Causality, Method, and Modality: Essays in Honor of Jules Vuillemin (The Western Ontario Series in Philosophy of Science)

CAUSALITY, METHOD, AND MODALITY

THE UNIVERSITY OF WESTERN ONTARlO SERIES IN PHILOSOPHY OF SCIENCE A SERIES OF BOOKS IN PHILOSOPHY OF SCIENCE, METHODOLOGY, EPISTEMOLOG LOGIC, HISTORY OF SCIENCE, AND RELATED FIELDS

Managing Editor ROBERT E. BUTTS

Dept. of Philosophy, University of Western Ontario, Canal

Editorial Board JEFFREY BUB, University of Western Ontario L. JONATHAN COHEN, Queen's College, Oxford WILLIAM DEMOPOULOS, University of Western Ontario WILLIAM HARPER, University of Western Ontario JAAKKO HINTIKKA, Florida State University, Tallahassee CLIFFORD A. HOOKER, University of Newcastle HENRY E. KYBURG, JR., University ofRochester AUSONIO MARRAS, University of Western Ontario .rORGEN MITTELS1RASS, Universitiit Konstanz JOHN M. NICHOLAS, University of Western Ontario BAS C. VAN FRAASSEN, Princeton University

VOLUME 48

CAUSALITY, METHOD, AND MODALITY Essays in Honor of Jules Vuillemin With a Complete Bibliography of Jules Vuillemin

Edited by Gordon G. Brittan, Jr. Regents Professor of Philosophy, Montana State University

KLUWER ACADEMIC PUBLISHERS DORDRECHT I BOSTON I LONDON

Library of Congress Cataloging-in-Publication Data Causality. method. and modality: essays in honor of Jules Vui11emin I edited by Gordon G. Brittan. Jr. p. cm. -- (The University of Western Ontario series in philosophy of science; v. 48) Essays in English or French. Includes bibliographical references. ISBN 0-7923-1045-4 (a1k. paper) 1. Phi losophy. 2. Vui 11emin. Jules. I. Vui 11emin. Jules. II. Brittan. Gordon G. III. Series. B29.C336 1990 110--dc20 90-19840

ISBN 0-7923-1045-4

Published by Kluwer Academic Publishers. P.O. Box 17.3300 AA Dordrecht. The Netherlands. Kluwer Academic Publishers incorporates the publishing programmes of D. Reidel. Martinus Nijhoff. Dr W. Junk and MTP Press. Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers. 101 Philip Drive. Norwell. MA 02061, U.S.A.

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Table of Contents

PREFACE

vii

Jules Vuillemin / "Ma vie en bref'

1

Patrick Suppes / "Indeterminism or Instability, Does It Matter ?"

5

Erhard Scheibe / "Covariance and the Non-Preference of Coordinate Systems"

23

Pierre Laberge / "Kant's 'Platonic' Argument in Behalf of the A Priori Character of the Representation of Space"

41

Duchesneau / "The Sense of the A Priori Method in Leibniz' s Dynamics"

53

Gilles Gaston Granger / "Methode axiomatique et idee de systeme dans l'oeuvre de Jules Vuillemin"

83

Gordon G. Brittan, Jr. / "Algebra, Constructibility, and the Indeterminate"

99

Fran~ois

Karel Lambert / "On Whether an Answer to a Why-Question Is an Explanation If and Only If It Yields Scientific Understanding"

125

Ruth Barcan Marcus / "Some Revisionary Proposals About Belief and Believing"

143

Brian Skyrms / "Quantification, Modality, and Semantic Ascent"

175

David Wiggins / "Temporal Necessity, Time and Ability: a philosophical commentary on Diodorus Cronus' Master Argument as given in the interpretation of Jules Vuillemin"

185

Jules Vuillemin / "Replies"

