific content of the special sci14

ence! Even when it is argued that mathematicians, theologians, and the like would be best equipped to answer questions such as: what is mathematics? what is theology? and the like, this does not abolish the truth that the answer which they give precedes their work in the special science they practice. The measure is not who gives the definition, but: what is the nature of the definition! This state of affairs confirms the indissoluble coherence which exists between every special science and its philosophical foundational questions. We could also describe this state of affairs as follows: there are basically two kinds of science, (i) the kind of science which, when it discusses itself, transcends its own limits, and (ii) the kind of science which, when it discusses itself and the general question of the nature of science, remains within its own limits. The first option indicates a special science, and the second indicates philosophy. In this sense philosophy is the science of sciences, which is engaged inter alia with the philosophical foundational questions of the special sciences. Philosophical Foundational Questions in the Special Sciences Philosophical presuppositions entail foundational questions such as: what is the coherence and structure of the diverse aspects of reality which serve as gateways for our experience of concrete phenomena? Does each of these aspects have a unique and irreducible own nature, or can all aspects be reduced to/explained in terms of a few aspects (as suggested by the many philosophical and special scientific isms: rationalism, idealism, universalism, individualism, irrationalism, realism, nominalism, physicalism, materialism, vitalism, historicism, psychologism, aestheticism, moralism, pietism, and the like)? Can facts and norms (or in the natural sciences, facts and laws) be divorced, or are they irreducible correlates? What is freedom? Does causality and freedom mutually exclude each other? What is the nature of being human (and is there a fundamental difference between being human and being an animal)? What is the relation between individual and society? In what does the creational diversity find its consummation and central focus, and what is its origin? To illuminate the influence of some of these foundational philosophical problems in the natural sciences more closely we will critically analyse some examples of dominant (and sometimes conflicting) paradigms from the fields of mathematics, physics, and biology. In all of them the inevitability of modal distinctions will surface – thus underscoring our claim that modal abstraction indeed constitutes the distinctive feature of scholarly scientific activities.

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Paradigms in Mathematics, Physics, and Biology: Their Philosophical Roots

Chapter II Foundational Philosophical problems in Mathematics

Introductory remarks Let us start our reflections by considering one of the standard arguments against the possibility of divergent standpoints in mathematics: Are you sure 3 + 4 = 7? This kind of question is more or less a standard argument used by secular humanism in an attempt to defend the supposed neutrality of scholarly activities: the statement 3 + 4 = 7 apparently concerns a fact which is true irrespective of our being Christian, Jewish, atheistic or communistic. Suppose now that I would object by saying: 3 + 4 = 5! One way to ‘convince’ me that I am wrong, would be to count some of your fingers. My response then would be: your ‘finger counting’ operation indeed clarifies the initial statement – it concerns numerical addition. However, I had something different – but equally legitimate – in mind: geometrical addition. Just think about a person starting from a specific point: He then walks 3 miles to the north and afterwards 4 miles east. How far would that person be away from his point of departure? 5 miles! Clearly we are now confronted by two different kinds of facts: a numerical fact (3+4=7) and a geometrical fact (3+4=5) – adding distances is mathematically accounted for by the theory of vectors: a vector has distance and direction – which actually means that we should have used a no17

tation to acknowledge this (for example by using pointed arrows above the ‘3’, ‘4’ and ‘5’ of the vector-sum). The point of our two examples is that ‘facts’ are not simply ‘facts’ – they are always structured or qualified. In our case they are qualified by the aspects of number and space. Consequently, in order to distinguish between different kinds of facts an awareness of the order-diversity of reality is presupposed. But exactly at this point the views of mathematicians depart. Different schools of thought in modern mathematics arose on the basis of alternative views with respect to the nature of and the coherence between the aspects of number and space. For example, intuitionistic mathematics constructed a whole new mathematics introducing concepts and methods not found in classical mathematics. Stegmüller strikingly remarks: “The special character of intuitionistic mathematics is expressed in a series of theorems that contradict the classical results. For instance, while in classical mathematics only a small part of the real functions are uniformly continuous, in intuitionistic mathematics the principle holds that any function that is definable at all is uniformly continuous” (1969:331).

From the seemingly ‘innocent’ and ‘neutral’ statement of ‘fact’ (3+4=7) we are not only immediately entangled in foundational questions of mathematics as a special science but also confronted with a serious challenge regarding the supposed ‘exact’ nature of mathematics as an academic discipline! A number of years ago a well-known mathematician, Morris Kline, wrote a whole book dealing with the way in which the classical ideal of mathematics as an exact science with certainty as its guiding star was undermined. He remarks: “The developments in the foundations of mathematics since 1900 are bewildering, and the present state of mathematics is anomalous and deplorable. The light of truth no longer illuminates the road to follow. In place of the unique, universally admired and universally accepted body of mathematics whose proofs, though sometimes requiring emendation, were regarded as the acme of sound reasoning, we now have conflicting approaches to mathematics. Beyond the logicist, intuitionist, and formalist bases, the approach through set theory alone gives many options. Some divergent and even conflicting positions are possible even within the other schools. Thus the constructivist movement within the intuitionist philosophy has many splinter groups. Within formalism there are choices to be made about what principles of metamathematics may be employed. Non-standard analysis, though not a doctrine of any one school, permits an alternative approach to analysis which may also lead to conflicting views. At the very least what was considered to be illogical and to be banished is now accepted by some schools as logically sound” (1980:275-276)

It is indeed strange that the history of mathematics explored the dual one-sidedness of an arithmeticistic (founded by Greek mathematicians and again enthroned during the past hundred years) and a geometricistic approach (dominant during the intermediate period) and never ventured to explore the following obvious third possibility: acknowledge both the uniqueness and the mutual coherence of number and space as aspects of the richly varied creational order-diversity. 18

We now proceed with a sketch of major historic and systematic analyses in order to elucidate the complex interrelations between number and space. First we mention some modern definitions of the nature of mathematics. From these definitions we deduce the importance of the notion of infinity for an assessment of the nature of mathematics – supported by our historical overview. This overview, then, is followed by a brief assessment of some prominent issues, and we close our discussion by a succinct treatment of an alternative systematic perspective. Definitions of mathematics Though it may seem natural to relate mathematics as a special science to the aspects of number and space in the first place, the way in which most modern mathematicians define their subject matter does not explicitly refer to these aspects. Logicism, for instance Russell, wants to stress that mathematics is not concerned with quantity, but with order. Already W. Hamilton defined algebra – in a work from the year 1833 – as the “science of pure time or order in progression” (quoted in Cassirer, 1957:85). Cassirer himself continues this line of thought, actually dating back to Leibniz, in his own way. Smart points out that according to Cassirer the main purpose of the critical study of the history of mathematics “is to illustrate and confirm the special thesis that ordinal number is logically prior to cardinal number, and, more generally, that mathematics may be defined, in Leibnizian fashion, as the science of order” (1958:245). Works on the foundations of set theory and dealing with the philosophy of mathematics often refer to mathematics as “the science of formal systems”. To those who are inclined to an axiomatic approach this statement means the same as “mathematics is set theory” (cf. Meschkowski, 1972:356). In spite of its recent origin as a mathematical discipline, set theory has from the very start been confronted with basic trends running through the history of mathematics. We only have to refer to the tension of uncompleted infinitude and completed infinitude – a contrast which has been familiar since Greek philosophy in terms of the opposition between the potential infinite and the actual infinite. The uncompleted infinite is used to indicate the conception that the infinite is literally in-finite, i.e. without an end. The completed infinite, again, is seen as a quantity which is determined in all its parts while it simultaneously exceeds every finite quantity. As founder of set theory, Cantor was convinced “that Set Theory deals with the actual infinite” (Robinson, 1967:39). By using the completed infinite Cantor, in 1874, proved that the set of all real numbers cannot be enumerated in the manner of the set of all natural numbers, i.e. that real numbers are non-denumerable (we will return to this matter). But exactly in this proof, which uses the completed infinite, H. Meschkowski (1972b:25) sees the “foundation of set theory”.1 1

Although one could consider this statement also in view of developments in the field of category-theory and topos-theory, this falls beyond our current concerns.

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Whoever therefore defines mathematics as set theory has thereby already placed the problematic relationship between uncompleted infinity and completed infinity at the centre of the definition. In modern mathematics there are special scientific points of view which diverge exactly in terms of the infinite: some in the direction of the uncompleted infinite and other in the direction of the completed infinite. Hermann Weyl (1966:89) comments strikingly in this regard: “If in conclusion one would want to provide a brief slogan which would indicate the living centre of mathematics, one would be able to say: it is the science of the infinite”. In view of what we have already considered, we would however immediately have to add the words of perhaps the greatest mathematician of the 20th century: “The infinite has moved the human mind like no other question since the earliest times; the infinite has brought about mental stimulus and fruitfulness like virtually no other idea; the infinite however needs clarification like no other concept” (Hilbert, 1925:163). Clearly, no account of mathematics can escape from assessing the nature of the infinite. However, the question of how we gain insight into the nature of the infinite appeals to the foundational question regarding the unique domain of investigation of mathematics. It is clear that it is insufficient to delimit mathematics as a discipline in terms of “formal systems”, if only because other disciplines use extensive abstractions without being mathematics (e.g. philosophy). A prominent mathematician such as Gottlob Frege even questioned the notion that the concept of abstraction is of use in the determination of the nature of mathematics. When, for instance, we begin with the moon as an entity, we arrive through abstraction only at concepts such as “companion of a planet”, “celestial body without its own light”, “celestial body”, “body” and “object” – nowhere in this row does the number 1 appear (Frege, 1884:44)! Frege is of course reacting against the view that number should be seen as a set of units [pure ones] which we gain out of our experience of concrete things via abstraction. Angelelli is even of the opinion that Frege's critique in this regard is devastating: “by abstracting from the particular differences and natures of the given objects no plurality can be attained, but only one thing (the concept cat, for example)” (1984:467). In a comment on Cantor's definition of a subset (cf. Cantor, 1962:283), Zermelo also refers to the effort to introduce the concept cardinal number by means of a process of abstraction, which would imply that a cardinal number should be seen as a set of “pure ones”.1 Immanuel Kant already perceived that a purely logical synthesis could never provide a new number (Kritik der reinen Vernunft, 1787:15). In a different way Frege emphasizes the same point: concrete (or: entity-directed) abstraction can only continue and arrive at ever more general entity concepts – it can never arrive at number as such. The fundamental question, however, is: is it possible to distinguish different characteristics/properties of one and the same entity? In terms of Frege's example of the moon we can be more specific: does the moon have any numerical features? This new perspective con1

An “aus lauter Einsen zusammengesetzte Menge” (a “set constituted by pure one's”) – cf. Zermelo's first comment in Cantor, 1962:351.

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cerns the question of how many moons the earth has – and obviously the answer in terms of current knowledge can only be: one. This introduces us to a distinct kind of abstraction which differs fundamentally from concrete (entity-directed) abstraction as this emerges in our everyday concepts (such as human, tree, horse, and the like), namely ‘characteristic-abstraction’. Another way of indicating this possibility, is to indicate it as modal abstraction, which should actually be seen as the distinctive characteristic of scientific thought. We shall pursue this further later when we give systematic attention to the relation between the two fundamental kinds of infinity which have always confronted mathematics. By first drawing out a few historical threads we can attempt to gain some initial illumination on infinity and the critical turning points in the development of mathematics. We must at the same time keep in mind that all Western thought on infinity and continuity has been decisively influenced by Aristotle – that is, until Cantor fundamentally questioned it.1 The Infinite in Greek Thought In the Milesian philosophy of nature we find a philosophically expressed answer to their deepest search for the Arche (origin, beginning) of all temporal things. Anaximander was the first to choose the infinite (unlimited) as Arche: “the Arche of the existing things is the apeiron (the infinite-unlimited)” (Diels-Kranz, 1959-69:B Fragments 1).2 He adds to this that the “apeiron is ageless” (Fr.2) and “the apeiron is without death and transience” (Fr.3). Where the divine Arche has previously always been identified with fluid elements (e.g. water by Thales, air by Anaximines and fire by Heraclitus), we find Anaximander (remarkably!) seeing the Arche as something ageless, deathless, and intransient – i.e. something standing in opposition to all fluidity! It is probable that Anaximander was on the track of an essential element and characteristic of the infinite – but we will return to this when we mention Cantor's description of the completed infinite. We encounter the bipolar nature of the infinite strikingly in Zeno's arguments against motion. Aristotle mentions Zeno's four arguments in his Physika (cf. 233 a 13ff. and 239 b 5ff.). We will refer to only two particularly illustrative arguments, namely that of Achilles who can never catch up with the tortoise (since the tortoise has always again established a lead by the time Achilles has caught up with him), and the argument that it is impossible to move from point A to point B. In order to do so, after all, it is first necessary to complete half the 1

2

“Die entscheidende Erkenntnis des Aristoteles war, dass Unendlichkeit wie Kontinuität nur in der Potenz existieren, also keine eigentliche Aktualität besitzen und daher stets unvollendet bleiben. Bis auf Georg Cantor, der in der 2.Hälfte des 19.Jahrhunderts dieser These mit seiner Mengenlehre entgegentrat, in der aktual unendliche Mannigfalgtigkeiten betrachtete, ist die aristotelische Grundkonzeption von Unendlichkeit und Kontinuität das niemals angefochtene Gemeingut aller Mathematiker (wenn auch nicht aller Philosophen) geblieben” (Becker, 1964:69). Subsequently we refer to Diels-Kranz's B Fragments with the abbreviation Fr. (=Fragment).

21

distance, thereafter half of the remaining distance, and thereafter half again of the remaining distance – ad infinitum (cf. Dielz-Kranz, 1959-60 B Fr.3). Zeno concludes: an infinite number of spatial sub-intervals must be crossed to move from A to B and this is impossible in a finite period of time. It is after all impossible to actually exhaust the infinite. Therefore motion is impossible. This is an instance of the contradiction between the uncompleted infinite and the completed infinite. We must therefore differ from Titze's statement that the “numerically-infinite was inconceivable in Greek philosophy” (1984:141). Anaxagoras without doubt already had a conception of the potentially-infinite, while we obviously find an initial conception of the infinite divisibility of a continuum in Zeno's thought (cf. his B Fr.3 – explained below). Since Aristotle provided the classical formulation of the notion that a whole is more than its parts (in his Politeia:1253a19-20), this idea has exerted an inextinguishable influence in the history of philosophy and the various special sciences – in modern times often defended in the form of the statement that a whole is different from the sum of its parts. It would however appear as if we need to return to Zeno for the first insight into the divisibility of a spatial continuum. Zeno is best known in the literature for his four arguments against multiplicity and motion as this is rendered in Aristotle's Physics (cf. 233a13ff. and 239b5ff.).1 The peculiar sense of his third Fragment lies exactly therein that it explicitly explores both sides of the whole-part relation – apparently for the first time in the history of philosophy (and mathematics). Let us note his formulation: When multiplicity exists, then necessarily only as many (things) exist as what are actually there, no more and no less. When there however are as many as what exist, then it (the number thereof) must be limited.

In this first half Zeno therefore argues from multiplicity to limitation. Exactly the opposite happens in the last half: When multiplicity exists, then that which exists (the number thereof) is unlimited. Because continually other ones exist in between those which exist and again others between these. Thus that which is (the number thereof) is unlimited.

Although both main parts of this Fragment begin with “when multiplicity exists”, Zeno reaches opposite conclusions in them – in the first instance the initial comment implies that the number of existing things are limited, and in the second instance that it is unlimited. The static spatial terms which Parmenides and his school uses suggest the possibility that Zeno is indeed exploring the two sides of the spatial whole-part relation (we shall return to the nature of the whole-part relation in a later context). 1

It is true that we have of Zeno himself only the following striking formulation in his fourth and last transmitted Fragment: “That which moves, moves neither in the space it occupies, nor in the space it does not occupy”.

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If multiplicity in the initial comment indicates a multiplicity of parts (of the world) then their sum total must be limited (simultaneously constituting the world-whole). If, alternatively, one starts with the world-whole in terms of which to account for the parts, then it would indeed be possible to localize the multiplicity of parts in such a way that there would always be further parts present in between – an argument which of course could be continued infinitely with regard to all parts.1 The discovery of the whole-part relation was therefore indissolubly linked to the development of a notion of infinity in Greek philosophy, since it is concerned with the infinite divisibility of the (world-) whole. To trace the steps of the notion of infinity further we would however have to return to the Pythagoreans. One of the foundational characteristics of the earliest Pythagorean school was the arithmeticist statement: “everything is number” (cf. Thesleff, 1970:82). The apparent possibility to arithmetize musical consonants lead to the general theorem: if any two things in their relation to each other appear as two numbers, then they themselves actually are covert numbers (cf. Scholz and Hasse, 1928:6). The interest of the Pythagoreans in the form of figures (including form-congruence/uni-form-ity) apparently stimulated the proof of Pythagoras’ familiar theorem, namely that in any rectangular triangle the square of the diagonal side equals the sum of the squares of the two rectangular sides. In Babylonian texts the following figure appears which already suggests the theorem of Pythagoras. According to Euclides’ Elements the original arithmetical proof of the theorem of Pythagoras is founded in the congruence of figures: Hippasus of Metapont (450 BC) probably already discovered that this proof is not generally valid since it proceeds from the presupposition that the ratios of all line stretches stand in relation to one another as integers – i.e., can be represented in the form a/b where a and b are normal natural whole numbers. The pentagram2 convinced Hippasus of the falsity of this presupposition. Consider the following pentagram: If a1 and a0 have a common length, a1 could be precisely (without remainder) divided into a0. If this division yields an infinitely continued fraction, this would mean that the two line-stretches 1 2

Herman Fränkel even explicitly uses the whole-part relation when he analyzes Zeno's Fragment 3. Cf. Fränkel 1968, pp.425ff., especially p.430. That is a regular pentagon. The Pythagoreans used a regular pentagon of which the sides where extended to the points of intersection (cf. Moritz Cantor, 1922:178).

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are incommensurable due to the absence of a common logos or ratio – which necessarily yields the discovery of irrational numbers. From the adjacent figure follows: ao = a1 + a2 Therefore: ao/a1 = 1+ a2/a1 and: a1 = =a2 + a3 implying: a1/a2 = 1 + a3/a2 Similarly a2/a3 = 1/a1/a2 and: a3/a2 = 1/a2/a3 etc. From the preceding it follows: 1 + 1 1 + 1 1

a

b

f d

e

c

a1 a0 a1

a1

+ 1

a2

a2

which confirms the existence of irrational numbers

a3

The simplest example of an irrational number is the oblique side of a rectangular triangle where both the rectangular sides measure 1. Apparently a proof of the fact that the square root of the number 2 is irrational was already known to Pythagoras. In his dialogue Theaitetos (147d) Plato mentions that the Pythagorean Theodoros had proven further irrationalities.1 The nature of irrational numbers, which cannot be indicated in terms of the relation (ratio, logos) between two whole numbers (integers), was the core of the crisis in Pythagorean mathematics. Only when Greek mathematics is understood in terms of the deepest motive active at the root of all Greek thought, does it become clear why this discovery was experienced as so central a crisis. The formative and delimiting function of number was however undermined by the mentioned discovery of irrational numbers (incommensurability – cf. Von Fritz, 1945:242-264), since it appeared that the formatively-delimited oblique side, e.g. of a rectangular triangle with two rectangular sides with a length of 1, in itself (from an arithmetical perspective) contained an infinite (unlimited) sequence. In other words, in this case the apeiron abrogated the delimiting function of the peras! To avoid this consequence all algebraic problems2 were translated into spatial terms – hence the geometricization of mathematics.3 The foundational motive of Greek thought, namely the motive of the limited and unlimited (transient and intransient, or, in Aristotelian

1

2 3

“Our friend Theodoros was proving to us something about square roots, namely that the sides (or roots) of squares representing 3 square feet and 5 square feet are not commensurable in length with the line representing 1 foot; and he continued thus, taking each case in turn up to the root of 17 square feet.” Of course the Greeks did not yet know any algebra. Cf. Boyer, 1956:8ff.

24

terms: the motive of form and matter), therefore integrally determined the entire direction of Greek mathematics!1 The discovery of irrational numbers constitutes the first foundational crisis of mathematics. To overcome this crisis Eudoxos devised a method which approached the modern differential and integral calculus, but due to an excessive adherence to the spatial perspective the most important discovery had to wait until the 17th century A.D.2 Apart from the contribution of Zeno's B Fragment 3 to our understanding of the nature of the spatial whole-parts relationship, we find statements in Anaxagoras regarding the nature of spatial continuity which are still actual today. He says: In that which is small there is no smallest, since there always exists something smaller. That which is can never cease to exist through further division, no matter how far we continue this division (B Fr.3). And since no smallest can exist, it also cannot insulate or contain itself, but must, as in the beginning, exist with everything else (B Fr.6).

This simultaneous existence suggests the coherence of spatial continuity which includes all (material) things – a continuity which is not, however, the co-ordination of discrete (separated) parts, as if separated with an axe (B Fr.9). With these characterizations Anaxagoras reaches forward not only to the view of Aristotle, but even to the position of intuitionist mathematicians in the 20th century (namely Brouwer and Weyl).3 In Anaxagoras’ course of thought it is clear that the infinite should not be understood only externally as spatial extension without limits, but also internally (i.e. inwardly) as infinitely divisible spatial extension. One of the categories which Aristotle distinguishes is quantity. As higher genus proximum quantity encompasses both number and space as differentia specifica. That is, when one relinquishes the specific differentiating (distinctive) characteristics of number and space, both belong under the higher concept of kind: quantity. “Quantity is either discrete, or continuous” (Categoriae, 4 b 20). “Number, ... is a discrete quantity” (Cat., 4 b 31). The parts of a discrete quantity have no common limit, while it is possible in the case of a line (as a continuous quantity) to find a common limit to its parts time and again (Cat., 4b 25ff., 5 a 1ff.). 1

2 3

P.A. Meijer is of the opinion that the most appropriate indication of the Greek foundational motive is to be found in the Greek hankering after the intransient (cf. 1968:207, and note 15 on p.206). Bram Bos reconsiders form and matter to signify the foundational motive (Dooyeweerd) by using the alternative designation of the “titanic meaning perspective” (1986). The tension between becoming (e.g. of the seasons) and the underlying quest for constancy however remains central to this motive. Compare Bos 1994:220. We shall return to this second foundational crisis in the history of mathematics shortly. H. Weyl points out the significance of Greek thought: “Yes, exactly now we are being brought everywhere to return directly to the Greeks in the foundations of mathematics” (1931:1).

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With Zeno Aristotle denies the possibility of the actually infinite (completed infinite). He believes that his particular view of the potentially infinite (the uncompleted infinity) overcomes the problems of Zeno. While a particular line-stretch is infinitely divisible, this divisibility is only a potentiality, a possibility, which can never actually (in reality) be carried through. When something moves it does not, according to Aristotle, move in a counting manner, since then Zeno's antinomy (that in the covering of a limited spatial distance an actually infinite sequence of numbers would simultaneously have to be “counted through”) would be valid. Aristotle confronts Zeno's problem with the following argument: “In the act of dividing a continuous distance into two halves, one point is used twice, since we make it the starting-point and the end-point: ... But if such divisions are made, neither the distance, nor the motion would be continuous: ... and although that which is continuous contains an infinite number of halves, these are not actual but only potential halves” (Physica, 263 a 23ff.).

Aristotle rejects the existence of the actually infinite on two grounds (cf. Physica, 204 a 20ff., Metaphysica, 1066 b 11ff., and Metaphysica, 1084 a 1ff.): (i) if the actually infinite consists of parts then these parts must themselves be actually infinite, which would imply the absurdity that the whole is no longer larger than a part; and (ii) If it consists of finite parts, this would imply the impossibility that the infinite can be counted, or there would have to be transfinite (cardinal) numbers which are neither even nor uneven. It is understandable, therefore, that the formative deity of Aristotle (the nous as thought of thought – Metaphysica, 1074 b 34-35) is finite. According to Aristotle, only that which is limited can be known (conceived), and he consequently does not hesitate to conclude from the unlimited nature of matter that matter as such cannot be known (Metaphysica, 1036 a 8-9). A few further contours from the history of the infinite Under Aristotle's influence Origines taught in the 3d century A.D. that God could not be infinite, since if he is infinite (unlimited) no limits exist and he would not be able to delimit or conceive himself – which would imply that God could not know himself! With Plotinus, however, we find a return of appreciation for the infinite since he characterizes both the One (out of which everything arises) and the contrasting matter as infinite (cf. Enneads II,4,4; II,4,10; II,4,15; VI,7,32), although the term infinite is used in a dialectically opposed manner with regard to the One and matter: (formless) matter receives form (as a permanent substratum) – the (formless) One gives form (cf. En. VI,7,17). This re-appreciation is related to Plotinus’ view of infinity as the timeless present (cf. the whole En.III,7), which simultaneously exerted a considerable influence on the conceptions regarding infinity of Boethius, Augustine (Confessiones XI,11,13; De Trinitate XII,14), Thomas Aquinas (Summa Theologica I,10) and Schilder (1948:61). 26

Augustine went further than Plotinus and stated explicitly that our inability to understand the infinite should not be used as a measure for God, since God in his omniscience understood every infinity – also the completed infinite set of all numbers – without any passage of thought, at once, without before and after. Therefore God can also know his own completed infinite being.1 Creation, however, is finite. At the end of the middle ages and the beginning of the modern era Cusanus changed this view with his doctrine that God is actually infinite while reality is only endless. Linked to his conviction that the infinite line is simultaneously a triangle, circle and sphere (De Docta Ignorantia, I,13-17) Cusanus taught that of God, as the actually infinite, one could in a certain sense say everything and nothing at all (he is e.g. the biggest and the smallest – De Docta Ignorantia, I,5) since all contradictions are resolved in him (coincidentia oppositorum) (De Docta Ignorantia, I,22; De Coniecturis II,1 and II,2).2 Descartes turns the classical view on its head with his view that the infinite is complete and the finite incomplete, so that the finite should actually be referred to as the non-infinite. Since Spinoza identified God with nature (Deus sive natura), he also saw the universe as completed infinite. Galileo discussed the remarkable relation between square numbers and the sequence of all numbers in dialogue form in March 1638: All numbers are not square numbers (like 1, 4, 9, 16, 25... ). The combination of all numbers, i.e. square and non-square numbers, are certainly more than the square numbers on their own. From 0-100 there are only ten squares (100 = 102). That means only one tenth are squares; from 0-10000 there are only 100 squares, i.e. 100/10000 = one hundredth; from 0-1000000 there are only a 1000 squares, i.e. one thousandth, and so forth. If we however ask how many square numbers exist, we can answer: as many as there are square roots, since every square has a root and every root has a square. Then, however, there are as many squares as the combination of all numbers! 12 22 1 2

32 3

42.......... 4...........

Bernard Bolzano built on this in a posthumously published work by considering an infinite set characterized by the fact that the whole set can be matched element by element (in the case of Galileo's example: 1 with 12, 2 with 22, 3 with 32, and so forth) with a true subset (the set of squares is a subset of the set of natural numbers) (Bolzano, 1920, par.20:27ff.). In the case of an infinite set the whole is therefore equal (correctly stated: equivalent) to a part – this in opposition to Aristotle's mentioned conviction that the whole is always greater than a part (cf. in this regard Strauss, 1987)! Before we continue with Cantor, we return for a moment to the historical relation between irrational numbers and the infinite (among the Greeks this relation led to the geometrization of mathematics). In 1790 S. Maimon argues that 1 2

Cf. Augustine: De civitate Dei, Book XII, Chapter 19; and cf. Heimsoeth, H. (n.d.:68). According to K. Kremer Plotinus and Proclus actually already taught that all contradictions are resolved in God (1966:354).

27

an essential difference exists between a so-called irrational number and the approximating sequence of rational numbers employed to expresses it.1 In an effort to handle the problem of infinity, Maimon distinguishes between the limited human mind2 and the absolute unlimited mind: “an infinite number cannot appear otherwise to us (since our perception is bound to the form of time) than as an infinite succession in time (which consequently cannot be thought of as completed). In the case of an absolute mind, on the other hand, the concept of an infinite number, without any passage of time, can be thought of at once. Thus that which the mind in its limited form considers as a mere idea, is in terms of its absolute existence a real object” (Maimon, 1790:228). Infinitesimals and the second foundational crisis of mathematics Newton discovered his first “calculus” in 1665-1666. It is also historically determined that Leibniz between the years 1673-1676 made the same discovery independently of Newton.3 Yet he only published his discovery in 1684 and 1686. This “calculus” was known as infinitesimal analysis and was subdivided into differential calculus and integral calculus. In differential calculus it is possible, for instance, to determine the inclination of a given curve at any point (the inclination is trigonometrically indicated by means of the tangent). Integral calculus, which was practically developed as the inverse of differential calculus, enables the mathematician to determine surfaces and volumes of figures (partially) delimited by curves. The spatial representation of the inclination of a curve at a particular point is only the correlate of problems with bodies in motion, which was the actual starting-point of Newton's discovery. Suppose the equation y = x2 indicates the prescription for the motion of a body (where x is the time in seconds needed to move y feet). The problem is to determine the speed of the body at a given moment. An analysis of this problem gives the following result: after 1 second the body moves 2 feet per second, after 2 seconds 4 feet per second, after 3 seconds 6 feet per second, and so forth. Thus an equation providing the speed at any given moment (y' = 2x, with y' referred to as the derivative) is derived from the original equation. E.T. Bell comments: “Newton's first calculus, of 1665-6, seems to have been abstracted from intuitive ideas of motion. A curve was imagined as traced by the motion of a ‘flowing’ point. The ‘infinitely short’ path traced by the point in an ‘infinitely short’ time was called the ‘momentum’ and this momentum 1 2 3

The irrational number “square root of 2” ( 2) e.g. is approximated by the sequence of rational numbers: 1, 1.4, 1.41, 1.414.... According to the Kantian conception it is bound to time – as a so-called a priori form of intuition. Hawking points out that although it is clear that both discovered this subsection of mathematics independently, a furious argument nonetheless broke out over who discovered it first. Newton made the discovery before Leibniz but published it later. Most of the articles defending Newton's position were however written by Newton himself and published under the names of his friends (1987:182)!

28

divided by the infinitely short time was the ‘fluxion’” (Bell, 1945:151). This “fluxion” of Newton is nothing other than the speed at a given moment in our example above. The development of differential and integral calculus paid virtually no attention to its foundations, until it became clear at the beginning of the 19th century that the use of the infinitesimal (the infinitely small) causes a great many problems. Slowly but surely this brought about an arithmetically founded consideration of limits. In 1770 A.G. Kästner described a limit as follows: “A magnitude approximates a value infinitely, when the difference between the approximating and approximated value is less than any specifiable magnitude. This value is then referred to as the limit”.1 Even Cauchy (1789-1857) still characterizes the derivative of a given moving point as the “derniére raison des différences infiniment pitites Dy et Dx” (as the “extreme ratio of the difference between the infinitely small Dy and Dx”). Cauchy also uses an unwarranted transition from the ratio of infinitesimally small numbers to the ratio of normal numbers which are sufficiently small. Although Cauchy still considers Dx and Dy as the variables (i.e. magnitudes which take on successively changing values), he nonetheless provides a relatively clear definition of a limit in his Textbook of Analysis (1821): “When the successive values assigned to a variable indefinitely approaches a fixed value to the extent that it eventually differs from it with as little as one wishes, then this last (fixed value) can be characterized as the limit of all the others”.2 Consider for instance the sequence 1/n with the number 0 as limit when n increases indefinitely through the sequence of natural numbers 1, 2, 3, .... The sequence 1/n then furnishes 1/1, ½, 1/3, 1/4,.... By merely choosing a sufficiently large n it is therefore possible to bring the value of 1/n as close to the limit-value 0 as one wishes. Despite the firmer foundation which Cauchy provided for analysis, he still did not provide a sufficient foundation for the introduction of real numbers, and exactly real numbers are vital for the firm development of analysis. Cauchy believed that irrational numbers can be defined as the limits of converging sequences of rational numbers. Consider for instance the following interesting sequence of fractions (probably already known to the Greeks): 1 , 3 , 7 , 17 , 41 , 99 , 239 , 577 , 1393 , ×××× 1 2 5 12 29 70 169 408 985 1

2

“Eine Grösse nähert sich einem Werte unendlich, wenn ihr Unterschied von diesem Werte kleiner als jede Grösse werden kann, die sich angeben lässt. Der Wert heisst alsdann ihre Grenze” (1770:1). D'Alembert in particular exercised sharp criticism of the notion of infinitely small magnitudes (quantités infiniment petites) and brought to the attention of European mathematics the limit concept as the central concept of analysis (cf. Robinson, 1966:268-269). “Lorsque les valuers successivement attribuées à une même variable s'approchent indéfiniment d'une valuer fixe, de manière à finir par en différer aussi peu que l'on voudra, cette dernière est appelée la limite de toutes autres” – quoted by Robinson, 1966:269.

29

The sequence is calculated as follows: the denominator (under the line) of every subsequent fraction equals the sum of the numerator (above the line) and denominator of the previous fraction, while the numerator of every subsequent fraction equals the sum of its own denominator and that of the previous fraction. The sum of the numerator and denominator of 1/1 equals 2 – the denominator of the second fraction – while the sum of the first two denominators (i.e. 1+2) equals 3 – the numerator of the second fraction. In the same way the denominator of the third fraction equals the sum of the numerator and denominator of the second fraction (i.e. 2+3=5) and the numerator of the third fraction equals the sum of the denominators of the second and third fractions (i.e. 5+2=7). This sequence of fractions approaches 2 alternately from both sides, namely: 1 7 41 239 1393 577 99 17 3 < < < < < ×× < 2 < ×× < < < < 1 5 29 169 985 408 70 12 2

To the left and right of 2 we find two sequences of rational numbers which both approximate 2 as their limit. Since a limit is itself defined as a number (!) approximated by the terms of a sequence in such a manner that the difference between the terms of the sequence and the limit-value can be made arbitrarily small (i.e. smaller than an arbitrary rational number Î > 0, as it was later formulated), it is clear that the numerical character of 2 cannot be defined by means of the limit concept, since the limit concept presupposes that whatever functions as limit must already be a number. Even Cauchy was still of the opinion that irrational numbers must be considered as the limits of converging sequences of rational numbers. He is therefore caught in the same circular argument since the presence of an irrational limit presupposes its existence as a number, which means that the numerical nature of irrational numbers cannot be defined in terms of limits. In 1883 G. Cantor expressly rejected this circle in the definition of irrational real numbers (1962:187). The eventual description of a limit still found in textbooks today was only given in 1872 by E. Heine, who was a student of K. Weierstrass with G. Cantor (cf. Heine, 1872:178,182). In 1887 Cantor, however, pointed out that the core of the ideas in Heine's article were borrowed from him (1962:385). Furthermore Cantor himself wrote an article on trigonometric series in 1872 (Mathematische Annalen, Volume 5) in which he gave an equivalent description of a limit with reference to convergent sequences of rational numbers (1962: 93). A few pages later he also describes a limit-point of an infinite set of points (i.e. a point each environment of which contains an infinite number of points of the original set) (1962:98).1 At the same time his description of the environment of a point provided an important starting-point for modern topology. Already in the approach of Weierstrass a new presupposition penetrated analysis, namely the presupposition that limits can be defined in terms of a static 1

In general a number l is called the limit of the sequence (xn), when for an arbitrary 0 < e a natural number no exists such that |xn – l| < e for all n ³ n0.

30

numerical domain encompassing all real numbers. Boyer refers to this as follows: “In making the basis of the calculus more rigorously formal, Weierstrass also attacked the appeal to intuition of continuous motion which is implied in Cauchy's expression that a variable approaches a limit. Previous writers generally had defined a variable as a quantity or magnitude which is not constant; but since the time of Weierstrass it has been recognized that the ideas of variable and limit are not essentially phoronomic, but involve purely static considerations. Weierstrass interpreted a variable x as simply a letter designating any one of a collection of numerical values. A continuous variable was likewise defined in terms of static considerations: If for any value x0 of the set and for any sequence of positive numbers d1, d2, ..., dn’ however small, there are in the intervals x0 – di, x0 + di others of the set, this is called continuous” (1959:286).

The use of the completed infinite in the mathematical approaches of Weierstrass, Dedekind and Cantor did indeed succeed during the last three decades of the 19th century in a brilliant way to give shape to the mathematical handling of the completed infinite. With reference to Aristotle's Physica 208a6 Cantor (1962:396) distinguishes between a[peiron dunavmei and a[peiron wJ" ajfwrismevnon: – the former is the potential (non-actually) infinite and the latter is the actual infinite. Cantor describes the potentially infinite (in our provisional notation: uncompleted infinity) as follows: “The potential infinite is preferably indicated where an indefinite variable of finite magnitude occurs, which either increases beyond all finite limits ..., or decreases beyond all finite bounds”. Under the “actually infinite, though, is understood a quantum, which on the one hand does not change, but which rather is set and determined in all its parts, a true constant, but simultaneously on the other hand exceeds in magnitude every similar finite magnitude”(1962:401).

The uncompleted infinite is linked to the nature of a variable and the completed infinite to the nature of a constant. Anaximander however already described the infinite (apeiron) – which he considered to be the Arche – as intransient (i.e. constant), and set this against all fluidity (i.e. transience)! Cantor himself appeals inter alia to Augustine's view of the sequence of all numbers as an actually infinite quantum (cf. his De civitate Dei, book 12, chapter 19). Augustine's explanation that God conceives the actually infinite sequence of all numbers at once (without any process of thought or before and after), was influenced by the view of Plotinus of eternity – linked to the timeless present. This heritage we still find (via Cusanus) in the 18th century when Maimon teaches that an “absolute mind” thinks the “concept of an infinite number at once without any passage of time”. Cantor and Weierstrass avoids the circle in Cauchy's definition of irrational numbers through the use of the actual infinite. Weierstrass simply defines the completed infinite set of numbers 1,1.4,1.414, ... as the “square root of two” ( 2). Cantor also defines irrational numbers as completed infinite sets of rational numbers.1 Cantor explicitly comments in this regard that a real number b can1

Cantor refers to these as “Fundamentalreihe” (1962:186ff.,410).

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not be defined as the limit of the members of a fundamental sequence (an), since this leads to the logical error of presupposing the existence of the limit (1962:187). He adds to this “that the irrational number by means of the nature which has per definition been ascribed to it possesses as much of a particular reality in our mind as the rational and even the integers and that the irrational numbers are not obtained only by means of a limit-process, since one is on the contrary already convinced previously, due to their possession, of the practicability and evidence of limit-processes in general” (1962:187). Conversely Cantor defends his doctrine of transfinite numbers (developed on the acceptance of completed infinity) by means of irrational numbers: “One can say without more ado: the transfinite numbers stand or fall with the finite irrational numbers; they concur with regard to their innermost being; since both are particular delimited expressions or modifications of the actually infinite” (1962:395-396). Paul Lorenzen describes this modern conception of real numbers in terms of the completed infinite in a way which strikingly reflects the age-old tradition on which it rests: “and thus every real number as an infinite decimal fraction is already represented as if the infinite quantity of numbers all existed at once (auf einmal existierten)” (1968:100).

Cantor and Aristotle In the elaboration of his set theory Cantor takes critical distance from Aristotle's radical rejection of the use of the actually infinite, although he otherwise essentially retains in his description of the continuum the two stipulations which Aristotle posited for the nature of continuity. (a) Aristotle's objections against the actual infinite Aristotle's first objection, namely that the whole could not in the case of the actually infinite exceed the part in magnitude, had already been used by Bolzano (analogously to Galileo) exactly as criterion for infinite sets: a set is infinite if and only if the whole set can be mapped one-to-one with a true subset thereof. In this characterization lies the answer to the first part of Aristotle's second objection, the objection namely that it is impossible to count the infinite. Cantor however only uses the characterization: denumerable. Any set which can be correlated one-to-one with the natural numbers (0), 1, 2, 3, 4, 5, ... is denumerable. The true meaning of the indication “denumerable” however only became apparent when Cantor proved that there does indeed exist non-denumerable transfinite numbers.1 When one relinquishes the nature and all relations of the elements of a set (particularly also the ordering which might exist among the various elements), then the power (or: cardinality) of different sets can be compared. Two sets M and N are referred to by Cantor as equivalent when their elements can be mapped one-to-one (“wenn sie sich gegenseitig eindeutig Element für Element einander zuordnen lassen”) (1962:387). It can be indicated with the term 1

Cantor considers it meaningless to reject references to an infinite number and to speak only of infinite sets – both are according to him indissolubly linked to each other (1962:394).

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order that for every two elements of a set the first precedes the second, while the second follows on the first. If A and B are two ordered sets, and if an order-preserving relation exists among their elements, both have according to Cantor the same order-type, while the sets are referred to as similar (ähnlich) (1962:297). “A set is called well-ordered, if it meets the requirement that every subset (Teilvielheit) has a first element”.1 In view of this Cantor defines an ordinal number: “The order-type of a well-ordered set F we call its due ordinal number” (1962:321). In view of the preceding we can state that the transfinite ordinal number (w) must meet the following four conditions (strictly speaking the last condition is equivalent to the second): (i) It has a first element; (ii) each element has an immediate successor; (iii) each element except the first element has an immediate predecessor; and (iv) no last element exists. The remarkable characteristics of this transfinite ordinal number w (w indicates the set of natural numbers in their natural order: 1, 2, 3, 4, 5, 6, ...), enables Cantor to answer Aristotle's last objection (namely that actually infinite numbers would have to be neither even nor uneven) (cf. Cantor, 1962: 178-179). Within the system of natural numbers, which is closed under addition and multiplication (i.e. adding and multiplying natural numbers in every instance renders natural numbers), the commutative law is valid with regard to both these operations. In other words, with regard to two natural numbers a and b, a + b = b + a and ab = ba. This commutative law is not, however, generally valid with regard to transfinite ordinal numbers. We can illustrate it as follows (note however that Cantor in this example understands in the product ba that b is the “multiplier” and a the “multiplicandus” – 1962:178): Take the set A = (1,2) and B = (1, 2, 3, 4, ,5 6, ...) and consider the (lexicographically) ordered product AB; i.e. {(1,1),(1,2),(1,3), ...,(2,1),(2,2),(2,3), ..} or w + w = w.2 (i.e., w ¹ w.2)

The ordered product BA however gives us: {(1,1),(1,2),(2,1), (2,2),(3,1), ..} This product clearly meets the four requirements, which implies that 2.w = w. From the product AB we saw that w ¹ w.2: The number w can therefore be represented as 2.w and also as 1 + 2.w – but never the other way around, since w ¹ w.2 and w ¹ w.2 + 1. The conclusion is clear: w is even (namely 2.w), as well as uneven (namely 1 + 2.w) and simultaneously w is neither even (namely ¹ w.2), nor uneven (namely ¹ w.2 + 1 – cf. Cantor, 1962:178-179). Compare this with the notion of Cusanus that God as the actually infinite is the union of all opposites: the coincidentia oppositorum – he did after all recognize something essential about the infinite! 1

This description is to be found in a letter to Dedekind (28 July 1899), contained in Cantor, 1962:444 (cf. also Cantor, 1962:312).

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(b) Continuity in Aristotle and Cantor-Dedekind Aristotle considers it impossible to explain the continuity of a straight line in terms of the (infinite) number of its points. If “that which is infinite is constituted by points, these points must be either continuous or continuously in contact with one another” (Physica, 231a29-31). Points are however indivisible (a point has no parts), while “that which exists between two points is always a line” (Physica, 231b8). According to Aristotle it is clear that “everything which is continuous is divisible into divisible parts which can be divided infinitely: since if it was divisible into indivisible parts, we would have the divisible and indivisible in contact since the limits of continuous things are one (i.e. the same – DFMS)” (Physica, 231b15ff.).1 In the 19th century a new arithmeticistic tendency came to the fore in mathematics which would appear to be intent on the arithmetical definition of spatial continuity. Bernard Bolzano already illuminates this tendency in par.38 of his (quoted) work on the paradoxes of the infinite. He mentions the objection that a circle would appear to be hidden in the attempt to build extension out of parts not themselves extended, but is of the opinion that the problem disappears when it is realized that “each whole” exactly “has numerous properties absent in the parts” (1920:72). The question however is whether the relationship: whole-parts is originally arithmetical in nature?! (We will return to this again.) The criterion which Bolzano sets for a continuum is that “a continuum is present there, but also only there where a set (Inbegriff) of simple objects (of points ...) finds itself, which is situated in such a way that every single object has at least an environment (Nachbar) (of points – DFMS) in this set for every distance (Entfernung) however small” (1920:73). Cantor criticizes Bolzano's criterion as insufficient, since a set constituted e.g. by distinct continua (and therefore being as a whole discontinuous) would still be continuous in terms of Bolzano's definition (the end points of each distinct continuum would after all still contain further points of the particular set in an arbitrarily small environment – cf. Cantor, 1962:194). Cantor declares that he has no other choice but to posit a “general purely arithmetical concept of a point-continuum” with the help of the way in which he defined real numbers (1962:192). A point-continuum he defines as a perfectly coherent set. A set is perfect when every point of the set is a limit-point and when all limit-points of the set belong to the set. He calls “T a coherent point set, when for every two points t and t’ of this set, at a given arbitrarily small number e there are always a finite number of points t1, t2, t3, ..., tn of T’ present in multiple ways, so that the distances tt1, t1t2, ...tnt’ are all together smaller than e” (1962:194). 1

Von Weizsäcker says that the domain in which figures are defined displays, when compared with natural numbers, the property of continuity. In this context he calls upon Aristotle's view of continuity: “Continuity is defined by Aristotle as that which could be divided endlessly in similar parts. The parts of a continuum cannot be counted; but one can measure continua” (1993:115).

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Comment: Cantor's definition of coherence concerns a metrical characteristic of the continuum. In modern topology, however, continuity is described in terms of open sets (abstracted from the characteristics of a metrical space – cf. Willard, 1970:16-19). P.S. Alexandroff defines the continuum as a non-empty compactly coherent set (1956:163ff., 201ff.). A set is compact if every infinite subset has at least one limit-point. (A point x is a limit-point of a set A when every environment of x contains at least one point of A different from x). This implies that a set in a Euclidean space is only compact if it is delimited. In terms of Alexandroff's definition an infinite straight line is therefore not continuous, while it is for Cantor (also cf. Meschkowski, 1967:55). Dedekind follows the continuity of a straight line arithmetically by time and again introducing new numbers: “If now, as is our desire, we try to follow up arithmetically all phenomena in the straight line, the domain of rational numbers is insufficient and it becomes necessary that the instrument R constructed by the creation of rational numbers be essentially improved by the creation of new numbers (namely irrational numbers – DFMS) such that the domain of numbers shall gain the same completeness, or as we may say at once, the same continuity, as the straight line” (Dedekind, 1901:9)

On this foundation Dedekind describes his well-known cut notion which characterizes continuity: when “all real numbers break up into two classes U1, U2, such that every number a1 of the class U1 is less than every number a2 of the class U2 then there exists one and only one number by which this separation is produced” (1901:20).

Dedekind's notion of a cut is dealt with in analysis textbooks in such a manner that the real number which brings about the “split” is greater than or equal to every element in the one set and smaller than or equal to all the elements of the other set (cf. e.g. Bartle, 1964:51). Cantor himself refers to the relation which exists between his view of a perfect set and Dedekind's cut theorem (1962:194). G. Böhme strikingly shows in an article how Cantor's definition of the continuum contains two stipulations which both meet the Aristotelian definition of a continuum, namely coherence and a characteristic which ensures the existence of dividing points for infinite division (1966:309). By means of only allowing a Dedekind-cut at divisions, Böhme justifies his statement as follows: “when a Cantorian continuum as such is divided in two by means of the indication of a point so that the one set contains those points which are in numerical value greater than or equal to the indicated point, while the other set contains those points of which the numerical value are smaller than or equal to the numerical value of the indicated point, both parts are again continuous. Such divisions are possible into infinity (due to the perfection of the continuum), and the parts are still coherent in the Aristotelian sense (i.e. their limit-points are the same)” (1966:309).

This is a remarkable situation: the Cantor-Dedekind description of the continuum presupposes the use of the completed infinite (in particular by using the 35

completed infinite set of real numbers), but nonetheless meets Aristotle's two requirements for a continuum – and this while Aristotle exactly rejects the completed infinite and recognizes only uncompleted infinity! Did Aristotle actually use the completed infinite implicitly, or is the Cantor-Dedekind definition in the last instance not purely arithmetically founded? We shall shortly attempt to find an answer to this question. Non-denumerability: Cantor's Diagonal Proof A set is called (d)enumerable when its elements can be correlated one-to-one with those of the set of natural numbers, i.e. any set the elements of which can be arranged in a natural sequence of 1, 2, 3, 4, 5, 6, ... It is clear that the integers are denumerable: 0, -1, +1, -2, +2, -3, +3, ... Since all rational numbers can be depicted by two integers in the form of a/b (with b ¹ 0), it is clear that they also can be denumerated. Notice the course of the arrows in the following depiction: 1 1

1 2

1 3

1 ............. 4

2 1

2 2

2 3

2 ............. 4

3 1

3 2

3 3

3............. 4

4 1

4 2

4 3

4............. 4

...........................

Even all algebraic numbers are denumerable.1 In a letter of 29 November 1873 Dedekind mentions to Cantor that he had proven that all algebraic numbers are denumerable (cf. Meschkowski, 1972b:23). Dedekind does this by defining the height h of an algebraic number x satisfying a polinomial equation anxn + an-1xn-1 + ...+ a1x + a0 = 0 as follows: h = n – 1 + |a0| + |a1| + .......... + |an| Since the coefficients an are integers, only a finite number of algebraic numbers belong to each height h. Since every finite quantity is denumerable, the algebraic numbers as such are also denumerable (cf. Meschkowski, 1972b:24). In 1874 however Cantor proved that the real numbers are not denumerable (i.e. are non-denumerable). Only in 1890 does he provide his diagonal-proof, which we use in our explanation below (cf. Cantor, 1962:278-281). A one-to-one correspondence could be established between all real numbers and the set of real numbers between 0 and 1. Furthermore, every real number in 1

Algebraic numbers are the roots of algebraic equations.

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this interval can be represented as an infinite decimal fraction of the form xn = 0.a1a2a3a4 ... (numbers with two decimal representations, e.g. 0.100000... and 0.099999 ... are consistently represented in the form with nines). Suppose a denumeration x1, x2, x3, ... exists of all the real numbers between 0 and 1, i.e. of all the real numbers in the interval 0 £ xn £ 1 (i.e. [0,1]), namely: x1 = 0.a1 a2 a3... ............ x2 = 0.b 1 b 2b3 ............ x3 = 0.c1 c2 c3 ........... ................................... If another number can be found between 0 and 1 which differs from every xn, it would mean that every denumeration of the real numbers would leave out at least one real number, which would prove that the real numbers are non-denumerable. Such a number we can construe as follows: y = y1y2y3y4 ..., with y1 ¹ 0, a1 and 9; y2 ¹ 0, b2 and 9; y3 ¹ 0, c3 and 9; and so forth. It is clear that y is a real number between 0 and 1 (i.e. 0 £ y £ 1). The number y does not have two decimal representations since every decimal number in its decimal development is unequal to 0 and 9. The number is also unequal to every real number xn since the decimal development of y in the first decimal place differs from the first decimal number x1, in the second differs from the second decimal number of x2 (namely x2), and in general from the nth decimal number of xn. It is clear from this that a denumeration of the real numbers will always exclude at least one real number (“miscount” it in the denumeration), which concludes Cantor's proof that real numbers are non-denumerable. Comment Although intuitionism accepts this proof as valid, it does so in a constructivist sense.1 All constructivist interpretations are however inadequate to reach a non-denumerable conclusion (cf. Wolff, 1971), simply because no constructive transition is possible from the potential infinite to the actual infinite. Our current discussion does not provide room for a broader indication that the approach of Brouwer (already in his 1907 dissertation) and Heyting indeed is ambiguous with regard to the role of the actual infinite. The same can after all be said of Poincaré – on the one hand he rejects the actual infinite (1910) and on the other he attempts in the same article to provide an alternative proof for the non-denumerability of real numbers – without realizing that non-denumerability can only be proven if the actually infinite is accepted. There is a further remarkable side to this result. In his original proof of 1874 (1962:115-118) Cantor first proved that all real algebraic numbers are 1

Cf. Heyting (1971:40), Fraenkel et al (1973:256,272), and Fraenkel (1928:239 note 1).

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denumerable. All real numbers, however, are non-denumerable, which proves the existence of a non-denumerable number of non-algebraic (also called: transcendent) numbers!1 The third foundational crisis in Mathematics The first two foundational crises in mathematics were the result of the discovery of irrational numbers and the founding of the so-called infinitesimal calculus.2 By 1895 Cantor discovered that his set theory contained anomalies. Cantor proved e.g. the proposition that for every set A of ordinal numbers an ordinal number exists which is greater than every ordinal number contained in the set. Consider however the set W of all ordinal numbers. Since this set is a set of all ordinal numbers, the foregoing proposition implies that an ordinal number exists which is greater than every ordinal number contained in W – but this is contradictory, since the set W is supposed already to contain all ordinal numbers! A similar antinomy is valid with regard to Cantor's cardinal numbers (cf. Meschkowski, 1967:144-145 and Singh, 1985:73). In 1900 Russell made public his antinomy which can be formulated in terms of the ABC of set theory. Consider the set C with elements A and the prescription that elements of set C may only be those sets A which do not contain themselves as elements. Thus C = (A/A ÏA). (The set of ten chairs is e.g. not itself a chair and does not contain itself as an element. On the other hand the set of thinkable thoughts is in itself thinkable and therefore does contain itself as an element.) Now suppose that C is an element of C (C ÎC). Every element of C, however, does not contain itself as an element – this, after all, is the requirement for being an element of C. This implies that if C is an element of C, it must also meet this requirement – but then C Î C implies C Ï C! Suppose on the other hand that C Ï C. Then C does meet the requirement for being an element of C, which means that C ÎC! In other words, C is an element of C if and only if C is not an element of C! C Î C Û C Ï C! In 1900 the French mathematician, Poincaré, made the proud claim that mathematics has reached absolute rigour. In a standard work on the foundations of set theory, however, we read: “ironically enough, at the very same time that Poincaré made his proud claim, it has already turned out that the theory of the infinite systems of integers – 1 2

Just by the way: Cantor developed a whole hierarchy of transfinite cardinal numbers on the foundation of his definition of ordinal numbers. Kline writes: “It was clear to the mathematical world of the late 18th century that proper foundations for the calculus were urgently needed, and at the suggestion of Lagrange the Mathematics section of the Berlin Academy of Sciences, of which he was the director from 1766 to 1787, proposed in 1784 that a prize be awarded in 1786 for the best solution to the problem of the infinite in mathematics” (Kline, 1980:149-150).

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nothing else but part of set theory – was very far from having obtained absolute security of foundations. More than the mere appearance of antinomies in the basis of set theory, and thereby of analysis, it is the fact that the various attempts to overcome these antinomies, ..., revealed a far-going and surprising divergence of opinions and conceptions on the most fundamental mathematical notions, such as set and number themselves, which induces us to speak of the third foundational crisis that mathematics is still undergoing” (Fraenkel, 1973:14).

Divergence of opinion Although the divergent opinions which emerged cannot be seen simply as a mere reaction to the antinomies (since their roots reach much further back into history), the flowering of three schools – logicism, intuitionism, and (axiomatic) formalism – during the first decade of the 20th century are nonetheless inextricably linked to it. Implicitly or explicitly every point of view in mathematics must account for the relationships among the various aspects of reality, including number, space, movement, the logical and lingual facets. Bertrand Russell's logicism wanted to reduce all of mathematics to logic – for this reason he declares: “mathematics and logic are identical” (1956:v). A. Heyting expresses the exactly opposite intuitionist sentiment: “every logical theorem ... is but a mathematical theorem of extreme generality; that is to say, logic is a part of mathematics, and can by no means serve as a foundation for it” (Heyting, 1971:6). Even Hilbert's formalism must (under the influence of Kant) begin with the recognition of a more than logical diversity which implies that no science can be exclusively founded in logic: “Kant already taught – and it represents an integral part of his doctrine – that mathematics has a guaranteed content independently of all logic and can therefore never be grounded solely in logic, which implies that the efforts of Frege and Dedekind must fail. There is a further prerequisite for the application of logical conclusions and for the performing of logical operations, namely that something must be given in the conception: specific extra-logical concrete objects, intuitively present as immediate experience prior to all thinking” (Hilbert, 1925:170-171). According to the formalistic-axiomatic approach mathematics should completely relinquish the truth of the initial postulates (axioms) and only take account within the requirement of consistency (non-contradiction) that the valid theorems can be deduced analytically.1 Since, according to this point of view, it does not matter what we are talking about, as long as we are doing so consistently, Russell formulated his famous epigram: “Pure mathematics is the subject in which we do not know what we are talking about, or whether what we are saying is true” (quoted by Nagel, 1971:13). Already in 1926 P. Finsler showed that in a pure formal mathematical discipline, defined by axioms and rules of calculus, there are propositions which can be neither proven nor contradicted (cf. Mathematische Zeitschrift, 25 (1926); Finsler, 1975:1-49; as well as Heitler, 1972:50). But in 1931, at the 1

The distinction between analytical and synthetic judgments is used in a Kantian sense by the various approaches in mathematics. Cf. Kant, 1787:11.

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age of 25, K. Gödel shook the world of mathematics with an article on the formally undecidable propositions in the Principia Mathematica of Russell and Whitehead, and related systems. Gödel showed that a proof of the consistency of arithmetic cannot be reflected in the formal deductions of arithmetic itself – the consistency of arithmetic, that is, cannot be proven in terms of the axioms of arithmetic. In a formal axiomatic system Z there always exists a statement A which can be neither proved nor disproved with the aid of axioms of Z. In other words, to prove that the conclusions reached from certain axioms are consistent, it is not possible to use the method in question. In principle every axiomatic system in mathematics is incomplete – it requires and presupposes insight into its content which transcends its own formalism. H. Weyl comments strikingly in this regard: “It must have been hard on Hilbert, the axiomatist, to acknowledge that the insight of consistency is rather to be attained by intuitive reasoning which is based on evidence and not on axioms” (1970:269). Although this divergence in modern mathematics relates to the third foundational crisis of mathematics, its philosophical sources are much older. Historically seen, Brouwer, Gödel, and Hilbert derive their philosophical points of departure from the three main parts of the Kritik der reinen Vernunft (1781, 2nd edition 1787) of Kant: Brouwer from the transcendental aesthetics, Gödel from the transcendental analytic, and Hilbert from the transcendental dialectic. This explains why even in its second and third foundational crises mathematics still could not escape from the fundamental philosophical problems already present in the foundational crisis of Greek mathematics. Problems surrounding differential and integral calculus (the so-called infinitesimals) led to the reformulation of the limit concept, which in its turn brought about the use in set theory of the actually infinite, with the eventual exposure of the antinomies in the naive set concept of Cantor. Hilbert and Bernays point out that the failure of Frege's logicistic project particularly exposed the problems in the presupposition of the totality of numerical sequences (Grundlagen, I, 1934:15). Exactly in this regard neo-intuitionism chose for the uncompleted infinite, while axiomatic set theory (fathered by Zermelo and Fraenkel) attempted by means of all sorts of limitations to protect set theory against antinomies without sacrificing the actual infinite. We shall argue below that the use of the uncompleted infinite presupposes the numerical order of succession on the law-side of the numerical aspect and that when this numerical time-order is disclosed by anticipating forward to the spatial order of simultaneity any uncompleted infinite set (such as the set of natural numbers, whole numbers, or rational numbers) can be viewed as if all the elements are at hand at once as a complete totality (‘fertige Gesamtheit’ in the words of Hilbert and Bernays). The relationship between these two kinds of infinity, which has demanded attention anew since the third foundational crisis of mathematics (it had already been at issue in Greek thought), ultimately appeals to one of the fundamental philosophical issues of mathematics as a special science: what is the relation and coherence between the numerical and spatial aspects of reality? The emphasis in reformational philosophy on 40

the cosmic impact of the creational principle of sphere-sovereignty therefore retains its relevance for mathematics with regard to the mutual irreducibility of the numerical and spatial aspects. Within the ambit of this study the purpose has only been to illuminate the importance of philosophical reflection also in mathematics, with reference to the foundational crises in mathematics. Particularly against this background the following statements gain significance: “from the earliest times two opposing tendencies, sometimes helping one another, have governed the whole involved development of mathematics. Roughly these are the discrete and the continuous” (Bell, 1965:12);

and “Bridging the gap between the domains of discreteness and of continuity, or between arithmetic and geometry, is a central, presumably even the central problem of the foundation of mathematics” (Fraenkel, A., et al., 1973:211).

Rucker elsewhere remarks: “The discrete and continuous represent fundamentally different aspects of the mathematical universe” (Rucker, 1982:243).

Also Moore discerns two ‘clusters’ of concetps which dominates the history of the notion of infinity. In the first cluster the following terms are categorized: “boundlessness; endlessness; unlimitedness; immeasurability; eternity; that which is such that, given any determinate part of it, there is always more to come; that which is greater than any assignable quantity” (1990:1). Within the second cluster he mentions: “completeness; wholeness; unity; universality; absoluteness; perfection; self-sufficiency; autonomy” (Moore, 1990:1-2).

An interesting development deserves to be mentioned before we conclude this discussion. A. Fraenkel indicated the infertility of the idea of “infinitesimals” (the infinitely small) and its rejection by Cantor and the mathematical world at large in the fourth printing of his Abstract Set Theory (Amsterdam 1968: 120-123). In the meantime, however, Abaraham Robinson developed a new and fertile use of the infinitesimal exactly on the basis of the use of actually infinite sets (transfinite cardinalities) by Cantor. A number a is called infinitesimal (or infinitely small) if its absolute value (that is its value regardless of the plus or minus sign) is less than m for all positive numbers m in Â(Â being the set of real numbers). According to this definition 0 is infinitesimal. The fact that the infinitesimal is merely the correlate of the transfinite numbers, is apparent in that r (not equal to 0) is infinitesimal if and only if r to the power of minus 1 is infinite (cf. Robinson, 1966:55ff). By means of the infinitesimal it is possible to meaningfully define limits, derivatives, etc. Robinson points out that Cantor's treatment of infinite sets is necessary: “... abstract set theory forms a historical background to the free and easy handling of infinite sets that is required in Non-standard Analysis” (p. 279). To this he adds that “whatever our outlook ...., it appears to us today that the infinitely small and infinitely large numbers of a non-standard model of Analysis are neither more nor less real than, for example, the standard irrational numbers” 41

(p. 828).1 And rightly so, since both infinitesimal and irrational numbers after all exist by the grace of the deepened sense of number which, as we shall presently argue, provides (at the law-side to the quantiative aspect) mathematical meaning to the completed infinite. The consideration of these fundamental problems will always remain relevant, as is confirmed by the statements of Fraenkel et al. and Kline at the beginning of our exposition. Herman Weyl posits against this that “it appears from the intuitionist perspective that (complete) induction2 protects mathematics from being entirely tautological and characterizes its assertions as (non-analytically) synthetic” (1966:86). Following the project of Kant's critique of Pure Reason (1787:19) Weyl here defends the existence of synthetic judgments a priori in mathematics. Admittedly he (and Brouwer) accept only Kant's notion of time. Number, according to Kant, is due to the schematization of the logical category quantity in time as a form of intuition (1787:182); it belongs to the synthetic judgments of mathematics a priori. At his promotion in 1885 D. Hilbert defended the a priori character of numerical judgments: Thesis II: “the objections against Kant's theory of the a priori character of arithmetical judgments are unfounded” (cf. Reid 1970:17). Brouwer bases his mathematics on the primordial intuition of continuity and discreteness, a possibility of thinking together multiple units combined by an ‘in-between’ which cannot be exhausted through the ‘inter-positioning’ of new units.3 Brouwer's description of the primordial intuition already contains the rejection of the actual infinite by intuitionism – our next theme. Questioning completed infinitude In intuitionist mathematics the infinite is taken literally as without an end, never to be completed and continually becoming. Already in his letter of July 12, 1831 to Schumacher Gauss stated that “in this manner I protest against the use of an infinite magnitude as something completed, which is never allowed in mathematics” (referred to in Becker 1964:180). The early intuitionist, Leopold Kronecker, a contemporary and opponent of Cantor, radically rejected the completed infinite and even attempted to base all of mathematics in 1

2 3

As mentioned, Cantor also referred to the irrational numbers with regard to his transfinite numbers: “One can simply say that the transfinite numbers stand or fall with the finite irrational numbers; they are similar in their innermost being” (GA, pp. 395-396). If an assertion is true for a number 1 and also for n + 1 if it is true for n, then it is generally true (a principle first demonstrated by Blaise Pascal, according to Freundenthal, 1940). “... als het van qualiteit ontdane substraat van alle waarneming en verandering, een eenheid van continu en discreet, een mogelijkheid van samendenken van meerdere eenheden, verbonden door een tusschen, dat door inschakeling van nieuwe eenheden zich nooit uitput” (1907:8). Both this mathematical primordial intuition of Brouwer and the immediate extraand pre-logical intuitive experience of Hilbert is seen in a Kantian sense as a priori of science in general and mathematics in particular. Brouwer however differs from Hilbert's distinction between formalized mathematics and the intuitive interpretation thereof (the latter is also referred to as metamathematics) (cf. Beth, 1965:94). Admittedly an intuitive mathematics is needed even to define a formal mathematics (cf. Kleene, 1952:62)!

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the finite natural numbers (cf. Scholz 1969:293-294). The early French intuitionist, H. Poincaré (known in life as the greatest mathematician of his age until his death in 1912 when Hilbert took over his laurels), also expressly rejected the completed infinite. The totality of ordinal numbers of the smallest transfinite cardinal number (aleph-null: Ào) were used by Cantor to obtain the next transfinite cardinal number, aleph-one (À1): “With regard to the second transfinite cardinal number aleph-one, I am not entirely convinced of its existence ... (and) whether we can speak of its cardinality without contradiction. The actual infinite in any case does not exist” (Poincaré, 1910:48). Also Brouwer (who identifies existence and constructibility and denies the validity of the logical principle of the excluded third with regard to the infinite) rejects the completed infinite.1 With this Brouwer rejects the transfinite number theory of Cantor. The notion of countability after all only becomes particularly relevant after the demonstration of the existence of non-denumerable cardinalities. Did Cantor not demonstrate that the set of real numbers is non-denumerable? In Cantor's diagonal proof it is assumed that all (i.e. the completed infinite set of) real numbers are correlated one-to-one with the set of natural numbers, after which it is demonstrated that a further real number can be presented which differs from each of the counted real numbers (in at least one decimal place), from which the non-denumerability of the real numbers is concluded. The validity of this conclusion depends, however, on the acceptance of the completed infinite. Someone who recognizes only the uncompleted infinite can never accept this conclusion, since the diagonal method then only proves that for a given constructible sequence of countable sequences (i.e. decimal expansions of real numbers) of natural numbers, yet another different countable sequence of natural numbers can be construed. Becker states this in the following way: “The diagonal method demonstrates, strictly speaking, the following: when one has a counted (law-conformative) sequence of successive numbers, a sequence of successive numbers can be calculated which differs in every place from all the previous ones” (1973:161 footnote 2). In this interpretation there is nowhere mention made of non-denumerability! A mathematical proof which apparently takes an “exact” course therefore comes to conflicting conclusions depending on the presuppositions (namely completed infinity or uncompleted infinity) from which one proceeds! Fraenkel points this out emphatically: “Cantor's diagonal method does not become meaningless from this point of view, ... the continuum (i.e. the real numbers – DFMS) appears according to it as a set of which only a countable infinite subset can be indicated, and this by means of pre-determinable constructions” (1928:239 footnote 1).

Whoever rejects the completed infinite cannot accept the description of real numbers given by Dedekind, Weierstrass, and Cantor. Paul Lorenzen explained that a real number can be presented in terms of the completed infinite 1

“immers de intuitionist kan geen andere, dan aftelbare wiskundige verzamelingen construeeren” (1919:24).

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as an infinite decimal fraction of which the infinite multiplicity of numbers all exist at once (i.e. as an infinite totality). Cassirer already fiercely indicated in 1910 that the concept of a cardinal number depends on a simultaneously given totality.1 Felix Kaufmann is however of the opinion that “it is generally recognized that an infinite decimal fraction signifies nothing else but a sequence of natural numbers, where ... by a sequence is meant not an infinite totality, but merely the domain of a particular relation ” (1968:122-123). With regard to an irrational number Ludwig Fischer writes: “Every representation of 2, whatever its nature might be, can only be taken as an infinite and completely ‘unfinishable’ sequence of rationally approximating values. Only once the in itself contradictory fiction of the completed infinite is added, can the infinite decimal fraction be considered an instance of 2. Without the contradictory notion of the completed infinite the concept of an irrational number cannot be formed” (1933:108). Brouwer and Weyl define real numbers in terms of the uncompleted infinite, namely as uncompleted infinite choice sequences or as a medium of free realization. Weyl describes a real number as follows: “A single real number can be described as an infinite sequence of fractional intervals of growing magnitude, where each is contained in the following one in the sequence” (1966:74-75). Fraenkel, Bar-Hillel, Levy and Van Dalen comment: “The conception of the continuum as an aggregate of existing points (members), which is at the bottom of nineteenth century analysis and of Cantor's set theory, is replaced by an aggregate of parts which are partially overlapping and which are so to speak the manifestations of real numbers still to be generated” (1973:256). Intuitionism also sees the continuum differently: “In agreement with intuition Brouwer sees the essence of the continuum not in the relation of the element to the set, but in that of the part to the whole” (Weyl 1966:74). Following Aristotle Weyl also claims that: “It rather belongs to the essence of the continuum that each of its parts are infinitely divisible” (1921:77). In opposition to the notion of a continuum of points Weyl indicates that the concept of environment must still be used to salvage continuity: “To render a continuous coherence of points analysis has up to today (since it decomposed the continuum into a set of isolated points) found refuge in the concept of environment” (1921:77). It should therefore come as no surprise that even a non-intuitionist such as Paul Bernays – the well-known colleague of David Hilbert – sharply rejects the supposed successes of arithmeticism in mathematics. 1

“Wenn in der Theorie der Ordnungszahl die Einzelschritte als solche festgestellt und in eindeutiger Folge entwickelt wurden, so tritt jetzt die Forderung ein, die Reihe nicht nur nacheinander in ihrer einzelnen Elementen, sondern als ideelles Ganzes zu erfassen. Das vorangehende Moment soll durch das folgende nicht in ihm aufbehalten bleiben, so dass der letzte Schritt des Verfahrens zugleich alle vorhergehenden und das Gesetz, das die wechselseitig verknupft, in sich fasst. Erst in dieser Synthese vollendet sich die blosse Folge der Ordnungzahlen zum einheitlichen, in sich geschlossen System, in welchem jedes Glied nicht nur für sich steht, sondern zugleich den Aufbau und das formale Prinzip der Gesamtreihe repräsentiert” – 1910:55.

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Elsewhere Weyl declares: “The sequence of numbers which grows beyond any stage already reached by passing to the next number, is a manifold of possibilities open towards infinity; it remains for ever in the status of creation, but is not a closed realm of things existing in themselves. ... Brouwer opened our eyes and made us see how far classical mathematics, nourished by a belief in the absolute that transcends all human possibilities of realization, goes beyond such statements as can claim real meaning and truth founded on evidence”. Earlier on the same page he clearly states: “Brouwer made it clear, as I think beyond any doubt, that there is no evidence supporting the belief in the existential character of the totality of all natural numbers” (1946:6). In intuitionist mathematics many subdivisions of classical mathematics no longer serve. Hilbert values e.g. Cantor's transfinite number theory “as the most wonderful flourishing of a mathematical spirit and as such one of the highest achievements of pure human intellectual activity” (1925:167), while A. Heyting considers transfinite number theory as no more than a phantasm (1949:4). In his work: Das Kontinuum, Weyl worked on the foundations of analysis and reconstructed large parts in terms of intuitionism, although he had to admit that he had to sacrifce the theorem “every delimited set of real numbers has an upper limit” (1932:23-24). Intuitionism in fact constructed an entirely new mathematics: “The intuitionists have created a whole new mathematics, including a theory of the continuum and a set theory. This mathematics employs concepts and makes distinctions not found in the classical mathematics” (Kleene 1952:52). E.W. Beth also comments: “It is clear that intuitionistic mathematics is not merely that part of classical mathematics which would remain if one removed certain methods not acceptable to the intuitionists. On the contrary, intuitionistic mathematics replaces those methods by other ones that lead to results which find no counterpart in classical mathematics” (1965:89). The influence of intuitionism on the approach of Dooyeweerd Both Brouwer and Dooyeweerd only acknowledge the potential infinite – as a law of progression: “A set is a law on the basis of which, when a new value is chosen repeatedly, a determined sequence of signs is generated for each one of these choices ... with or without terminating the process” (Brouwer, 1925:244). Accordingly, also Dooyeweerd considers an infinite sequence of numbers as only determined “by the law of arithmetical progression.” This makes it possible “to determine the discrete arithmetical value in arithmetical time of any finite numerical relation in the series. For the rationalist conception of law this is sufficient reason to attribute actual completed infinitude to the series as a totality” (NC, II:92). The basic mistake in the idea of the actual infinite, according to Dooyeweerd, is a confusion concerning the distinction between law-side and factual side: “Numbers and spatial figures are subject to their proper laws, and they may not be identified with or reduced to the latter. This distinction is the subject of the famous problem concerning the so-called ‘actual infinity’ in pure mathematics. The principle of progression is a mathematical law which holds for an infinite series of numbers or spatial figures. 45

But the infinite itself cannot be made into an actual number” (1997-I:98-99 note 1).1 Brief systematic assessment of the relationship between the potential and the actual infinite In spite of the fact that Dooyeweerd found the idea of the actual infinite unacceptable, his philosophical theory of the modal aspects of reality does provide the starting-point for a new and non-reductionistic explanation of the nature of this kind of infinity. According to his general theory of modal law-spheres, each aspect has a law-side and a correlated factual side. Within the modal structure of an aspect their are ‘modal meaning-moments’ that establish the coherence with earlier aspects (called retrocipations) and ‘modal meaning-moments’ maintaining the coherence with later aspects in the cosmic order (called anticipations) (1997-II:75). To this original conception Dooyeweerd adds his equally unique idea of time. Time is no longer identified with the physical aspect of reality but is seen as a distinct dimension guaranteeing the temporal order of succession between the different (modal) aspects of creation. Furthermore, according to Dooyeweerd, time expresses itself within the boundaries of every aspect by ‘taking on’ the nature of the aspect concerned. It is thus differentiated into a modal time-order and a modal time-duration (1997-I:28). Every structural element of each modal aspect is qualified by its ‘meaning-nucleus’ or ‘primitive meaning’ which also stamps the way in which cosmic time evinces itself within the aspect concerned. The anticipatory analogies of an aspect need to be opened up, to be disclosed. Within the not-yet-opened-up meaning of the numerical aspect we discover the original and basic meaning of infinity as it manifests itself at the law-side of this modality: infinity taken in the literal sense of endlessness. This primitive meaning of the infinite is an expression of the arithmetical time-order of succession which not only lies at the basis of the principle of [mathematical] induction but which also determines every countable (denumerable) endless (i.e., potentially infinite) succession of numbers (be it the natural numbers, the integers, or the rational numbers). In stead of using the characterization ‘potential infinite’ we would prefer to use an expression that would reflect the determining role of the numerical time-order of succession at the law-side of the arithmetical aspect: the successive infinite. Intuitionism only acknowledges this primitive meaning of the infinite. It is seen as the product of the free and creative power of the mathematician (Brouwer, 1952:140-142). As soon as the successive infinite meaning of the numerical time-order is disclosed through the anticipation from number to space, the meaning of successive infinity is deepened by the meaning of the spatial time-order of simultaneity. Any successive sequence of numbers could then, under the guidance of this anticipatory hypothesis, be viewed as if its elements are given all at once. The deepened and disclosed meaning of the infinite encountered here could be designated as the at once infinite. Under the 1

De Swart remarks that for Brouwer's intuitionistically constructed spreads different conditions hold which are not valid for other sets (1989:41).

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guidance of this hypothesis the initial successive infinite sequences of natural numbers, integers and rational numbers could be viewed as actual infinities, i.e., as infinite totalities given at once. Ultimately, the spatial order of at once (simultaneity) forms the basis of the age-old legacy concerning the actual (at once) infinite as something without before and after (Augustin), connected with the timeless at once (Maimon) or with the ‘all existing at once’ (Lorenzen). Cantor referred to a constant ‘fixed and determined in all its parts’. An intuitive example might help to clarify what is at stake in the idea of the at once infinite. Employing the idea of the at once infinite one can, in an anticipating way, correlate the natural numbers with the points of a straight line – for example, between 1 and 0, with the aid of the rational number sequence 1/n where n ranges over the set of all natural numbers. This mode of speech only has meaning under the guidance of the regulative hypothesis of the original spatial time-order of simultaneity by means of which the number concept of the successive infinite is deepened and disclosed to the number idea of the at once infinite.1 In Kantian fashion David Hilbert even remarks: “The role that remains for the infinite to play is solely that of an idea – if one means by an idea, in Kant's terminology, a concept of reason which transcends all experience and which completes the concrete as a totality – that of an idea which we may unhesitatingly trust within the framework erected by our theory” (1925:190 – this article is translated in Benacerraf & Putnam, 1964 – compare p.151). Modern set theory claims to define the ‘continuum’ purely in arithmetical terms, i.e., in terms of the actually infinite set of real numbers which is, due to Cantor's well-known diagonal proof, non-denumerable. In opposition to this Ludwig Fischer declares: “Between distinct (non-coalescing) points continuity is present ... under all circumstances, therefore, for each individual point, the law holds: continuity-point-continuity” (1933:86-87). Since the measure of each single point is zero, it apparently speaks for itself that every denumerable set of points would have measure zero. However, in the case of non-denumerable sets, addition is not defined (if one cannot enumerate the elements of a set, one cannot add them). This seemingly allows for the arithmeticistic claim that the uncountable set of real points could constitute a positive measure, larger than zero! Thus, so Cantor and the modern mathematical measure theory holds, a complete arithmetization of the ‘continuum’ is achieved! Without questioning the spectacular achievements of 20th century mathematics in the domains of measure theory and integration, it still must be pointed out that their arithmeticistic claims cannot be justified. There is simply no constructive way in which one can bridge the gap between denumerable and non-denumarable infinity. Grünbaum, ardently advocating the conception of the continuum as constituted by an aggregate of non-extended elements (‘degenerate intervals’), remarks: “The consistency of the metrical analysis which 1

I have analyzed the distinction between concept and idea in my PhD dissertation: Concept and Idea (Begrip en Idee), Assen: Van Gorcum, 1973.

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I have given depends crucially on the non-denumerability of the infinite point-sets constituting the intervals on the line” (1952:302). It is only possible to demonstrate non-denumerability if one proceeds from the assumption of the at once infinite. This need for the at once infinite explains why one cannot, constructively, proceed from denumerability to non-denumerability (cf. Wolff, 1971:399-400). In addition to this we have seen that the primitive meaning of number does not furnish any grounds for introducing the at once infinite since it is strictly bound to an endless succession of numbers (determined by the numerical time-order of succession at the law-side of the arithmetical aspect). Only by considering the anticipation from number to space do we discern the meaning of the at once infinite – completely dependent upon the irreducibility of the spatial order of simultaneity. In other words, without the nature of spatial simultaneity this supposition of an at once infinite set has no foundation. It is an anticipatory spatial analogy within number. Since the at once infinite presupposes the irreducible, unique nature of the spatial aspect it cannot be used subsequently to reduce space to number (a distinct number of points) in terms of a non-denumerable set of real points. This reductionist attempt is antinomical and implies the following contradiction: space can be reduced to number if and only if it cannot be reduced to number (i.e. if and only if the at once infinite is used, which presupposes the irreducibility of the spatial aspect)! We can therefore fully subscribe to the words of the well-known mathematician, Paul Bernays (co-worker of David Hilbert) where he writes: “The arithmetizing monism in mathematics is an arbitrary thesis. That the field of research of mathematics exclusively proceeds from representations of number is nowhere demonstrated. Much rather concepts such a continuous curve and a surface, which are particularly developed in topology, could not be reduced to number concepts” (1976a:188). We should add to our antecedent brief argument that the spatial aspect (with the primitive meaning: continuous extension) not only differs from the numerical aspect but also cannot exist without cohering with it. It indeed belongs to the very nature of a spatially extended continuum that anyone of its parts allows of a successive infinite divisibility. Clearly, this divisibility presupposes the meaning of the numerical aspect – evident in the qualification: successive infinite. Remarkably enough, the set of rational numbers is called ‘dense’ because the numerical difference between any two rational numbers (fractions) could be ‘divided’ indefinitely. This feature manifests an anticipatory coherence between number and space at the factual side. Since the infinite divisibility of any spatial subject (functioning at the factual side of the spatial aspect) in itself entails a retrocipation to the numerical time-order of succession (at the law-side of the numerical aspect), we are justified to see in the system of rational numbers an anticipation to a retrocipation. The ‘reflecting-back’ character of this anticipation could best be captured by speaking about the semi-disclosed nature of number. After all, the divisibility at stake in 48

this context remains denumerable – just as the set of rational numbers in the first place could be described in terms of the successive infinite. This feature enabled Brouwer to introduce an intuitionistic theory of the continuum. He abstracts fully from any measure concept by focussing only on a fundamental, completely ordered and overall dense sequence with a first and last element: “The second act of intuitionism creates the possibility of introducing the intuitionist continuum as the species of the more or less freely proceeding convergent infinite sequences of rational numbers” (Brouwer, 1952:142). It is evident that this intuitionistic position is dependent upon the mentioned numerical anticipation to a retrocipation. Thus it remains within the confines of a semi-disclosed handling of the successive infinite in terms of which the real numbers are approached. Under the pretense that it proceeds purely in arithmetical terms, Cantorian (and the subsequent axiomatic formalistic) set theory in fact provides us with a deepened number theory, i.e. with a number theory that is disclosed by the use of the at once infinite as an anticipatory hypothesis within the enriched meaning of number. Lack of realizing what is done caused the mathematical legacy to reduce the original meaning of space into this anticipatory sphere of number – explaining why mathematicians are inclined to identify the notion of the ‘continuum’ with the real numbers. Intuitionism, on the other hand, reduces the meaning of space to the semi-disclosed meaning of number. In both cases, however, and contrary to their true intentions, essential elements of the aspect of space are used. In the case of intuitionism, the infinite divisibility of a spatial continuum is ‘borrowed’, and in the case of axiomatic formalism the spatial order of at once is ‘borrowed’ in their usage of the at once infinite. This new perspective simultaneously explains why both Aristotle and Cantor advanced similar criteria for continuity. Since Aristotle, on the one hand, approached continuity from the perspective of the spatial aspect, with its characteristic endless divisibility, he clearly only had to use the successive infinite. Cantor, on the other hand, took the avenue of the numerical aspect, and the only way to ‘get at’ continuity was in terms of the numerical anticipation to space given in the idea of the at once infinite.1 Systematically seen, we therefore have to distinguish between the number concept of denumerability on the one hand, and the number idea of denumerable and non-denumerable transfinite numbers on the other hand. A fully disclosed treatment of the real numbers, i.e., a treatment using the at once infinite, thus raises mathematics to a level of increasing thought economy (by allowing, amongst other things, indirect existence proofs and the free use of the logical principle of the excluded middle) shedding a new light on the unnecessary complexity and limitations of intuitionistic mathematics. The primitive term in Zermelo-Frankel set theory, namely set / element of, reveals the 1

Not without reason Becker remarks: “Thus the Aristotelian theory of the infinite and the continuum, in its peculiar problem-setting, is still of actual importance to a genuine adequate foundation of higher [mathematical] analysis” (1965:xii, note 2).

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implicit dependence of set theory (as a spatially disclosed number theory, anticipating the spatial whole-parts relation and the spatial order of at once), on the irreducible nuclear meaning of space.1 The acceptance of the integral biblical account of creation guided our preceding analysis. On the basis of a theoretically articulated account of the order-diversity within creation (acknowledging the principles of ‘spheresovereignty’ and ‘sphere-universality’ – entailing anticipations ans retrocipations) our focus ought to be on the mutual coherence and irreducibility of the aspects of number and space. It should also be clear that in pursuing this avenue (as we have remarked in the beginning) a third alternative emerges, side-stepping the extremes of a geometricized and an arithmetized approach.

1

Logicism had to concede that it failed in providing a successful reduction to logic of the notion of infinity. Myhill remarks: “the axioms of Principia [Mathematica] do not determine how many individuals there are; the axiom of infinity, which is needed as a hypothesis for the development of mathematics in that system, is neither provable nor refutable therein, i.e., is undecidable” (1952:182). We add the words of Kline, stating that Hilbert “did agree with Russell and Whitehead that infinite sets should be included. But this required the axiom of infinity and Hilbert like others argued that this is not an axiom of logic” (Kline 1980:246).

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Lawside Numerical time-order of succession

Spatial time-order of at once

The Successive (potential) Infinite

The At Once (actual) Infinite

(The primitive meaning of infinity at the lawside of number)

(Anticipation from number to the lawside of space) Anticipation

Primitive meaning:

tion cipa retro retroc ipatio Re n t

Number

discrete quantity

law roc sid ipat e o ion f n to N (natural numbers) um the be Z (integers) r

Q (fractions) (Fractions represent an anticipation to a retrocipation)

dimension

Space

Primitive meaning:

magnitude continuous extension

Spatial figures (the whole-part relation) successive infinite divisibility

(Semi-disclosed)

R (real numbers) Anticipation

needs the at once infinite (The real numbers represent the fully spatially disclosed structure of number)

Factual side The mutual coherence and irreducibility of number and space

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Paradigms in Mathematics, Physics, and Biology: Their Philosophical Roots Chapter III Basic questions in Physics The Prejudice against Prejudices Perhaps one of the distinctive features of Western civilization is that it experienced an Enlightenment during the 18th century. As a result of the increasing trust in the rationality of the human being historians assess the 18th century as the era of Enlightenment. In itself this characterization harbours a relatively positive evaluation. As a result it would be difficult to have an eye for the tremendous blocking out of insight it also brought about. In order to discern this shadow side of the century of Enlightenment we only have to refer to what Gadamer called the prejudice against prejudices! It was undoubtedly Immanuel Kant, the famous philosopher of the 18th century, who enthroned theoretical reason. Even law and faith were challenged by this judge. In the Preface to the first edition of his Critique of Pure Reason (1781) Kant declares: Our age is, in every sense of the word, the age of criticism, and everything must submit to it. Religion, on the strength of its sanctity, and law on the strength of its majesty, try to withdraw themselves from it; but by doing so they arouse just suspicions, and cannot claim that sincere respect which reason pays to those only who have been able to stand its free and open examination (A12 – translation F.M. Müller).

It was Kant’s intention to set a limit to the applicability of (natural) science since he did want to leave open a domain for practical reason transcending the sphere of sense-perception and logical understanding. However, as we have seen in Chapter I, by the end of the nineteenth century and the beginning of the twentieth century positivism emerged as a philosophical trend with the explicit purpose to abolish whatever supersedes sense perception. We have seen that it was foremost the neo-positivism of the Vienna Circle which, during the second and third decades of the 20th century, advocated with great enthusiasm that the positive (empirical) natural sciences should attain the leading role in the further development and unfolding of society. These positive sciences were (pre-)supposed (!) to operate without any pre-suppositions. 53

In his sharp criticism Karl Popper realized that this understanding of science entails serious difficulties. He rejected the generality (universality) of empirical testing (so-called verification) in favour of falsification – his newly introduced criterion of demarcation. W. Stegmüller unequivocally takes the position that a person first has to accept something before something else could be justified in terms thereof. In doing this he implicitly rejects the prejudice against prejudices characteristic of the Enlightenment: A Self-guarantee of human thinking, within whichever domain, does not exist. One has to believe in something in order to be able to justify something else in terms of it (1969:314).

Discrepancy between philosophers of science and the practitioners of science Although the development of contemporary philosophy of science during the past three to four decades does harbour many points of difference, there is perhaps one general consensus in the unanimous rejection of the positivist denial of unavoidable prejudices in science. At the same time there continues to exist a pertinent discrepancy between the practitioners of science in various academic disciplines (who are still continuing this outdated positivist view of science) on the one hand, and the current situation in the area of philosophy of science. About a decade ago I had the opportunity to address a number of natural scientists who still wished to maintain that only what is accessible by sensory experience qualifies for natural scientific investigation – only that which van be weighed, counted and measured falls within the domain of science. This reaction immediately reminded me of the mocking words of the American sociologist, McIver, who referred to a supposed “pre-suppositionless” and “unbiased” positivist attitude as follows: The following seems to be the chief tenets of their creed. First, I believe in facts, and to be saved I must discover new ones. Second, when I have discovered them, I must if possible measure them, but, failing that consummation, I must count them. Third, while all facts are sacred, all theories are from the devil. Hence the next best thing, if one can’t discover new facts, is to refute old theories (1967:21).

The problem with the positivist preoccupation with facts is given in the unavoidability of doing science without employing theoretical terms. Let us call these terms provisionally property terms. The unique power of science is precisely given in its ability to grasp in a systematic fashion experiential data of an apparently widely diverging nature in the systematic grip of a specific universal perspective. For example, within the domain of modern physics many different sorts of entities are encountered – from elementary particles and atoms up to macro-processes and macro-systems. However diverse these entities and processes may be none of them escape from the integrating and universal perspective of the core physical discipline known as thermodynamics. The laws of thermodynamics – such as the law of energy constancy or the law of non-decreasing entropy – are after all applicable to all possible physi54

cal entities and processes there may be, regardless of the particular natures involved. The questions to be addressed to positivism at this point are the following: how do we account for this universality? and: what is the empirical status of a property term such as energy-constancy? By pausing for a moment at the development of the concept of matter since ancient Greece we shall try to answer these questions. Property terms – the Achilles’ heel of positivism In the previous Chapter we have seen that the Pythagoreans adhered to one statement above all else: everything is number. After the discovery of irrational numbers – revealing within the seemingly form-giving and delimiting function of number the formless – Greek mathematics as a whole was transformed into a spatial mode (the geometrization after the initial arithmetization). As a consequence material entities were no longer described purely in arithmetical terms. Space now provided the necessary terms used to characterize material entities. This spatial angle of approach remained in force until the rise of modern philosophy, since philosophers like Descartes (1596-1650) and Kant (1724-1804) still saw the ‘essence’ of material things in extension. It was due to Galileo and Newton that the main tendency of classical physics eventually caused a shift in modal perspective by trying to describe all physical phenomena exclusively in terms of (kinematical) movement.1 Writing about the foundations of physics, David Hilbert2 refers to the mechanistic ideal of unity in physics but immediately adds the remark that we now finally have to free ourselves from this untenable ideal. It is therefore strange that the contemporary physical scientist from Cambridge, Stephen Hawking, still writes: “The eventual goal of science is to provide a single theory that describes the whole universe” (1987:10). Since the introduction of the atom theory of Niels Bohr in 1913, and actually since the discovery of radio-activity in 1896 and the discovery of the energy quantum h, modern physics realized that matter is indeed characterized by physical energy operation – the physical aspect of reality must be seen as the qualifying function of matter. This brief sketch of the genesis and growth of the concept of matter illustrates in which way different (modal) property-terms served to characterize matter – starting with the perspective of number and then proceeding to the aspect of space, the kinematical aspect and eventually the physical aspect of reality. What is important to realize is that the description of matter was decisively dependant upon a particular theoretical view of reality (Kuhn would have used the expressions paradigm or disciplinary matrix) which is entailed in the preference which is assigned to specific property-terms. Is it possible to account for this foundational choice in an empirical way? Is it possible to perceive the 1

2

The British philosopher, Thomas Hobbes (1588-1679), was familiar with the mechanics of Galileo enabling him – as opposed to Descartes – to employ the basic concept moving body as descriptive tool. Perhaps the greatest mathematician of the 20th century.

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numerical aspect? Can we weigh the spatial aspect? Can we determine the volume of the kinematical aspect? Can we ‘measure’ the ‘distance’ between the spatial aspect and the physical aspect? The obvious absurdity of these questions not only illustrates the untenability of the positivistic faith in facts, but at once point at a crucial distinction operative throughout the history of the special sciences, namely the distinction between aspects and entities. Reformational philosophy pointed out that these aspects enable our scholarly reflection to establish a universal coherence between different kinds of entities – just recall the universal scope of the fundamental laws of thermodynamics (which hold for all possible physical entities). In general an implicit choice on this level of scientific convictions cause a divergence between special scientists. The question concerning the relationship and coherence between the different aspects of reality (in terms of which we can describe anything) simply cannot be settled with the aid of the positivistic method of (empirical) perception and verification. Positivism did realize that we can only discover the structural nature and the laws holding for physical entities by investigating the lawfulness (law-conformity) they evince. However, precisely the difference between the universality of God’s law and the unique instances empirically tested in experimental settings once again unveils the untenability of a positivistic position. A limited number of experimental instances could never warrant the claim of universality contained in law statements. In its materialistic variant positivism reveals even further inconsistencies. Let us look at the typical claim that matter is all there is: atoms, molecules, and macro-molecules in interaction. This statement claims that there is nothing beyond matter – but what about the statement making this claim!? Is it true? If so, then there is something immaterial (truth). And what about the natural laws holding for material things? They condition being material but are not themselves material! Thus both with respect to the truth-value and the universal validity of natural laws the basic claim of positivistic materialism is self-defeating! The measurement of time and modal time orders Physicists normally claim that time is an exclusively physical phenomenon. As a consequence, they also hold that only physicists are competent to speak about the nature of time. Now suppose we visit, accompanying the Historical Association, a historically significant farm. Upon arrival we notice that the old couple still live as they did almost fifty years ago when they moved to this farm – it looks as if nothing changed during this period, as if time came to a stand-still. What does it mean to say during the past fifty years time came to a stand-still? If we had physical time in mind obviously this statement would have been meaningless, since physical time continues without any interruption! However, when we realize that the mentioned statement concerns our historical awareness of time, no absurdity will be observed. To phrase this insight in terms of the general pattern of cultural development, taking into ac56

count the ascending line from the stone age, the bronze age, the iron age, and so on, then it is perfectly meaningful to say that within the 21st century there are communities still living in the stone age (an era more or less dated between 2 million and 10 000 years ago)! Let us consider another example. Modern governments are entitled to promulgate laws with a retroactive effect! This reality would be impossible if legal science had to operate with a physical concept of time. One cannot escape from the reality of such a law by viewing it as a mere legal fiction. Surely, physical time is irreversible and therefore flows in one direction only. Even if we raise the argument that historical and legal time can only exist on the basis of physical time, it does not cancel the uniqueness (and irreducibility) of these other modes of time. The important systematic conclusion to be drawn from the given examples is that no single experiential mode of reality can exhaust the full meaning of time. This novel proposal was first made by the Dutch philosopher, Herman Dooyeweerd. Time should be seen as a unique dimension of reality. Within the diversity of modal aspects time expresses itself in accordance with the unique nature of each aspect. Interestingly, Stephen Hawking (1987:8), the well-known contemporary physicist, correctly emphasizes (with Augustine) that time itself is a creature and it does not exist from eternity. By looking at the history of time-measurement significant points of connection are found for distinguishing the first four modes of reality in terms of their respective time orders. It belongs to our general awareness of time: earlier and later, simultaneity, time-flow and irreversibility are well-known modalities of time. In his work on the foundations of physics (1980:16) Stafleu remarks: This is most clearly shown by an analysis of the historical development of time measurement. Initially, time measurement was simply done by counting (days, months, years, etc.) Later on, time was measured by the relative position of the sun or the stars in the sky, with or without the help of instruments like the sundial. In still more advanced cultures, time was measured by utilizing the regular motion of more or less complicated clockworks. Finally, in recent developments time is measured via irreversible processes, for example, in atomic clocks.

What is striking in this whole development is that different time orders are used, the one after the other: the numerical time order of succession,1 the spatial order of simultaneity,2 the kinematical time order of constancy and the irreversible physical time order, expressed in the relationship of cause and effect. On the one hand we always encounter time as time order – in which case it appears at the law-side/norm-side of reality – and on the other hand as time-duration (at the factual side of reality). By virtue of the cosmic time order there 1 2

In Chapter II we related this to our most basic awareness of infinity, namely the successive infinite. Compare the at once infinite discussed in Chapter II.

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also exists a time order of succession between the various aspects. Time in the aspects of number and space Mathematicians who are only acquainted the dominant trend in modern mathematics, namely the axiomatic formalist standpoint, will claim straight away that time does not have a place in mathematics. However, those who took notice of neo-intuitionist mathematics (discussed in die previous Chapter) exemplified in the work of L.E.J. Brouwer and his successors (amongst whom are people like H. Weyl, A. Heyting, D. van Dalen, A. Troelstra and to a certain extend also P. Lorenzen), will realize that this school explicitly proceeds from the assumption of an original intuition of time. In this intuition, according to Brouwer, continuity and discreteness coincide giving birth to the primal awareness of one, another one and son on – a process that, through the endless addition of new units could never be exhausted. In other words, this process is literally infinite, without an end. This intuitionistic conception of time is historically dependent upon the philosophy of Immanuel Kant who saw time as one of the psychical forms of intuition of being human.1 What intuitionism identifies as the intuition of one, another one and so on, relates to the arithmetical time order of succession at the law-side of the numerical aspect. It belongs indeed to the time intuition of every person since without this numerical time order one of the cornerstones of our modern civilization will collapse, including our measurement and calculation of (physical) time. Put differently: our experiential intuition of numerical relations provides us with an insight into the original (ontically given) numerical time order of succession. In mathematics this time order lies at the foundation of the principle of (mathematical) induction – first introduced by Pascal. It simply says that if a statement is valid for the number 1 and, subsequently, if it could be shown that whenever it holds for a number n it also holds for the number n+1, then it obtains universally. According to Weyl already this principle is sufficient to safeguard mathematics against becoming a mere tautology, in other words to prevent that a set of formal axioms be the basis of mathematics in stead of a basic insight that can not be formalized. The most primitive correlate of the numerical time order of succession is given in the sequence of natural numbers: (0), 1, 2, 3, 4, 5, 6, 7, ... (without an end, endless, infinite). Axiomatic set theory sometimes attempt to define order. This is done, for example, by introducing the concept of an ordered pair. Even in the standard work on Set Theory (second, revised edition, 1973) Fraenkel et al. there suddenly appears an unexpected petitio principii in this regard. Without explaining the technical detail it is sufficient to take note of

1

According to Kant the concept number originates through a schematization in time of quantity as category of our understanding.

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the remark added to their example about an ordered pair (derived from Kuratowski): “Taken in that order”!1 Within the aspect of space (cosmic) time expresses itself in the spatial time order of simultaneity, correlated with factual spatial extension. Already this insight cancels the misconception that time is spaceless and that space is timeless. The awareness of simultaneity (that which exists at once) belongs to our basic intuition of space. When the arithmetical order of succession at the law-side of the aspect of number is disclosed under the guidance of the theoretical insight into the nature of the spatial order of simultaneity we discover the regulatively disclosed idea of infinity, namely the idea of the actual or completed infinity – designated by us as the idea of the at once infinite. The kinematical and the physical time order Since the development of Galileo’s mechanics classical physics attempted to understand all bodies in terms of the denominator of mechanical movement. From Newton up to the beginning of the 20th century this mechanistic tendency stamped the main development of modern physics. Max Plank,2 characterized this mechanistic orientation as follows in 1910: The conception of nature that rendered the most significant service to physics up till the present is undoubtedly the mechanical. If we consider that this standpoint proceeds from the assumption that all qualitative differences are ultimately explicable by motions, then we may well define the mechanistic conception as the conviction that all physical processes could be reduced completely to the motions (the italics are mine – DFMS) of unchangeable, similar mass-points or mass-elements (1973:53).

In kinematics all processes are reversible in principle. This reversibility concerns the kinematical time order. It is analogous to the numerical and the spatial time orders which are also reversible. The reversibility of the numerical time order first of all flows from the reversibility of the + and – directions in the system of integers. Although concrete events in physical reality are unidirectional, the time order within the numerical aspect could be experienced both in the positive and the negative directions.3 Already in 1824 Carnot discovered fundamentally irreversible physical processes. The implications of this discovery was further developed simulta1 2

3

The ordered pair (a,b) is “defined” as the class (a,b) containing the classes (a) and (a,b) “taken in that order” as elements! As mentioned above, he discovered the quantum of energy h (6.62 10-34 joule sec) – portraying the fundamental discontinuity of energy. In order to account for the discrete nature of the omission or absorbtion of energy, Planck postulated that radiant energy is quantized, proportional to the frequency v in the formula E = hnv – where n is an integer, v the frequency, and h the quantum of action (Wirkungsquantum) with the value 6.624 10-34. It may take 5 minutes (physical time duration) for 100 students to enter a class successively. At the end of the class these students may leave the class in a reversed order, and this may take one minute only. Although the physical time duration took place in one direction only, the numerical time order is reversed at the end of the class.

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neously by Clausius and Thompson in their formulation of the second main law of thermodynamics.1 In 1865 Clausius introduced the term entropy. This law accounts for the irreversibility of physical processes – it determines the direction of a physical (or chemical) process in a closed system.2 Thus the law of non-decreasing entropy was established as the second main law of thermodynamics. At the same time the classical mechanistic reduction to pure motion was uprooted. Justifiably therefore Max Planck (in his mentioned article from 1910) remarks that the “irreversibility of natural processes ” confronted the “mechanistic conception of nature” with “insurmountable problems” (1973:55). Consequently, whereas the time order in the first three aspects is reversible, it is irreversible in the physical aspect. This is easily seen in the a-symmetrical relation of causality: it stands to reason that the cause precedes the effect! Since the discovery of radio-activity it turned out that within micro-structurse themselves there are irreversible processes present proceeding spontaneously in one direction only. In addition this state of affairs straightaway confirms the irreducibility of the physical aspect to the kinematical aspect (with its reversible time order). Already in his Isagogè Philosophiae from 1930 Vollenhoven distinguished between the mechanical and the physical aspects. However, in the edition of 1936 this distinction no longer appears. Dooyeweerd, on the contrary, initially maintained the order numerical, spatial, physical – thus identifying the kinematical aspect with the physical aspect. Round about 1950 he realized that this distinction is necessary to account for the fact that kinematics (phoronomy) can define a uniform motion without any reference to a causing force (compare Galileo’s law of inertia). The uniqueness of Constancy and Dynamics Perpetual motion From antiquity there have been attempts to make a machine which, once set in motion, would continue this motion perpetually without using an external source of energy. At the beginning of the seventeenth century Fludd designed a closed-circuit water mill. This initially appeared quite feasible, but every effort to actually make it work practically failed. Already in 1775 the French Academy for Science and Art decided to pay no further attention to purported designs of “perpetuum mobile”. In England also all claims to the patent rights on such machines were subjected to the provi1 2

The first law is the law of energy conservation. Suppose an ideal gas in a container is connected with another one in which a vacuum is present. The internal energy will not change. However, the gas molecules will fill the entire space spontaneously. This indicates an increase in entropy. Viewed statistically an increase in entropy will always reflect the occurrence of the most probable state. For that reason, within a closed system, there will always only occur an increase or constancy of entropy – but never a decrease of entropy.

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sion of a working model – to no positive effect. The question is: why doesn’t it work? To understand why this sort of perpetual motion machine cannot work, we must refer to the first main law of physics. The underlying idea of perpetual movement, after all, is that useable energy would be produced without using any energy. Practically, this means that energy would have to be created. What does this first law say? Stimulated by German natural philosophy at the beginning of the 19th century (especially the ideas of the philosopher Schelling), German natural scientists searched for a unifying law which would encompass all physical phenomena in a single perspective. The physicists Heimholtz and Mayer and the chemist von Liebig defended the notion of the indestructible character of matter even before experimental evidence confirmed their view. In 1847, at the youthful age of 26, Helmholtz presented a formulation of his first main law of physics (actually thermodynamics) to the Physics Society of Berlin. He began by pointing out that nobody had succeeded in building a successful perpetual motion machine. This was a logical consequence of the indestructibility of energy. Till the present physicists recognize this law as the law of energy conservation which means that energy cannot be created or destroyed.1 In view of the law of energy conservation it is quite clear today that the construction of such a machine is impossible in principle, since it would mean that useful (newly created) energy would be released without using any energy! Comment: A second such sort of machine had also been imagined – a machine which would draw heat from its environment and then convert this entirely into work. The impossibility of such a machine is evident in view of the second law of thermodynamics, that of non-decreasing entropy. Statistically this means that in any closed system the most likely situation would occur. Owing to the difference in temperature in the environment it required to convert heat into work, the second law implies the impossibility of this type of machine.

These two main laws of physics are fundamental insofar as they are universally applicable to all physical entities. Laws which indiscriminately hold for all entities, must completely ignore the typical differences between such entities. Such modal laws indicate the fundamental ways of being or modi of such entities. To deduce universal modal laws requires that scientific activity of analysis which we have called modal abstraction. Closer reflections on constancy and change To grasp the physical modality (way of being) of physical entities, it is necessary to ignore their non-physical aspects. Amongst other things, this implies that it is essential to distinguish clearly between the physical aspect of energy-operation and its founding kinematic aspect – that is, the aspect in which 1

This law does not exclude the fact that one energy form can be transformed into another form of energy.

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we only refer to uniform movement without referring to the cause of motion. Movement – as the mode of constancy or uniform flow – is an original given, just as number, space, the economic or the ethical. For this reason Galileo’s law of inertia implies that we may at most speak of the origin of a change in motion! All change presupposes a continuing basis. If you do not remain yourself (constancy), you would not be able to age (change)! The importance of our understanding of constancy and change (dynamics), justifies a closer discussion of their nature and origin – which would also enable us to demonstrate further structural characteristics of modal aspects. The core of Einstein’s theory of relativity Einstein’s theory of relativity is well-known. A physicist of his stature lends credit to the popular view linked to his theory, namely that everything is relative and changeable. Remarkably, Einstein’s theory rests on a fundamental presupposition which is the opposite of all relativism. Einstein had to start with the idea of an order which is uniform and constant – which means that everything which he has indicated to be relative is only relative in relation to this constant order. That this is the case is evident from his postulate that the speed of light is constant in a vacuum. Einstein worked from the presupposition that a particular light signal would have the same constant speed (c) in relation to all possible moving systems. It was not even necessary for his theory for such a signal to actually exist. The fact that later experimentation proved experimentally that the speed of light does indeed conform to Einstein’s postulate, is – as the physicist Stafleu puts it – relatively irrelevant! One has to keep this in mind in connection with contemporary discussions regarding the changing speed of light The crux of Einstein’s theory of relativity is therefore to be found in the nature of the order of constancy which it presupposes.1 We are familiar with the numerical order of succession which founds every counting activity: one, another one, another one, and so on. Just as familiar is the spatial order of simultaneity. In distinction from the numerical order of succession and the spatial order of simultaneity, we experience the order of constancy in the kinematic aspect of movement. This means that Einstein’s special theory of relativity of 1905 is a purely kinematic theory.2 Einstein’s theory therefore did not primarily develop a theory of relativity, but rather one of constancy. Galileo already discovered the particular nature of the kinematic order of time, as it was revealed in his law of inertia. In terms of this law a body in mo1

2

Spielberg and Bryon correctly emphasize that it is about “invariance” – i.e. constancy – although they unfortunately thereby confuse the terms absolute and unchanging: “Indeed, Einstein originally developed his theory in order to find those things that are invariant (absolute and unchanging) rather than the relative. He was concerned with things that are universal and the same from all points of view” (1987:6). The term unchanging is simply the denial (negation) of change – a physical term. The term absolute cannot really be applied to anything in creation, that is, not if one wants to avoid the idolization of created reality. The irreducible nature of the kinematic time order is imported with the help of a subject which moves at a constant speed.

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tion would continue its movement without stopping – unless something else (a force or friction) influences it. That means that our insight into the nature of movement does not depend on a causal power. The term “cause” belongs to the physical aspect of our experience where we come across the effects of energy-operation. It cannot be sufficiently emphasized that we can never talk of a cause of movement, but rather only of a cause of a change in movement – thus once again acknowledging the modal difference between the kinematical and physical aspects of reality.1 The unique nature of constancy (that is, the irreducibility of the kinematic aspect) is the foundation of all references to dynamics or change. Without a constant basis all talk of change is senseless. For this reason physics cannot link any meaningful content to a discontinuous change of movement – change of movement (acceleration and deceleration) is always continuous, since a discontinuous change would require a physically impossible infinite force.2 Consequently, we can only establish change on the basis of something continuous. An alternative formulation of the first main law of thermodynamics This foundational position of the aspect of movement enables us to philosophically find a formulation of the first main law of thermodynamics which is true to reality. The physical aspect must not only be distinguished from its foundational kinematic aspect, since there is also an indissoluble coherence between these two aspects. For this reason we shall find in the physical aspect a structural moment which reminds us of the foundational kinematic aspect. Constancy appears in the physical aspect as a structural reminder of the meaning of motion. In philosophical terms we may say that we find an analogy of the kinematic aspect at the law side of the physical aspect. A formulation of the first main law which intends to be true to reality would therefore have to refer to energy constancy. Strictly speaking the use of the term “conservation” is inadequate, since the activity of retention itself requires an input of energy – as in the case of thermodynamic “open systems” (or “steady states”). The law of energy constancy illustrates not only the distinct uniqueness of the kinematic and physical aspects, but, taking into account the distinction between law side and factual side, also the indissoluble coherence between them: without the foundational position of the kinematic aspect in the order of the various cosmic aspects we would have no grounds for discerning an analogy of the aspect of movement in the physical aspect, that is, the analogy of energy constancy. The theory of relativity and relativism In modern times there is virtually no science (including theology) not beset with attempts at historical relativism. Historicism, after all, claims that every1

2

In his mentioned article from 1910 treating the classical mechanistic view of nature, Max Planck sharply and correctly distinguished between a “mechanical” and an “energetical” view of nature (1973:65). Janich stresses a “strict distinction between phoronomic (hereafter named kinematic) and dynamic statements” (1975:68).

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thing changes all the time, that nothing remains the same – moral standards, religious convictions, legal opinions, economic practices – all things continue to change. The pitfall in this argument is already evident in the fact that every indication of change is inevitably accompanied with kinematic constancy terms such as “continually”, “still”, “always”, “incessantly”, etc. This implies that we may not identify constancy with something static, but that we should much rather evaluate it positively as the foundation of all dynamics! At the same time, however, we should leave aside the one-sided and excessive concern with dynamics which is set against all forms of constancy.1 Such an approach only leads to an unjustified dialectical tension: that which is the condition and prerequisite of dynamic change – that is, something constant – is seen as its opposite pole and enemy. The remarkable coherence between the terms constancy and dynamics not only enlightens us regarding the natural scientific basis for the use of these terms, since it also emphasizes the insight that the way in which we talk about everyday occurrences can never escape from the inevitability of having a perspective on particular aspects. Determinism and indeterminism Using energy constancy as formulation of the first main law of thermodynamics implicitly presupposes the distinction between a law (as order for) and whatever is subjected to and correlated with that law. It belongs to the nature of a law that it determines and delimits that which is factually subjected to it. Conversely, factual reality is determined and delimited by a correlating law – and in its orderliness/law-conformity factual reality shows this subjectedness. Even with regard to the supposed “initial state” of the “big bang” Hawking explicitly states that one has to assume “that there are also laws governing the initial state” (1987:11). What is striking is that Hawking does not reflect at all on the origin of these laws! Werner Heisenberg points out that the development of quantum theory led to the formulation of physical laws in statistical terms. Moreover, with his well-known principle of uncertainty, Heisenberg established that “it will never be possible to determine both the position and velocity of an atomic particle with an arbitrary precision” (1956:11).2 The then prevailing conception of physical causality was convinced that physical subjects ought to be seen merely as the extension of physical laws determining their existence exhaustively – explaining why this view is also known as determinism. In his mentioned work Heisenberg explains that according to this deterministic ap1 2

Where a few a decades back one would still refer with the highest regard to a resolute or principled person, today it is fashionable to speak of a dynamic person. In April 1927, before he made known his relation of uncertainty, Heisenberg (in a personal conversation) said to Von Weizsäcker: “I believe I have disproved the law of causality” (Von Weizsäcker, 1993:132, note).

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proach exact knowledge of nature or a particular section of it will suffice to determine the future. He continues: If one interprets the word causality in such a strict sense, one also speaks of determinism and means by that that there exist laws of nature determining unequivocally from the present the future condition of a system (1956:25).1

Determinism holds that every effect is strictly determined by a cause. In reality this point of view represents a deification of the law-side of the physical aspect – explaining why it sees physical entities (subjects) merely as an extension of physical laws. Determinism reduces the factual side to the law-side. Stafleu correctly points out, by contrast, that the causal relation at the law-side of the physical aspect simply states: nothing happens without a cause – but what the effect of a specific cause may be need not to be fixed in advance. This formulation side-steps the one-sidedness of both determinism and indeterminism: it grants determinism that the concept of a cause is meaningful and should not be discarded as it is claimed by indeterminism; and it grants indeterminism that the effect need not to be fixed in advance (just think about the half-value of radio-active elements), thus highlighting the untenability of determinism in this regard. The principle of uncertainty of Heisenberg had the effect that the paths of great physicists of the 20th century parted – concerning the question whether or not the concept of causality ought to be maintained in the further development of physics. Planck and Einstein wanted to uphold the claims of determinism whereas Heisenberg and Bohr (the Copenhagen interpretation of quantum physics) opted for the other extreme: indeterminism. If it is the case that determinism absolutizes the law-side of the physical aspect, then we have to say that indeterminism absolutizes the factual side of the physial aspect. The alternative approach advanced here is to view law-side and factual side as irreducible correlates – an alternative implicitly supported by the necessity to employ statistical laws in physical theories (cf. Stafleu, 1968:304). Order and delimitation in physics In spite of the contemporary interest in chaos it is really not strange that physicists are still looking for an “underlying order in the world” (cf. Hawking, 1987:13). Chaos theory, in the final analysis, wants to unveil a more complex and overarching order in what appears to be disorderly and chaotic. The newly discovered complex patterns, however, always point to a determining and delimiting law, as we have seen. Once this has been realized, another related issue may gain in prominence – the question concerning the boundaries of science and reality. Already in the earliest phases of the Western scientific legacy scholarly knowledge was confronted with the quest to determine the limits of scientific thinking. In practical terms this urge coincided with determining the limits of the universe itself. 1

Max Planck actually adhered to this deterministic understanding of causality. Compare his 1932 article on Causality in Nature (Planck, 1973:252).

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The finite and limited cosmos in Greek culture It may be surprising to us that Greek thought apparently found a point of rest in the delimitation provided to their world picture by the “large world-sea”, the Okeanos. According to their understanding the earth is a circular slice delimited and surrounded by the Okeanos. It seems strange that the Greeks did not move beyond the boundaries of the Okeanos. Our own modern acquaintance with the idea of infinity almost automatically forges this question. In terms of the Greek mind this was impossible. The Okeanos was one of the primal forces subdued when the Olympic gods started their reign. The ordered cosmos owes its form, measure, harmony and determination (concept) to these gods. Whatever finds itself outside this limit does not display any form-delimitation and can therefore not be thought or conceptualized. As a consequence Aristotle does not acknowledge an abstract or empty space. He lacks our modern concept of space. According to the mature Greek understanding space does not exist, only place. Place is a property exclusively attributed to a concrete body. In the absence of a body there is no subject for the predicate place. From this it naturally follows that an “empty place” is the place of nothing – in other words, it is no place at all! The possibility to understand (and encompass) the ordered cosmos flows from its finite and limited nature – for that reason science is restricted to this finite and limited, ordered cosmos. This orientation in different ways caused tensions within Greek science and culture. In Chapter II we already pointed at the problems caused by the discovery of irrational numbers by the Pythagoreans – eventually leading to the geometrization of Greek mathematics. However, the counterpart of the question concerning our knowledge of the limits of the cosmos is given in the question whether “space” allows for a continued division or whether the process of division is blocked by last (smallest) indivisible units? The Greek atomists, Leucippus and Democritus, were convinced that there indeed are such last indivisible units, which they called atoms. Since Descartes modern conceptions switched to the conviction that physical space is both continuous and infinitely divisible.1 By the end of the 19th and the beginning of the 20th century, however, the following distinction turned out to be necessary: that between mathematical space and physical space. Whereas the former – in a purely abstract and functional perspective – is both continuous and infinitely divisible (cf. Chapter 2), physical space is neither continuous nor infinitely divisible. Since it is bound to the quantum structure of energy it cannot be subdivided ad infinitum. Energy quanta indeed represent the limit of the divisibility of energy. 1

In Chapter II we have pointed out that this property represents an essential feature of spatial extension. This characteristic provides the basis for the (semi-disclosed) intuitionistic mathematics of Brouwer, Weyl and their successors.

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Are there inaccessible limits in the natural sciences? One can say that energy quanta represent an accessible (lower) limit. Are there truly inaccessible physical limits? The second main law of thermodynamics, the law of non-decreasing entropy (practically simply stating that within any closed systems there will always be an inclination towards the most probable condition), provides us with two good examples in this regard. This law entails that no single machine can be so efficient that while it produces energy no energy is lost. For this reason the classical ideal of perpetuum mobile is unattainable. Amongst other things it also implies that the lower limit (absolute zero point: -273,16o) is inaccessible. Although the physicist Kurti managed to reach a temperature as close as one millionth from the zero point, it still remains impossible in principle to bridge this last tiny gap – simply because it will require an optimally effective machine, capable to convert all the energy it needs into usable energy! It is precisely this that is forbidden by the second main law. The unlimited but finite universe in Eintein’s theory of relativity Einstein’s theory of relativity advanced remarkable perspectives in this regard. Since all celestial bodies are subjected to the effect of gravity Einstein introduced the notion of the “curved space of the universe” (thus employing non-euclidean geometry). On the one hand it still suggests that the universe is unbounded, i.e., one can move in any direction beyond all limits. However, since world space is curved, on the other hand, it entails that eventually one will end up where one started – showing that in spite of being unbounded the universe is still finite! In the final analysis his entire theory rests on the another inaccessible limit: the velocity c of light in a vacuum. This velocity is a true constant – such that whatever moves is moving relatively to this element of constancy.1 On the basis of these considerations we have already pointed out that strictly speaking Einstein’s theory – in its dependance upon this “upper limit” of mois actually a theory of constancy. Complementarity – limits to experimentation There are also remarkable limits to physics in the sense of experimental exactitude and determination. By introducing his principle of uncertainty Heisenberg showed that it is impossible simultaneously to measure the impulse and position of an electron. The Copenhagen interpretation of quantum physics employs the notion of complementarity in order to account for the impossibility to establish both at once - thus allowing for two irreducible (and complementary) modes of description, in terms of “place” and “impulse” respectively. In following some ideas of Mario Bunge the physicist Henry Margenau defends a so-called “moderate reductionism.” He takes this the be “the strat-

1

Sometimes Einstein uses the term invariance (cf. Schilpp, 1951:56). Often he explicitly refers to the constancy of light in a vacuum (cf. Schilpp, 1951:54, 56).

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egy consisting of reducing whatever can be reduced without however ignoring emergence or persisting in reducing the irreducible.1 Entities with a physical qualification Although science tends to be occupied primarily with universal properties or with the specified universality of types no academic discipline – and therefore neither physics – can escape from concrete reality where we also experience the individual side of things and events. The history of academic reflection knows many examples where an attempt is made to account for the individuality of things in terms of a particular property they may have. Sometimes individuality is related to matter – as in the case of Aristotle and Thomas Aquinas (matter as principium individuationis). However, the brief remarks made earlier in this Chapter and the previous Chapter, already showed that it is impossible to revert to a view which tries to consider the aspect of number as stamping matter (the error of the Pythagoreans). We have also pointed out that both the spatial and the kinematical aspects fail to provide us with a qualifying aspect of material things and events. The only candidate left is the physical aspect of ernergy-operation. In order to account for the type-law of physically qualified entities we have to explain their foundational function. We have to keep in mind that any entity of whatever kind functions typically with all aspects of reality, in other words, every entity in a very concrete and plastic way displays a typical function within the (universal) modal structure of the various aspects of reality. Just consider the general focus of thermodynamics – a strictly modally delimited discipline with a universal scope in which the typical features of different kinds of physical entities are disregarded. In thermodynamics it does not matter whether we are talking about the solid state, the fluid state or the gaseous state – the specific weight and heat remain the same. As soon as we take into account the relationship between micro-structures and macro-structures (such as within the confines of statistical physics) then the formerly neglected nuances do matter, because the specific heat or weight is specified differently in each of the three mentioned states (solid, fluid, gaseous). An analysis of material things, furthermore, cannot escape from the functional interrelations present between the different modal aspects in which material things function in a concrete way. The interrelations are first of all accounted for in terms of what in reformational philosophy is called modal analogies (anti- and retrocipations) – structural moments within each aspect reflecting the coherence between the aspect concerned and other aspects. But entities and events always have typical functions within modal aspects. What is typical about these functions is that they are evincing the effect of the qualifying function of the entity under consideration. This implies that the typical function of material entities within the first three modal aspects of reality will always point towards (anticipating) the qualifying physical aspect of matter. Universal modal properties are specified in a typical way. This calls for the 1

Cf. Margenau, 1982:187, 196-197.

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following terminology: we may speak of anticipatory forms of modal specificity. For example, qualified by the physical function of energy-operation we meet the modal specificity of the quantum of energy h which it – for the physicist – a typical number. (h = 6.62 x 10-34 joule sec). Sometimes this kind of type specificity is designated with the use of the term: constants. Similar to the constant h in quantum mechanics, c as the velocity of light (in a vacuum) serves as a constant in Einstein’s theory of relativity.1 By contrast, the atomic number (equal to the number of protons present in the nucleus of an atom) characterizing a chemical element, is another example of a typical number. Even a biotically qualified entity – such as a cell with a nucleus – presents itself with typical spatial relationships explicable in a numerical constant K. This constant K is designated as the “nuclear plasmic index” since it refers to the ratio between the volume of the cell-nucleus (Vn) divided by the volume of the whole cell (Cc) from which the volume of the cell-nucleus is subtracted. Natural and artificial crystals are classified in systems on the basis of symmetry properties – where each system of crystal is characterized by a peculiar axis system. The arrangement occurs according to the diminishing number of symmetry-elements. The seven systems are known as cubic, hexagonal, trigonal, tetragonal, orthorombic, monoclinic, and triclinic. When the various combinations possible between these symmetry elements are considered, 32 possible symmetry classes could be designated – all of them spatial form types encompassing all the possibilities of physically qualified crystallization.2 In other words, these 32 classes of crystals represent typical spatial relationships. We now have to make explicit an important distinction – although it played a hidden role in our discussions up to this point. Firstly, there are universal modal laws – such as those treated above (like the law of energy constancy and Galileo’s law of inertia). Discerning modal laws requires the distinctive feature of scholarly activities as described in Chapter 1 – modal abstraction. It is only on the basis of our integral (multifaceted) experience of reality that we gain theoretical access to the underlying modal structure of it. For this reason we may call this method of articulating modal properties transcendental-empirical. Traditionally, especially since Kant’s Critique of Pure Reason, the word transcendental is employed to account for that which provides the basis of all experience in the sense that it makes possible what we experience. Unlike Kant, however, we don’t want to assume that the transcendental conditions of experience are in advance (i.e., a priori) contained in the formal structure of the knowing person (Kant’s forms of intuition and thought categories). Much rather, we proceed from the conviction that the modal condition for experiencing physical phenomena is given in the universal modal structure of the physical aspect of reality. With this approach we in1 2

In this case, however, the speed of light is taken in an unspecified universal modal kinematical sense. Von Federov and Schönfliess, independently of each other, came to this classification already in 1890 and 1891.

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tend to claim that the physical aspect lies at the foundation of whatever we can experience in a physical sense. An analysis of the distinct modal aspects of reality therefore rests upon a transcendental-empirical approach. Secondly, we have to acknowledge that there are entitary laws for different types of entities – succinctly designated as type laws. The existence of type laws enable us to classify physical entities and place them in various categories. The typical nature of an entity specifies1 the modal meaning of the aspects in which it functions. These typical natures of entities provide a peculiar “colouring” to their modal functions. But most importantly, type laws do not hold for each and every possible kind of entity – they apply to a limited class of entities only. Stafleu explains this distinction as follows (1980:11, cf. pp.6 ff.): Hereby we distinguish laws which are valid for a limited class of subjects (typical laws) from those which are valid for all kinds of subjects (modal laws). Typical laws, in principle, delineate a class of subjects to which they apply, describing their structures and typical properties. Examples of such laws are the Coulomb law (applicable only to charged subjects), the Pauli principle (applicable to fermions), etc. Often the law describing the structure of a particular subject (e.g., the copper atom) can be reduced to some more general laws (e.g., the electromagnetic laws in quantum physics). On the other hand, modal laws are those which have a universal validity. For example, the law of gravitation applies to all physical subjects, regardless of their typical structure. We call them modal laws because, rather than circumscribing a certain class of subjects, they describe a mode of being, relatedness, experience, or explanation.2

It is well-known that Immanuel Kant launched his epistemology by asking the question: How are synthetical propositions a priori possible? (1787:19). In Chapter 1 we have mentioned his view on the thought categories of our understanding and his claim that (in a formal sense) these categories are not derived from nature but are prescribed to nature in an a priori way (1783 par.36). Although misdirected by the rationalistic assumptions of his epistemology, Kant, in his search for the synthetic a priori, actually struggled with the nature of modal universality. To appreciate Kant’s position better in this regard we have to return to the differnce between modal laws holding for whatever there is and type laws applicable to a limited class of entities only. Whoever modally abstracts a particular aspect in transcendental-empirical manner gains access to the (unspecified) universality of modal-functional relationships. Since modal aspects are not concrete entities or events they cannot be treated as if they are entitary in nature, because this would simply amount to a reification of modal functions. If one really wants to gain an understanding of the type law of any particular 1 2

Take note that we do not say individualizes because universality does not exist on one end of a continuum with individuality at its other end. The fact that modal laws – such as those of quantum physics – hold for all possible “objects” is clearly seen by Von Weizsäcker: “Quantum theory, formulated sufficiently abstract, is a unversal theory for all Gegenstandklassen (classes of objects)” (1993:128). When he explains, on the next page, that one cannot deduce the kinds of entities of experience from the universal scope of quantum theory, he has in mind what we are calling type laws.

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kind of entities one has to investigate those entities empirically. One cannot derive the typical nature of different kinds of physical entities from modal analysis or abstraction – what is required is empirical testing through experimentation. This explains why even Kant was compelled to make a distinction between his (supposedly universally valid a priori) thought categories on the one hand and so-called empirical laws of nature on the other hand: We rather have to distinguish empirical laws of nature, which always presuppose particular perceptions, from the pure or general natural laws, which, without having a foundation in particular perceptions, only contain the conditions of their necessary connection in an experience. In respect of the latter nature and possible experience are entirely the same; and since within these the law-conformity of the necessary connection of appearances in an experience (without which we are totally incapable of knowing any object of the world of the sense), actually is based upon the original laws of the understanding, so it initially does sound strange, but it is nonetheless certain, when I state with respect to the latter: understanding creates its laws (a priori) not out of nature, but prescribes them to nature (1783 par.36:320).

This distinction runs parallel with the one which we have drawn between modal laws and typical laws (type laws). Whereas Kant ought to receive credit for wrestling with the dimension of modal universality, positivism and neopositivism ought to be acknowledged for their emphasis on experimental testing (not the same as: verifying!). Only through studying the orderliness or law-conformity of entities is it possible to arrive at an understanding of the type laws holding for that limited class of entities conforming the their peculiar type laws. In the case of physics it requires empirical research through experimentation. Of course this does not free physics from an overarching and underlying paradigm (theoretical perspective) in which modal properties are also accounted for. Sometimes this dimension the theory formation is implicitly acknowledged when reference is made to theoretical terms which cannot directly be tested against actual experiences. By making an appeal to Dilthey’s sketch of seeing a natural scientific theory formation as constructing reality via logical mathematical elements of consciousness (and thus asserting the power over nature of this sovereign consciousness as an effect of the autonomy of the human intellect),1 Weyl wants to follow the conception of Hugo Dingler regarding the principle of symbolical construction. Weyl is convinced that the “constructive character of the natural sciences, the situation that their individual propositions do not have a verifiable meaning in intuition (Anschauung), but that truth builds a system which can only as a whole be assessed” (1966:192) has been explained by him. Max Planck states a similar perspective in a concise way: “Strictly seen it is totally impossible to find any physical question which can be assessed directly through measurements without the aid of a theory” (1973:341). 1

Weyl refers to the second volume of the 1923 edition of Dilthey's Collected Works (p.260). Cf. Weyl 1966:192.

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Weyl affirms the correctness of Dingler’s definition of physics as that discipline in which the principle of symbolical construction is fully carried through and then adds a statement once again making an appeal to the above-mentioned distinction between modal universality and typicality: “But what is connected with the a priori construction is experience and an analysis of experience through the experiment” (1966:192). Discussing the nature of an a priori synthetic element in the “empirical sciences,” Stegmüller raises the following possibility – also alluding to the same issue (1969:316): Surely, this cannot imply that the totality of law-statements present in a natural science could be of an a priori nature. Much rather, such an apriorism should limit itself to the construction of a limited number of a priori valid law relationships, while, furthermore, all more specific laws of nature should be dependent on empirical testing.1

Keeping in mind that we must distinguish laws in an ontical sense from our hypothetical law statements in scientific formulations, we also have to note the similarity between the just-mentioned statement of Stegmüller and the following explanation of Stafleu (related to the distinction between modal laws and typical laws): Whereas typical laws can usually be found by induction and generalization of empirical facts or lower level law statements, modal laws are found by abstraction. Euclidean geometry, Galileo’s discovery of the laws of motion ..., and thermodynamic laws are all examples of laws found by abstraction. This state of affairs is reflected in the use of the term “rational mechanics”, in distinction from experimental physics (Stafleu, 1980:11).

It must be clear that what is intended with the distinction between modal and typical laws indeed has captured the reflection of prominent thinkers. To mention one last example: C.F. von Weizsäcker. He says that although the basic assumptions of quantum theory could be written down on one page (for the mathematically trained reader!), the number of known experiences conforming to this theory runs into billions – and not a single one is found contradicting quantum theory in a convincing way. He then says, alluding to the universal validity of Kant’s thought forms: “I use an idea of Kant and conjecture that quantum theory therefore holds universally in experience, because it formulates the conditions for possible experience.”2 In order to speak about entities the modal aspects have to be used as points of entry. Even when we refer to the totality-structure of an entity the employed terms stem from a unique modal aspect: the spatial mode. The term totality is 1

2

Fales entertain, in a different conext, “the possibility that there are synthetic a priori truths; truths about abstract entities may express facts which are not merely the result of linguistic convention” (1990:148). Von Weizsäcker, 1993:93. “Laws capable of mathematical formulation finally forms the hard core of the natural science: not the important detail, but the form of universal validity” (1993:113). In another context he writes that the quantitative results of astronomy are based upon physical laws and that we postulate, as a working hypothesis, a universal validity for these laws (1993:25).

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after all simply synonymous with the terms coherence and wholeness (and it implies a multiplicity of parts and – at least in the case of spatial continuity – the whole is infinitely divisible). Although Kant did not foresee that his wrestling with the nature of the synthetic a priori actually bears upon the issue of modal universality which forms the counter-part of typicality, the effect of this distinction is still immensely important for a meaningful understanding of physical reality and for the discipline studying it: physics. We conclude our reflection with a brief analysis of the structural uniqueness of an entity and with a succinct statement of what the idea of a physical qualification of material things entail. The unity and identity of an entity One of the basic problems of theoretical reflection – occupying philosophers throughout the centuries – is given in the question how we account for the experience of identity which we attach to different things in the world. What makes it possible to recognize a changing and ageing human being as the same human being over time? Are we justified in saying that a tree – with its different appearances in summer, autumn, winter and spring – is always the same (identical) tree? Plato wrestled with the problem and eventually formulated his speculative theory of static super-sensory ideal forms, albeit on the basis of the lasting insight that change requires constancy as its basis. Perhaps it is correct to say that Plato stumbled upon the law for entities in his quest to attain knowledge of changeful things. The type law of and entity may indeed be seen as the condition for the durable identity of an entity – underlying all the changes and alterations it may experience. The identity of an entity, however, can only be approached through the point of entry provided by its different modal aspects. For that reason we already had to use terms coming from the kinematical and physical aspects even to formulate our problem – as can clearly be seen when we say that changes (physical point of entry) could only be established on the basis of something relatively constant (kinematical point of entry). In continuing its identity every single entity factually displays an orderliness correlatively reflecting the order for its existence to which it is subjected. Phrased differently: in its orderliness and law-conformity an entity in a universal way evinces that it is subjected to the universally conditioning law for its existence.1 The said order for, as law for being an entity differs in principle from the static eidos construed by Plato as the super-sensory essence of things. Similarly, also Aristotle did not escape the one-sidedness of his epistemological ap-

1

Universality and individuality are always strictly correlated at the factual side: this tree (individual side) is a tree (universal side).

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proach which identified knowledge with conceptual knowledge.1 As a strictly individual entity Aristotle’s primary substance is unknowable. This caused Aristotle to introduce his secondary substance as the universal substantial form of things – in order to save the possibility of conceptual knowing! Our knowledge of the individuality of entities closely coheres with the way in which we experience the identity of those things. This identity is something given to us in our experience and can therefore never be construed afterwards in terms of the various modal aspects by means of which we gain explanatory access to it. Being bound to these points of entry the only alternative is to set apart what is known as the typical foundational and the typical qualifying function of an entity. But even this approach cannot replace the given identity and unity of an entity – something that we can only approximate in knowledge which is of a concept-transcending nature, in other words in idea-knowledge. This entails that our (constitutive) concept of the order for and the orderliness of entities is always (regulatively) based upon the idea of the temporal unity, individuality and identity of an entity. Physically qualified entities Although the history of philosophy and the natural sciences have tried for long to find a qualifying qualification for material things in one of the first three aspects of reality, it was only at the beginning of the 20th century that general natural scientific consensus was reached concerning the energetic qualification of material things (elementary particles, atoms, molecules, macro-molecules, macro-systems). We have seen that the Pythagoreans wanted to reduce everything to number. The discovery of irrational numerical relationships led in the school of Parmenides (to which Zeno with his arguments against movement and multiplicity also belonged) to the geometrization of Greek mathematics and to the conviction that all physical things are spatially characterized. This spatial orientation lasted for more than two thousand years! The father of modern philosophy, Descartes (1596-1650), divided reality into the two spheres of an extended and thinking “substance” (res extensa and res cogitans): “the nature of body consists not in weight, hardness, colour, and the like, but in extension alone” (Principles, Part II, IV). Even until the 18th century this view exerts its influence unchanged. As we have seen Kant says that when we remove everything which the mind conceives of in the representation of the body (like substance, strength, divisibility, etc.) as well as everything which belongs to our 1

Concept-formation always occurs on the basis of universal properties. For this reason the individual side of things is conceptually speaking unknowable! Those who acknowledge conceptual knowledge only cannot account for the knowledge we have of things (and ourselves!) in their (our) individuality. We prefer to call those who identify knowledge with (universal) conceptual knowledge rationalists and those who reject conceptual knowledge while holding on the that kind of knowledge with which we know things in their individuality as irrationalists.

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awareness of the body (like impenetrability, hardness, colour, etc.), then all that remains is extension and form (Ausdehnung und Gestalt) (CPR, B:35). In connection with the nature of constancy and change we saw that the main tendency in classical physics (since Newton) was mechanistic – in other words, it was believed that all physical processes can be reduced to (mechanical) movement. The last great representative of this mechanistic approach was probably Heinrich Hertz – the German physicist who did experimental work about electromagnetic waves more than a hundred years ago.1 We have mentioned Planck’s article from 1910 where he clearly stated that the “irreversibility of natural processes” confronted the “mechanistic conception of nature” with “insurmountable problems” (1973:55). It is clear that every attempt to find an arithmetic, spatial or kinematic qualification for physical entities necessarily runs into theoretical antinomies. Let us consider the nature of an atom for a moment. Besides the arithmetic function which an atom has (think about the atomic number), it also possesses a clear spatial function: it is characterized by a particular spatial configuration – the nucleus of an atom with peripheral electron systems. According to wave mechanics, we find quantified wave movements around the nucleus of the atom – the kinematic function of the atom. Already in 1911, in Rutherford’s atomic theory, the hypothesis was posed that atoms consist of a positively charged nucleus and negatively charged particles which moved around it (a view which was inspired by the nature of a planetary system). In the following year (1912), Niels Bohr set up a new theory which contained two important new ideas: (i) the electrons move only in a limited number of discrete orbits around the nucleus and (ii) when an electron moves from an orbit with a high energy content to one with a low energy content, electromagnetic radiation occurs. In 1925, Pauli formulated his exclusion principle (Pauli-exclusion).2 According to the division of charges of electrons, corresponding electron-shells exist, and in each peel there is room for a “maximum” number of electrons. This maximum number is given by the simple formula: 2n2. In the first peel (known as the K-peel) there is room for 2 electrons; in the following L-peel, there is room for 8; in the M-sheel for 18; in the N-sheel for 32; and so on. Within a sheel with a quantum number n, (where there is room for 2n2 electrons) 1

2

This work not only established him as the founder of wireless telegraphy and the radio, but also immortalized his name in the unit of frequency (Hertz) named after him. Soon after his death in January 1894 his large theoretical work appeared: “The Principles of Mechanics developed in a New Context (Die Prinzipien der Mechanik in neuem Zusammenhange dargestellt (312 pp.).” Restricting himself to the first three modal aspects only (represented by the concepts time, space, and mass) he rejected the concept force (a physical concept) as something inherently antinomic (cf. Katscher, 1970:329). Thus we can see how consistently he carried through the mechanistic approach. It applies to fermions, i.e., elementary particles with a 1/2 spin (1/2, 3/2, 5/2, etc.) for which the statistical laws of Fermi-Dirac are formulated.

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sub-orbits are identified so that each sub-orbit with a quantum number l has room for 2(2l+1) electrons. It is already obvious from these facts that the distinct number of elementary particles in the internal atom structure are joined into a typical spatial order of electronic orbits which configure the atom as an individual physical-chemical micro-totality. The special spatial configuration which is manifest in the internal build of an atom, reflects the typical foundational function of atoms.1 The wave particle duality and the idea of the typical totality structure of an entity After Einstein reverted to a particle theory regarding the nature of light,2 it turned out, on the basis of interference phenomena,3 that it is always possible to ascribe a wave-character to elementary particles. Conversely, the Comptoneffect – regarding the interaction of a photon and an electron – supplied evidence to support the idea of distinct particles. De Broglie broadened the perspective by showing that with each and every moving particle (atoms, molecules and even macro-structures) one can associate a wave (cf. Eisberg, 1961:81, 151). Although it turned out to be impossible to establish experimentally at the same time both the particle and the wave nature Bohr claims that these two perspectives are complementary (cf. Bohr, 1968:411 ff.). In the light of the generalization provided by De Broglie one may ask: if it is possible to describe/explain entities qualified by energy in terms of two mutually exclusive experimental data, namely as particles and as waves, is it then still meaningful to speak about a unitary structure? This question puts the finger exactly on that point where the special scientific description reaches its limits and needs to fall back upon a perspective transcending the confines of special scientific inquiry. What is here required is some or other philosophical account transcending the mere combination of one or more (modally delimited) special scientific points of view. We have seen that the idea of the unity and identity of an entity could never be provided to us by theoretically explicating various modal functions, simply because this underlying unity is presupposed in all theoretical explanations. In a strict and technical sense this idea of an entity in its totality – preceding the analysis of its modal aspects – 1 2 3

Dooyeweerd initially thought – 1935-1936 – that natural things do not have a typical foundational function. In 1950 he relinquished this position (Cf. 1950:75 note 8). Light quanta are called photons and similar to the neutrino they possess a zero mass. Interference phenomena were established after Michelson – round 1880 – designed an interferometer capable of cutting light and afterwards recombining it. Thus one ends up with the same light beam – with slightly less energy. The remarkable result was that the sum did not produce light but darkness! However, when one of the two halves was blocked with a piece of black paper the other halve did appear. Seemingly the only way to explain what happened here is to assume that the interference of the split light-waves cancel out each other when reunited.

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refers to an individual whole embedded in the inter-modal and inter-structural coherence of reality, to an entity emerged in the dept-layer of an all-embracing temporality transcending genuine concept-formation and only to be approximated in a concept-transcending idea. A deepening of this basic (transcendental) idea occurs when – through theoretical reflection and investigation – the dimension of micro-structures is unveiled (the micro-world with atoms and sub-atomic particles). It is important in this context, however, to realize that concepts such as particle, field, and wave are not type concepts but modal functional concepts (sometimes referred to as elementary basic concepts of physics). Consequently, the terms particle and wave analogically reflect retrocipatory structural moments within the structure of the kinematical aspect, namely movement multiplicity (numerical analogy) and movement extension (spatial analogy). These facets are deepened in physically qualified entities and could be approximated in physical theory from the perspective of mathematical anticipations to the physical aspect – compare Shrödinger’s wave function formulated in terms of differential equations. Since number, space and movement remain irreducible aspects regardless of the nature and type of entities functioning in them (their modal universality), it is also from this perspective understandable why the functionally distinct concepts particle and wave cannot be reduced to each other – a state of affairs supported by experimental data. Physically qualified structural interlacement All entities with a physical qualification belong to the realm of material things. Atoms represent a radical type within this realm. What is the nature of the relationship between atom and molecule? Is it possible to see a molecule in an atomistic sense as the external linking of atoms actually continuing to exist? But what then about the obvious totality-properties of molecules? Are we not compelled, on the basis of the latter, to conclude (in a holistic sense) that the nature of a molecule is such that its contains and embraces the constitutive atoms in a transformed way – as integral parts of a new whole? Van Melsen says that in “most forms of Atomism it is a matter of principle that any combination of atoms into a greater unity can only be an aggregate of these atoms.” By contrast, he refers to holistic tendencies: “In modern theories atomic and molecular structures are characterized as associations of many interacting entities that lose their own identity. The resulting aggregate originates from the converging contributions of all is components. Yet, it forms a new entity, which in its turn controls the behaviour of its components” (1975:349).

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Chemical bonding is accounted for in terms of the electron shells.1 Normally only the outer electron shells are responsible for chemical bonding. Ions,2 which are normally designated with a plus (+) or minus (-) sign3 constitute ionic bonding in such a way that the positive ions are balanced by negative ones.4 Coulomb forces keep ions in chemical bonding together.5 Non-ionic (covalent) bonding takes place when certain atoms share specific electrons. This type is also known as covalent bonding or electron pair bonding not only because the pair of electrons spend much time in the space between atoms but also because they are paired their spins point in different directions. A third type of chemical bonding is found in common metals, known as metallic bonding. Most organic crystals are kept together by Van der Waals forces, but because they are to weak they should not be seen as true forms of chemical bonding. We may now return to the apparent extreme possible positions regarding the relationship between atoms and molecules in chemical bonding. From the fact that chemical bonding is chemically accounted for in terms of the (outer) electron shells, it is clear that the nucleus of the atom maintains its internal integrity in the chemical bonding. The nucleus of the atom is not simply an accidental feature of the atom but indeed that central part of the atom which determines its place in the periodic system. For example, in a crystal lattice individual atoms still serve as sources of radiation when subjected to Röntgen rays. Another consideration is that the chemical bonding does not affect the radio-activity of atomic nuclei. These kinds of considerations suggest that atomism is correct – in the quoted words of Van Melsen: “it is a matter of principle that any combination of atoms into a greater unity can only be an aggregate of these atoms.” However, this does not tell the whole story, since there are equally forceful arguments in favour of the view that the molecule, in the final analysis, forms a new unity fully encompassing the atoms as parts of an integral whole. Biochemistry discovered many isomeric forms, that is, they have identified chemical structures which are constituted by the same atoms, viewed purely numerically, but that nonetheless, owing to different spatial arrangements, differ chemically. The formula C3H6O may yield the following (chemically distinct) structures: CH3.CH2,CHO or CH3.CO.CH3. Another example is C4H4O4. 1 2 3 4 5

Weiniger points out that regarding the relationship between the classical concept of molecular structure and quantum-mechanics there are still serious unresolved problems (1984:940). These are atoms with a positive or negative charge depending on whether or not they have acquired or lost an electron. For example Na+ for a nitrogen ion and Cl- for a chlorine ion. Sodium chloride consists of a lattice such that each sodium ion (Na+) is surrounded by six chlorine ions (Cl-) and such that each chlorine ion is surrounded by six sodium ions. Distinct from electrons with a negative charge the nucleus of the atom is positively charged. The latter is constituted by neutrons (electrically neutral) and protons with a positive charge.

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H

COOH

H

COOH

H C

C

C

C COOH

COOH Maleic acid cis

H

Fumaric acid trans

From a neo-Thomistic perspective P. Hoenen, for example, defends a view of a molecule in line with Aristotle’s approach. He accepts Aristotle’s substance concept which holds that only the combination of form and matter can yield a substantial unity. Any atom maintaining its actual existence within the molecule would jeopardize the substantial unity of the molecule. Those features apparently suggesting the actual existence of atoms after their bonding in molecules ought to be seen as mere virtual characteristics. In connection with the problem of the structural interweaving of entities, Dooyeweerd developed a theoretical approach which accounts for the continuation of the internal nature of entities which are interwoven (cf.1996-III:627 ff., 694 ff.). When the internal nature of an interwoven entity is retained, Dooyeweerd speaks of enkapsis. When the structure of one kind of entity is foundational for the structure of another kind of entity, it is referred to as a one-sided enkaptic foundational relationship. With regard to the infinite divisibility of a spatial whole, there are important limits in the unqualified use of the spatial whole-parts relation. The nature of enkaptically interwoven forms illuminate further limits in this regard. The interweaving which exists, for example, between the sodium and chlorine atoms which are found in table salt, is in no way given account for with the help of a whole-parts perspective. Every division of table salt must – that is if we still want to be working with real parts of salt – still possess the same chemical structure (NaCl). The critical question is whether sodium and chlorine have each individually got a salt structure? Are sodium and chlorine true parts of salt? The answer is obvious: No, because neither one has a NaCl-structure on its own! This simple example already uproots the unqualified way in which, especially in modern system theory, literally everything in reality is spoken of in terms of a whole and parts (systems and subsystems) (cf. my criticism of this in Strauss, 1985). We have mentioned that within the realm of physically qualified entities we encounter different geno-types. Different bonds of the same atom display a 79

number of variability types. When an atom engages in chemical bonding, we encounter an enkaptic structural totality: besides an entity’s internal structural working sphere there is an external enkaptic sphere of operation – a sphere in which the enkaptically-bound structure stands in service to the enkaptically encompassing totality. A water molecule, e.g., can exist as a structural whole on the basis of the geno-type of the bond of the oxygen and hydrogen atoms. Without atoms, there can be no mention of a molecule – thus the indication: unilaterally founded. Does this imply that the atoms totally become part of the chemical bond which exists in the molecule? Not at all, because the bond applies only to the bonding electrons and not to the whole atom. Besides, as we have noted, the atom nucleus is not just a specific characteristic of the atom, but precisely that nuclear part of an atom which determines its physical-chemical geno-type (compare the atomic number = the number of protons of the nucleus), as well as the atom’s place in the periodic table. The fact that the atom nucleus remains structurally unchanged in the chemical bonding, guarantees the internal sphere of operation of the atom. Because the electrons cannot be disengaged from the atom nucleus, the atoms function as a whole in the water molecule. Note that we cannot say that the atoms function in a chemical bond. The bonding does not encompass the atomic nuclei. Nonetheless the atoms (with their nuclei, electron shells and bonding electrons) are present as a whole in the water molecule which encompasses them enkaptically. The indication: enkaptically encompassed, shows that the atoms, retaining their internal nature, is externally serviceable to the water molecule as a whole. The enkaptic interweaving of the atoms in the molecule does not make them intrinsically part of the molecule, since this would abrogate the internal sphere of action of the atoms. The external enkaptic function of the oxygen and hydrogen atoms in the water molecule indicates the functioning of the atoms in the molecule as totality via the chemical bond. This presents us with three facts: (i) First of all, we must distinguish the internal sphere of action of the atom. (ii) Secondly, we find the chemical bond which leaves the atom nucleus unchanged because it only reaches the outer electron shells, so that the atom nuclei can in no way be part of the chemical bonding.1 (iii) Thirdly, we find the enkaptic structural whole of the water molecule which enkaptically encompasses the atomic nuclei and bonds and ascribes to each its structural typical place. This theory of enkaptic interlacement enables us to side-step the one-sidedness in an atomistic and a holistic structural theory of chemical bonding within a molecule – and it also naturally reconciles apparently contradictory experimental data, since it accounts both for the continued actual existence of atoms in molecules (the point of orientation of atomism) and for the typical unitary character of the molecule (the emphasis of holism) as a new totality enkaptically founded in the structural nature of atoms. 1

Holism is therefore mistaken in its claim that the atom “loses” its identity in the molecule. Cf. Van Melsen, 1975:349.

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By briefly returning to the original and analogical meaning of spatial extension we can demonstrate the effect a mistaken conception of space had on the intepretation of data. Whereas the mathematical space – in a purely abstract and functional perspective – is, as we have observed, both continuous and infinitely divisible, physical space (by being bound to the quatum structure of energy) is neither continuous nor infinitely divisible. For a number of years a controversy existed between Millikan and Ehrenhaft. The former received in 1923 the Nobel prize for physics for his work which established that the electron is the fundamental and invisible unit for negative electrical charge. The latter believed to have observed electrical chrages smaller than the electron. Cushing remakrs that is appears as if Ehrenhaft misinterpreted the data because he still believed that electrical charge is continuously divisible (Cushing, 2000:10).

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Paradigms in Mathematics, Physics, and Biology: Their Philosophical Roots Chapter IV The Mosaic of philosophical stances in modern biology Introduction Of course the decisive basic philosophical problem facing biological scientific thinking throughout its history has been that concerning the relationship between the biotical aspect and the physico-chemical aspect of reality in which the former is founded. Just as mathematics is often reduced to either the arithmetical or spatial extreme, the history of biological thought is marked by the tension between mechanistic and vitalistic approaches. Do biotically qualified entities really exist, or can one exclusively and completely describe such things in terms of their constitutive physico-chemical components? If the latter point of view is correct, then one has to ask whether the distinction between ‘life’ and ‘death’ still makes any sense: if everything is determined by the interaction among lifeless material constituents then the difference between being alive and being liveless fades to an illusory peripheral phenomenon of the physical mass of reality. Hans Jonas once strikingly typified the monistic forms of vitalism and mechanicism. Unlike dualists, monists do not attempt to reduce reality philosophically to two fundamental principles, but rather posit a single all-inclusive and universally explanatory principle. We may therefore just as well speak about pan-vitalism and pan-mechanicism. Already in Greek philosophy we come across hulèzoism (zoè = life; hulè = matter): one of the indirectly preserved aphorisms of Thales supposedly was that everything lives. From this perspective it is unimaginable that ‘life’ may not be the universal rule. Jonas comments: “In such a world view death is a riddle confronting one, a contradiction of the natural, self-explanatory and understandable, of the common life” (1973:20).

The paragraph in which Jonas makes this statement treats pan-vitalism and the problem of death (1973:19ff). On the other hand, people who think panmechanistically emphasize the notion that living phenomena are peripheral in an encompassingly homogeneous physical world. Quantitatively negligible in 83

the immeasurable expanse of cosmic matter, qualitatively an exception to the rule of material characteristics, scientifically inexplicable in an explicable physical natural reality, “life” becomes an insurmountable obstacle for pan-mechanicism: “Life as problem here indicates recognition of its strangeness in the mechanical world, which is the real world; to explain it means – on this level of the universal ontology of death – to deny it, reducing it to a variant of the possibility of the lifeless” (1973:23). This paragraph treats pan-mechanism and the problem of life (1973:22ff).

A first step out of this dilemma is to be found in making a distinction between different modal aspects. The fundamental modal character of the physical and biotical aspects remains only a functional condition for concrete entities which continue to function in these (and other) aspects of reality in a typical way . What is at issue here is the basic distinction between the aspects of reality and the dimension of entities – a distinction continually disregarded by the different points of view in biology which time and again speak of modal functions as if they are concrete entities (thus the habitual reference to the origin of life, rather than to the origin of living things).1 As an aspect of reality life has to do with the how of entities, not their concrete what. Phenomena of life are always linked to living entities which – as entities – can never be encompassed by their biotical aspect. This has been a problem especially in the vitalistic tradition – which absolutises life and sees it as varieties of an immaterial vital force. That it is impossible to consider the biotical aspect of living things detached from the intermodal coherence within which it finds itself is confirmed repeatedly by the inherent analogies in the structure of the biotical aspect. Even the expression vital force, so often used by vitalism (although often replaced with other terms like Gestaltungsfaktor or Zentralinstanz), can never indicate the distinctiveness of the biotical aspect – simply because it is unmistakably a physical analogy in the modal structure of the biotical aspect. The term force reveals the original (non-analogical) modal meaning of the physical aspect of energy-operation. Biotically qualified entities Biotically qualified or characterized entities belong to the realm of plants. The distinction between the physical and biotical aspects is foundational to the entity-structural distinction between the realm of physical things and the realm of biotically qualified entities – i.e. the plant realm. In the absence of the necessary modal distinctions it is still common for supporters of differing biological points of view to use expressions like: living matter and dead matter. Material things, however, are exclusively physically qualified and can therefore not simultaneously have an internal biotical qualifying function.

1

The latin word for a thing, “res,” renders a service to the way in which one can designate the attempt to treat a modal function as if it is an entity: this fallcy is known as reification.

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It is not meaningful to refer to matter as being “dead”1 – for strictly speaking only something that once was alive could later on be called “dead.”2 What guarantees the identity of living things? Already the indication that certain things are alive implies their active functioning in the biotical aspect of reality – i.e., their biotical subject function. The fact that living things must, in a thermodynamic sense, be considered open systems, moreover indicates that every living thing in distinction from its qualifying biotical aspect also has a physical aspect. This is the topic of the well-known book of Erwin Schrödinger: What is life? The physical aspect of the cell (1955). Any living entity does have subject functions in the first three aspects of reality as well – namely the aspects of number, space and movement. For example, linked to the question whether living things can move by themselves we find another important distinction in biological systematics, namely that between plants and animals.3 The continuity (endurance) of life of a plant can be determined in coherence with the kinematic function of living entities. Apart from the proportions or spatial form of living things, their spatial function is also prominently exhibited in expressions like bio-milieu or Umwelt. The term Umwelt gained prominence especially owing to the biological thinking of Jacob von Uexküll (cf. e.g. Von Uexküll & Kriszat, 1970). A living thing is furthermore a unity in the diversity of its organic life processes – if these various processes are not bound together as a unity, the living entity disintegrates and dies. Since living things maintain, in thermodynamic terms, a flowing equilibrium in which order is withdrawn from the environment (to which Schrödinger refers as negative entropy), it can be said that living things maintain themselves in a state of high statistical improbability: in typical growth processes living entities even continually increases their internal order. Of course it cannot be considered to be the distinctive characteristic of living things, since several non-living entities and porcesses – including flames and glaciers – are also open systems in a thermodynamic sense. Only when we take into account the qualifying biotical subject function of living things can we uncover their distinctive characteristic in comparison with material things. This qualifying function determines the biotical identity of living things. According to the mechanistic point of view in biology, however, living things are only complex interactive systems in which, in accordance with the nature of open systems, continuing metabolic processes (anabolism and catabolism) occur. Remark: Since Descartes modern philosophy and biology is acquainted with a machine model. Although we may think that this model consti1 2 3

Oparin, for example, following the dialectical materialistic approach of Engels, does that without hesitation. Von Weizsäcker explicitly highlights this insight – 1993:32: “Die Steine sind unbelebt. Man sollte aber nicht sagen, sie seien tot. Tot sein kann eigentlich nur etwas das gelebt hat.” This question regarding their mobile capacities appeals to the typical function of plants or animals within the kinematic aspect.

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tutes a straightforward reduction – even of the human being – to “nature,” the implicit technicistic undertones of this model are lost sight of. The nature of a machine ought te be scrutinized first, because a machine only originated in the course of human civilization. Van Weizsäcker correctly says: “Thinking nature – and with it the human being – as a machine, subjects nature and with it the human being to a specific industiral mode of thought, that of designability. Not the reduction of the human being to nature is the mistake here, but the reduction of nature to the structural properties of a very specific human artefact.”1 Thus from a mechanistic point of view a living thing has a physico-chemical identity constituted by its atoms, molecules, and macro-molecules. Which of these physico-chemical components should however be considered constitutive of this supposed physico-chemical identity of living things: currently present atoms, molecules, and macro-molecules, those present years ago, or those which will be present a few years hence!?2 When living things are physicalistically reduced to their material constituents, their biotical identity is necessarily lost – since the supposed elements of identity continually vary. Once the biotical function of living things is taken into account, it is even possible to claim that a living thing, biotically considered, is in a stable state (referred to as health), while simultaneously claiming – without any contradiction – that physico-chemically considered (with a view to the flowing equilibrium of its physical-chemical constituents) it exists in an unstable state. If the physical-chemical substratum of living things approaches a state of higher statistical probability, biotical instability increases as a sign of the final process of dying. From the perspective of his organismic biology von Bertalanffy strikingly indicates the cul-de-sacs of the mechanistic point of view which eliminates the biotical function of life processes: “These processes, it is true, are different in a living, sick or dead dog; but the laws of physics do not tell a difference, they are not interested in whether dogs are alive or dead. This remains the same even if we take into account the latest results of molecular biology. One DNA molecule, protein, enzyme or hormonal process is as good as another; each is determined by physical and chemical laws, none is better, healthier or more normal than the other” (1973:146). 1

2

“Die natur und dann damit den Menschen als Machine zu denken, unterwirft die Natur and damit den Menschen einer spezifisch industriellen Denkweise der Planbarkeit. Nicht die Reduktion des Menschen auf die Natur ist hier die Fehler, sondern die Reduktion der Natur auf the Struktureigenschaften einbes sehr speziellen Menschenwerks” (Van Weizsäcker, 1993:38). Jones et al points out that all “the atoms of our body, even of our bones, are exchanged at least once every seven years. All the atoms in our face are renewed every six months, all our red blood cells every four months and 98% of the protein in the brain in less than a month. Our white blood cells are replaced every ten days and most of the pancreas cells and one-thirteenth of all our tissue proteins are renewed every 24 hours” (1998:40).

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The origin of living things – a biological boundary question Although the view that living things could spontaneously emerge out of lifeless matter (generatio spontanea) has been known since Greek antiquity, it is no longer accepted in modern times by any natural scientist – inter alia. due to the work of Pasteur. Nonetheless the mechanistic (or rather: physicalist) perspective must make at least one exception: the origin of the first living entity under circumstances entirely alien to those known to us today. The oldest known fossils of living entities are those of unicellular algae – found near Barberton in South Africa. By means of the half-life of radio-active substances the age of these Archaeosphairoïdes barbertonensis have been calculated as approximately 3 100 million years (cf. Schopf, W. & Barghoorn, 1967:508ff). Since living entities, considered physico-chemically, function on the basis of both (enzyme) protein and nucleic acid (DNA), the mechanistic point of view is obliged to presume that initially there must be a close relationship between protein and DNA. Already in 1971, however, Orgel and Sulston comment in this regard: “This approach leads to new difficulties so severe that it has never been carried very far” (1971:91). They continue with the striking observation that “progress” can only be recorded in this regard when characteristics are attributed to protein and DNA “which have not been demonstrated experimentally, and which usually seem implausible” (1971:91). These comments actually refer back to ideas initially (and independently) developed by Haldane (already in 1928) and the Russian Oparin (cf. 1953, chapters 4-7: pp.64-195). The assumptions of the Oparin-Haldane approach eventually turned out to be questionable. That the initial atmosphere of the earth was mainly composed of hydrogen, methane, ammonia and water vapor. In particular Oparin holds that carbon “made its first appearance on the Earth’s surface not in the oxidized form of carbon dioxide but, on the contrary, in the reduced state, in the form of hydrocarbons” (1953:101-102). Silver points out that there is at present “no evidence that the atmosphere was reducing (methane and hydrogen)” and remarks that “the prevalent opinion at the moment is that the Earth’s atmosphere, at the time that life emerged, was mainly carbon dioxide and nitrogen” (1998:344). The role of methane is also unacceptable in the Oparin story since it is one of the components of natural gas which is produced by the “effect of millions of years of pressure and heat acting on prehistoric plant material” (Silver, 1998:344). Although the Haldane-Oparin conjecture was kept alive for a considerable time, supported by the experiments done by Stanley Miller (from Chicago) in 1953, it does not bring us closer to an understanding of the mystery of the genesis of the living cell. With regard to Miller’s experimentation Silver remarks: “The Haldane-Oparin hypothesis is out of fashion. Of the forty or so simple molecules that would be needed to form a primitive cell, the experiment produces two. It is worth bearing in mind that glycine contains only ten atoms and alanine, thirteen. The simplest nucleotide contains thirty atoms. The probability that a given large molecule will be produced by chance from small molecules, by sparks, falls drastically as the molecular size increases. It has to be 87

realized that even if heat, radiation, and lightning, on the young Earth, had produced all the amino acids and nucleotides needed for present forms of life, the gap between an aqueous solution of these emolecules and a living cell is stipendous. It’s a question of organization: in the absence of a guiding intelligence, presentday scientists are not doing very well. For the moment, let’s show the Miller experiment to the side door and see who is next in line in the waiting room” (Silver, 1998:345).

In neo-Darwinist thought natural selection receives much prominence.1 A similar story is used to explain the origin of the first living entities: by means of selection the accidental emergence of organic combinations (amino acids, nucleic acid, enzymes, etc.) supposedly gave rise to the formation of reproductive units, virus-like forms, proto-organisms and eventually true living cells. In view of physical laws, Von Bertalanffy, amongst others, also questions this construction: “In contrast to this it should be pointed out that selection, competition and ‘survival of the fittest’ already presuppose the existence of self-maintaining systems; they therefore cannot be the result of selection. At present we know no physical law which would prescribe that, in a ‘soup’ of organic compounds, open systems, self-maintaining in a state of highest improbability, are formed. And even if such systems are accepted as being ‘given’, there is no law in physics stating that their evolution, on the whole, would proceed in the direction of increasing organization, i.e. improbability. Selection of genotypes with maximum offspring helps little in this respect. It is hard to understand why, owing to differential reproduction, evolution should have gone beyond rabbits, herring or even bacteria, which are unrivalled in their reproduction rate” (1973:160-161).

Those who have respect for scientific modesty may do well to reflect upon a remark made by Haldane in discussion with Silver: “I had a long conversation with J.B.S. Haldane, which started off with politics and ended with science. When I questioned him about evolution, one of his remarks sparked my interest, and sent me to the library that evening: ‘Evolution’s not the problem. Life is’ Then he said, ‘Oparin and I once had an idea about that, but we’ll never know the real answer’ ” (Silver, 1998:353). Are viruses a transitional form between material and living entities? Viruses consist of nucleic acid (either RNA or DNA) housed in a mantle of protein and occasional lipids. In 1935 W.M. Stanley succeeded in purifying and crystalizing the tobacco mosaic virus. Viruses are only able to multiply (and in the process act in a deformative way as parasites) in living cells. We know nothing about the actual origin of viruses, which means that the possibility that they could be reduced micro-organisms, or genes dismantled from the cell-structure, or products of cell metabolism, remain speculative. Thus the supposed in-between-position of viruses remains problematic. What is remarkable about the “in-between-position-hypothesis” is that it starts off from the distinction: material things and living things. That is why 1

Darwin already developed the theory that since far more descendants are born than could survive, a continual struggle exists in which only the best equipped “organism” makes the grade – out of which gradually new kinds emerge.

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the question is thus posed: is a virus living or is it only a macro-molecular material structure? Since it can reproduce inside a true living cell, it is suggested that it could maintain an in-between-position. In reality, however, we once again run into the presuppositions of scientific distinctions. All scientific distinctions, after all, presuppose the whole human being with all one’s pre-scientific experience of reality. The diversity implied in this experience is only made explicit by means of modal abstraction, e.g. when we distinguish between the physical and biotical aspects of reality. This distinction, however. is embedded in our pre- (or rather non-) scientific experience of reality – and particularly in the difference encountered between material and plantlike things (by means of entity-oriented abstraction). Before scientists (whether philosophers or special scientists) can investigate the nature and characteristics of plants, they must first, by means of their non-scientific experience, gain insight into the qualitative difference in kind between plants and material things. The question: what is botany? after all is a philosophical presuppositional question of botany as a special science. Although we would be inclined to claim that botanists are indeed able to say what a plant really is, this privilege actually must ultimately be denied them, since, if they did not already in their non-scientific concrete experience of reality have had the ability to distinguish (e.g. by focusing on the differences between matter, plants, and animals), then they may well have been studying material things or animals in the mistaken belief that they were studying plants! There can be no denial that scientific thought rests on our non-scientific understanding of distinctness. Without this foundation it simply cannot function. The important implication of this insight into the founding role of our non-scientific understanding of distinctions in this regard is this: a particular entity can only be either (non-living) material in nature or (biotically) alive. This of course bears decisive implications for the question of whether a virus is living (i.e. plantlike) or non-living (i.e. material). Stating the question in this manner already implies that the answer cannot be ambivalent – and in both cases there can be no question of any in-between position. This “transition problem” between non-living and living things points towards the far greater problem caused in modern biology by the evolutionary theory of Charles Darwin. Since Darwin’s notions began to gain biological support, it has become a matter of course in modern biological literature to talk of evolution in the sense of an all-encompassing development across all borders and distinctions which can supposedly still be found today in the living world. The categorizations to which we can come by only paying attention to currently living plants and animals is known as the natural system (abbreviated as NS). The central philosophical question is whether the NS can be used as the foundation for some sort of evolutionary theory or whether one or another evolutionary theory should not rather be used as the foundation for a reasoned categorization of the NS. Reflection on this foundational issue however soon comes in contact 89

with a traditional philosophical conflict in which modern biologists have largely taken sides – in favour of nominalism. Nominalist structural understanding in modern biological literature When G.G. Simpson distinguishes in one of his works between the physical sciences and biology he characterizes the former as largely typological and idealistic: “the physical sciences are for the most part typological and idealistic. I mean by that, that they usually deal with objects and events as invariant types, not as individuals with differing characteristics” (1969:8).

This approach is according to him completely inadequate for the study of phenomena belonging to the biotical levels: “for phenomena special to the biological levels” (1969:8). What strikes one in this statement by Simpson, is that he distinguishes between two types of phenomena, namely physical and biotical phenomena (although the latter is mistakenly referred to as biological). In order to arrive at an identification of biotical phenomena where a typological (and even idealistic) method would be of no use, Simpson uses exactly a typological method – a striking internal contradiction: biology can function nontypologically if and only if it is typologically founded! The fundamental principle of intra-biological research is formulated by Simpson as follows: “Organisms are not types and do not have types” (1969:8-9). The main ground for this statement is the claim that organisms are individuals “and no two are likely ever to be exactly alike” (1969:9). This argument already betrays Simpson’s view with regard to the relationship between physical law and physical individuality. Since physical subjects are according to him studied in physics only as “objects and events” with “invariant types” (1969:8), it follows that biology, which studies organisms in their individuality only, cannot make use of any typological method. Simpson’s approach typically descends from classical mechanistic physics (see the previous chapter in this regard). In classical physics physical subjects had been consistently reduced to the law-side of reality (in a rationalist manner). There is nonetheless no conflict between what modern physicists know and the attribution of individuality to physical subjects such as atoms. Should one maintain that physically-characterized entities have individuality although they are correlated with universally valid physical laws, the question arises why this could not be the case with biotically and psychic-sensitively characterized entities (such as plants and animals). Apparently without noticing Simpson contradicts his own view that organisms do not belong to any type when he reflects on the typical characteristics of a human being. In this regard he refers, without reserve, to mouseness and man-ness (1969:88). Mouseness and man-ness, however, allude to the structural preconditions which entities must meet before they can be known as mice and human beings. Structural preconditions are no less real because they do not themselves have a concrete individual identity.1 1

The conditions for being green are not themselves green – green things simply meet these conditions by being green.

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Since Darwin however many biologists are inclined to dismiss any idea of structure and universality. This is done since the history of the origin of the plant and animal world must supposedly be encompassed in a structureless evolutionary continuum. Even the distinction between plant and animal is considered a mere convention, merely arbitrary names (nomina) given to an unlimited number of concretely individual living entities. The universality implied by these names have no foundation in things outside the human mind – this universality is purely a product of human thought. The epistemological position of nominalism rests on cosmological presuppositions found already in Greek philosophy, but which only became dominant in modern biological thought since Darwin. Over and against the nominalist techniques of classification used in many biology texts, mention is sometimes made of the older and supposedly outdated idealistic morphology of Ray, Linnaeus and others. There are nonetheless still important 20th century representatives of this morphology, including E. Dacqué (cf. 1935, 1940, 1948), W. Troll (1949, 1951 and 1973), K. Lothar Wolf (1951) and W. Leinfeller (1966). According to Troll the foundation of comparative morphology is to be found in ideas (in the platonic sense) which serve as ordering “inner articulations of our intuition” by means of which types as “Urbildliche Einheiten” (primal imagery units) become study subjects (cf. Ungerer 1966:232). Troll partially reaches back to the thinking of J.W. Goethe – the Romantic poet and natural philosopher. In his biological investigations, largely concerned with morphology, Goethe emphasized the character of “Gestalt” – form in an almost platonic sense – although he shifted the emphasis to the factual side of reality, since he did not see the “Gestalt” as rooted in the law, but rather the law in the “Gestalt.” In idealist morphology a primal leaf or primal plant is designed in which certain basic typological characteristics are present. Zimmerman engages in a dialogue with this idealist morphology in his Evolution und Naturphilosophie (Berlin 1968). He points out that Troll continues to believe that morphology determines the possibility of the descent and not the other way around: “It is not the descent which is decisive in morphology, but rather the opposite: morphology has to decide about the possibility if descent.”1

It is possible to believe that the problems of the NS should serve as the foundation of any possible theory of descent without being a supporter of idealist morphology. Portmann comments for instance that “few biologists still consider that systematics is the foundation of evolutionary theory, that this is the certain, that which we know, while evolutionary theory is what we suspect” (1965:10). It is possible, without supporting either the argument or point of departure of idealist morphology, to agree that the existing structural diversity of our contemporary experience of reality is of decisive importance with regard to the 1

“Es is nicht die Dezendenz, welche in der morphologie entscheidet, sondern umgekehrt: die Morphologie hat über die Möglichkeit der Dezendenz zu entscheiden” (Zimmermann 1968:49).

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question of descent of any existing living entity. Exactly in this sense the present is the key to our understanding of the past. The geologist J.R. Van de Fliert even goes so far as to say that the doors to the past can only open to the extent that the keys of the present fit them. He mentions the example of a number of old fossils of the pre-Cambrium (more than 600 million years ago): “These fossils consisted of imprints of animals, which probably had not possessed any hard parts, and which in part could be determined because of close resemblance with the structure of living jelly-fish, worms, and other animals. Some of these fossil structures, however, are so far completely unknown in living animals or plants and as a result they are enigmatical, ‘problematica’. In the absence of any structural link with the present they could not be attributed to any known phylum.”1

The existing structural diversity in the plant and animal realms would clearly limit the construction of family trees in evolutionary theory. Whoever relativizes the NS in a nominalist sense to a structureless line of descent, however, chooses in principle for a chaotic brew in which every taxonomy of living things becomes either impossible in principle, or at least entirely arbitrary. But the plant and animal realms are entirely meaningfully divisible, barring only the instances where our scientifically developed criteria for distinction are still insufficient. (Think for instance of the protista, which includes algae, fungi, mucus fungi, and protozoa, of all of which it cannot be said with certainty whether they are plants or animals. These problems do not however relativize the fact that each of these protista is either plantlike or animal-like). Structureless continuity versus structural discontinuity A core concept in nominalist evolutionary theory is summarized in the word variability. Think for instance of the comment by Simpson, mentioned above, that physics is concerned with invariant types while biology is concerned with changeable individual organisms. The fundamental question concerns the relationship between constancy and variability. The stronger tendency in evolutionary circles is to choose for a complete variability rather than any structural constancy. In idealist morphology a choice is apparently made for a (platonically influenced) understanding of constancy on the opposite extreme end of the spectrum of options. The problem, however, is that the concept variability only makes sense when delimited by some or other typicality or constancy. Here we come across a further analogy of the indissoluble coherence between the kinematic and physical aspects of reality. Biotic constancy and biotic dynamics are ontologically considered equitable – to emphasize the one over the other would entail disregarding the uniqueness of either the kinematic or the physical aspects (cf. the more extensive founding of this insight in the previous chapter). Eisenstein correctly points out that the term constancy encompasses the concept of variability also in biology in the sense that variation is only possible within limits (1975:278). Van de Fliert comments in addition that the quanti1

The Christian and Science, unpublished presentation given at Calvin College, Grand Rapids, 1969, pp.26-27.

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tative can only be understood as something determined by the qualitative: “More or less of this remains this and does not become that” (1969:28). Whoever emphasizes variability has difficulty answering the following question: If living entities during the past three thousand million years have been governed by a universal evolutionary law so that they developed in a globally progressive way towards the human being, it cannot be explained why there still are, apart from the highly evolved animals, such primitive entities as bacteria, algae, mosses, amoebae, worms, etc. – why did the evolved animals not also remain stuck on these original levels? Eisenstein writes: “The simultaneous co-existence of the greatest variety of life forms, from amoeba to man, anyway proves that from the perspective of nature these are all equitable and equally viable (existenzfähig = able to exist), without any necessity of further development” (1975:245). The zoologist W. H. Thorpe comments: “[i]t seems to me that there is an outstanding problem raised by our discussion – namely the problem of fixity in evolution. What is it that holds so many groups of animals to an astonishingly constant form over millions of years? This seems to me to be the problem now – the problem of constancy; rather than of change. And here one must remember that the genetic systems which govern homologous structures are continually changing. Thus the control system is continually changing but the system controlled is constant, and constant over millions of years. This problem seems to me to stick out like a sore thumb in modern evolutionary theory.”1 Within the framework of the reformational philosophical tradition an attempt is made to avoid the partialities of both nominalist and idealist (realist) structural ideas. Just as little as a modal physical and typical entity structural law should be confused with any subject function or concrete physical subject, just as little should the structural types of plants and animals be confused with particular concrete plants or animals. All plants and animals belong respectively to the realms of structurally either biotically or sensitive-psychically directed ordered types. As true law types these belong to the law side of created reality, in which admitedly a relatively constant dynamics finds expression, but which as law types cannot be reduced to or equated with transient individual living plants or animals. These relatively constant ordering types can only be realized in the course of time in transient individual creatures which as correlate is subject to the ordering types. The recognition of this correlation of the law and factual sides of reality is a stumbling block for the structereless evolutionary continuum construed in a nominalistic fashion. To form a better image of the nature of the divergence between different directions in modern biology we must consider the factual limitations set to the religion of continuity by paleontology. 1

A discussion comment after the contribution of L. von Bertalanffy (Change or Law) in the collection: Beyond Reductionism, edited by A. Koestler and J.R. Smythies, London 1972:77.

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Continuity of descent? Since the first appearance of Darwin’s notorious writings much faith has been placed in the substantiatory power of paleontological fossil findings – a certainty had grown that the necessary missing links would eventually be found. It was a conviction that paleontology would provide direct access to the key moments in the evolutionary history of plants, animals, and human beings. The paleontologist D.B. Kitts points out, however, that the spatial distribution and temporal sequence of organisms with which paleontology works is founded in the ordering principles of geology, and can therefore not be encompassed in any biological theory: “Thus the paleontologist can provide knowledge that cannot be provided by biological principles alone. But he cannot provide us with evolution. We can leave the fossil record free of a theory of evolution. An evolutionist, however, cannot leave the fossil record free of the evolutionary hypothesis” (1974:466). The danger continues to exist that biologists are convinced of the acceptability of the evolutionary hypothesis by a theory which is already inherently evolutionistic: “For most biologists the strongest reason for accepting the evolutionary hypothesis is their acceptance of some theory that entails it” (Kitts 1974:466). A core paleontological problem facing the various trends is the presence of striking gaps and discontinuities in the fossil record. G.G. Simpson openly mentions this factual state of affairs: “moreover, it is a fact that discontinuities are almost always and systematically present at the origin of really high categories” (1961:361) and he emphasizes several pages later “the point that for still higher categories discontinuity of appearance in the record is not only frequent but also systematic. Some break in continuity always occurs in categories from orders upwards” (1961:366).1 The quest for missing links has always been a hopeful glance at paleontology on the part of evolutionary theory – an expectation which has apparently not remained entirely unrequited, since the much sought after forms appears to have been found between the different classes of vertebrates. Four forms are of importance in this regard. The link between certain fishes (Crossopterygii) and amphibians is looked for in the Ichthyostega (first finds in Greenland in 1931) which belong to the Tetrapoda (quadruped vertebrates, including the amphibians, reptiles, birds, and mammals) but still have a true fish tail (cf. Kuhn-Schnyder 1967:350-352). Since the Ichthyostega are quadrupeds, they are placed with the amphibians (they are fishlike amphibians). Over and against this bottom end of the amphibians we find at the top end reptile-like amphibians, the Seymoria (first finds 1904). D.M.S. Watson comments: “The whole effect of its structure is that of a mosaic of separate details, some completely amphibian, some completely reptilian, and very few, if any, showing a passage leading from one to the other.”2 1 2

The basic systematic classification consists of the following: realm, phylum, class, order, family, genus, and species. Quoted by Kuhn-Schnyder, e.: 1967:357.

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A further form, the placement of which brought about interesting problems, is the Ictidosauria (discovered in 1932) which has both reptilian and mammalian characteristics. Such an Ictidosaurier can be seen in the Bloemfontein museum (described by A.W. Crompton as Diarthrognathus broomi). Its skull has both a reduced articulare-quadratum-joint and a dental-squamosum-joint. The presence of a dental-squamosum-joint is a typically mammalian characteristic and caused this form to be thought of as a reptiloid mammal. The classification of Hopson and Kitching revised this classification since they grouped the Ictidosauria with the Cynodontia (a group developed mammaloid reptiles of the Permian and Triassic eras). In the class of reptiles (Reptilia) we therefore meet the order of mammaloid reptiles (Therapsida), the suborder Cynodontia, the family of Tritheledontidea (a group of highly developed, carnivorous small cynodonts including the Ictidosauria), and the genus Pachygenelus which is the same as the Diarthrognathus broomi of Watson of the late Triassic (red riverbed and cave sandstone strata in S.A.) (cf. Hopson and Kitching 1972:76). The provisional result is therefore that we are still dealing with a reptile, although a mammaloid reptile. Widely differing evaluations have been the fate of especially the Archaeopteryx (discovered already in 1861), which has both reptilian and avian characteristics. Although G.G. Simpson and O.H. Schindewolf largely concur with regard to the discovered state of affairs, they approach the factual information from radically divergent points of departure. Schindewolf is of the opinion that the transition from the class of reptiles to the class mammals found expression in the appearance of Archaeopteryx.1 This animal was a bird with wings which could fly, the first representative of a new class – the Aves (birds). In this regard Simpson comments: “Schindewolf disposes of it by saying that it is ‘a true bird’ and so cannot close the discontinuity between reptiles and birds. But if we did not know that Archaeopteryx had feathers, or if we found its last featherless ancestors, then of course we would have ‘a true reptile’. The break can be maintained in words even if it is closed by specimens” (Simpson 1960:370, cf. p. 342). M. Grene typifies Simpson’s approach as follows: “Simpson says Archaeopteryx was a species like any other, originating by normal speciation from other reptilian species; only when we look back over the whole vista of evolution do we say, this particular species was the first of what turned out to be a new class” (1974:130). Exactly in view of this apparently inevitable presence of gaps in the paleontological record Stephen Gould and his followers have begun during the past two decades to scramble backwards with their theory of punctuated equilibria.2 All the supposed transitional forms mentioned above therefore do not qualify as true transitional forms in the sense of an entirely continuous evolutionary 1 2

Schindewold is a great German paleontologist, whose main work is Grundfragen der Paläontologie 1950, cf. 1969. As an aside we must mention just how major a problem the development of reptilian scales into bird feathers is for evolutionary theory. In addition no fossils have been found which can be considered as ancestors of currently living birds, while the fourth Archaeopteryx was discovered in 1956, namely Archaeopteryx lithographica.

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process. That is why D.B. Kitts can still say that “Evolution requires intermediate forms between species and paleontology does not provide them” (1974:467). He points out that Darwin hoped that continuing fossil finds would fill the gaps and then remarks: “But most of the gaps are still there a century later and some paleontologists were no longer willing to explain them away geologically” (p.467). This fact gives much room for the acknowledgment of structural discontinuities in the paleontological record – the paleontological data is entirely reconcilable with the hypothetical presupposition of fundamental (relatively constant) types of order within the plant and animal realms which as true law types make variability possible in the first place. The current evaluation of the supposed transitional forms is even more complicated by the existence of living “transitional forms.” One example is the well-known platypus of eastern Australia, Tasmania, New Guinea, and the Salawati islands. Mammalian characteristics (Theria) are decisive in the placement of these animals in the subclass Prototheria and the order of Monotremata. The platypus is one of two families in the order of Monotremata and consists of only one species of platypus, namely Ornithorhynchus anaticus. What is so remarkable about this kind of mammal is that apart from mammalian characteristics (such as the dental-squamosum-joint, enucleate red corpuscles, presence of a diaphragm, only left aorta, hair, milk glands, three ear ossicles), it does not only have reptilian characteristics (eggs with yolk and shell, no ear muscle, etc.), but also avian characteristics (similarly to the platypus birds lay eggs, have a beak and a cloaca into which the intestine, urine and genitalia discharge). As is often the case with living fossils platypi are highly specialized in certain particular characteristics while remarkably showing no further developmental trend towards either birds or more typical mammals. According to Eisenstein platypi therefore have a right and potential to existence equal to that of birds and other mammals (1975:251). In this regard we must point out furthermore that various paleontologists are struck by the fact that the so-called intermediary forms are by no means truly intermediate since various typical characteristics are present intact next to one another (compare Watson’s comment on the Seymoria above). Schindewolf refers to these as mixed types (Mischtypen), while G. de Beer in honour of D.M.S. Watson refers to the mosaïc figure of these forms in terms of the Watson rule. This rule states that in the transitional area between two levels of development mosaïc figures appear in which each organ appears to have an evolutionary tempo of its own and in which the relevant characteristics develop sharply independently of one another (cf. Kuhn-Schnyder 1967:362). The formulation of this rule however presupposes evolutionary transitions even though the tale of continuity remains threatened since true intermediate forms simply do not fit into the picture it paints. In this context a distinction is made between levels of development (Stufen) which must be distinguished in phylogenetics from true lines of descent (Ahnenreihe). In the strictest sense of the word it is impossible to conclusively and exactly prove any line of descent. Even if it is taken into account that the 96

“primeval bird” Archaeopteryx is similar to modern birds in its possession of feathers, in other regards similar to true reptiles (e.g. in its possession of a reptiloid tail), and that it appeared paleontologically at a moment which provides a link with comparable reptiles (Archaeopteryx is about 30 million years younger than the comparable Pseudosuchier) while occurring relatively shortly before the appearance of birds in the Crustacean, there is still no conclusive proof that contemporary birds have descended from Archaeopteryx.1 According to Walter Zimmerman it is more possible rather to show that certain fossils are not true ancestors of later forms. This is true even of what appears to be the most solidly founded lines of descent (1967:100). The general conclusion reached by Zimmerman is formulated by him as follows: “In short, the ancestral and typical lines of phylogeneticists are not only on occasion but always what O. Abel on occasion referred to as ‘level lines’. The fossil forms which we encounter in the past, as well as current organisms represent in the characteristics which interest us the developmental level which the particular ancestor involved had reached.”2 The implication is that the fundamental discipline of evolutionary theory in this context is the phylogenetics of distinctive characteristics (Merkmalsphylogenie), which provides the foundation of the supposed lines of descent (Sippen-phylogenie) (cf. Zimmerman 1967:103). The contribution of Darwin’s original work, which is concerned exactly with the ‘origin of species’, is clearly threatened when A. Meyer draws the radical conclusion that “There is no phylogeny of species, but a phylogeny of the typological characteristics of the species” (1964:60). Before we consider the foundational philosophical question of the basic denominator in modern biological thought, we must first cast a cursory glance at the remarkable structural interlacement between the physical-chemical constituent substances of living entities and that which we will describe below as the living organism of e.g. a living cell. Since the cell is the smallest viable entity known, it is a good starting point for this reflection. The structure of a nuclear living cell The uncritical scientific use of the term ‘life’ – as if it is a concrete quidity – denies the modal nature of the biotic aspect of reality. After all, the earliest fossil which appears on the paleontological horizon is by no means ‘life’, since algae (or algae-like living things), have a biotical aspect amongst other aspects. If the inarticulate practice of referring to life as an entity is consistently carried to its conclusions, it would imply that an entity must be living in all its articulations. This is the consistent point of view of the neo-Thomist Hoenen. The Aristotelian-Thomistic substance concept requires of Hoenen to reach this conclusion, since the substantial unity of a living thing would be 1 2

Walter Zimmerman openly acknowledges as much – cf. 1967:100. “Kurz, die Ahnen- und Artenreihen der Phylogenetiker sind nicht nur gelegentlich, sondern stets das, was O Abel ‘Stufenreihen’ genant hat. Die fossilen Formen, die wir in der Vergangenheit auffinden, sowie die heutigen Organismen Repräsentieren in den uns interessierenden Merkmalen die entwichlungsstufe, die damals der betreffende Ahn erreicht hat” (1967:102).

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suspended if independent non-living ingredients (with a matching substantial form) were present therein. In Hoenen’s view the supposed substantial vital unity of a living thing could never be a mixture of life and non-life. Physical-chemical constituents in the living cell Organic chemistry, but especially recent developments in biochemistry, conclusively determined over the past three decades that there are all sorts of macro-molecular material structures present in a living cell. A brief synopsis of the pertinent information which must be taken into account in this regard follows. The chemical components of protoplasm (nucleus and cytoplasm) are of primary interest. Although we find extremely complex and labile organic compounds in both the nucleus and the cytoplasm, it is striking how relatively few elements are used. The main ingredients are the four so-called organic elements: hydrogen (H), oxygen (O), nitrogen (N) and Carbon (C). Apart from these the following inorganic elements can also be found: phosphorus, magnesium, calcium, potassium, sodium, sulphur, iodine, iron, cobalt, manganese, and zinc. A percentage indication of the various compounds in which these elements are to be found looks as follows: water (on its own and in compounds) 85-90%, protein 7-10% (albumin, histone, protamine, and nucleoprotein), lipids (e.g. fats) 1-2%, other organic substances (carbohydrates) 1-1,5%, and inorganic substances 1-1,5%. In most cells colloidal systems are found which represent a mixture of substances with chemical characteristics midway between true solutions and suspensions. These surfaces have an enormous electrical charge which quickly registers changes in temperature and electrical charge. A more liquid situation is referred to as a sol state, and a more solid situation as a gel state. The plasma of almost all cells is covered with a three-dimensional network of pockets linked with a system of membranes. These pockets appear most commonly in the form of cysts or tubes – thence the so-called alveolar system (cysts = alveola).1 In 1896 the Buchners discovered alcoholic ferments which serve a catalytic function in cells, initially referred to as ‘zymase‘, it gradually became apparent that it is a mixture of enzymes and co-enzymes.2 Protein refers to macro-molecules consisting of 20 different amino acids. When an amino group (NH2) of one amino acid is linked with a carboxyl-group (COOH) of another amino-acid, a peptide bond (NH-CO-) is formed – coupled with the release of water (H2O). Multiple amino acids are bonded in this way into a macro-molecule – a polypeptide. Enzymes have a protein structure built up out of amino acids and occasionally occurs in their thousands in a particular cell. This promotes chemical reactions in the cell, although each kind of enzyme catalyses only a limited number of reactions. Enzymes are very sensitive to abnormally high temperatures – unlike inorganic catalysts, who normally perform better under warmer con1 2

Of course the membrane functions as an organ of the cell. Co-enzymes are organic compounds which play an essential role in reactions catalyzed by enzymes, although it lacks the protein structure of enzymes.

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ditions. The entire metabolism of the cell depends on the functioning of enzymes. In the nucleus of the cell nucleotides are formed through the bonding of a sugar and a nitrogenous base on the one hand and a phosphorous acid-remnant on the other. In this way polinucleotide chains are formed. In the nucleonic acid DNA (desoxyribonucleic acid) four nucleotides are found, namely Adenine (A), Guanine (G), Cytosine (C), and Thymine (T). These spontaneously associate in the links A-T and G-C. Out of this mutual attraction emerges two polinucleotide-strings with various possibilities. A series like ATGACGT is complemented by a series TACTGCA. The so-called genetic code concerns the rule in terms of which a polipeptide series is linked to a given polinucleotide series. This linkage is made possible by RNA – a nucleonic acid differing from DNA in that the T is replaced by U(racil). To transfer the matrix of DNA to protein it appears that a combination of three letters is necessary in the DNA for every amino acid to be formed.1 This means that some amino acids are correlated with more than one triplet of nucleotides – i.e. different triplets are occasionally attached with only one amino acid. The triplets UAA, UAG, and UGA appear to be inoperative, since they are not correlated with any amino acids. The two strings of nucleonic acids are shaped in a double helix structure (according to the model of Watson and Crick, 1953), and have the ability of duplicating. When duplicating the two strings come apart and each nucleotide attracts its counterpart out of the free nucleotides present in the environment, so that the two new DNA spirals which come into existence are exact duplicates of the single original. Due to chemical influences, Röntgen- or cosmic radiation it is possible that one or more nucleotides can be added or left out, which change the genetic information of the DNA-molecule. This ‘mistake’ can then be exactly copied – bringing about a ‘mutation’. Such mutations could take the form of changes in single genes, in chromosomes, or even in a number of chromosomes, and almost always has negative consequences. In view of these negative consequences of mutations, neo-Darwinism was forced to still make use of Darwin’s original notion of natural selection. When climactic or other natural conditions change significantly, it is conceivable that the otherwise disadvantaged mutant member of a species could turn out to be advantaged under changed circumstances. In this manner nature selects those living things which has the better chance of success in the struggle for survival, due to mutations. The geneticist Th. Dobzhansky summarizes this theory as follows: “Mutation alone, uncontrolled by natural selection, could only result in degeneration, decay and extinction” (1967:41). Further non-living ingredients of the cell includes the already mentioned genes (localized in the chromosomes).2 During cell division a reduction in nu1

2

There are 20 amino acids and if we consider only two possible combinations of 4 DNA nucleotides only 16 amino acids can be explained: 42 = 16. The mentioned nucleotides of A, G, C, and T can however be arranged in 64 combinations of three: 43 = 64. Chromosomes are threads in which the colourable substance in the nucleus of the cell visibly contracts during the process of cell division (chroma = color).

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cleonic plasma takes place, twice dividing the generative cells, while the chromosomes only divide once. This process is called meiosis.1 Similar ingredients are the hormones and the ‘resting nucleus’, as well as the vacuoles, which contains cell-sap and is delimited by a membrane. The latter does not occur in the cells of bacteria and blue-green algae. With regard to the unique manner in which the living cell functions in the physical aspect of reality, Karl Trincher2 mentions the following four macroscopic characteristics (1985:336): 1) spatial macroscopy which defines the cell as a spatially delimited surface; 2) temporal macroscopy, which determines the finite time in which the energy cycle of the cell occurs; 3) the isothermic nature of the cell, which is responsible for the constancy of temperature throughout the cell; 4) the persistent positive difference between the higher internal temperature of the cell and the lower external temperature of the environment adjacent to the cell surface. Organelles – the different organs in the cell The different organs in the cell can be considered true parts of the cell organism. Although the whole-parts relationship is a typically spatial relationship, it receives in all living entities a typically biotical qualification. The distinctive form characteristics of cells which emerge in the different ways in which the cellular nucleus relates to its surrounding cytoplasm, were already studied and classified in the thirties of the 20th century by the German biologist R. Woltereck.3 The cellular nucleus is generally the (either round or ovoid) site of the DNA, and serves, despite the mutual dependency of nucleus and cytoplasm, to initiate the metabolism of the cell. The phenomenon of dual or multinucleic cells does not in the least diminish the centeredness of the structure of cell – in many protozoa this is only a passing stage related to procreation, serving the same function as cellular division among metazoa. Those protozoa distinguished by cilia are known as Ciliophora and have a double nucleus – a macro- (somatic) and a micro- (generative) nucleus. Bacteria and blue-green algae have a diffuse nucleonic sphere rather than a proper nucleus.4 Bacteria have no definitive nucleonic membrane between the cyto- and caryoplasm. The centriole is a cell organ present in animal and a few lower plant cells. In some cells the centriole tends to be positioned at the geometric centre of the cell. In general it is however displaced by the nucleus and cytoplasmic products. When a cell is not engaged in mitosis, centrioles generally appear in 1 2 3

4

The number of chromosomes are for instance reduced from 46 to 23. Dept. of medical physiology, University of Vienna. Compare his Grundzüge einer allgemeinen Biologie, in which he distinguishes among hylocentric, morphocentric and (in the case of animals) kinocentric structures in which the typical centrality of a cell is expressed (1932:323-329). Green algae of the genus Cloadophora have multinucleic cells.

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pairs. Before initiating cellular division centrioles must first themselves divide. Ribosomes, which are mainly to be found in the cytoplasm, are the main site of protein synthesis. The genetic message of the chromosomes is transferred to the RNA of the ribosomes which is ultimately responsible for the production of enzyme protein. Lysosomes – a term first used in 1955 – are granular subcellular organs with an encompassing membrane, and contain particular hydrolitic enzymes. When a cell is for instance damaged, this enzyme is released and breaks down the protein and nucleonic acid so that adjacent cells can use these for the repair of such damage as occurred. In any process of dying (autolysis) cells and tissue are broken down by means of the lysosomic enzymes. Certain fibrous microtubes often play an important role in the formation of cells, while microfilaments are organelles involved with the mobility of cells. Mitochondria are granular cell organs of which the inner membrane functions to “breathe” – making mitochondria the power stations of the cell. Energy in foods (captured inter alia by means of the citric acid and Krebs cycles) is recaptured by the mitochondria and transformed into adenosine triphosphate (ATP) by means of phosphorilization. In this way energy is produced for various cellular functions. Bacillary bacteria are strikingly similar in shape to mitochondria, which has prompted suggestions that mitochondria might initially have been independent prototrophic cells (cf. the comments of Roodyn and Wilkie 1968:53-57). This suggestion is much relativized by the fact that the function of mitochondria is dependent on the central and directive function of the cellular nucleus. After 1965 direct biochemical research confirmed the universal presence of DNA in mitochondria (MDNA). The basic composition of MDNA is more homogeneous than that in the cellular nucleus, and appears to be relatively genetic self-sufficient. Isolated mitochondria can synthesize DNA and MDNA is even passed on the daughter cells without being broken down. Despite this apparent independence, sufficient evidence remains for substantial nucleonic control over the generation and collection of the ingredients of mitochondria. The Golgi-complex, which is rich in lipids, apparently contains the secreting function in the cell organism. Plastides contain a pigment and/or food reserves, and differ significantly from one kind of cell to another, although it is absent in bacteria, blue-green algae, and fungi. Nucleoli, discovered by Fontana in 1781, have a high protein content – especially phosphoric protein. Other organelles include ‘fagozomes’ and ‘peroxyzomes’. The quest for a basic denominator Any consideration of the various schools of thought in modern biology is made more difficult by the immense variety of disciplines and huge volume of information relevant to such a consideration. Nonetheless, no school of thought escapes certain fundamental structural requirements which determines and makes possible scientific thought in the first place. We discovered already in the first chapter that scientific thought is a particular kind of 101

thought, namely deepened (disclosed or modally abstractive) logical thought. The key moment of the logical aspect of our experience of reality is to be found in the character of identification and distinction (identifying distinction/distinguishing identification). A scientific, subjectively logical thoughtact is by nature dependent on information about reality which can be identified and distinguished, and is as such determined and delimited by logical norms which must be honoured in all logical thinking activities. The diversity of reality cannot be encompassed by the logical aspect of reality – scientific thought is always an engagement with a cosmic diversity which is translogical. There are therefore apart from the logical norms for scientific thought also cosmological norms, such as the norm of excluded antinomy which demands that this diversity be honoured if such thought is not to succumb to antinomy (anti-normativities, in distinction from mere logical contradictions). It appears as if the idea of continuity in the dominant biological schools of thought in itself already provides a denominator for consideration. Many phylogeneticists who argue with a consistent nominalism would argue in the first place that classificatory delimitations are entirely artificial since the actual line of descent consists of a structureless continuity. According to these thinkers the recognition of certain structural arrangements (Gefügeordnungen) would not be in conflict with the nominalist conviction with regard to the artificiality of particular delimitations. W. Zimmermann comments in this regard that “In phylogenetic development the origin of a structural arrangement and the emergence of limits to groups of organisms do not coincide in a single phase. One may well recognize a structural arrangement while nonetheless being convinced as a ‘nominalist’ of the artificiality of limits. In lines of descent (Ahnenreihe) it is entirely unnecessary for limits to appear. The processes bringing about structural arrangements (with regard to a coherence of descent) and those bringing about the current limits among living organisms, may well be millions of years removed from each other. Whoever fails to observe the distinction between these two phases, has not yet grasped philogenetics” (1967:98). Although Schindewolf recognizes only individuals as really existing entities, apparently in line with the nominalist point of departure with regard to the supposition of a structureless continuum he is nonetheless of the opinion that the types systematically distinguished on all levels are not mere arbitrary fictions, but are rather general concepts founded in objective factual data. He appeals to the fact that among living things successive levels exists in accordance with the degree of generality and the degree of similarity, in terms of which it is possible to comparatively coordinate and subordinate the groupings of these layers in terms of the characteristic combinations present. Apart from this he also appeals to the transitionless discontinuities among such structures.1 On occasion he identifies his thought as idealistic morphology, even though his emphasis on the temporal succession in the emergence of organizational forms in the history of the earth indicates a distance in principle with regard to the metaphysical primeval forms of idealistic morphology. On 1

Cf. the exposition by Ungerer 1966:233 ff.

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the other hand Schindewolf remains a hardened opponent of the idea of a phylogenetic systematics. Exactly in this regard a comparison with Simpson is worthwhile. Simpson considers phylogenetics to be the basic discipline of biology within which he then places the evolutionary structureless continuity (with the eventually artificial classification). According to Schindewolf the more general systematic category appears first, and all differentiation and specialization can only take place within this category. Schindewolf makes use of the presupposition of discontinuous macro-mutation, the notion that nature is able to bring forth truly new types, which he then elaborates in his theory of typostrophism which appeals to paleontologically determined trends. The emergence of new structural types1 Schindewolf calls typogenesis. In the typical development of different levels typogenesis is generally followed by a a period of steady differentiation and transformation, which leads to a directed (orthogenetic) development of the particular structural type which Schindewolf calls typostasis (the flourishing of the type). Eventually a period of degeneration and eventual extinction follows – typolysis (cf. Ungerer 1966:235-236). Conflicting views despite “the same facts”! M. Grene points out that Simpson and Schindewolf accuse each other of essentially the same or similar mistakes, making use of unnecessary and mystifying presuppositions. She believes that each accepts as premise the negation of the other’s conclusions – while hardly if at all differing with regard to the facts: “Simpson, wedding paleontology to the statistical methods of population genetics, sees a gradual change in populations such that the sharp divisions of traditional morphology become false. Schindewolf, basing his theory on the logical priority of morphology, concludes that gradualist, statistical picture of neo-Darwinism is false. To put it very schematically; Simpson argues: the neo-Darwinian theory is true; morphology implies that neo-Darwinism is not true; therefore morphology is wrong. Schindewolf argues: morphology must first be accepted as true; morphology implies that the neo-Darwinian theory is wrong; therefore the neo-Darwinian theory is mistaken. Or to put the matter another way, they agree on the major premise: traditional morphology and neo-Darwinism are incompatible” (1974:132). Referring to the theory of Schindewolf, D.B. Kitts writes: “It permits an explanation of the fossil record as adequate as any other” (1974:469).2 From the disagreement between Simpson and Schindewolf it is therefore clear that there are trends in biology emphasizing either continuity or discontinuity. The Noble laureate (1973) Konrad Lorenz rejects the mechanistic postulate of continuity as sharply: “From events in the atom to those in the history of humanity inorganic as well as organic developments occur in leaps. Even though 1 2

According to Schindewolf the Archaeopteryx is an example of such a new structural type, since he considers it to be the first exemplar of a new class of vertebrates, namely birds. Kitts refers to Simpson in this regard as follows: “Simpson did not provide compelling support for synthetic theory against Schindewolfian or Lamarckian, or any number of other theories both evolutionary and non-evolutionary” (1974:468).

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some quantitatively summarized processes in this course of events might superficially appear continuous, eventually it turns out to be as discontinuous as the major qualitative changes in organic evolution, first clearly understood by Hegel ...” (1973:186). This situation implies that the search for a common denominator needs to continue in this direction: under what denominator is both this continuity and discontinuity discussed? Neo-Darwinism The dominant neo-Darwinist synthetic evolutionary theory in principle chooses for a physical basic denominator, even though increasing efforts are made from this perspective to account for the qualitative differences which emerged in the course of the continual evolutionary process. J. Huxley warns against the “nothing but” trap into which many evolutionary and natural scientific explanatory techniques fall: .".. if sexual impulse is at the base of love, then love is regarded as nothing but sex; if it can be shown that man originated from an animal, then in all essentials he is nothing but an animal. This, I repeat, is a dangerous fallacy. We have tended to misunderstand the nature of the difference between ourselves and animals. We have a way of thinking that if there is a continuity in time there must be a continuity in quality" (1968:137). Simpson also distinguishes between non-biotic and biotic levels (of organization) and is convinced that it is preposterous “to base ... a concept of scientific explanation wholly on the non-biological levels of the hierarchy and then to attempt to apply it to the biological levels without modification” (1969:8). Any treatment of this problem would according to Simpson have to avoid the extremes of both vitalism and ‘physicism’ (p.21). Against an extreme physicalist reductionism he openly states: “I think it fair to say that in this respect, as truly biological investigation and an attempt to explain vital phenomena, unmodified reductionism has failed” (1969:26). Because of this he remains convinced that evolutionary organismal biology cannot be reduced “to a philosophy taking account only of the physical, non-biological aspects of the universe” (1969:7). Simpson rejects an extreme reductionism (physicalism), and speaks of the physical and biological aspects of reality. Does this mean that he implies with this distinction an irreducibility in principle between the physical and biotical aspects? Apparently not, since when he says that the principles of evolutionary biology (which otherwise do not contradict anything in physics) transcend the principles which can be deduced from non-living atoms and molecules, he still adds “but without becoming anything other than naturalistic” (p.7). Only the concept of organization in the end indicates that in which living and non-living things differ: “It is the complexity and the kind of structural and functional assembly in living organisms that differentiate them from non-living systems” (1969:7). In Simpson’s view the biotical aspect emerges out of the organizational complexity of natural systems, which actually implies that the term “biotical aspect” cannot be understood in the sense of irreducible ontic mode. Although not stated in extreme reductionistic, or unmodified reductionistic 104

terms, Simpson still defends a form of physicalism, even if a physicalism in which it appears as if the differences among various levels of organization are taken into account. Vitalism Only in vitalism is a principled choice made for an alternative basic denominator, namely the biotical – even though not in the sense of what is referred to in this text as an aspect or modality of reality. The father of neo-vitalism, Hans Driesch, speaks of an immaterial vital force (to which he refers as entelechie or psychoide), which would be far more than just the biotical aspect of reality. Without surrendering the validity of the mechanistic analysis of matter, and without denying the causal claims of the classical humanistic scientific ideal with regard to nature, Driesch tried to apply the concept of natural law (in just as deterministic a sense) to biotical phenomena. In agreement with Driesch Rainer Schubert-Soldern defended the vitalistic position with a range of biochemical arguments. As the functional and formal unit of life the existence of the cell would according to Schubert-Soldern depend on the actualization of a double potential: “(a) the ‘form’ or order of the cell, and (b) the chemical laws governing molecules. ... This principle of order may be called the ‘active potentiality’ of the material parts” (1962:102). His view of the principle of order returns to Aristotle: “Hence the Aristotelian concept of entelechy corresponds exactly with the principle of order, which we see at work making the cell into a whole. It is a principle of wholeness which forms a unity from parts which would otherwise go their separate ways. Thus a hologenous system is born” (1962:113). Where Aristotle, Thomas Aquinas, and even Driesch still account for individuality in terms of the material components, Schubert-Soldern chooses another way: “Since the form brings about the individualization of something which previously had been poli-substantial or poli-individual, it must be the form, which expresses the individuality, which itself must be the individuality” (1959:285). In his view the form of a body “brings about a real entity with a non-material character, concerning a substance which in its essence possesses its dynamic character” (1959:286). Simpson chose the term organization to indicate the essential distinctive characteristic of living things. In neo-vitalist circles organization is understood in terms of their particular understanding of form (order). The botanist E.W. Sinnott, for example, writes “Uexküll and others have emphasized this idea and regard organic form as essentially an independent aspect of an organism, parallel with its matter and energy. ... Indeed, the concept of organization as something independent of the inner and outer environment implies that form must be a basic characteristic of all living things” (1972:51). Against mechanistic atomism Sinnott emphasizes in neo-vitalist manner the dynamic-creative and indivisibly continuous form of living things: “Form, ... is changing and creative. ... It is a category of being very different from matter” (1963: 199). 105

The neo-vitalist biologist J. Haas emphasizes the obedience of every living thing in the elaboration of the course of its life to an inherent law or programme, which he prefers to indicate as its life plan: “The life plan contains as components the blueprints of each of its expressions; the genetic plan for their succession; the functional plan for carrying out its activities; the behavioral plan for all its ‘acts’” (1974:336). Life plans have (similar to norms and laws in general) an ideal being (ideales Sein) in Haas’s view (p.338), and cannot be explained physically-chemically: “Physical-chemical forces and laws are in themselves unable to bring forth the structures of meaning which we identify as the life plan, and even less can it produce a non-material bearer of life plans” (1974:355). Following the (idealistic-morphological) Austrian botanist Wilhelm Troll (cf. his standard text Allgemeine Botanik 1973:19 ff.), Walter Heitler speaks of a Zentralinstanz which must exist in every organism (1976:6). Heitler uses this expression in the context of the following hypothesis which he would like to defend (against a consistent physicalism): “The organism has its own laws, which partly displaces the laws of physics and chemistry with something more general” (1976:3). He believes an important point of departure for his argument to be the fact that neither physics nor chemistry knows or uses a true concept of Gestalt or Ganzheit. The analytical treatment of these sciences disturbs the Gestalt. This happens because physical analysis can only be expressed in the systematic measurements of length, time, weight, and temperature (the so-called c.g.s. system). Due to this “merely analytical methodology the laws are differential, i.e. it makes direct statements only about the behaviour of objects for immediately neighbouring points in time and space. By means of integration one is able to obtain statements concerning the entire relationship (e.g. the form of planetary orbits), but these must follow from the differential elements” (1976:5). The Gestalt of a cell (or of the paw of a cat) transcends all the descriptive possibilities of the c.g.s. system. For such descriptions it is not rich enough. After all, if one only used differential laws, such as those of physics, cells would have to divide ad infinitum without the emergence of a cellular complex. In these terms the expression of a cat’s paw is unimaginable (1976:5-6). The central instance directing the eventual teleological activities of living things, is referred to by Heitler as the biologischen Instanz, who also specifies the following sub-instances (Unter-Instanzen): organs, cells, organelles (1976:16). Related to vitalism one finds the organismic biology founded by L. von Bertalanffy and developed into a general systems theory in which the terms whole and totality are central, with organization similarly functioning as a key term. Von Bertalanffy considers the organismic world view to be a step beyond the mathematical more geometrico ideal and also beyond the mechanistic world view: “First came the developments of mathematics, and correspondingly philosophies after the pattern of mathematics – more geometrico according to Spinoza, Descartes and other contemporaries. This was followed by the rise of physics; classical physics found its world-view in mechanistic philosophy, the play of material units, the world as chaos ... Lately, biology 106

and the sciences of man come to the fore. And here organization appears as the basic concept – an organismic world-view taking account of those aspects of reality neglected previously” (1968:66). M. Beckner elsewhere comments that “Even though in fact many biologists agree with the organismic position, they will say they disagree” (1971:60-61). Holism Vitalism consists mainly of an attempt to exalt life as an immaterial substance influencing as an ordering form a constellation of matter, or elaborating a life plan within such a constellation. In the holistic biology of A. Meyer however, an attempt is made to place the biotical aspect so centrally that in principle the physical can be reduced to biology. J. Needham summarizes the position of Meyer: “Thus Meyer, in his interesting discussion of the concept of wholeness, maintains that the fundamental conceptions of physics ought to be deducible from the fundamental conceptions of biology; the latter not being reducible to the former. Thus entropy would be, as it were, a special case of biological disorganization; the uncertainty principle would follow from the psycho-physical relation; and the principle of relativity would be derivable from the relation between organism and environment” (1968: 27 note 34). The key notion of holism (already introduced in 1926 by genl. J.C. Smuts), is that of the whole (Ganzheit, Greek: to holon). Meyer defines a whole by drawing a sharp distinction between parts (teilen) and articulations (Gliedern): “Ganzheit ist, was nie aus Teilen besteht, sondern stets in Gliedern ensteht und nur gegliedert existiert” (1949:284). Without going into the basic principles of holistic biology we refer only to Meyer’s evaluation of the construction of trees of descent. We indicated above that phylogeny is not ultimately a phylogeny of kinds, but rather of typological characteristics. With the aid of extensive empirical information Meyer formulates the following remarkable ‘basic typological law’: “There is no group of existing organisms belonging to any taxonomical category of the Natural System, whose members possess all group characters in their most primitive or in their most progressive phases only. Rather are primitive, intermediate and progressive character phases thus combined with each other in each real member of a group that an organismic holism suited for living in any real existing ecological biotope results from it. Forms which possess all their morphological characters in their primitive or in their progressive phases only are neither living holisms nor suited for existence in ecological biotopes and are, therefore, but purely ideal constructions. ... Therefore, the existence of all so-called phylogenetic trees, which make use of such, always hypothetical stem-forms, have become dubious” (1964:59-60). On page 113 Meyer writes: “But all these phylogenetic trees begin with purely idealistic constructions.” In an earlier work Meyer commented “that all of the phylogenetic tree construction to date is impossible since it depends on entirely utopian pre-suppositions” (1950:8). How can development be imagined without trees of descent? “Only as discontinuous, quantum-like (quantenhafte) development” 107

(1950:12), Meyer replies. It is therefore necessary to continue to take into account the poliphyletic origin of new kinds: “New types are as a matter of fact not potentially present in one or more kinds of the functioning type, but in its combined representatives still on hand. Out of this there suddenly breaks through a new type with primevally sudden force – paleontologists rightly speak of revolutions, and not initially in only one kind which must then develop at a snail’s pace, but immediately in a wholeness of new kinds and forms” (1950:14). Meyer is in conclusion of the opinion that from the starting point of the holistic idea, namely that the a-biosphere must be considered as a simplification of the biosphere, a superior dialectical synthesis (cf. Hegel) is possible between mechanism and vitalism (1964:162). Against the background of his idea of a quantum-like, discontinuous development, Meyer considers phylogeny as the history of life through emergence evolution (cf. 1964:147). Emergence evolutionism Emergence evolutionism in general attempted to take seriously the qualitative differences which indicate the irreducibility in principle of various evolutionary levels, while simultaneously retaining the conviction that higher evolutionary levels emerged out of lower ones. The great emergence evolutionists openly admitted that this position contained an inner antinomy. R. Woltereck does so in his Ontologie des Lebendigen (1940:300ff.), while M. Polanyi writes: “We have reached the point at which we must confront the unspecifiability of higher levels in terms of particulars belonging to lower levels, with the fact that the higher levels have in fact come into existence spontaneously from elements of these lower levels. How can the emergent have arisen from particulars that cannot constitute it” (1969:393).

Th. Dobzhansky calls arrival at a new level “evolutionary transcendence” (1967:44). “The flow of evolutionary events is, however, not always smooth and uniform; it also contains crises and turning points which, viewed in retrospect, may appear to be breaks of the continuity. The origin of life was one such crisis, radical enough to deserve the name of transcendence. The origin of man was another” (1967:50). Although Dobzhansky himself went as far as to acknowledge that different levels are subject to typical laws valid for it, he remained of the opinion that an irreducibility in principle of these laws is unnecessary: “The phenomena of the inorganic, organic and human levels are subject to different laws peculiar to those levels. It is unnecessary to assume any intrinsic irreducibility of these laws, but unprofitable to describe the phenomena of an overlying level in terms of those of the underlying ones” (p.43).

Pan-psychism In the end the notion of continuity relativizes Dobzhansky’s recognition of different kinds of laws. The acceptance of the postulate of continuity does not however necessarily imply the choice of a physical-chemical basic denomina108

tor. A remarkable position is taken within the deterministic scientific ideal by the German zoologist Bernard Rensch. Although Rensch accepts the postulate of continuity of the science ideal, he explicitly distances himself from both the mechanistic and vitalistic points of view (the former deals with continuity in terms of a physical-chemical denominator and the later in terms of a biotical denominator). Although he accepts the validity of the natural scientific causal analytical method, Rensch rejects every monistic theoretical picture of reality which attempts to reduce all of reality to a single principle. According to him world events are governed by multiple basic laws: “Depsite all evidence in favour of the monistic principle, the primal ground of world events is pluralistic” (1971:33). Rensch refers in particular to .".. the causal law, universal constants, the law of conservation, the principles of symmetry, and the logical laws" (1971:33). Rensch characterizes his own position as ‘panpsychistic’ and ‘identistic’ – that is, all events are founded by something which is neither psychic nor material, but which has psychic and material characteristics (1971:159). It implies considering the evolutionary continuum in terms of a psychic basic denominator. If no discontinuities exist in the evolutionary line of descent, then lower animals, plants, and even the inorganic sphere should exhibit certain corresponding “psychic” components – a consequence drawn by Rensch: “According to our previous findings and discussions we are justified in assuming ... psychic (parallel) processes of some kind in all living beings” (1959:352). ‘Psychic’ continuity also bridges the transition from living to non-living: “Here again it is difficult to assume a sudden origin of first psychic elements somewhere in this gradual ascent from nonliving to living systems. It would not be impossible to ascribe ‘psychic’ components to the realm of inorganic systems also, i.e. to credit nonliving matter with some basic and isolated kind of ‘parallel’ processes” (1959:352). Rensch believes that such a panpsychistic approach has the advantage of not having to assume that the psychic, as something basically distinctive from the material, appeared on our planet at some stage after the emergence of living creatures. As a substitute for the assumption that psychic phenomena appeared suddenly after an astronomic and geological prehistory of millennia, Rensch considers it far more conceivable and acceptable to link the evolution of the psychic to the evolution of the material (anzufügen), i.e. to ascribe a protopsychic nature to matter (1969:134-135). Metabolism as first level of freedom The modern anthropocentric or humanistic science ideal, emerging during the time of Descartes out of the modern human quest for autonomous freedom (the personality or freedom ideal) as an instrument of control with the aid of which all of reality could be brought in the grip of the natural sciences, has threatened the humanistic freedom ideal from its inception, exactly because a closed causally-determined natural order leaves no room for genuine human freedom. Just as Rensch retroprojects psychic characteristics to the realm of material things, H. Jonas is ‘forced’ in the interest of the primacy of the free109

dom ideal to ‘recover’ freedom on the level of the material: “Our position is in actual fact that it is possible to observe freedom already at the level of metabolism – yes, even that it is the first form of freedom” (1973:13). According to Jonas “life manifests this polarity in a durable fashion in the fundamental antithesis in between which it existence weaves itself: the antithesis of existence and non-existence, of self and world, of form and matter, of freedom and necessity” (1973:15-16). A new mechanistic approach Supporters can still be found in modern biology of the classical mechanistic science ideal which wants to consider all natural phenomena in terms of a kinematic (movement) denominator. As we saw in Chapter 3, this was the typifying characteristic of classical physics. Although Eisenstein acknowledges that sensory experience provides us with qualitatively different things (in terms of which he develops many cutting arguments against the evolutionistic continuity of descent), he remains of the opinion that the inherent scientific tendency towards uniformity (Vereinheitlichung) carries abstract thought to a level transcending the qualitative expressions of things, where everything which appears to be qualitative, is reduced as far as possible to dynamic processes of degrees of speed differing only quantitatively (1972:256): “At the highest level of scientific abstraction we do not therefore think of things as isolated, essentially differing existences, but, since they have been brought under a common denominator, we consider things indissolubly linked in the coherence of universal motion” (p.256).1 Structural diversity founds structureless fantasies From this synopsis it is clear that modern biological literature hosts various diverse schools of thought. The first facet bringing about a provisional division is the problem of continuity and discontinuity – a conceptual contradistinction originally founded only in the spatial aspect of reality (see the previous chapter in this regard). Eventually this spatial analogy finds a closer specification in the actual denominator under which the concerned biological position considers the identifiable and distinguishable diversity in reality: in the case of Eisenstein under the classical mechanistic denominator of motion; among supporters of the general synthetic theory of evolution in principle under a physical denominator in which apparent (but not principled) recognition is given to higher structural levels; in vitalism, holism, and organicism under a biotical denominator; in the pan-psychistic identism of Rensch under a sensitive-psychic denominator, and in the personality ideal-oriented thought of Jonas under the denominator of freedom. Emergence evolutionism wanted to 1

Eisenstein links up with the dynamic theory of Constantin Brunner (cf. i.a. Brunner's Materialismus und Idealismus, 2nd impression 1962), which ends up in a quasi-Hegelian dialectical synthesis in which all finite contradictions are reconciled in the infinite totality of so-called absolute being: “From the higher and encompassing perspective of dynamic theory all things in the nature of things have diverse origins and merge in the infinite totality. In the end ... all types of existence are equal manifestations of the one absolute being” (Eisenstein 1975:265). Cf. also the obituary article which Eisenstein wrote in memory of the death of Brunner in Philosophia Naturalis (1987:346-349).

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have its cake and eat it by recognizing both a continuity of descent and a (quantitative) discontinuity of being. The choice of a denominator implies (with cosmological necessity) that all other facets of the diversity of reality must be reduced to the chosen denominator which as an absolutized perspective encompasses as aspects all other dimensions of reality. What is particularly striking, is that all the diverse approaches mentioned continue to be confronted with the diversity of reality which can be identified and distinguished. No single understanding of continuity denies the differences among matter, plant, animal, and human, or the differences among the aspects of movement, the physical, biotical, the sensitive-psychic, and the post-psychical – they simply describe these different facets and structures as non-essential since it can apparently be reduced to one or another denominator. The basic question remains whether this diversity of choices in denominator has any “objectively factual” foundation. It cannot be denied that the inherent diversity in reality offers a point of departure for this diversity in perspectives, but the belief that all of this diversity can be reduced to one particular facet which would as basic denominator encompass all others doubtlessly indicates fundamental theoretical presuppositions – theoretical-philosophical presuppositions which exist since theoretical logical thought by nature requires an idea of the diversity in reality, while as theoretical presuppositions themselves being directed and determined by ultra-theoretical convictions. No single perspective in modern biology can be released from one or another central foundational motive which determines its course as an ultra-theoretical dunamis. While most modern biologists in one way or another support nominalism, it is remarkable to note – as we saw in Chapter II – that most mathematicians of our day in principle reject nominalism in their discipline. Platonism in mathematics is described by P. Benacerraf and H. Putnam as follows: “In general, the platonists will be those who consider mathematics as the discovery of truths about structures which exists independently of the activity of thought of mathematicians” (1964:15). Paul Bernays is of the opinion that the use of platonism is so common in mathematics “that it would be no exaggeration to say that platonism currently dominates mathematics” (1976:65). It is a peculiar situation that the dominant directions in modern mathematics and modern biology are directly opposed in terms of their theoretical points of departure! We conclude this chapter by once again returning to the structural character of the cell – in terms of the theory of enkaptic structural wholes which we already considered in chapter II. Structural dimensions of the cell – an enkaptic structural whole A remarkable tendency among the different biological approaches is that virtually all of them in their own way speak of living and non-living matter. Although this expression means something different in each case, the usage nonetheless reflects unresolved problems for each of these points of view. 111

For the mechanistic (-physicalistic) approach everything is in principle material, physically determined – which implies that any term which appeals to the biotical aspect of things is actually problematic. On the other hand it is vitalism which attempts to find the essence of “life” in immaterial life plans, gestalt-expressive factors or central instances. This means that it is also problematic from that perspective to speak of living matter – a problem Haas obviously perceived with his emphasis of the fact that physical substances maintain their “being and function” also “after their assimilation” into living things. Understandably Haas himself opposes the usage of “living matter” – according to him biochemistry and cell physiology knows of no “living matter” with “mysterious vital characteristics” (1968:24). He prefers to speak of the material substratum of organisms (1968:20-40). This approach of Haas rejects what he considers to be the “monistic vitalism” of Aristotle. At the same time he draws the conclusions of his own position: “Organisms essentially consist of two distinctive realities, a material and a non-material component, therefore having in ontological terms a dualistic constitution” (1968:39). Atoms, molecules and macro-molecules are not alive – they are physically qualified material structures. Just as Heisenberg’s uncertainty principle form the lower limit to physical determination, N. Bohr formulates his so-called biological uncertainty principle in his articles from the thirties (which Heitler uses in 1976 as point of departure), which indicate the upper limit to physical determinacy. This upper limit actually indicates that a biotically qualified entity such as a cell directs the functioning of its basic constitutive substances towards the existence of the living unit as a whole, which implies that the actual material structures – apart from this biotical serviceability – only comes in view once the living cell dies. The material substances simply do not have a biotical subject function. Apart from the four (physical) macroscopic characteristics referred to by Trincher (cf. 1985:336) the typical biotical qualifying function of the cell is structurally expressed in the physical aspect in the typically centered (i.e. biotically organized) manner in which the cell functions. Driesch had no brief for the typical individuality of living things, since he was of the opinion that the material components with or without entelechie did not differ. Furthermore he failed to describe the influence of the immaterial entelechie on the material components of living things otherwise than in terms appealing to the physical aspect. He could not see that scientific conceptualization necessarily utilizes modal points of entry – even preferring to see entelechie as a system of negations which could not be determined positively: it is non-spatial, non-mechanical, indivisible (cf. Sinnott and Haas) and non-energetic (1931: 297). By means of a theory of the enkaptic structural whole this question is placed in a different context. In the first place this structural theory provides a peculiar perspective on the diverse points of view of organic chemistry and biochemistry. In the nature of things entity structures and their interlacement transcends any particular scientific perspective. When organic chemistry (from a physi112

cal-chemical perspective) investigates the nature of molecular and crystalline structures, this does not mean that the biotically directed nature of the cell can be reduced to this. Apart from the excretory products of living cells, many substances are produced in cells which have e.g. regulatory, inductive, organizing or catalytic functions, without these substances being externally excreted. The attempt to reveal the structure of such substances is most commonly attempted today by biochemists. Such substance structures actually, however, fall within the research ambit of organic chemistry, since it retains a physical directive function as macro-molecular structures. Consider for instance the study of the chemical structure of enzymes, which is today considered one of the most distinctive facets of biochemistry – which actually principally falls within the ambit of organic chemistry. Biochemists would probably protest such a point of view, since they do not clearly distinguish between the structure of the mentioned material elements in the cell and their biotically directed functions. As physical-chemical substructure the living cell organism these material building blocks found in the cell are not entirely self-enclosed, since they remain entirely open, dynamic, and labile through being disclosed and made serviceable to the subjective biotical function of the living organism. And it is towards these biotically disclosed and directed physical functions of the substances in the cell which biochemistry should direct its investigatory efforts. The typical metabolic functions of the cell certainly occur on the foundation of its physical-chemical constitutive substances, but can nevertheless not be detached from their disclosed directedness towards the qualifying biotical aspect of the cell. Since the molecular and crystalline structure itself already exhibits the form of an enkaptic structural whole, we are in this case dealing with a complex enkaptic form. The entity structure of the cell represents a unilateral enkaptic foundational relationship: without physical-chemical constitutive substances there could be no cell, without these substances therefore participating in the subjective life function which qualifies the cell. (we must repeat that it would therefore be self-contradictory to speak of molecular biology or a bio-molecule, since both expressions suggest that physically qualified entities at the same time have an internal biotical qualifying function). The biochemical constellation therefore begins exactly where the focus shifts from the molecular or crystalline structures of the organic substances to the actual biotically disclosed and directed functions of these substances. In the biochemical constellation the essential character of the so-called organic substances is neither revoked nor excluded, since they are only enkaptically (i.e. with the retention of their internal physical-chemical structure) made serviceable to the typical biotical functions of the cell. The disclosure of organic chemistry, which placed biochemistry as an independent discipline in the encyclopaedic coherence of all the sciences, at the same time confirmed in a unique way the philosophical dependency of these particular sciences, since only in close mutual interaction with organic chemistry can biochemistry properly fulfil its task. In the same way as the physi113

cal-chemical structure of constitutive substances is foundational to their enkaptic (i.e. biotically directed) functions, organic chemistry ought to be foundational to biochemistry, which should focus on the disclosed enkaptic functions of the substance structures which organic chemistry reveals. This foundational relationship confirms the close interlacement of the structure and functions of the constitutive substances of living things. Within the context of the ordered (centered) structure of the cell, we do however (from a biotical perspective) come across the various organs (organelles) which are true parts of a living whole. Since the cell is built up of non-living material components we cannot simply say that organelles are parts of the cell. To indicate the biotical subjectivity of the cell Dooyeweerd uses the term: cell organism. In other words, the various organs of the cell are all part of the cell organism. The different organelles referred to above exist only on the foundation of physical-chemical constitutive substances – this is the meaning of the unilaterally enkaptic foundational relationship. The cell organism therefore is a specifically biotically qualified structure which can only exist on the foundation of enkaptically bound physical-chemical constitutive substances. Since these substances are not themselves biotically qualified, but nonetheless function in the living cell, we are forced to distinguish a structural trio if we wish to account for the complex structure of the living cell. (i) In the first place there are the physical-chemically qualified constitutive substances which themselves already represent enkaptic structural wholes. (ii) Secondly we encounter the living organism of the cell as a biotically qualified substructure which can only function on the foundation of the enkaptically bound substructural substances. (iii) Finally we find the body of the cell as structural node which enkaptically encompasses both the previously mentioned substructures. Although the cell organism is living in all its articulations, it cannot exist without the enkaptically contained substances, and can only be realized in consequence in the enkaptic structural whole of the body of the cell. Since the body of the cell as enkaptic structural whole necessarily also enkaptically encompasses the nonliving substances, the cell cannot be entirely living. In plant structures the living organism is therefore only a qualifying substructure of the living body of the cell, which exists in a unilateral foundational relationship with its molecular substance structure. At this point we must again clearly distinguish the theory of the enkaptic structural whole from a traditional universalistic scheme of a whole with parts. Only with regard to the cell organism can we speak of a true biotically qualified whole-parts relationship (the whole cell organism which in its typical structural centeredness possesses different sub-organs). Physical-chemical constitutive substances, which as such can never be biotically qualified, can therefore neither as such be part of the biotically qualified cell organism. It remains only enkaptically bound ingredients in the actual parts. Macro-molecular and quasi-crystalline substance structures remain physical-chemically 114

qualified and cannot as such be alive. Nonetheless such substance structures are present in the body of the cell, since without them the cell organism cannot live.

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Paradigms in Mathematics, Physics, and Biology: Their Philosophical Roots Chapter V Remarks about the mystery of being human1 In the previous chapter we have dealt with a few aspects of the problems present in the evolutionist account of the origin of living entities with special reference to the assumed evolutionary transition from the non-living to the living. Surely, this transition is as difficult as it is crucial for the whole evolutionist picture of the human being as an extension of the animal realm. At the same time, in order to postulate the origin of the first living entity, a ‘jump’ is needed just as big as the one from the level of uni-cellular life to that of being human. Dobzhansky says: “The origin of life and the origin of man are, understandably, among the most challenging and also most difficult problems of evolutionary history” (1967:459). Lately, the connection between the molecular level and the human level is once more emphasized by developments in the study of the relationship between humans and the anthropoids (cf. Chiarelli, 1985; Schwartz, 1985). Continuity or discontinuity between the various levels? The tremendous differences between the various ‘levels’ of the evolutionary ‘path’ seem to be so impressive, that various evolutionists tend, in stead of advocating a simple gradualist perspective, to support a more articulated ‘emergentistic’ approach. Th. Dobzhansky introduces a term borrowed from Paul Tillich: evolutionary transcendence (1967:44): “The origin of life and the origin of man were evolutionary crises, turning points, actualizations of novel forms of being. These radical innovations can be described as emergences, or transcendences, in the evolutionary process” (1967:32, cf. 50). Given this attempt to acknowledge something ‘novel’ in the diversity of reality, it may still be somewhat surprising to see the following statement of 1

In what follows we leave aside the so-called moral issues following from the view that being human fundamentally does not differ from any animal. A remark from Azar will suffice in this context to point at the obvious inconsistencies present in these current reductionistic neo-Darwinistic views: “In a word, if Ruse sees no fundamental difference between man and the other animals, why should he condemn genocide? We certainly slaughter animals every day. If we enjoy filet mignon or fried chicken, why object to killing people?” (1986:233).

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Dobzhansky: “Stated most simply, the phenomena of the inorganic, organic, and human levels are subject to different laws peculiar to those levels” (1967:43). This could have been said by anyone working within the tradition of reformational philosophy! We may even go further and quote a statement from Simpson emphasizing the difference in kind between being human and being an animal: “Man has certain basic diagnostic features which set him off most sharply from any other animal and which have involved other developments not only increasing this sharp distinction but also making it an absolute difference in kind and not only a relative difference of degree” (1971:271). From this statement, however, one can sense the subtle emergent evolutionist undertones: although human beings did evolve from other animals, this development established “an absolute difference in kind”. The same supposition is present in Dobzhansky’s words about “different laws peculiar to those levels”, mentioned above (1967:43), since, in the very next sentence, he proceeds by saying: “It is unnecessary to assume any intrinsic irreducibility of these laws, but unprofitable to describe the phenomena of an overlying level in terms of those of the underlying ones”. This mode of expression came to the fore when, in the the 19th century, certain biologists and philosophers could no longer find peace of mind in the age-old controversy between mechanism and vitalism. According to Passmore (1966:269) this conception of ‘emergence’ probably goes back to G.H. Lewis’s work on “Problems of Life and Mind” in 1875. Although Lloyd Morgan was not a realist, he continued this emergent evolutionism in his Gifford lectures of 1923 (‘Emergent Evolution’) and 1926 (‘Life, Mind and Spirit’). In his “Process and Reality”, first published in 1920, A.N. Whitehead also chooses for an emergent evolutionist approach. The German biologist, Richard Woltereck, continues the emergentistic trend in his ‘Ontology of the Living’ (1940, 300 ff.). The same applies to the well-known German Philosopher of the natural sciences, Bernard Bavinck (cf. his work of 1954). The contemporary systems philosopher, E. Laszlo (1971),1 as well as the philosopher-chemist, M. Polanyi, also adhere to this tradition. The German biologist Walter Zimmerman once (implicitly) formulated this idea as follows: “Without any doubt organisms today possess a typical nature distinct from all other (non-living – D.S.) things in the world. However, this typical nature emerged through evolution” (1962:202-203). On the one hand, the striking differences between distinct kinds of entities seems to be so impressive that these thinkers have to respect them by distinguishing different levels/laws of being; on the other hand they still want to uphold their commitment to a continuous line of descent. In other words, they want to have it both ways: “genetic continuity” and “existential discontinuity”. What is more, as we have seen earlier, some of them are well aware of the tension between these two aspects of their views. Just recall what Woltereck 1

The fact that Laszlo is directly influenced by the line from Lloyd Morgan, S. Alexander and A.N. Whitehead, is shown by Pretorius (1986:29-37).

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and Polanyi say (Woltereck, 1940:300 ff.): “How can the emergent have arisen from elements that cannot constitute it?” (Polanyi, 1969:393).1 Although we have shown that there are still different trends of thought present in 20th century biology, both evolutionist and non-evolutionistic, those oriented to the neo-Darwinistic tradition are not at all willing to even consider an alternative approach. In the Introduction to the publication of the Proceedings of an International Conference on Human Evolution, Tobias, after referring to the presence of differences of opinion between some participants, writes: “This is perhaps a good moment to reaffirm that nothing in human biology makes sense except in the light of the evolutionary concept” (1985:iv). Remark: He proceeds: “To speak of the concept or hypothesis or theory of evolution is, in turn, often seized upon by anti-evolutionists as a sign of weakness in the evolutionary doctrine. Evolution, they are liable to declare, is only a theory. Thereby, of course, they are betraying their ignorance of the way in which science works – by the creating of hypotheses, the testing of them and the refuting or confirming of them. This approach, which is the essence of the scientific method, is no sign of weakness; it is surely the very strength of science”. Evidently, Tobias is not at all aware of the fact that he still adheres to the presently out-dated positivistic philosophy of science – seemingly without any awareness or knowledge about the revolution in the area of the philosophy of science that took place since the 1960s! It is perhaps not farfetched to require that a person adhering to the “evolutionary doctrine”, with its fundamental emphasis on change, should take notice of the far-reaching changes affecting the area of the philosophy of science during the past two to three decades – reflected in the work of Popper, Toulmin, Polanyi, Kuhn, Lakatos, Feyerabend, McMullin, Stegmüller and others. At this stage of our discussion, we must ask the following question: if the human being is evolutionistically considered to be nothing but an extension of (and a higher development within) the animal realm, what should be made of the distinctive features of being human? Or are there no such features? First of all we shall succinctly deal with the available fossil data related to the supposed evolutionary descent of humankind, and afterwards discuss some of the most prominent ‘candidates’ for being uniquely human characteristics. Is the fossil-record conclusive? With the announcement of the discovery of the Taung child skull by Raymond Dart in 1925, designated as Australopithecus africanus, a new picture of human origins took shape. For some time, however, the Piltdown hoax complicated the matter. Found in a gravel pit on the Sussex Downs of England between 1908 and 1913, these remains, in the words of Tobias “showed the astonishing combination of a large-brain cranium, or rather modern aspect, with 1

We must note in this context, that the analysis of Polanyi in 1967 and in 1968 exhibits the same ‘emergentistic’ ambiguity. With reference to the ideas of Polanyi and Laszlo, Hart also pays attention to this problem: “If orders of kinds are irreducible, can things of certain kinds still arise from things of other kinds?” (1984:121).

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an ape-like jawbone (now known to have belonged to an orangutan – Lowenstein, this volume) and lower canine tooth. As long as Piltdown was accepted as genuine and considered an ancient human precursor, it was impossible to accept that Australopithecus was ancestral to man” (Tobias, 1985a:37). Remark: The story about the Piltdown “man” is not a good one for the scientific reliability of evolutionary scientists (cf. Weiner, 1955). During the twenties strong claims were made by prominent scientists as to the reliability and belonging together of the jaw and the skull of the Piltdown “man” (like the anatomist, Arthur Keith, and anthropologist George G. MacCurdy from Yale University). Without acknowledging at all that this forgery simply showed that evolutionary authorities can fantasize what they want to find (by ignoring what they don’t want to recognize), Tobias simply writes: “When the hoax had been perpetrated more than 40 years earlier, its features had been in conformity with the then fixed ideas about human evolution” (1985a:38). If, at a certain stage, it was possible for a forgery to ‘fit’ “then fixed ideas”, how certain are we that, at another stage, we are not the victims of a “theoretically forgerous” interpretation ‘fitting’ the then known ‘facts/fossils’? By the early fifties, according to Tobias, almost all obstacles to the acceptance of the Australopithecus disappeared, since it gained pretty well universal acceptance as a member of the hominids “and as a genus, one of more whose species were on the direct lineage of modern man” (Tobias, 1985a:38). In the fifties and sixties this meant that the evolutionary line proceeded from the Australopithecines and via the Java- and Peking Ape-men (currently classified as belonging to Homo erectus) to Homo neanderthalensis and to Homo sapiens (cf. Le Gros Clark, 1964:168). During the sixties and early seventies L.S.B. Leakey (working near Lake Rudolph in East Africa together with his son Richard), discovered a new species, called Homo habilis.1 Similarities with modern2 human beings caused Leakey to reject Homo erectus as a human ancestor (1970:172). At the same time, he argues that one cannot see the Australopithecines as ancestral to Homo habilis since they were for the greater part contemporaries!3 1

2

3

This name was proposed by Leakey, Tobias and Napier in 1964. The term ‘habilis’ means that this creature not only was better equipped and more suitable than the Australopithecines for the usage of tools, but that it was also capable of making stone tools (cf. Gieseler, 1974:486). The numerous differences separating Australopithecus and Homo habilis from Homo sapiens are described in detail by Henke & Rothe (1980:80 ff.). In order to meet the objections of Le Gros Clark to the classification of Homo habilis separately from the genus Australopithecus, Leakey, Napier and Tobias had to introduce a new definition of the genus Homo, rejecting cranial capacity as a defining feature of the genus Homo (cf. Leakey & Goodall, 1970:161). We should mention in this connection that T.C. Bromage mentions the fact that “the teeth of Australopithecus resemble the great apes more closely than modern Homo” (1985:243).

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Perhaps the most remarkable finding in this category is a skull which was given the registration number 1470 at the National Museum of Kenya. Eventually this skull was classified as belonging to Homo habilis (cf. Henke & Rothe, 1980:95). Leakey remarks: “after its careful reconstruction, it is the most complete specimen of its type: its cranium and face are virtually intact, but the lower jaw (the mandible) is missing” (1978:52). According to the description of this specimen by Richard Leakey in the well-known Journal “National Geographic” (June 1973), which estimated its age at 2,8 million years,1 it “leaves in ruins the notion that all early fossils can be arranged in an orderly sequence of evolutionary change. It appears that there were several different kinds of early man, some of whom developed larger brains than had been supposed” (1973:819).2 If we do not take skull 1470 into account, it seems reasonable, for a number of evolutionist paleontologists, to see a line from the Australopithecines, via Homo habilis to Homo sapiens.3 What is rather astonishing to me, is that no one participating in the conference on Hominid Evolution, organized by Tobias in 1985 in Southern Africa, even mentioned the works of Richard Leakey or referred to skull 1470! Is it because his interpretation rules out the possibility of linking the Australopithecines with the human lineage? In terms of Leakey’s interpretation they were, after all, contemporaries (cf. 1978:52)! Furthermore, in Leakey’s case, the speculative common ancestor should be pushed back to at least 14 million years (Kenaypithecus wickeri – found near Fort Kernan in East Africa), providing the starting-point for two lines of development: (i) the one leading to Homo sapiens while (ii) the other (including the Australopithecines) became extinct (Richard Leakey, 1973:829). Seemingly in order to transcend these problems, some scholars have recently been focussing their attention in more detail on the (mentioned) possibilities of establishing relationships between human beings and their supposed rela1 2

3

In his work of a few years later, Richard Leakey's references to the age of skull 1470 stick to an age of “close to two million years” (cf. 1978:53). It is noteworthy to note that the erectus forms, which are estimated to be 1 million years old (Java- and Peking ‘Ape-men’), have a cranial capacity similar to that of skull 1470. The latter, however, is not only much older than these erectus forms, but its morphological features are also much more similar to that of Homo sapiens. The cranial volume of skull 1470 is 800cc; that of the erectus forms varies between 700cc-1100cc; the earliest finds of Homo habilis were about 650cc; while that of the Australopithecines is 500cc. It should also be noted that skull 1470 lacks the prominent eyebrow-ridge of Homo erectus – which shows a closer affinity to Homo sapiens than the latter. The Black Skull (compare Faul, 1986:10) throws, according to Richard Leakey, “cold water on the notion that as recently as 3 million years ago there was only one species (of early man) which gave rise to the others”. The brain of this new extremely primitive hominid skull is about the size of a modern ape's and less than a third of the size of the human brain – indeed the smallest of any hominid measured to date. Cf. McHenry, 1985:222, where he claims, due to 19 traits in which Australopithecus afarensis resembles Homo habilis more closely than any other species of Australopithecus, that “this species of Australopithecus is the immediate ancestor of Homo” (cp. Figure 2 of Clark 1985:75).

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tives on the basis of molecular and chromosomal evidence. However, also on this level we can discern serious difficulties. First of all, Schwartz (1985:268) points out that chromosomal phylogenies and some molecular and chromosomal evidence support the relationship between the human being and the orangutan – a perspective which is, according to him, also consistent with morphology. This means that, according to this analysis, the large Hominids differentiate into human/urangutan and chimpanzee/gorilla sister groups (Schwartz, 1985:268). In the same volume, however, we read the following conclusion from Chiarelli in connection with a figure which shows the number and types of chromosome mutations detectable in the karyotype of the different apes compared to the human being: “The type and number of changes, up to now detected, demonstrate that the orangutan is the most conservative and the most unrelated to man, among apes, while the African apes (especially the chimpanzee) share a number of derived changes with the human karyotype” (1985:400). With reference to different investigations, these two scholars indeed reach directly opposite conclusions: the first one relates humans to the orangutan (explicitly rejecting the chimpanzee as a candidate), and the second one relates them to the chimpanzee! Immuno-biological evidence (blood antigen studies) and protein homologies provide another indirect way to relate humans and animals. Nevertheless, both the direct and the indirect methods of analysis and comparison only give rise to what Henke & Rothe indicate as a ‘Similarity-phenogram’: “Since biochemical analyses do not provide the time factor necessary for any construction of a phylogenetic tree” all “attempts until now, trying to establish phylogenetic trees on the basis of biochemical evidence, are not satisfactory in view of the numerous and not yet proven presuppositions made in connection with the tempo of evolution in the molecular field” (1980:17). What is even more important, is that they “show important deviations from those phylogenetic trees which are constructed on the basis of morphological criteria” (Henke & Rothe, 1980:17). There are even well-known and important sholars which deny the justifiability to work at all with a genetic mode of expression in paleontology and in the construction of phylogenentic trees. Schindewolf states that the introduction of a genetic reasoning in phylogeny is not justified simply because all the necessary presuppositions are absent (1969:69). He also rejects Simpson’s notion of “quantum evolution” (explosive development), since we have no certain knowledge about the adaptive zones or the “everything-or-nothing-reactions” (1969:69). The crucial point in mentioning these data and differences of opinion, is to show that there are extreme difficulties and problems present in the attempt to come to a coherent and rationally justified picture of human origins even if one accepts the assumptions of neo-Darwinism. Remark: That theoretical presuppositions are inevitably part and parcel of the science of paleontology and the construction of phylogenetic trees (just compare Grene’s analysis of the radical opposition between 122

Simpson and Schindewolf, 1974:130), is explicitly conceded by Schwartz in the final paragraph of his mentioned article: “Sophisticated technology does not provide more accurate phylogenies than conventional means. Phylogenetic interpretation is ultimately a reflection of the theoretical predisposition of the investigator” (1985:268). The biologist, P. Overhage, goes even further by emphasizing that such an essential and penetrating question as that concerning the origin of human beings, by its very nature, reaches into the sphere of our world and life view. Therefore, also the answers given to questions like these are necessarily co-determined by pre-suppositions and pre-decisions which are non-scientific in nature. Especially natural scientists misled many with their supposed ‘objectivity’ and ‘unprejudicedness’ by accusing alternative conceptions of evolution as being restricted by a world and life view. Precisely these convictions, however, make it very difficult for these scientists to realize that mostly the opposite is the case. So many diverging interpretations of fossil findings and so many differences in the evaluation of phylogenetic coherences, evinced foremost in the “trees of descent”, are not explainable purely in terms of the current state of affairs (A. Meyer straightforwardly disqualifies “all these phylogenetic trees” which proceed from “purely idealistic constructions”, 1964:113, cf.59-60). Much rather, it makes an appeal to fundamental convictions and suppositions which influence theory construction from the underlying philosophical and world-and-life-view attitude, as well as from the tradition within which the scientist is working (1959:287). To illustrate this point we mention some differences of opinion regarding Homo habilis. Whereas Clarke (1985:296) emphatically claims that “all indications are that Homo habilis probably developed into Homo erectus some time before 1.5 m.y.”, Jelínek argues that the difference from Homo sapiens to Homo erectus is not on the species level, but on the subspecies level, implying that the correct name should be Homo sapiens erectus (1985:345). Aguirre also writes: “The separation between Homo sapiens and ‘Homo erectus’ vanishes. The authors propose that all populations from the Far East, Africa and Europe, currently referred to as ‘Homo erectus’, should be considered Homo sapiens” (1985:328). One of the crucial questions is whether we can really rely on anatomical and morphological studies to explain the differences between humans and their supposed Hominid ancestors.1 It frequently happens that recourse is taken to the presence of tools in order to determine the human nature of fossil findings. But if we consider archaeological evidence as an aid to interpret fossil findings, are we still working within the framework of paleo-biology? Schinde1

In passing we may note the striking circularity present in some of the arguments evolutionists use to ‘prove’ the existence of a specific organ in terms of its selective value. Gehlen points out (1971:124) that the usefulness of every existing human ability serves as a proof of its selective value: the existence serves as the proof, in stead of showing (as was envisaged), that a specific ability emerged as a result of its selective value!

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wolf warns us that obviously the paleontologist should ‘disregard’ the “technical and cultural achievements of man” because considering them would take us “outside a biological approach” (1969:67). Seemingly without being aware of the fact that they are transcending the limits of biological research, as the archaeologist Narr establishes, even scholars inclined to follow a natural scientific approach now once more start looking for the line between humans and animals where signs of the typical human spirituality are seen in cultural activities (1959:393). The Swiss biologist, Portmann, warns that, in order to get a better understanding of the origin of humankind, we should dispense of the unwarranted and unproven assumption that human spirituality is a late phenomenon in the development of the human body. If this assumption is rejected, however, and human nature is considered in its totality, then the distance between the human being and animals will come to the fore in its full magnitude (1965:57-58). To this we may add his acknowledgement of the fact that his own investigations into the ontogenetic uniqueness of humankind are “guided by the conviction that which can biologically be grasped is essentially co-determined by those aspects of humankind, which have to be investigated with methods different from those employed by the experimental biologist” (1969:23-24). The anthropologist, A. Gehlen, also points out that a total view on being human functions as the guiding philosophical view-point in his research – and this total-view cannot be deduced from the view-point of any special science (1971:13). In one of his earlier works, P. Overhage displays a similar sensitivity: “To reduce the whole question about the human origins simply to the biotical-bodily (morphological-anatomical) facet, witnesses an astonishingly one-sided approach and imply a radical simplification of the total depth of the problem” (1959a:5). Is there anything distinctive to human tools? Originally it was thought that the human being is the only creature that can use tools. When it turned out that animals are also capable of doing this, Overhage emphasizes the human ability to produce tools (1973:359). We have mentioned the fact that the name Homo habilis was introduced to indicate that this species was able not only to use tools, but also to produce them (cf. Gieseler, 1984:486). Although Y. Coppens tries to ascribe some of the oldest flaked stone tools of Omo to the Austrolopithecines, Jelínek says that the “whole situation is still far from clear” (1985:343, cf. Clarke, 1985:287). According to him, archaeologists do accept the view that stone tools can be up to two million years old (1985:343, or even 2.6 million years – cf. Narr, 1974:107). Early Acheulean artifacts, possibly about 1.6 million years of age, are associated with Homo habilis (Clarke, 1985:297). Jane Goodall observed that Chimpanzees are able to construct two kinds of tools: (i) “they crushed wads of fresh leaves lightly between their teeth to increase their absorbent quality and then dipped such a wad in water to use as a sponge” and 124

(ii) “they prepared slim sticks or the stalks of coarse grasses and used these objects to probe into termites’ nests. The termites would sieze the intruding objects in their jaws and then be pulled out to be eaten by the chimpanzee” (Reed, 1985:90-91). Sometimes the chimpanzees would even prepare their termite-sticks before they are used at the mound, an ability until then thought to be uniquely human. Reed remarks that the current usage of the term ‘tool’ simply indicated an “object purposely made and used in a certain way” (1985:91). This definiton caused new problems. If a termite-stick is a tool, why can’t the same be said about the nest of a chimpanzee? Beck proposed that to be a tool, in the words used by Reed in his exposition, “an object must be free of any fixed connection with the substrate, must be outside the user’s body at the time of use, and the user must hold or carry the tool during or just prior to use” (1985:92). Reed himself opts for a new approach. He starts from the physical level where energy and its utilization are the key factors. Thus he reverses the evaluation of tools. Instead of starting with a contemplation of humankind and its works, the evolutionary perspective requires that one should start with what Reed calls the “primary energy-traps”, such as the cellular or protoplasmic part of any living being (1985:93). Secondary energy-traps are A) protoplasmic secretions – their use as energy-traps is extra-cellular (either used within or outside the body); B) habitat features; and C) the use of other organisms (symbiosis). Tools are defined as the forth type of secondary energy-traps: “A tool is a particular kind of secondary energy-trap, an object or a controlled chemical process (fire for example) in the production of which the environment is modified” (1985:95). How wide this definition actually is, is seen from the fact that in terms of it even one-celled animals (from the protozoan family Difflugiidae) are ‘tool-making’. Reed proceeds: “Yes, one group of one-celled animals make tools, and so do thousands of other kinds of animals; all nests are tools, and so are the burrows produced by physical removal of substrate (alteration of the environment). .... Tool-making is and has been of two kinds, instinctive and cultural. The latter, the learning anew of tool-making by each individual of each generation, represents the activities of only a few kinds of primates and perhaps elephants” (1985:96-97). In the final analysis, a distinction is introduced between instinctive and cultural, without paying much attention to the way in which the latter is defined. Are there really no criteria available to distinguish between animal tool-making and human tool-making? Henk Hart is convinced that, in this connection, we may discern a low level category of “formative control”: “The sensitivity of lower animals found in drive and instinct is opened up in higher animals to conscious and purposively directed behaviors which differ in principle from the mechanical and automated structures of behavior” (1984:181). To support this view, he mentions nests, ant hills and beaver dams, which “require skill and control in the molding and shaping of materials” (1984:179). Examples of “resourceful behav125

ior/creative behavior” are mentioned to substantiate these moulding and controlling skills of animals further (Hart, 1984:180). Reed holds the opposite opinion: “Otherwise all tool-making is instinctive, even the seemingly purposive and complex hive-building by some bees and wasps and dam-building by weavers” (1985:97). It seems to me that Hart does not distinguish sufficiently between sensitive intelligence and rational intelligence on the one hand, and between the differences present in animal and human tool-making on the other hand. We shall return to the first distinction presently. At the moment we want to focus our attention on the uniqueness of human tool-making. In the first phase of the paleolithicum (i.e., the early stone age), Von Königswald claims to see evidence of a true invention (1968:167). Narr is more articulate, since he distinguishes three criteria which demarcate and qualify typical human tools (cf. Narr, 1973:61-62, 1974:105-107): a) The form of the tool should not be suggested: The distinctive features of the kind of tool in making should not be exemplified (vorgebildet) by the form of the raw material – for example a stick which could only be freed from obstructing branches and leaves. The final product should still be ‘concealed’ within the raw material. It amounts to abstracting the form “to be brought to the fore” from what is given to the senses. b) The function should not be suggested: Tools are not projections of human organs. They are not to be seen as a strengthening, elevation or extension of bodily organs. Think about a chopping stone which strengthens the fist, or about a stick extending the reach of the arm or the fingers. When a tool is used to cut, it is performing a novel function which is not suggested by the function of any of our bodily organs (this function of cutting must be distinguished from scratching with the nails or from tearing apart with the teeth). In this sense, tools are the product of genuine inventions in the context of creating a new principle of technics and manipulation on the basis of a true insight into the nature and relationships between things. c) The way of production should not be suggested: Tools should not be manufactured simply by using the natural organs of the body (hands, teeth). It must be created with the aid of existing (for instance, chopping) tools, although it is not strictly necessary that the latter themselves should represent artificial products. Note that these criteria deal with objects which are qualified by the formative (cultural) mode of reality, i.e. technical tools, which must be distinguished from other cultural objects which qualified by different (non-formative) functions (such as musical instruments – aesthetically qualified; money – economically qualified; and son on). Clearly, these criteria explicitly presuppose our typical and uniquely human freely varying control guided by our formative fantasy. Kant defined our fantasy as the ability to represent an object without its presence to the senses (1781:B151). Narr goes further by emphasizing that the human formative fantasy must be able to invent something different from what is present to the senses. Furthermore, he also requires that truly human 126

tools must be made with the aid of (formed or unformed) tools. Even the making of simple stone tools as such requires “tool-making tools” (“das Werkzeug zum Werkzeugherstellen”): “In this we see a trait transcending the known and expected behaviour of animals: It presupposes possibilities and achievements which we may view as essentially and specifically human in nature” (Narr, 1973:62). Another way of formulating this perspective is to say that it is typical of the most basic human tools that their ‘end’ is to be a ‘means’! They are formed in order to produce something else. This approach is on a par with the systematic characterization Van Riessen gives for a human tool: it is historically (culturally) founded and qualified (1948:509). Schuurman continues this classification in terms of a cultural foundational and qualifying function: “All technical objects are exceptional in the sense that both their foundational and qualifying function are cultural or technical in nature” (1972:16). Instead of referring to the formative or cultural aspect of reality, we prefer to speak about the technical aspect. The structural conditions of this modal aspect require the accountable freedom and inventive imagination of the human being, the only creature capable of acting responsibly within the matrix of normative conditions. Therefore, I can see no reason to accept that both the human being and animals subjectively function in this aspect of reality, as it is claimed by Hart (1984:179 ff.). In fact, the criterion he uses is just as broad and non-distinctive as the definition which Reed provides for tools as secondary energy-traps (Reed, 1985:95). This definition enables Reed to interpret the form-products down to the level of unicellular animals as tools. The only distinction which he employs on the basis of his definition is that between “instinctive and cultural” tool-making (1985:96-97). However, what he considers to be ‘cultural’ includes both the unique technical abilities of humankind and the examples of animal behaviour which show that there are animals which are not totally determined by their instincts. However, this non-instinctive dimension of animal behaviour (sometimes referred to as sensitive intelligence) should not be confused with the technical-formative mode encompassing all truly cultural activities of the human being. Henk Hart does explicitly speak about animals as “sentient creature(s)” (1984:180), but he does not want to explain the instinctive formations of animals as formative. In order to qualify as ‘formative’ he requires ‘resourceful’ or ‘creative’ behaviour (1985:180). His examples, nevertheless, do not show that animals are capable of performing tasks evincing a responsible, normatively qualified technical inventiveness. It merely shows that animals can act in sensitively intelligent ways. Elsewhere he correctly states that all tools are “technical objects” which are “shaped and designed according to a human plan” (1984:239). By considering the tool-making abilities of human beings and animals, one more and more comes under the impression that the relationship between anatomical and morphological features on the one hand, and particular behavioural patterns on the other hand, are not to be seen in a strictly one-to-one correlation. Narr considers it to be one of the most important results of contempo127

rary ethology that it showed that closeness in terms of the zoological system does not guarantee similarities in behaviour. The reverse is also true: in Primates which are systematically wide apart from each other, corresponding behaviour does occur (1974: 109)! If we add this insight to the uncertainties attached to the fossil evidence concerning human origins, it does not seem unreasonable to support the “docta ignoratia” proposed by biologists such as Haas, Overhage and Portmann.1 Just compare the following remarkable statement of Overhage: “Even if, more or less exactly, the fossils could be shown which represent the stadia through which the human body passed during a long process of evolution, the problem of the origin and the phylogenetic becoming of humanity would still not be solved. Whatever the value of this knowledge may be, it provides only a partial perspective on the genesis of humankind, since it cannot encompass the total life-form of being human. Because, in the fossil findings, the boundaries between the bodily morphological features of being human and that of an animal fade away, and since the spirituality of being human does not express itself univocally in the mere form, we no longer have any certainty about the question whether somatic characteristics – such as the erect gait, the free hand and the cranial capacity (all of which are exactly determinable in fossil findings) – are standing in a strict correlation to the spiritually stamped behaviour of the human being, such that it allows us to make claims about the nature and way in which the primates experienced and viewed the world. .... The riddle of human genesis is not solvable simply by transforming and recombining animal proportions” (1973:374-375).

Do animals share the dimension of (human) logicality? The famous German zoologist, Bernard Rensch, extensively argues, with numerous references to experimental data, that we must assume the presence of “a-verbal concepts” (1971:9, 197, 242, 245), for instance in the anthropoid apes. He does qualify this mode of referring to animal behaviour immediately by stating that the inference of psychical processes of any kind in animals must be formulated in an “as if” way: “it is more correct to say that chimpanzees act as if they were abstracting and generalizing, as if they had formed a-verbal concepts (i.e., concepts not connected with words), and as if they were proceeding with intention and foresight” (1971:242; cf. the contrasting view of Koehler, 1973:199). This should be kept in mind when he discerns “concepts of value”, a “more or less distinct concept of the self” (243) and even “primitive aesthetic feelings” (244) present in anthropoid apes. Experiments done with cats show that they are also able to make a choice between what Rensch calls “two patterns corresponding to an abstract concept of ‘unlike and like’” (245). These achievements are not so much dependent on the development of the cortex of the forebrain. With their forebrain only slightly developed, even fishes can “learn to grasp the significance of up to five pairs of patterns, and they can retain a pair of patterns for some months and recognize them even when these are considerably altered” (246). 1

Portmann also stresses the fact that it is no longer tenable to uphold the conviction of a direct descent of human beings from the anthropoids (1977:461).

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Clearly, the way in which sentient creatures orient themselves in the world does show many similarities with human behaviour. The logical point, however, is that all similarities imply and presuppose differences. In the absence of differences, we meet identity, not similarities! What are the differences, if any, in this context of behavioural similarities? Is it enough to say that we “are not just sensitive to goals and aware of consequences” since “we foresee them and plan ahead” (cf. Hart, 1984:181)? If it is possible to show that animals can locate similar entities and afterwards act accordingly, are we then justified to conclude that they have a-verbal concepts? What do we mean with the term: concept? This question becomes more urgent when Rensch stresses that his use of the term ‘a-verbal’ is meant to emphasize that “a-verbal concepts” in animals do not proceed from logical operations (1973:118). For this reason, he claims that the trait determining the gap between humans and the anthropoids is logical thinking (1968:147). Rensch says that although animals approach causal ‘concepts’ (concerning relations in different situations), humankind is transcending it in its unique ability to grasp truly logical relations, expressed in concepts like “as a result of”, ‘because’, “in case of”, and so on. Cassirer argues that the determination of a concept as “unity in multiplicity” belongs to the classical legacy of logic and philosophy as such (1929:339). Whatever is logically grasped cannot fully prescribe in which way the multiplicity of features should be united in the unity of a concept (Cassirer, 1928:134), because it is also a result of the real creative element in our thinking: the power to discern/observe.1 Logical concept formation is always aimed at discerning the multiplicity of universal traits of that which is conceptualized. As such it is subject to universal (modal) logical norms, such as the principles of identity and non-contradiction. Although the construction of each concept is dependent on logical subjectivity - since only a human being is able to respond with normative freedom to these normative conditions for logicality - no concept is exclusively the product of our subjective logical functioning. Leakey’s announcement that the “ability to see commonalities between objects of the same type – classes such as trees, fruit, predators, birds, etc – is a crucial step in creating conceptual order in what otherwise might be an overwhelming perceptual chaos” (1978:204) ascribes, in a typical Kantian fashion, a truly formal creativity to concept formation. The capacity of the anthropoids to recognize perceived objects (in animals limited to a small number of them) and to associate them with each other (even in different contexts), does

1

It is difficult to find a suitable English equivalent for the German word used by Cassirer: “...der Aufmerksamkeit als dem eigentlichen schöpferischen Vermögen des Begriffsbildung” (1910:31). “... the power of observation as the truly creative ability of concept formation”.

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not provide conclusive evidence that they are able to function subjectively in the analytical aspect of reality.1 The perception of a multiplicity of objects, the sensitive delimitation of particular perceptual objects or events (capable of exerting a controlling influence on behaviour in later situations – such as avoiding fire), due to the continuity provided by the associative ability of animals – all of this are still enclosed within the domain of sensitively qualified beings. Precisely because our human capacity to judge is foundational to every act of identification and distinguishing, it differs in principle from a merely sensitive delimitation and association.2 A correct logical concept must entail the multiplicity of identified characteristics united in the concept in such a way that logically justified judgements could be inferred from it. Anything logically known is, by means of subjective logical conceptualization, logically objectified.3 Objectification is always a subjective activity, conforming to or contradicting the applicable logical principles. Only this perspective can explain why we consider it to be illogical (i.e., logically incorrect) whenever judgements explicate conceptual elements which are not analytically implied in the unity of the concept concerned. Let us take an example. Does the concept of a chair only appeal to its opened up logical features, or must we assume that all the non-logical characteristics are implied by opening up the logical object-function of a chair? If these non-logical characteristics are not implied, an unbridgeable gap, plainly, exists between the logical subject-object relation and the non-logical aspects of a chair (in connection with the structure of analysis, cp. Strauss, 1984). We have to conclude, therefore, that in making the logical object-function of a chair patent (manifest), the non-logical (modal) characteristics (specified according to the typical entitary uniqueness of the chair) are also logically objectified. Therefore, the multiplicity combined in the unity of the concept of a chair enables us to make predications like: this chair is beautiful (aesthetic feature); this chair is expensive (economic); this chair is big (spatial); this chair is heavy (physical); and so on. If these (modal) characteristics were not analytically implied in the correct concept of a chair, all these mentioned statements (explicating them in distinct judgements) would, in a logical sense, be contradictory. In other words, if the correct concept of the chair does not imply these characteristics in an analytical way to begin with, they cannot af1

2

3

Leakey also refers to the use of signs by chimpanzees and gorillas. With the aid of different ‘sign-labels’ these animals are supposed to be able to generalize to cognitively economical concepts (essential for language): “For instance, Lucy calls a watermelon a drink fruit; Washoe refers to ducks as water birds, and she invented the name rock berry for a brazil nut when she first encountered one” (1978:202). O. Koehler explicitly claims the contrary: “In the absence of verbal language, we call such an operation with representations, concepts and judgements, founded on intuitions without bearing any names, unnamed thinking” (1973:119). This objectification is nothing but the opening up of the logical object-function of entities.

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terwards be predicated of the chair, except illogically: From: P is non-Q; one cannot infer: P is Q. But if it is granted that animals cannot argue and cannot infer in a logical or an illogical way, shown in judgements which (do not) conform to logical principles (such as the principium identitatis and the principium contradictionis), are we not justified in maintaining that they do function up to the level of (a-verbal) concept formation? This question amounts to the following one: Are there non-logical concepts? and: are animals capable of forming illogical concepts?! We only have to think about the well-known example of Bertrand Russell: a square circle. Our lingual ability to designate entitary analogies, as distinct from modal analogies (such as the modal difference between mathematical space – which is continuous and infinitely divisible – and physical space – which is not continuous, since it is bound to the quantum structure of energy and therefore is not infinitely divisible), is known to us in the form of metaphors (viz. “foot of the mountain”). In our case we only have to think about the nature of a “boxing ring”. If the sign-mode of reality was not distinct from the logical mode, then this metaphor would have been an assertion that a square circle exists! Remark: In spite of the fact that Henk Hart has a clear view on the nature of modal analogies, he does not distinguish between modal analogies (anti- and retrocipations) and entitary analogies (designated by metaphors). His initial thesis is that an analogy is an interfunctional relationship (Hart, 1984:87, cf. 153). True examples of interfunctional analogies are mentioned by Hart – such as analogies of growth (158) and the modal difference between social distance and spatial distance (171, cf. 205). However, he constantly equates them with ‘metaphors’ (cf. 1984:153, 156, 158, 160). A metaphor can be replaced by another one totally different from it. This is not possible with modal analogies: every replacement simply turns out to be synonymous with the original one! This concept is illogical because both the identification and the distinguishing are not conforming the relevant logical principles, viz. the principle of identity and the principle of non-contradiction: (in Euclidean space) a circle is a circle (correct identification), and: a circle is not a ‘non-circle’ (such as a square – correct distinguishing). Bernard Rensch mentions the fact that, in Münster, experimentalists have tried for half a year to teach chimpanzees to copy a given drawing of a square or triangle, but without any success (1968:148). My question is: if these animals are not even able to draw these figures, how are we going to be convinced that they truly have the concept of a square or the concept of a triangle? We must remember that a concept is something different from a sensory picture which can be associated with something else (cf. Overhage, 1972:252) – as is the case in the so-called ‘name-giving’ mentioned by Leakey. The decisive point, however, in showing that they do possess these concepts, would be to show that they can think illogically by forming, for example, the self-contradictory concept of a “triangular square” or a “square triangle”! This has 131

never been shown by any of the experiments referred to by the mentioned authors. In other words, animals are simply not able to function subjectively in the analytical aspect of reality. Consequently, it should not be surprising that they are unable to think – be it in a logically correct way or illogically! Does this imply that we cannot ascribe any form of intelligent behaviour to animals? Buytendijk refers to ethological research in order to substantiate his conclusion that the animal world merely shows gradual differences in this respect. “Every species has its own practical intelligence, limited by disposition and experience” (1970:98). This conclusion presupposes his basic distinction between animal and human intelligence. When, in a given situation, human beings and animals will pursue a similar goal, they will experience similar emotional drives. However, what is absent in the case of the animal, is action on the basis of judgements: “Therefore, one defines animal intelligence as the concrete experiential and senso-motoric structuring of practical behaviour, whereas human intelligence displays itself as a rational-logical, categorically judging conceptualization of the task-setting nature of the concrete situation and the discovery of a solution which does not follow from the immediate sensory effect of the situation” (1970:97). Overhage, rejecting the anthropomorphic mode of speech present in the writings of Rensch, Koehler and Lorenz, emphasizes that animal form-perception does not result in genuine concept formation, since it remains enclosed within the sensory-perceptive sphere (1965:307; cf. Overhage 1972:251-276). The human being as “Homo symbolicus”? The uniqueness of being human is sometimes sought in morality, in the ability to commit suicide, in the consciousness of death (Dobzhansky) or in the lingual potential of being human (Cassirer, Von Bertalanffy). Cassirer (cf.1944) introduced the well-known distinction between signals and symbols. The former belongs to the physical world of being and the latter is a part of the human world of meaning, the world of human culture. Von Bertalanffy says that symbolism “if you will, is the divine spark distinguishing the most perfectly adapted animal from the poorest specimen of the human race” (1968:20). In order to identify symbols, he uses three criteria: (i) symbols are representative, i.e., the symbol stands in one way or the other for the thing symbolized; (ii) secondly, symbols are transmitted by tradition, i.e., by learning processes of the individual in contrast to innate instincts; (iii) finally, symbols are freely created (1968:15, cf. 1968a:134). Helmut Plessner wants to transcend the self-contradictory notion of an ‘entelechie’, presented to him by his tutor Hans Driesch. As an alternative, he introduces the notion of positionality. Physical entities are delimited by the surrounding environment. In the case of organic entities, this delimitation belongs to the entity itself (for example, the membrane), and thus evinces positionality (1975:291). This concept provides the possibility to view humankind as belonging to the last level of living beings. Animals are considered to be closed and centric, distinguished from the human being as an eccentric (and relatively ‘Weltoffen’) living being (1975:292). The first anthropo132

logical ‘Grundgesetz’ (fundamental law) mentioned at the end of his Book: “Die Stufen des Organischen und des Menschen” (1928, reprint 1965) states the “vermittelte Unmittelbarkeit” (mediated immediateness) valid for all eccentric positions (cf. 1975:297). Language positions itself in between the grasp of the hand and the view of the eye – the eye as the “organ of making-something-immediately-present”. Thus, in various respects, the hand and the eye become dispensable (cf. Hofer, 1972:203). Animal communication, according to Plessner, does not know a “mediation through objects” (1975a:380, cf.379). Surely, this phenomenon is particularly remarkable, since, in the domain of human sensitivity, the sense of seeing and of the sense of touching dominate that of smelling (cf. Haeffner, 1982:16). Precisely by means of the mediated immediateness of language, human beings possess an awareness of the past and the future – an awareness taking the limited life-span of being human into consideration. This explains the uniquely human awareness of death as well as the possibility to commit suicide. The communication of animals does not refer to the distant past or remote future – it is restricted to the immediate needs of the animal. As a consequence, the ‘signs’ used by animals (signals, in terms of Cassirer’s distinction), are strictly univocal. Just compare the remarkable dance of the bees where the (i) tempo, (ii) the direction and (iii) the tangent is constantly associated with the (i) distance, (ii) location and (iii) the course to be followed in order to reach the detected source (cf. Overhage, 1972:220 ff.). Lingual signs, on the contrary, presuppose choice and therefore require interpretation (cf. Nida, 1979:203; De Klerk, 1978:6). Furthermore, it is striking that the typical human lingual ability is dependent on specific anatomical conditions absent in the anthropoids. The anatomical conditions of human speech Ever since Descartes it was believed that the uniqueness of the human brain is responsible for human language. The result was that anatomists insisted that anthropoids also have the ‘machinery’ available to articulate speech. The order of primates – which includes human beings according to the prevalent classification – is nevertheless, of course with the exception of humans, unable to vocalize. The ability to reproduce human speech sounds as it is found in birds is totally absent in the mammals. The vocal potential of the gorilla and urangutan is exceptionally poor. The chimpanzee is somewhat better off, and the gibbon can produce sounds covering almost an octave. All these anthropoids, however, completely lack the playful sounds produced by the human suckling. The unprecedented possibilities of human sound production transcend that of the anthropoids by far. In addition since this human sound production also displays an exceptionally rich modifiability (Overhage, 1972: 242). Post mortem studies of the upper respiratory tract in mammals as well as cineradiographic studies have shown that the position of the larynx is crucial in 133

determining the way in which an individual breathe, swallow and vocalize (Laitman, 1985:281). This implies that there are certain anatomical peculiarities which go hand in hand with the contribution of brain functioning in the production of human speech, in particular the gradual descent of the larynx after the post-natal period (cf. Portmann, 1973:423). The failure of the anthropoids to imitate human sounds follows from the totally different structure of their larynx. In all anthropoids it is positioned extremely high in the neck. Laitman remarks: “This high position permits the epiglottis to pass up behind the soft palate to lock the larynx into the nasopharynx, providing a direct air channel from the nose through the nasopharynx, larynx and trachea to the lungs. ..... In essence, two separate pathways are created: a respiratory tract from the nose to the lungs, and a digestive tract from the oral cavity to the esophagus. While this basic mammalian pattern – found with variations from dolphins to apes – enables an individual to breathe and swallow simultaneously, it severely limits the array of sounds an animal can produce. ... While some animals can approximate some human speech sounds, they are anatomically incapable of producing the range of sounds necessary for complete, articulate speech” (1985:282, cf. Goertler, 1972:249). In order to provide the newborn human suckling with a milk tract separate from the respiratory tract, the position of the human larynx at birth is the same as in that of the mammals. In the period between the first and second year this highly positioned larynx starts its descent in the neck. This downward movement creates the pharynx cavity, necessary for the articulation of the richer voice disposition present in human beings. Laitman declares that the precise time this shift occurs, as well as the physiologic mechanisms which underlie it are still poorly understood (1985:282). As soon as the larynx reaches its destined low position, it can no longer lock into the nasopharynx. Consequently, in human beings, the respiratory and digestive pathways cross above the larynx. This creates the possibility to suffocate, which surely is, evaluated in itself, something negative. However, it is precisely this expanded pharynx which provides the human being with its unique potential to produce a rich variety of speech sounds. The palate between the mouth and nose cavities serve as resonance basis for the produced sounds. Goerttler even mentions the fact that in the third month after conception a distinctively human structural element develops (it is called the vocal chord ‘blastem’ – 1972:250). It is interesting to note in this connection that Laitman informs us that the basicranial similarities between the Australopithecines and extant apes suggest that their upper respiratory tract was also similar in appearance. Consequently, as with the living nonhuman primates, the pharynx portion available for sound modification in these early hominids would have been greatly restricted: “As a result, these early hominids probably had a very restricted vocal repertoire as compared with modern adult humans. For example, the high larynx would have made it impossible for them to produce some of the universal vowel sounds found in human speech patterns” (1985:284). His conjecture is that the first instances of full basicranial flexion similar to modern humans 134

do not appear until the arrival of Homo sapiens (estimated by him at 300,000 to 400,000 years ago): “It may have been at this time that hominids with upper respiratory tracts similar to ours first appeared” (1985:286). Do human beings have ‘speech-organs’? This question points at another astonishing feature of human speech production. If we define a speech-organ as that bodily part which exists solely in service of the production of speech sounds, then we are in for a surprise. Let us enumerate possible candidates: the lungs, larynx, mouth cavity, palate, teeth, lips and nose cavity. Without an exception, all these organs perform primary functions which would normally proceed even if a person never utters one word (Overhage, 1972:243)! Human language simply takes hold of all these different organs in the production of speech sounds. This highly developed and subtle cooperation, especially of three organs so heterogeneous in character as the mouth, the larynx and the brain integrated in the production of speech sounds, makes it rather difficult, if not hopeless, to provide us with an evolutionistic causal explanation of this astonishing phenomenon. The question arises what number of miraculous changes should have occurred to produce the articulation conditions necessary for truly human language formation. “Such an unfathomable process of change affecting so many differently structured organs and organ complexes, closely correlated with each other, should have proceeded harmoniously as a total change, if it was to come to the unprecedented perfection of human speech” (Overhage, 1972:250). Does human experience of the world differ from that of the animals? In the course of our discussion we often referred to the role of instinct in animal life. Adolf Portmann is convinced that animals are actually determined by their instincts and that they are restricted to a particular ambient (1969:86). The way in which animals experience the world is completely determined by their natural dispositions. They are only concerned with that which has a direct physical, biotic and sensitive meaning to them. Consequently, they experience reality in terms of places suitable for walking or flying (physical accessibility), in terms of sex partners and other animals belonging or not belonging to the same species, in terms of what can be eaten and what not (biotical interest), and in terms of things or events which are causing anxiety or which may be comforting (sensitive concern) (cf. Landmann, 1969:162 ff.). The instinct determinedness of animals function in a remarkable way. Due to inherited coordinations (concerning the motoric dimension) and inborn “triggering-off” mechanisms (Auslösemechanismus; concerning the domain of receptivity), particular animals, in given circumstances, can act instinctively and in predetermined ways. These instinctive behaviour patterns are inherited and not learned. Eibl-Eibesfeldt gives the example of a squirrel with its act of burying a nut. Normally, these animals bury collected acorns or nuts individually in the ground. In doing it, they perform certain typical actions such as running around, laying down the nut and burying it covered with soil. When a squirrel is raised in an artificially isolated cage without ever being given a nut, 135

the mature squirrel will, when presented with one, exactly execute the typical ‘burial ceremony’ (Eibl-Eibesfeldt, 1972:5). This demonstrates that a ‘programmed series of actions’ could be triggered off by a specific stimulus. The world of different animals differs, and so differs the world of animals and human beings. Von Bertalanffy explains one of the examples delineating the ambients of various animals as follows: “Take, for instance, a tick lurking in the bushes for a passing mammal in whose skin it settles and drinks itself full of blood. The signal is the odour of the butyric acid, flowing from the dermal glands of all mammals. Following this stimulus, it plunges down; if it fell on a warm body – as monitored off by this sensitive thermal sense – it has reached its prey, a warm-blooded animal, and only needs to find, aided by tactile sense, a hair-free place to pierce in. Thus the rich environment of the tick shrinks to metamorphize into a scanty configuration out of which only three signals, beaconlike, are gleaming which, however, suffice to lead the animal surely to its goal” (1973:241). Another well-known example, given by Von Uexküll, concerns an oak tree. Different kinds of animals ‘dissect’ their own different ambients (Umwelten) from the tree – constantly enclosed within the above mentioned parameters concerning their physical, biotical and sensitive needs (Von Uexküll, 1970:98, 100). He discusses the life-worlds of animals like the jackal, squirrel, owl, ant and beetle. Although human beings also have access to these dimensions, one cannot say that his experience is closed by or limited to these perspectives. Human functionality encompasses, but also transcends the physical, biotical and sensitive aspects of the tree. To the botanist the tree may be an analytical object of scientific investigation; a person going for a walk may experience its beauty; a criminal can use it as a hiding place, the carpenter may use its wood to manufacture furniture, and so on. Therefore, human beings are able to experience the tree in a variety of ways which are inaccessible to animals. Human functioning is neither completely determined by instincts, nor is it limited to only one ‘Umwelt’, simply because the whole bodily existence of human beings is directed towards and is guided by normatively qualified view-points. The tremendous flexibility of human functioning executed within these normative aspects of reality, makes it possible for human society to develop up to a level with far-reaching forms of differentiation and specialization, expressed in the mutlifarious roles which any person in such a society can assume. Even Simpson stresses this insight: “Such specialization, which is non-genetic, requires individual flexibility and could not occur in a mainly instinctive animal” (1969:90). Hart states it with concise clarity: “A worker ant is just that – and all its functions are geared to being a worker ant. A human being, on the other hand, has multiple roles to play and is not exhausted in any of them” (1984:146). However, this human-spiritual specialization and flexibility is dependent on a relatively unspecialized bio-psychical basis and foundation. Because Van Uexküll extended his ‘Umweltlehre’ also to the level of human beings, Portmann points out that we should take caution in this respect. A comparison between the different ambients experienced at a tree and the 136

different domains of human functioning, does show the important differences still present. The societal structure of human life enables communal understanding between all the differentiated societal spheres, something lacking between the ambients of different animals (Preface in Von Uexküll, 1970:XIV). The unspecialized traits of the human body Seen from a morphological perspective, human beings lack the highly specialized organs necessary to be perfectly adapted to a particular habitat. Gehlen refers to the archaic (in the sense of primitive/unspecialized) features displayed by the human organs (1971:86 ff.). Human denture is remarkably unspecialized if compared to those of the higher animals: it is not specialized solely for the eating of plants or for the eating of meat. There are no gaps (diastema) between our teeth – something typical of the highly specialized nature of the more developed mammals. By contrast we only have to compare it with the distance between the eye-teeth and front molars of the anthropoids which is closely connected with the way in which the latter are specialized to become canine teeth (Gehlen, 1971:92). Similarly, the human hand (Gehlen, 1971:98) and foot (Gehlen, 1971:100) represent a primitive state in comparison with the anthropoids (such as the orangutan, gorilla and chimpanzee). Altner points out that the teeth of the anthropoids are also relatively unspecialized. Nevertheless, he does not deny the general tendency present in the phenomena lifted out by Gehlen (1972:199-202). It should be noted that Gehlen does not use the term primitive in the sense of lower but only in the meaning of unspecialized. According to the predominant neo-Darwinist evolutionist approach there merely exists unidirectional evolutionary change: from the less specialized to the more specialized. The Belgian geologist, Dollo, formulated this in terms of his law of irreversible specialization. But if this law is universally applicable to all evolutionary change, as it is still upheld by Simpson and other dominant neo-Darwinistic thinkers, how can we ‘save’ the anthropoids as candidates for being ancestral to human beings? Clearly, they are specialized. However, accepting Dollo’s law seemingly makes it impossible to deduce the unspecialized features of human beings from the specialized traits of the anthropoids. To satisfy these strange conditions, one has to find (or: hypothetically construe) a transitional form which should unite radically opposed traits: archaic human features and specialized animal characteristics. Gehlen remarks that this would produce such a monstrous and miraculous being that it should be awarded a fantastic and separate position in the total realm of animals (1971:87-88)! Notwithstanding these difficulties, two escape routes have been explored! (i) Adloff and Klaatsch constructed an hypothetical ‘primitive form’ which is so unspecialized that, on the one hand, it can serve as a basis and starting-point for modern human beings, and, on the other hand, may represent the root enabling the branch which developed into the specialization present in the anthropoids. Obviously, this construction is begging the question, since it actually starts from a form which is so ‘human’ that the 137

supposed order of descent is reversed: human beings are ancestral to the anthropoids (cf. Gehlen, 1971:95)! (ii) Another possibility is to ignore Dollo’s law of irreversible specialization by considering the phenomenon of neoteny, i.e. the persistence of larval features in the adult organism. Neoteny is found in various kinds of animals, such as worms, insects and amphibians. The Dutch anatomist, Louis Bolk, used this phenomenon to explain certain human features. The remarkable similarities between human beings and the chimpanzees concerning the unspecialized and archaic nature of human organs, inspired Bolk to explain human beings in terms of the notion of fetalization (stabilization of pre-natal traits) and the idea that human development underwent a certain retardation. In other words, in human beings the infantile characteristics of the ape became fixed because the mature ape-form was no longer reached, while maintaining even fetal marks. Consequently, in this approach, as Landmann remarks, human beings are not ancestral to the apes, since they are themselves nothing but infantile apes (1969:148)! Konrad Lorenz, the Nobel prize-winner of 1973 (together with Tinbergen and Von Frisch), added to this theory of Bolk the life-long curiosity of human beings which correspond with the youthful curiosity of the chimpanzees: “The constitutive hall-mark of human beings, their persistent, creatively active engagement with their ambient (Umwelt), is a phenomenon of neoteny” (1973:183-184). Nietzsche once remarked: in every adult human being a child is concealed. Lorenz reverses this statement: in every child a mature person hides, eager to do research. The curious child, which disappeared completely in the full-grown chimpanzee, is not hidden in adult humanhood, since it controls the latter fully (1973:184, cf.242). To this approach Lorenz adds another perspective, following from the study of domesticated animals, namely that of self-domestication. Domesticated animals differ in some typical hereditary features from wild-living forms. These characteristics emerged in the process of domestication. For example, all domestic animals show a measure of spots, display a shortening of the extremities and the basicranial structure, are inclined to fatten easily, while an important increase in the domain of variation of all possible features of the species appear. Already E. Fischer pointed out that the pigment-division present in human eyes which are blue or grey is, though totally absent in all wild-living animals, in a corresponding way fully present in almost all domestic animals. Among the changes caused by the self-domestication of human beings, Lorenz is counting both the retardation and fetalization mentioned by Bolk. According to Lorenz, cave inhabitation represents the desicive step in the process of the self-domestication of human beings. This account, however, is not in accordance with what we know. A number of the most typical features of “becoming domestic” are totally absent in the case of human beings. We only have to mention the early sexual maturity and the constant or decreasing brain development of domestic animals – both phenomena which are completely reversed in human beings (cf. Gehlen, 1971:121 and Overhage, 1967:3-4)! 138

Furthermore, an appeal to phenomena of domestication calls upon factors not operative in the domain of non-human living organisms (cf. Overhage, 1967:4). Becoming a domesticated animal presupposes the cultural care of human beings as human beings! The absurd implication of this argument about self-domestication is already formulated by Von Eickstedt more than fifty years ago: “Culture, then, must be older than human beings, since it influenced humankind in its bodily development. It was not humankind who formed culture, because culture moulded humankind” (1934:121)! In other words, this view on self-domestication reverses the cultural subject-object relation, by making the human being, the cultural subject, an object of cultural control and moulding. When the instincts of animals deteriorate, owing to the fact that they become accustomed to the care of human beings, they do not develop any compensating faculties. Landmann goes a bit too far when he emphasizes that the peculiar capacities of human beings do not need any instincts (1969:148), because we must acknowledge that human beings still have instincts, however poorly they are equipped with them if compared with the existence of animals that is secured by their instancts. However, Landmann is fully justified in saying that a ‘wild’ human form, dominated by instincts, never existed and never could have existed. Even if we want to reject Dollo’s law, and support this rejection with Lorenz’s theory of retardation, fetalization and neoteny, unanswered questions keep popping up. According to Gehlen it seems absolutely impossible to understand in what sense the development of thought and language would have endowed human beings with a selective advantage in comparison with the anthropoids. The question should also be asked: in a fight against what would the prolonged and helpless youthful period of human beings provide them with a selective advantage in stead of being a serious and life-endangering disadvantage (cf. Gehlen, 1971:125)? Traditionally it was thought that human beings possess something lacking in animals: intelligence (wisdom). This legacy is reflected in the currently still generally accepted (evolutionist) classification of human beings as Homo sapiens. If we compare human beings, however, with the high level of specialization present in the anthropoids, we must conclude that human beings are lacking something, namely specialization. Must we conclude, then, that the “structural design” of human beings show shortcomings? Is the human being to be seen as a deficient creature? This notion stems from Gehlen (cf. 1971:20, 30, 80, 354). Indeed, if we compare the natural predispositions of human beings with the countless possibilities at the disposal of different animals, is does look as if human beings are treated niggardly by nature. HuMan beings are much slower than many wild animals. They do not have natural protective hair covering his body. The human senses are truncated in comparison with the sharpness and alertness of the senses of the animals. Human beings do not have natural and dangerous weapons. They do not possess the muscle power, claws or jaws of any preda139

tor. Some animals can register supersonic waves, some can see ultra-violet rays as light, while there are fishes that can perceive electrical fields. Birds orient themselves, by means of their remarkable navigating systems, to the magnetic poles of the earth. And all these ways of experience are withheld from human beings (cf. Portmann, 1970:200 ff.). If we measure human beings with the yardstick of an animal, we are almost doomed to evaluate them as animals that failed! But as soon as we reverse the perspective and acknowledge the uniquely human features which differentiate human beings so clearly from animals, a different picture arises. Portmann explains this with the following example: “The narrow limitedness of animal interest is opposed to flexible freedom of choice present in human beings. An animal can transcend the bondage to its drives only to a limited degree, whereas I am able, in every moment and according to my total power to discern, encompassing my full inner-participating dedication, to pay attention to something, however minute and unimportant it may appear to be” (1974:102).

Why is it that some scholars emphasize the unspecialized nature of human beings and even call them deficient beings? Notions like the unspecialized nature of human beings and the qualification deficient being only play a role if we choose the natural disposition of animals as basis of comparison. If we loose sight of this implicit choice, the objection of Hans Freyer would have been completely valid. Initially, the fiction is postulated that human beings are animals and then, only afterwards, it turns out that in this capacity human beings would represent something highly insufficient which, as such, is an impossibility! The erect gait, the free hand with the strongly opposing thumb (serving the formative cultural fantasy mentioned earlier) and the spiritually stamped facial expression of human beings - all these features reveal the true nature of humankind - to take over an expression from Lorenz - as a specialist in being unspecialized. Gehlen is inclined to see the distinctively human functions of human beings as something compensating the lack of being instinctively secured. However, precisely the opposite is the case. The physical, biotic and sensitive dimensions of human existence are fully geared towards the normatively stamped cultural life of being human. This cultural disposition, which should be considered to be our first nature (and not our second nature, as Portmann believes), come to the fore in the ability to think, to conceptualize and to argue intelligently. It is also seen in the technical ability to manufacture tools after a free project manifesting a free formative fantasy, in the lingual competence to discern and articulate meaningful speech sounds and to interpret those produced by fellow human beings correctly, and so on. The analytical ability of a person, enabling that person to identify (i.e., the lifting out of certain features) and distinguish (i.e., by disregarding other features), is foundational to the technical functioning of human beings: to produce something fantasized after a free project, presupposes an analytical ability (including conceptualization) in the mentioned sense. Although Hart argues for a place of the formative/technical aspect before the analytical aspect 140

(1984:179 ff.), in a different context he does argue for the foundational role of concept formation: “Much reality contains concepts as constitutive elements of its nature. Almost all typically human products, nearly all of what we refer to as culture, cannot exist except through conceptualization. Without our having concepts of these realities we cannot produce them” (1984:411, note 28). This is a good argument in favour of placing the logical aspect before the technical-formative aspect in the order of cosmic modalities! Although it is not the place here to give elaborate arguments for it, I want to mention the fact that the same applies to the sign-mode of reality (called the symbolic aspect by Hart, which he also places before the analytical aspect – 1984:180 ff.). All typical semantic phenomena, such as synonymity, antonymity, redundance, metaphoricity, and so on, not only presuppose the analytical aspect but only have meaning if these two aspects are distinct and irreducible (cf. Strauss, 1981:5-32). The functioning of human beings in these different normative spheres of reality, sometimes highly differentiated in a social sense, cannot fully be accounted for in merely functional terms. Hart correctly states that the integral identity of the human person transcends human functionality. He then proceeds: “Human responsibility and accountability point to another dimension of human existence besides the functional dimension, namely, the spiritual. The spiritual in humanity cannot be fully understood as functionality, although it can be understood only if we understand it in terms of functionality” (1984:270). Whereas material things, plants and animals are respectively stamped and qualified by particular aspects (viz. the physical, the biotic and the sensitive), the uniqueness of being human is precisely seen from the fact that no functional human activity can ever enclose or encompass all of our human functionality. Once again Hart puts it well-formulated to us: “In the life of a person, there is not a single qualifying function that structurally unites and integrates all of human experience” (1984:276). This also explains why human functionality is open, through faith, to being committed to the acceptance of what lies beyond the limits of subjectivity (cf. Hart, 1984:277). I would prefer, as Dooyeweerd does in his A New Critique (cf. 1997-III: 87-89), to deny the possibility of a human realm or kingdom (Hart consistently affirms such a kingdom – cf. 1984:268 ff.). Of course this is dependent on the definition we give for a realm. Hart defines it by saying that realms are categories of existence according to principles of order (1984:268). This definition introduces something ambiguous between Hart’s different realms, because the three realms of material, vegetative and animal existence are all uniquely qualified by a single modal function, whereas the “human realm” is not qualified by any single function at all (cf. Hart, 1984:276-7). Before we consider a number of diverging perspectives on the nature of human freedom, we still have to pay attention to the uniqueness of our human functioning within the biotical aspect of reality. The relevant material for this analysis is amply provided by Portmann (1969). 141

The ontogenetic1 uniqueness of being human In order to compare the ontogenetic developmental nature of humans with that of the animals, Portmann distinguishes two distinct developmental types: Nesthocker and Nestflüchter. ‘Nesthocker’, on the one hand, encompass all those animals which are born with closed eyes, which are naked and helpless at birth, and which need the care of their parents in providing the necessary food as well as a prepared nest. ‘Nestflüchter’, on the other hand, are represented by all those animals which, at birth, are capable of moving similarly to their full-grown parents. At birth their eyes are open as well as their auditory canals, while their posture and bodily proportions exactly correspond with those of the mature members of the species. Mammals falling within the category of ‘Nesthocker’ are born after a short period of pregnancy – about 20 to 30 days. Portmann mentions insectivore (primitive insect eating mammals), gnawers and some predators. The number of new-born animals per litter is relatively large – from 5 to 22. The higher developed ‘Nestflüchter’ have a long period of pregnancy (more than 50 days and sometimes longer than 20 weeks), while, in most species, their new-born offspring are limited to 1 or 2 (seldom 4). Examples of this category are ungulates, sea lions, wales, horses and monkeys). Mammals with a less developed brain are born as ‘Nesthocker’, such as the squirrel, house mouse, red jackal, domestic cat and tiger. Mammals possessing a more highly developed brain, experiencing a long growth period within the mothers womb, enter the world as ‘Nestflüchter’, such as the pig, beast, horse, sheep, sea lion and wale) (Portmann, 1969 chapter II). In order to substantiate this classification, Portmann uses a number of different and independent criteria.2 For example, the growth of the brain is a very important element in the development of an individual. By comparing the rates of increase present in the mammals (with a factor of 5 functioning as the dividing line), Portmann shows that ‘Nesthocker’ and ‘Nestflüchter’ are uniquely differentiated – the latter group showing a factor greater than 5 and the former showing one less than 5 (cf. 1969:50)! The obvious question is: are human beings ‘Nesthocker’ or are they ‘Nestflüchter’? Human beings are born helpless, unable to care for themselves and unable to move like a young or mature human being. Just as the ‘Nesthocker’ it is necessary to care for them. Is this condition sufficient to classify them as belonging to the ‘Nesthocker’? No, the answer must be negative, because human beings are born with something typical of the ‘Nestflüchter’, viz. open eyes and an open auditory canal! But then, surely we have to classify them as ‘Nestflüchter’. However, nor that will do. In sharp contrast to the ‘Nest1 2

Haeckel's theory of recapitulation, stating that ontogeny is a succinct recapitulation of phylogeny, is shown to be scientifically completely unjustifiable (cf. Overhage, 1959c). As a result of solely concentrating on the head (partly caused by the fact that, in fossil findings, the skull is easier accessible), Bolk introduced his mentioned fetalization theory, postulating the persistence of pre-natal features in the human form. Portmann remarks that as soon as the bodily proportions of animals are taken into account, these assumptions are contradicted by the facts (1969:46).

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flüchter’, whose bodily proportions are similar to those of the full-grown members of their respective species, at birth human beings are completely ‘disproportionate’ in comparison with their mature state, once again shifting them back to the already unsuitable category of the ‘Nesthocker’! Consequently, the only reasonably justified conclusion, based on the defining features of these two categories of mammals, is that human beings belong to neither of them! Furthermore, at birth the mass of the human brain exceeds that of the anthropoids at least 2-times (about 370 gram, compared to the 150 gram of the orangutan or the chimpanzee brain). In comparison with the anthropoids, human beings are born too early – almost a year to soon, because only after an age of one year human beings reach a development similar to that of the typical mammals at their birth. This means that, for a genuine human-like mammal, i.e. “for a true animal-human or human-animal”, an extra year is needed (Portmann, 1969:58). Portmann is convinced that this “too-early-birth-stage” (physiologischen Frühgeburt) can only be understood in terms of a broader perspective. He speaks about an “extra uterine” period/year in the biotic development of human beings (1969:87 ff.). During the second part of the first year the typical human features of the developing baby start emerging, such as the erect posture, analytical insight, language use, free decisions, and son on. All these activities are developing within the cultural milieu of human society. The fact that these phenomena occur within the first post-natal year, corresponding with the period in which the higher mammals are still in the womb, shows that the “extra uterine year” of the human being is destined to be taken up in the typical human way of transferring culture. The retardation of the growth rate from the 2nd to the 9th year also reveals a directedness towards the complicated processes of learning and appropriation which enable a person to master the vast cultural legacy present in the society within which he is growing up. Corresponding to the relatively long youth period, the human being experiences also a relatively long period of maturity, providing time to transfer, to the next generation, the heritage of successive generations by means of educational processes and institutions. By and large, all these perspectives simply add weight to the conclusion that, in spite of the fact that we have to acknowledge that both human beings and animals function within the biotical aspect of reality, we must constantly keep in mind that our human biotical functioning is totally unique, clearly shown in his exceptional ontogenetic growth pattern. Portmann emphatically declares: “In strict correlation to the measure in which we see our form of existence more clearly, we become more certain that the question about the origin of humankind, as well as the equally difficult question about the rise of the big ‘form-spheres’ (Gestaltenkreise) of the living, are not answerable with the research tools at our disposal” (1969:163). 143

Human freedom – the predominantly negative approach of modern philosophy As is shown extensively and convincingly in the writings of Dooyeweerd, modern philosophy received its ultimate directional impulse from the dialectical opposition which prevailed between the ideal of an all-encompassing causal (natural scientific) explanation on the one hand, and the ideal of the human being as an autonomously free personality on the other hand – i.e., the motive of nature and freedom (cf. 1997-I:207 ff., 216 ff.). Autonomous freedom versus natural causality At the birth of modern philosophy, during and after the Renaissance, a new ideal of an autonomously free personality came to the fore, although its first aim was to master nature rationally with the aid of the newly developing natural sciences. Descartes’ emphasis on the maxim that our ideas should be clear and distinct (considering clearness to be more fundamental than distinctness – Principles, XLVI), is orientated towards mathematics as model of thought. Even the certainty that God exists is only accomplished by clear and distinct understanding – showing – in the final analysis, that he uses the idea of God in order to furnish his deified mathematical thought with the feature of certainty, thus stamping the infallibility of the new mathematical method of analysis. Having mentioned Galilei’s mathematization of nature and modern physicalistic rationalism, Edmund Husserl characterizes this new phase in modern philosophy as having given birth to a rationalistic ideal of science (rationalistischen Wissenschaftsideal – 1954:119). However, the modern freedom motive which, almost with an inner necessity, gave birth to the domination motive in the science-ideal (nature motive), finally came into conflict with itself. If the whole of reality, by means of “reconstructing creative thought” could be framed in terms of exact and inexorable natural laws of cause and effect (universal determinism), it stands to reason that the freedom of the supposedly autonomous personality is reduced to, and determined by, invariable causal laws of nature without any freedom at all! The science-ideal turned out to be a real Frankenstein – demonstrating the inherent dialectic between the freedom-pole and the nature-pole in modern philosophy. The subtle but basic distinction between ‘Erscheinung’ (appearance/phenomenon) and “Ding an sich” (thing in itself), which Kant uses, is completely in service of his fundamental aim to safeguard a separate (and super-sensory) realm for being human as an autonomous ethical being (Zelbstzweck). The category of cause and effect (together with all the other categories) is only applicable to appearances and not to things in themselves (such as the free will of the human soul). Kant realizes that an unlimited employment of the category of causality (understood in the deterministic and mechanistic sense of classical physics) inevitably implies the abolition of all freedom. Kant explains his basic problem as follows: 144

“Now let us suppose that the distinction, which our Critique has shown to be necessary, between things as objects of experience and those same things as things in themselves, had not been made. In that case all things in general, as far as they are efficient causes, would be determined by the principle of causality, and consequently by the mechanism of nature. I could not, therefore, without palpable contradiction, say of one and the same being, for instance the human soul, that its will is free and yet is subjected to natural necessity, that is, not free. For I have taken the soul in both propositions in one and the same sense, namely as a thing in general, that is, as a thing in itself; and save by means of a preceding critique, could not have done otherwise. But if our Critique is not in error in teaching that the object is to be taken in a twofold sense, namely as appearance and as thing in itself; if the deduction of the concepts of understanding is valid, and the principle of causality therefore applies only to things taken in the former sense, namely, in so far as they are objects of experience – these same objects, taken in the other sense, not being subject to the principle – then there is no contradiction in supposing that one and the same will is, in the appearance, that is, in its visible acts, necessarily subject to the law of nature, and so far not free, while yet, as belonging to a thing in itself, it is not subject to that law, and is therefore free” (1967-B:XVII-XVIII).

It is clear that Kant’s ultimate concern to safeguard the (autonomous) freedom of human beings necessitated this distinction between appearance and thing in itself. This is most evident from the entire Transcendental Dialectic. In his discussion of the solution of the third cosmological idea he once more explains that we are not allowed to ascribe any absolute reality to appearances: “The common but fallacious presupposition of the absolute reality of appearances here manifests its injurious influence, to the confounding of reason. For if appearances are things in themselves, freedom cannot be upheld” (1967-B: 564). The final remark in this subsection reveals the basic motive of Kant’s whole Critique of Pure Reason (1967-B:565): “My purpose has only been to point out that since the thorough-going connection of all appearances, in a context of nature, is an inexorable law, the inevitable consequence of obstinately insisting on the reality of appearances is to destroy all freedom. Those who thus follow the common view have never been able to reconcile nature and freedom” (I am emphasizing – DS).

Bridging the abyss teleologically This inherent dialectic, enclosed in the basic motive of nature and freedom, already in his Critique of Pure Reason brought Kant to a negative interpretation of human freedom: freedom is seen as being free from natural necessity (1967-B:651-652). In his Critique of Judgement Kant develops a most influential formulation of the way in which nature and freedom presupposes each other dialectically. Although the human understanding a priori applies the category of causality, as an inexorable law, to nature, Kant approaches organic nature teleologically. It means that nature is thus represented as if the multiplicity of laws present in it is contained in the unifying basis of an understanding (1968-B:VIII). The concept of a natural teleology is proposed by the capacity to judge, in order to function as a mediating concept between the concepts of nature and the concepts of freedom. However, the purposiveness of 145

nature only functions as a regulative principle to the (reflecting) capacity to judge (1968-B:LVI). As guiding principle, this natural purposiveness is never to be used in a constitutive way, since then our reflecting ability becomes a determining faculty of judgement, implying that once again we are introducing a new causality (a final cause; nexus finalis; cf. 1968-B:269) into natural science (1968-B:270).1 The teleological principle merely functions as a subjective maxim in judging nature. Therefore, it cannot be applied to the objective reality of things in nature. Consequently, the reconciliation between the causally determining and the teleologically reflecting view of nature is sought in the unity of a supra-sensory principle which is supposed to be valid for the totality of nature as a system (1968-B:304). This ‘solution’ did not really reconcile the opposing poles of nature and freedom, since it simply reinforces the basic dualism between natural necessity and super-sensory freedom – each with its own law-giver (cf. 1968-B:LIII-LIV). Fr. Schelling attempted a synthesis between nature and freedom. According to him, in the absence of the antinomy (Widerspruch) between necessity and freedom, not only philosophy, but also every higher will of the spirit will shrink into insignificance (1968:282). As a result of this commitment he believes that in nature itself a principle of freedom is concealed, while history is founded on a hidden principle of necessity. Clearly, the result is not a real synthesis or reconciliation, since it amounts to nothing but a duplication of the original dialectic: necessity is present in the domain of freedom, and freedom is present in the domain of necessity! Entelechie negatively described: the influence of Hans Driesch Without rejecting the classical mechanistic analysis of matter, Driesch, in his neo-vitalistic biology, extends the application of the deterministic concept of law to biotic phenomena. The traditional mechanistic approach is limited by him to the material basis of living things. We have seen that Driesch interpreted the regenerative phenomena discernable in living things in terms of his theory of living entities as “equi-potential harmonious systems” and in terms of his notion of an entelechie operating as a “totality-causal factor” (Ganzheitskausalität). The important contribution which Driesch made to the problem of freedom, is given in his notion of the ‘entelechie’ as something that cannot be determined in any positive sense. As such, he considers it to be a “system of negations” (1920:513; 459 ff.), i.e., it cannot be positively determined: ‘entelechie’ is something non-mechanical, it is not energy, not force, not a constant (1920:460) and non-spatial (1920:513). The difference between the atomistic ‘Einselkausalität’ and the holistic ‘Ganzheitskausalität’ is also framed in terms of the opposition ‘Ganzheit’ and ‘Zufall’ (totality and chance). In the thought of Driesch determination is opposed to genuine freedom. He declares that the question about freedom is to be considered as a metaphysical question of faith which cannot be answered by the science of 1

Excatly this was done in the neo-vitalistic biology of Hans Driesch. Cf. his notion of ‘Ganzheitskausalität’ (1920:416 ff., 542 ff.).

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philosophy (cf. 1931:93-122). Although Kant and Driesch differ in their view on the nature of philosophy, they agree that freedom is not a question of scientific proof, but one of (practical) faith. In his theory of the freedom of the will, Arnold Gehlen continues Driesch’s negative description of the ‘entelechie’. However, with an explicit appeal to the freedom idealism of Schelling, he immediately transforms it in order to provide a point of entry for freedom. At the same time he realizes that Driesch actually brought biotical phenomena under the reign of the deterministic classical ideal of science. Therefore, once again he wants to restrict causality to mechanical causality: “Since causality is only thinkable as mechanical causality, the entelechie is negatively free, i.e. spontaneous and primary in a sense which cannot be subjected to a closer determination” (1965:60). The tension between nature and freedom brought Max Scheler to his well-known characterization in terms of what he calls the ‘Weltoffenheit’ of human beings (1962:38, 40).1 Against this background Plessner developed his own perspective on the human being as an eccentric creature, while biologists and anthropologists such as Portmann, Overhage and Gehlen gave the notion of ‘Weltoffenheit’ a prominent place in their writings. Even theology took advantage of this notion. Wolfhart Pannenberg, for example, interprets it in terms of what he calls “der grenzenlosen Angewiesenheit des Menschen” (the unlimited dependency of the human being) while relating it to the “fundamental biological structure of being human” (1968:11; cf. also Scherer’s treatment of the ‘Weltoffenheit’ of the human being, 1980:79 ff.). Ultimately, this term ‘Weltoffenheit’ is used to embody the reaction against the claims of the science ideal, namely that the human being is determined in all respects. In the final analysis, the intention of these authors is to show that the human being is free from being determined by natural causality. In his Ph.D-thesis, dealing with philosophical aspects in the biology of Portmann, R. Kugler states that Portmann essentially understands the human being in terms of freedom (1967:75). At the same time, Portmann is well aware of the fact that, as a “philosophical idea”, freedom withdraws itself from a scientific grasp. Kugler places this approach within the “large tradition” of a “philosophical determination of the human being,” dating back to Immanuel Kant: “The innermost essence of the human being is freedom, it is the possibility of the human being to transform itself into that what it is” (1967:81). Compare this announcement with the following words of Plessner: “As eccentrically organized creature the human being must make itself into that what it already is” (1965:309). Gehlen points out that this mode of expression manifests the logical scheme present in a normal teleology. This tradition is influenced by Fichte: “I want to be free ... means: I want to make myself into that what I shall be before I am it, in order to be able to perform it” (cf. Gehlen, 1965:103-104). And we have seen that Fichte himself is dependent on Kant, who introduced teleology as a 1

In this work, Scheler sketches absolute being as an endless, reciprocal interpenetration of spirit (Geist) and drive (Drang) – the former has to guide and direct the latter, but only receives its power from this equally original life-drive.

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bridge to human freedom. The philosophical tradition in which “mechanical causality” and “teleology” (nature and freedom) is always dialectically related, inspires Ed. von Hartmann to remind natural scientists in the following way: “If our natural scientists were philosophically better trained, they would have been aware of the fact that the whole German speculation, from Leibniz to Kant and up to the present, equally decisively rejects a teleology separated from mechanical causality, as it does with a mechanical causality divorced from teleology” (quoted by Haas, 1959:456). Reinforced dialectics: Existentialism and Existential Phenomenology Notwithstanding the fact that various philosophical trends of the 20th century departed from the rationalistic philosophy of Kant, the underlying motivating power present in the “leitmotif” of nature and freedom remained in force. The existential phenomenological thinker, Merleau-Ponty, for a great part relying on the results of psychological and psycho-patological studies, understands the human being dialectically in terms of two basic denominators: being a body (taken in a biotical sense as an organism) and existence (interpreted as being historical in nature). On the one hand, together with Sartre, he accepts the thesis: “I am my body”. On the other hand, however, he also holds the opinion that one’s historical existence must repress the bodily organism down to the pre-personal level of an anonymous complex. Inspired by the nature-pole of the basic motive (ground-motive) of humanism, Merleau-Ponty writes: “I cannot understand the function of the living body except by enacting it myself, and except in so far as I am a body which rises towards the world” (1970:75). From the opposite motivation he states: “... so it can be said that my organism, as a pre-personal cleaving to the general form of the world, as an anonymous and general existence, plays, beneath my personal life, the part of an inborn complex” (1970:84). On the one hand I am my body, and on the other hand my body is seen as a pre-reflexive, pre-personal, anonymous complex by virtue of its being-in-the-world (1970:79, 80, 82, 83, 86). Nature and freedom reciprocally endanger and presuppose each other: “... for most of the time personal existence represses the organism without being able either to go beyond it or to renounce itself; without, in other words, being able either to reduce the organism to its existential self, or itself to the organism” (1970:84). The dialectical movement, to and fro, between these poles is best illustrated in his following words: “Man taken as a concrete being is not a psyche joined to an organism, but the movement to and fro of existence which at one time allows itself to take corporeal form and at others moves towards personal acts (I am emphasizing – DS)” (1970:88). Perhaps Karl Jaspers saw the impasse of this whole dialectical legacy most clearly. His confession reads: “Since freedom is only through and against nature, as freedom it must fail. Freedom is only when nature is” (1948:871). Freedom at the molecular level Sometimes it is striking to see what the effect is of the presumed continuous and uninterrupted line of ascent from molecules to the human being. Clearly, if one wants to ascribe freedom to the human being, the continuity of the pos148

tulated genetic process demands that nothing truly novel can arise somewhere on the line. Consequently, Hans Jonas, proceeding from the primacy of the freedom motive, is ‘forced’ to recognize freedom already at the molecular level! “Our position is indeed that already metabolism, the bottom layer of all organic existence, reveals freedom, yes, that, in itself, it is the first form of freedom” (1973:13). ‘Life’, according to Jonas, “constantly manifests itself in the dialectical antitheses between which its existence is stretched: the antithesis between being and non-being, between self and world, form and matter, freedom and necessity (I am emphasizing – DS)” (1973:15-16). Bernard Rensch is committed to exactly the opposite conclusion, although he shares the conviction about the continuity of the line from molecules to the human being: “According to our previous findings and discussions we are justified in assuming .... psychic (parallel) processes of some kind in all living beings” (1959:352). This ‘psychic’ continuity must also bridge the gap between the living and the non-living: “Here again it is difficult to assume a sudden origin of first psychic elements. It would not be impossible to ascribe ‘psychic’ components to the realm of inorganic systems also, i.e. to credit nonliving matter with some basic and isolated kind of ‘parallel’ processes” (1959:342). Thus matter receives a “proto-psychical nature” (1969:134-135). And since the universe is ruled by eternal basic laws, Rensch cannot accept any freedom of the will: “If ‘free will’ really existed it would have emerged in the head of the human being, thereby disrupting the causal law which governs the processes of the brain” (1971:211). The rejection of structural conditions: nominalism It is tremendously difficult for modern philosophy to accept constant and universal conditions underlying our human freedom. As a result of the overwhelming influence of modern nominalism, the universal creational order for and the (universal) orderliness of entities subject to the former conditions, are mostly rejected. Some of the most prominent trends in modern philosophy stress the ever-changing and contingent nature of the world in which we live. Rauche, for example, is convinced that a person’s “basic contingent experience of the world”, its being interwoven with life’s “changing conditions”, implies that the theories a person advocates “can never be conclusive”, and consequently, they “should be regarded as blueprints which attempt to give” one “meaningful guidance in the permanent flux of becoming” that one finds oneself “in and of which they are an integral part” (1985:11; cf.20, 75, 87, 96, 137). It is not necessary to relate the idea of constancy to ‘God’, ‘Being’ or a metaphysical ‘Absolute’ (cf. Rauche, 1985:12), since the first appeal should be to the conditioning order for creaturely existence. During a visit to South Africa Richard Rorty was even announced as a specialist on contingency! I do not want to deny the uniqueness, individuality and contingency indeed present in our experience of the world. However, I do think we have to be very cautious in order to avoid the pitfalls of modern nominalism. In this context, it will suffice to point out that all contingency and changes can only take place within the boundaries of structural conditions which are not only universal, but which are also constant. In another context, I 149

have analyzed some aspects of this problem (cf. Strauss, 1985:133 ff, 138 ff.; cf. also Strauss, 1984:36-37 and Hart, 1984:65 ff.). The fundamental point in connection with the problem of human freedom is to realize that human freedom should be evaluated in terms of the strict correlation between universal normative conditions and subjective response of humankind to them. Hart is fully justified in saying that he “will defend the view that being free is not opposed to being determined” since “only what is determined can be free and only what is free can be determined” (1984:298). Here ‘determined’ can mean nothing but “being subject to a universal conditioning order”. However, the ‘order-diversity’ in creation confronts us with differently structured subjective responses by creatures, and it is in terms of this perspective that we see human freedom as an outcome of the unique accountable ability of human beings to respond (‘response-ability’). What is unique about human beings is not that they are free from conditions, but that they in their subjection to them, actually are free to obey them in uniquely varying ways and even, ultimately as an effect of sin, has the temptation to disobey them. The history of arguing for a special place of human beings on earth is often dialectically motivated by the urge to see the human being, not as something conditioned, but, in opposition to nature, as the (autonomous) origin of conditions (cf. Hart, 1985:295). The notion of a uniformly moving body, underlying Galilei’s law of inertia, was formulated by him in terms of a thought-experiment, without taking account of any real sense-experience. This inspired Kant’s whole epistemology (cf. Holz, 1975:345-358). Von Weiszäcker frames Kant’s epistemological problem in terms of the question: What is nature, that it must obey laws which a human being could formulate with his/her understanding (1971:128)? Kant implicitly interprets the Galilean procedure as follows: Since the law of inertia is derived and prescribed to moving entities out of the pure understanding of the human being in its spontaneous subjectivity, Kant brings about the (historically crucial) Copernican turn in epistemology, in ascribing the primary no longer to the object, but to the subject. Kant draws the radical (rationalistic) humanistic conclusion – the laws of nature are a priori contained in the subjective understanding of the human being: “the categories are conditions of the possibility of experience, and are therefore valid a priori for all objects of experience” (1967-B:161); “Categories are concepts which prescribe laws a priori to appearances, and therefore to nature, the sum of all appearances” (B:163). This rationalistic inclination of Kant was eventually historicized. The manner in which Rauche frames the problem of truth clearly portrays Kantian undertones: “Truth is a matter of the mind. It is the translation of our sense-experience into rational terms or concepts, while the real is perceived through the senses and is yet still chaotic and unorganized by the mind” (1971:9). He no longer accepts the Kantian notion of understanding which is capable of a universally valid act of form-giving (ordering) – every person can only account for his/her own particular constituting activity. Our relative human perspectives are rooted in a finite and contingent world verified by our experience. 150

“It is the world of becoming and change, namely, the ever changing concrete objects, which represent the environment of human beings, causing them to feel uncertain and insecure, so that they are impelled to order it rationally. It is thus not a vague abstract ”something“ toward which we direct our intent, but it is a concrete situation that causes us to build our world, which in its stage of constitution must be peculiar, different and in this sense contradictory to the constituted world of our fellow human beings” (Rauche, 1966:99).

The Kantian notion of universal validity is fundamentally historicized – leaving us with nothing but competing and contradictory ordered worlds of different people in different historical situations (cf. Rauche, 1971:34). Although the rationalistic and irrationalistic trends in modern philosophy seem to divert radically, their common root in nominalism transcends this superficial divergence. Rationalism considers universals to be the only source of knowledge, thus leaving no room for knowledge of things in their individuality. Surely, concept-formation is always bound up with the universal order for, and the universal orderliness of things. This implies, as already discovered by Aristotle, that one cannot conceptually comprehend the individual side of an entity. Unfortunately, in a typical rationalistic way, he identifies knowledge with conceptual knowledge, implying that something individual cannot be known (cf. Metaf. l040 a 5 ff.). Contrary to this rationalistic position, we must emphasize that in fact we do have knowledge of things in their individuality, although this kind of knowledge is not conceptual. Much rather, it is of a concept transcending and approximating nature, referring to the individual side of things in terms of universal features. But this is precisely what idea-knowledge is all about – an idea concentrates a conceptual diversity upon (resp. refers it to) that which transcends the limits of all concept-formation. Therefore, rationalism leaves no room for idea-knowledge. Irrationalism, on the other hand, always wants to emphasize the contingent uniqueness of the individual side of entities or events transcending the limits of concept-formation. Consequently, irrationalism leaves no room for real conceptual knowledge. In respect of the typical structure of entities, nominalism does not accept any conditioning order (universal structures for), or any orderliness (universal structuredness of) such entities. Every entity is strictly individual. In terms of our distinction between rationalism and irrationalism, nominalism surely represents an irrationalistic view in connection with the nature of entities, since every individual entity is completely stripped from its universal orderliness (law-conformity) and conditioning order. This characteristic applies to both moderate nominalism, viz. conceptualism (Locke, Ockham, Leibniz and others), and to extreme nominalism, that rejects all general and abstract ideas and accepts only general names (Berkeley and Brentano). This irrationalistic side of nominalism, however, does not exhaust the multifaceted nature of nominalism, because universals are fully acknowledged in the human mind, at least as general words in the case of Berkeley’s and Brentano’s extreme nominalism. This restriction of knowledge to universals is typical of rationalism in the sense defined by us. Therefore, it is possible to see nominalism as being simultaneously rationalistic (in terms of the universals – concepts and words – 151

in one’s mind), and irrationalistic (in terms of the strict individuality of entities). The common root of diverging trends in modern philosophy This dual nature of nominalism forms the starting-point of two diverging philosophical developments in modern philosophy. (i)

On the one hand, it provided rationalism with the possibility to elevate human reason to the level of the creator of a rational order in reality. This follows from the fact that nominalism in fact transposes the universal side of entities into the human mind. But the universal side of entities is nothing but the manifestation of the conditionedness of entities by the relevant universal order for their existence. Consequently, if an entity is stripped of its orderliness (its universal side), it is simultaneously stripped of its being subjected to a universal creational order. What is left is factual reality in its unstructured, chaotic individuality and particularity (contingency) (cf. Rauche, 1966:97). Driven by the new motive of logical creation, this very feature of nominalism enabled modern philosophy from Descartes onwards to reconstruct all of reality in terms of natural scientific thought. Only the extreme consequences of this natural science-ideal, cancelling in principle also human freedom, were questioned by Kant. Within the (limited) domain of the science-ideal, however, Kant draws the ultimate rationalistic conclusion of nominalism. Indeed, Kant tries to consolidate and strengthen the preceding natural science-ideal, be it in the restricted form of the rationalistically elevated understanding which (though limited to sensibility in order to save a separate super-sensory domain for the practical-ethical freedom of autonomous humanity), is considered to be the a priori (formal) law-giver of nature! Nominalism created a vacuum by leaving factual reality in its individuality unstructured. In order to fill up the lack of determination thus created, Kant introduces human understanding to take hold of this vacant position. To be sure, Kant not merely transposes the universal side of entities into human understanding, since in fact he elevates human understanding to the level of the conditioning order for things. (ii) On the other hand, nominalism provided a starting-point for all those trends in modern philosophy which, in an irrationalistic fashion, want to take the unique and contingent character of (mostly designated as: historical) reality serious. This avenue opened up by nominalism was followed up by a variety of historicistic designs in modern philosophy, for example from the forth phase of Fichte’s thought up to pragmatism, existentialism and contemporary neo-Marxism. If reality is tripped both of its orderliness and of its being subjected to a conditioning universal creational order, it seems to be a “self-evident historicistic truth” that, ultimately, everything is historical and therefore taken up in the dynamic and ever-changing contingent flow of historical events. 152

However, this does not necessarily mean that the ordering function of understanding is cancelled, as is evident in Rauche’s conception of the human task of self-constitution in order to transcend pure contingency. At this point we can link up the influence of nominalism with the predominant neo-Darwinistic evolutionism. The remark of Simpson, referred to in an earlier context, namely that plants and animals are not types and do not have types, since everyone of them is unique (1969:8-9), is a fully-fledged nominalistic conviction. The genesis of plants, animals and human beings are taken up in a structureless continuum. Systematic distinctions, exemplified in different taxonomies, are nothing but arbitrary names (nomina) given to an immense number of individually different living entities. The universality implied in these names is a product of our constitutive human understanding without any foundation in the “things outside the mind”. Already Charles Darwin adhered explicitly to this view in his “Origin of Species”. He says “that no line of demarcation can be drawn between species” (1968:443) and proceeds: “In short, we shall have to treat species in the same manner as those naturalists treat genera, who admit that genera are merely artificial combinations made for convenience” (1968:456). Within the context of an evolutionary epistemology Van Huyssteen recently demonstrates a strange mixture of different positions at once. He is in search after a new (interdisciplinary) space for the interaction between theology and science. He proceeds from the assumption that evolution is a fact (1998:143) and that we have to “take very seriously the general conclusions and findings of general cosmology ” – “that is that this universe is evolving, that all that is within it has had a common physical origin in time, and that all it contains is in principle explicable by the natural sciences” (1998:75). However, he does not escape from fundamental ambiguities. The neo-Darwinian presupposition of continuity (cf. 1998:111) and chance are slowly but surely substituted with a mixture of emergent evolutionistic (cf. 1998:134, 151) and vitalistic (cf. 1998:37, 121, 125, 127) overtones – without evincing an awareness that these positions are alternative to neo-Darwinian theory and that they contradict its basic assumptions.1 An amazing return to the rationalistic position of Kant and modernity is seen in his identification of the structure of the universe with human rationality and mathematics: “What is astounding, however, is to what extent our world is truly rational, i.e., in conformity with human reason” (1998:68). While mentioning Davies he refers to the “fact that the rational nature of our universe is reflected in its basic mathematical structure” (1998:71). Van Huyssteen and the modernist (rationalistic) tradition on this point do not distinguish between 1

The following statements show his emergent evolutionistic position, continuity in ascent, discontinuity in existence: “Culture indeed has evolved, but the principles of culture are not the same as the principles we know from organic evolution” (1998:146); and in affirming the approach of Wuketits he says: “culture is not reducible to biological entities” (1998:157). On page 130 he explicitly employs the phrase “emergent evolution.”

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ontically given universal features of reality and the nature of concept-formation. From the fact that concepts are formed on the basis of universal traits it does not follow at all that these ontic porperties themselves are rational in nature! This position is taken while at the same time an equally forceful attempt is made throughout the work to hold on to a postmodern perspective! Human freedom: a subjective response to normative conditions What is at stake here, is a confrontation with historicism on the basis of accepting norms or principles in the following sense: a principle or norm is a universal and constant unit that can only be made valid (enforced) in different situations by a competent organ possessing an accountable will which provides the freedom of choice to establish a normatively correct or antinormative positivization (form-giving) of the possibilities contained in such a starting-point. Only a positivized principle possesses validity. It is therefore contrary to the very nature of a pre-positive principle, providing the starting-point for form-giving activities in all different situations, to characterize such a pre-positive starting-point as universally valid. The validity of any positivized principle is fundamentally restricted to the unique setting of a specific place at a particular time. Consequently, the natural law view is untenable, since it ascribes a validity to norms that hold for all times and places. Hart uses the example of expressing respect, which was instantiated in greeting rituals of various different kinds – from taking off one’s hat up to simply raising the hand. In spite of all that varies, he says, “something ‘in principle’ remains invariant through all this historical development” (1984:59), viz. showing respect. This principle should not be tied down to only one kind of response (lifting the hat or raising the hand): “The legalist who claims that those who just tip their hats are in principle not showing the proper respect is making the same mistake as the nominalist: he is failing to distinguish underlying principles in their invariance from the observable patterns of variant behavior” (1984:59). It is a pity that Kugel, who explicitly follows Groenman’s reformational model of being human (cf. 1982:135), only acknowledges four types of norms – viz. the economical, the juridical, the ethical, and the aesthetical (1982:280-283). It is not even enough to refer to the normativity of all the post-sensitive aspects, since, at the norm-side of each one of these modalities, every retrocipation and anticipation unveils a fundamental modal norm (I have treated this perspective elsewhere in more detail – Strauss, 1979: 254-264). The possibilities contained in any universal and constant starting-point function as the basis for specific acts of form-giving (positivization) in diverging unique historical situations. The rationalistic trait of natural law conceptions cannot account for this freedom to positivize contained in a principle. The irrationalistic nature of historicism, on the other hand, cannot do justice to the universality and constancy of such a starting-point which actually form the basis of ever-changing positivizations. This one-sidedness of both natural law and historicism is a direct consequence of the autonomy-theme in modern phi154

losophy mentioned earlier. The autonomy-ideal hypostatized the freedom to positivize – thus trying to eliminate the very nature of a principle as a universal and constant starting-point for human action. When positivizations are elevated to the level of being universally valid, we encounter rationalistic casuistry. And when the freedom to positivize is one-sidedly accentuated, we encounter an irrationalistic situational ethics. The way in which the majority of contemporary social scientists use terms like values, norms, beliefs (cf. Sorokin, Parsons, Znaniecki, and others), sometimes called the cultural system, does not allow for principles as universal and constant starting-points that ultimately condition human action in a task-setting way, since they identify these terms with the result of free and formative human actions – typical of historicism. The long-standing influence of nominalism in our modern Western culture has ultimately succeeded in ruling out the biblical view on the creational order for the existence of creaturely subjects. The relativistic and self-contradictory nature of historicism is simply a symptom of the contemporary world view. Any confrontation with historicism that does not penetrate into this pre-scientific root has not succeeded in unveiling its deepest motivation and impasse. What, in the final analysis, is therefore ultimately decisive, is the basic (pretheoretic) commitment to the modern historicistic world view with its autonomy-ideal1 and nominalistic theoretical articulations on the one hand, or, on the other hand, the commitment to a different world- and life-view, namely that of biblical Christianity, which does allow for the acceptance of universal and constant principles which (as creational order for) condition human subjectivity in a truly normative way and at the same time leave humanity with the accountable authentic freedom to positivize responsibly in changing historical situations. The order of creation indeed shows us the good direction towards obedience to the will of God. However, due to the radical nature of the fall into sin, this God-obedient direction was redirected in service of some or other idol borne from the apostate heart of humankind. The creational order still exercises its normative appeal to obey the will of God, but in order to accomplish this we must be freed from the effects of sin by the redemptive work of Christ. Only in Him and through the work of the Holy Spirit are we, in principle, freed from the apostate inclination of our hearts and redirected towards obedient service to God within the world-wide, all-encompassing Kingdom of God in Christ. Obedience to God-given creational possibilities is a positive task, not something structurally negative which we have to transcend. 1

The definition which Rousseau gives for freedom, demonstrates the humanistic ideal of self-determination (autonomy) explicitly: “obedience to a law which we prescribe to ourselves is liberty” (1966:16).

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Conclusion The human being is not simply an extension of the animal realm. By means of the preceding exposition, this conviction is substantiated with the aid of various arguments and in terms of diverse perspectives. The crucial ‘turning-points’ in the (neo-Darwinistic) account of the all-encompassing process of evolution, namely the origin of ‘life’ and the emergence of human beings, in a marked way highlight the inadequacy of this predominant mode of thinking. First of all we discerned a difference of opinion in connection with the notions of continuity and discontinuity. Secondly, modern biological thinking tries to subsume this problem of continuity/discontinuity under different basic denominators, such as the mechanical (Eisenstein), the physical (neo-Darwinism), the biotical – in different ways (neo-vitalism, holism, organismic biology), the psychical (be it monistic: Teilhard de Chardin – or pluralistic: Bernard Rensch’s pan-psychistic identism), while even freedom is chosen (Hans Jonas). Sometimes, the obvious structural discontinuities discernable between material things, plants, animals and human beings, caused an ambivalent (emergentistic) attitude, trying to have it both ways: genetic continuity and existential discontinuity (Lloyd Morgan, Whitehead, Woltereck, Bavink, Polanyi, Laszlo, Dobzhansky and even certain statements of Simpson and Julian Huxley). None of the treated scholars, however, asked the question why, in spite of the supposed continuous (and: structureless) change constantly occurring, the theories about these evolutionary changes remain confined within the mentioned modal diversity. Much rather, this state of affairs confronts modern biological thought with the inescapable conditioning role of this modal diversity for theorizing as such. The dialectical escape route which these trends try to pursue is to ignore this given diversity by arguing as if it does not exist. Contrary to this intention, nevertheless, every theory presented to us simply came up with an over-estimation of one of these modal perspectives, not realizing that it is only while working and thinking within the conditioning ‘order-diversity’ of reality, that we even can attempt ignoring this constant ‘order-diversity’! Ultimately, the choice of any specific basic denominator is fully in the grip of the underlying basic commitment of the thinker in question – a commitment transcending the realm of theoretical thinking as such. Uncertainties and even contradicting interpretations of these fundamental questions warn us to be modest in our often premature conclusions. Much rather, it is important to emphasize what one can know about the uniqueness of being human – both in terms of the exceptional ability to respond in normative freedom and in respect of the way in which human beings function distinctly in those aspects of reality which they share with other creatures. We have seen that the prevalent dialectical appreciation of the meaning of human freedom is the outcome of an underlying motive power operative in this philosophical legacy, which is, ultimately, apostate in character. The unsolvable tension between the poles of nature and freedom inevitably leads to a neg156

ative idea of freedom, dialectically opposed to nature. Furthermore, our human ‘natural’ features, such as the bodily configuration of the human being, our unique biotical developmental status, and the relatively unspecialized organs, are, together with the erect gait and spiritual expression of the face, all in service of our normatively qualified truly human responsibility to obey the universal conditions of God’s creational order. Though, in the present sinful dispensation, we shall always be tempted to disobey these normative conditions, in Christ we are in principle saved from this sinful inclination and freed to constantly act in more norm-conforming ways, showing, in anticipation to God’s coming Kingdom, that already now we share in the restored paradise order of obedience and peace.

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Index of Technical Terms A Adenine 99 adenosine triphosphate 101 aestheticism 15 albumin 98 algebra 14 amino acids 99 amphibians 94-95, 138 Anaxagoras 22, 25 Anaximander 21, 31 Anaximines 21 Angelelli 20 anthropoids 117, 128-130, 134, 137-138, 140, 143 anthropomorphic 132 anticipations 77 antinomy 102, 108, 147 antonymity 141 Archaeopteryx 96, 103 articulare-quadratum-joint 95 astronomy 73 at once infinite 58-59 atomic nuclei 78, 80-81 atomic number 69, 75, 80 atomistic 77, 81, 147 Australopithecines 120-122, 135 Australopithecus 121, 163, 165, 169 Australopithecus africanus 120, 165

B bacteria 88, 93, 100-102 biochemical constellation 114 biological – disorganization 107 – theory 94 bio-milieu 85 biotical instability 86 biotope 108 blue-green algae 100-102 bonding electrons 80

C calcium 98 carbohydrates 98 carbon dioxide 87 cell organism 100, 102, 113-115 cell-structure 89 central instance 112

chemical bonding 78, 80-81 chromosomes 100-101 cineradiographic studies 134 classes of vertebrates 94 Cloadophora 101 cobalt 98 colloidal systems 98 comparative morphology 91 Compton-effect 76 concept formation 130-132, 141 concept-transcending 74, 77 conceptual knowledge 74, 152 consistent physicalism 106 constancy and change 62, 75 context of discovery 6 context of justification 6 Coulomb forces 78 covalent bonding 78 critical rationalism 5 crystal lattice 78 crystalline structure 113-114 Cynodontia 95 cytoplasm 98, 100-101 cytosine 99

D deficient creature 140 dental-squamosum-joint 95 desoxyribonucleic acid 99 Diarthrognathus broomi 95 differential reproduction 88 Difflugiidae 126 disciplinary matrix 8, 56 discontinuity 59, 92, 94-95, 104, 111, 117, 154, 157 divisibility 67, 75, 79 dogmatism 4 double helix structure 99 duration 59

E electrical fields 140 electromagnetic waves 75 electron pair bonding 78 electron-shells 76 elementary basic concepts 77 elementary particles 55, 76 emergence evolutionism 9 enkapsis 79 enkaptic interlacement 81

171

enkaptic interweaving 80 enkaptic structural whole 81, 112-115 Enlightenment 53-54 entelechie 105, 113, 133, 147 entropy 55, 60-61, 67, 85, 107 epistemology 2, 70-71, 151, 154 equilibrium 85-86 erect gait 128, 141, 157 everything is number 9, 55 Evolutionary – continuum 91, 94, 109 – hypothesis 94 – theory 96-97, 104 experience 1-3, 7, 9, 11, 13, 15, 53-54, 70-74, 89, 92, 102, 110, 128, 132, 136-137, 140, 144-145, 150-151 experimental physics 72 experimentation 8, 62, 67, 71, 87 external enkaptic function 80

F factual side 74, 91, 94 faculty of judgement 146 faith in reason 5 falsification 7-8, 54 fetalization 143 fossil record 94, 104 free will 145

G generalizations 6 generatio spontanea 87 genetic systems 93 geno-types 80 gravity 2, 8, 67 Greek philosophy 83, 91 guanine 99

H histone 98 historicism 15, 155-156 holistic biology 107-108 hominids 120, 135 Homo erectus 121, 123 Homo – habilis 121, 123, 125, 160

– neanderthalensis 120 – sapiens 121-124, 135, 140 – symbolicus 132 homologous structures 93 human freedom 110, 142, 146, 148, 150-151, 153, 157 hydrogen 80, 87, 98

I Ichthyostega 94-95 Ictidosauria 95 idealism 15, 147 idealistic morphology 91, 103 individualism 15 individuality 70, 74, 90, 106, 113, 150, 152-153 inertia 2, 8, 60, 62-63, 69, 151 infinite divisibility 79 infinite regression 6 infinitely divisible 66-67, 73, 131 interference phenomena 76 interferometer 76 intermediate forms 96-97 intuitionism 58 iodine 98 iron 57, 61, 85, 98-100, 106-107, 125-126, 133, 136, 151 irrational numbers 55, 66 irrationalism 15, 152 irreversibility 57, 60, 75 irreversible specialization 138 isomeric forms 79

K kinematical time order 59

L larynx 134-135 law-conformity 56, 64, 71, 74, 152 law-side 58-59, 65, 90 lipids 88, 98, 102 logical probability 6

M macro-molecules 56, 74, 86, 99, 112 macro-mutation 103 magnesium 98 magnetic poles 140 mammals 94-96, 134, 136-137, 142-144

manganese 98 mass points 9 materialism 15, 56 mechanical movement 9, 59 metabolism 89, 99, 101, 110, 149 metallic bonding 78 metaphoricity 141 metaphysical 4, 103, 147, 150 metazoa 101 methane 87 micro-organisms 89 micro-structures 68, 77 middle ages 8 missing links 94 mitochondria 101-102 modal abstraction 8, 12-14, 61, 69, 89 modal universality 8, 71-73, 77 molecular biology 87, 114 Monotremata 96 moralism 15 more geometrico 107

N natural numbers 58 natural system 90, 105 neoteny 138-139 Nesthocker 142-143 neutrality 3 neutrino 76 nitrogen 78, 87, 98-99 nominalism 10, 15, 90-91, 102, 112, 150, 152-153, 156 non-contradiction 132 non-decreasing entropy 55, 60-61, 67 nucleo-protein 98 nucleotides 99

O objectification 3, 130 open systems 63, 85-86, 88 orderliness 64, 71, 74, 150, 152-153 organic – chemistry 113-114 – compounds 99 organismic biology 9, 86, 107, 157 Ornithorhynchus anaticus 96 orthogenetic 103

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P Pachygenelus 95 paleo-biology 124 paleontology 94, 96, 104, 123 pan-psychism 9 panpsychistic 109-110 pan-psychistic identism 111, 157 paradigm 5, 7-8, 56, 71 Pauli-exclusion 76 peptide bond 99 perpetual motion 61 persistent themes 8 pharynx cavity 134 philosophical foundations 3 phosphorus 98 phylogenetic trees 108, 122-123 physical entities 55-56, 61-62, 65, 68, 70-71, 75 physical laws 73, 88, 90 physical time order 59 physicalism 9, 15, 105-106 pietism 15 Piltdown hoax 120 platonism 10, 112 platypus 96 Plotinus 27, 31 polinucleotide 99 polypeptide 99 positivism 1, 3-5, 53-56, 71 postulate of continuity 104, 109 potassium 98 pre-natal traits 138 presuppositions 1-2, 15, 89, 91, 104, 111-112, 122-123 primary substance 74 principle of the excluded middle 169 property terms 54 protamine 98 protista 92 protoplasm 98, 125 Prototheria 96 protozoa 92, 101, 126 psychoide 105 psychologism 15 punctuated equilibria 96 Pythagoras 23-24

Q quadrupeds 95 quantum theory 70, 72-73

R radiation 76, 78, 99 radio-activity 55, 60, 78 rationalism 5, 15, 145, 152-153 rationality 5, 53, 154 realism 10, 15 redundance 141 Reformational philosophy 56 regulative principle 146 reification 71 reptiles 94-95, 97 respiratory tract 134-135, 164 retardation 138-139, 144 retrocipations 68, 131 Ribosomes 101

S science-ideal 5, 145, 153 second law of thermodynamics 61 secondary energy-traps 125, 127 self-domestication 139 set theory 14, 58 Seymoria 95-96 sodium 79-80, 98 structural constancy 92 structureless continuum 103

subjectivity 2, 114, 151, 156 successive infinite 57 sulphur 98 supersonic waves 140 survival of the fittest 88 synonymity 141

T theoretical terms 54, 71 theoretical thought 1-2, 9, 14 theory of relativity 9, 62-64, 67, 69 Therapsida 95 thermodynamics 55-56, 60-61, 63-64, 67-68 thought categories 70-71 Thymine 99 time-duration 58 topology 14 totality-structure 73 transcendental-empirical 69, 71 Tritheledontidea 95 type laws 70-71 typical – foundational function 76 – laws 70-72, 109 typogenesis 103 typolysis 103 typostasis 103 typostrophism 103

173

U ultimate commitment 5 ultra-violet rays 140 universal substantial form 74 universalism 15 Universality 10, 74, 168 unspecialized 137-138, 140-141, 157

V vacuoles 100 Van der Waals forces 78 variability types 80 verification principle 3-5 vitalism 9, 15, 83-84, 105, 107-108, 111-112, 118, 157

W wave – function 77 – particle duality 76 wave-character 76 Weltoffenheit 148 whole-parts relationship 100, 115 Wiener Kreis 3

Z zinc 98

174

Index of Persons A Adloff 138 Aguirre 159 Alexandroff 35, 159 Allesch 159 Altner 137, 159, 165 Anaxagoras 22, 25 Anaximander 21, 31 Anaximines 21 Angelelli 20, 159 Apolin 159 Aristotle 21, 34, 36, 45, 50, 66, 68, 74, 79, 105-106, 112, 152 Augustine 27, 31, 57 Azar 117, 159

B Bar-Hillel 45, 161 Bartle 35, 159 Bavink 157, 159 Becker 21, 50, 159 Bell 29, 41, 159 Bendall 159, 161, 163 Bernays 41, 45, 49, 112, 159 Beth 43, 46, 159 Boethius 27 Bohr 55, 65, 75-76, 113, 159 Bolk 143, 159 Bolzano 27, 32, 34-35, 160 Born 160 Bos 25, 160, 162, 165, 167 Boyer 24, 31, 160 Bricmont (vi), 168 Bromage 121, 160 Brouwer 43, 46-47, 49, 66, 160, 169 Bryon 62, 160, 168 Buytendijk 132, 160

C Cantor 19-21, 23, 32-33, 38, 42-45, 47-50, 160, 168 Carnot 60 Cassirer 19, 44, 130, 133, 160, 167-168 Cauchy 29-32 Chiarelli 117, 122, 160 Clark 121, 123, 125, 160, 164, 169 Clarke 123, 125, 160, 169 Clausius 60

Coley 2, 161 Crompton 95 Cusanus 27, 32, 34 Cushing 81, 161

D Dacque 161 Darwin 9, 88-89, 91, 94, 96-97, 100, 104, 117, 119, 123, 138, 153-154, 156-157, 159, 161, 163 De Broglie 76 De Klerk 134, 161 De Swart 46, 161 Dedekind 33-37, 39, 44, 161 Descartes 27, 55, 66, 75, 86, 107, 110, 134, 144, 153 Diels-Kranz 21, 161 Dingler 72 Dobzhansky 100, 109, 117-118, 132, 157, 161 Dollo 138-139 Dooyeweerd 25, 46, 76, 79, 114, 142, 144, 160-161 Driesch 105-106, 113, 146-147, 161 Duley 161

E Ehrenhaft 81 Eibl-Eibesfeldt 136, 161 Eigen 161 Einstein 1-2, 9, 62-63, 65, 67, 69, 76, 161, 163, 167 Eisberg 76, 161 Eisenstein 111, 157, 161 Euclides 23 Eudoxos 25

F Fales 72, 161 Faul 121, 161 Feyerabend 119 Fischer 44, 48, 139, 161 Fontana 102 Fraenkel 38-39, 41-42, 44-45, 59, 161 Frege 20, 39, 41, 159, 162-163 Freudenthal 162 Friedrich 1, 162, 164

175

G Gadamer 53, 161-162 Galileo (vi), 2, 8-9, 27, 32, 55, 59-60, 62-63, 69, 72, 162 Gehlen 124, 137-141, 147-148, 162-163 Gieseler 120, 125, 162 Goerttler 135, 162 Gould 96 Goulian 162 Greenberg 162 Greene 162 Greenfield 162 Grene 95, 103, 123, 162

H Haas 106, 112-113, 128, 148, 162 Haeffner 133, 162 Haldane 87-88 Hallonquist 162 Harrison 162 Hart 119, 126-129, 131, 137, 141-142, 148, 150-151, 155, 162, 169 Hasse 23, 162, 168 Hawking 28, 55, 57, 64-65, 162 Hebeda 163 Heberer 162-163, 170 Heimholtz 61 Heimsoeth 27, 163 Heine 30, 163 Heisenberg 1, 65, 67, 112, 163 Heitler 40, 106, 113, 161, 163 Henke 121-122, 163 Hentschel 163 Heraclitus 8, 21 Hertz 75 Heyting 38-39, 45, 58, 163 Hilbert 43, 50, 55, 163, 167, 170 Hippasus 23, 169 Hobbes 55 Hoenen 79, 98 Holz 151, 163 Howells 163 Husserl 145, 163 Huxley 104, 157, 163

J Jammer 163

Janich 8, 63, 163 Jansen 163 Jaspers 149, 163 Jevons 163 Jonas 83-84, 110-111, 149, 157, 163 Jones 86, 163

K Kant 2-4, 20, 40, 43, 47, 53, 55, 58, 69-71, 73, 75, 127, 130, 145-149, 151-154, 160, 162-164, 168 Katscher 75, 164 Kaufmann 44, 164 Keith 120 Kerkut 164 Kitts 104, 164 Klaatsch 138 Kleene 43, 46, 164 Kline 18, 38, 50, 164 Koehler 130, 132, 164 Kremer 27, 164 Kronecker 43 Kugel 155, 164 Kugler 148, 164 Kuratowski 59

L Lagrange 38 Laitman 134-135, 164 Lakatos 119 Landmann 136, 138-139, 164 Laszlo 118-119, 157, 164, 167 Le Gros 121, 164 Leakey 120-121, 130, 132, 164 Leinfeller 91, 164 Lenk 164 Levy 45, 161 Liebig 61 Linnaeus 91 Lloyd Morgan 118, 157 Lorenz 32, 44, 47, 58, 104, 132, 138-139, 141, 164 Lorenzen 32, 44, 47, 58, 164 Lowenstein 120

M MacCurdy 120 Maimon 28, 32, 47, 164 Malthus 164 Margenau 68, 165 McHenry 121, 165

McMullin 119, 165 Meijer 25, 165 Merleau-Ponty 149, 165 Meschkowski 19, 35, 37-38, 165 Meyer 97, 107-108, 123, 165 Miller 87, 165 Millin 81 Monod 165 Moore 41, 165 Munson 165, 168 Myhill 50, 165

N Nagel 40, 165 Narr 124-128, 165 Needham 107 Newton (vi), 8, 28-29, 55, 59, 75 Nida 134, 165 Nietzsche 138

O Oparin 85, 87-88, 165 Orgel 87, 165 Origines 26 Overhage 142, 148, 165-166

P Pannenberg 148, 166 Passmore 118, 166 Planck 63, 65, 72, 75, 166 Plato 8-10, 24, 73-74, 112 Plessner 133, 148, 166 Plotinus 27, 31 Polanyi 5, 109, 119, 157, 166 Pongratz 166 Pope (vi) Popper 4-7, 54, 119, 166 Portmann 92, 128, 143-144, 148, 164, 166, 170 Pretorius 118, 167 Pythagoras 23-24

Rosas 159 Rucker 41, 167 Russell 4, 19, 50, 131, 167

S Scheler 148, 167 Schelling 61, 147, 167 Schilder 27, 167 Schilpp 67, 167-168 Schindewolf 103-104, 123-124, 167 Scholz 23, 43, 162, 168 Schopf 87, 168 Schubert-Soldern 105-106, 168 Schuurman 127, 168 Schwartz 117, 122-123, 168 Silver 87-88, 168 Simpson 104-106, 118, 123, 137-138, 153, 157, 168 Singh 38 Sinnott 106, 113, 168 Smart 19, 168 Smuts 107 Sokal (vi),168 Spielberg 62, 160, 168 Spinoza 27, 107 Stafleu 57, 62, 65, 70, 72, 168 Stanley 87-88 Strauss 8, 80, 131, 141, 150, 155, 161, 168-169

T Thesleff 23, 169 Thomas Aquinas 27, 68, 106 Thompson 60 Thorpe 93, 169 Titze 22, 169 Tobias 120-122, 159-160, 162-165, 167-169 Trincher 100, 113, 169 Troll 91, 106, 169

V

R Rahner 165 Rauche 150-153, 167 Ray 91, 120 Reed 125-127, 167 Reid 42, 167-168, 170 Rensch 9, 109-111, 129, 132, 157, 167, 169 Rickert 167 Robinson 19, 29, 42, 167 Rombach 167

176

Van Dalen 45, 161 Van Huyssteen 154, 169 Van Melsen 81, 169 Van Peursen 169 Van Stigt 169 Vollenhoven 60 Von Bertalanffy 88, 107, 136, 167, 169 Von Eickstedt 139, 169 Von Fritz 24, 169

W Watson 95-97, 99 Weierstrass 30-32, 44 Weiner 120, 162, 170 Weiniger 78, 170 Weyl 25, 40, 42, 45, 66, 72, 170

Whitehead 50, 118, 157 Willard 35, 170 Wolf 38, 48, 91, 148, 170 Woltereck 101, 108, 118-119, 157, 170 Wundt 170

177

Z Zeno 21-23, 25-26, 75, 162 Zermelo 20, 41, 50 Zimmerman 91, 97, 102, 118, 170