207

List of the Publications of Jules Vuillemin, 1947-1989

225

Preface Deservedly so, Jules Vuillemin is widely respected and greatly admired. It is not simply that he has produced a large body of outstanding work, in many different areas of philosophy. Or that he combines to an unusual degree rigorous standards with a very wide perspective. Or even that in his path-breaking accounts of algebra, of !)escartes, of Kant and of Russell, he showed in new and profound ways how the histories of science and philosophy could be used to illuminate each other. It is also that he has pursued the application of formal techniques and the defense of liberal institutions with a rare singlemindedness and courage. In a time and place where the former were generally ignored and the latter often attacked, he carried on, at some personal cost, embodying a traditional and ideal conception of the philosophical life, bridging national differences. Those who know him also treasure his friendship. Always curious, he delights in new facts and new experiences, and continually heightens the perception of those around him. Almost yearly, at the College de France he introduced brand new courses always with fresh and fruitful inSights. Exceptionally solicitous, he follows the lives of the families around him in great detail. The devotion of his students is legend. His personal energy is also legend. Many of us have followed him bounding up the stairs two at a time or through the gardens of the Luxembourg, his wit and irony apace. Some of us have been fortunate enough to ski with him along the ridges of the Jura dividing France and Switzerland, arriving back at his mountain home exhausted and famished, to enjoy the unending hospitality that he and his wife Gudrun, herself an established scholar, provide. The participation in this volume d'hommages of well-know philosophers from widely diverse areas of interest and a number of different countries testifies to the importance and influence of Jules Vuillemin's work. We hope that this volume, presented to him in his vii

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PREFACE

seventieth year and upon retirement from a distinguished career at the College de France, will underline the importance and extend the influence. In the Vie de Rance, Chateaubriand wrote: "Tout est fragile; apres avoir vecu quelque temps, on ne sa it plus si on a bien ou si on a mal vecu." Tout est fragile. But from a provisional point of view, who would not agree that Jules Vuillemin has tees bien vecu.

MA VIE EN BREF

I was born the 15th of February, 1920, at Pierrefontaine-IesVarans, a village in the Doubs. My father had entered the civil service, in the reconquered province of Lorraine, first at the prefecture of Metz, then at the sous-prefecture of Chateau-Salins, a small countrytown at that time both comfortable and prosperous, where I passed my childhood. Of my secondary and preparatory studies at the College de jesuites in Metz, then at the Lycee Louis-Ie-Grand in Paris, I have retained the memory of a liberal, but diSciplined instruction, isolated from the quarrels and noises of the outside world, solid in ancient languages, in French, in history, in mathematics - the ax had not yet been applied to separate letters and sciences - behind the times in physics: I heard the special theory of relativity mentioned as a curiosity in 1936. My baccalaureat in hand, I made a trip to Berlin in August, 1936. It opened my eyes to the imminence of hostilities. France, caught up in class struggles, strikes, local disorders, in the conflict in Spain, engaged in speeches instead of acting. A few weeks before the AnglO-French declaration of war, I entered the Ecole Norrnale Superieure for a period of four years. The circumstances seemed hardly favorable for philosophical studies. Nonetheless, I remember how it felt: the "dr6le de guerre," the mobilization of my class in May, 1940, the overwhelming defeat, several months in the "Camps de jeunesse" of Marshall Petain, the return to an occupied Paris, a failed attempt to get to England, far from undermining the scholastic background necessary for philosophical reflection, gave to it a kind of seriousness and urgency

G. G. Brittan (ed.), Causality, Method, and Modality © Kluwer Academic Publishers 1991

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which are ordinarily missing in peace time. At the Ecole Normale, Cuzin served as my mentor and introduced me to Kant. At the Sorbonne, I took the courses of Bachelard, Brehier, Gouhier, and Cavailles. Cuzin and Cavailles were shot for the part they played in the resistance. Having received the agregation in philosophy in 1943, I was named a teacher at the Lycee de Besan~on, not far from the village where my family had located. Teaching is the best way of learning, for the student stops at the point where the master has not understood. I did a survey, during my Besan~onian year, of the shadowy zones heretofore covered over by scholarly rhetoric. The war, however, pressed on both town and school. Four of my students were arrested. One of my friends was deported to Buchenwald. He was declared "disappeared" the day the camp was liberated by the American army. Guerillas organized. I rejoined my village group. The ebb-tide, with its dramas, began. Happily, the army, disembarked at Toulon, hurried on its way. It happened in a night. In the morning we were liberated. When armored cars and trucks left for the East, I thought the war was over. I was mistaken by more than six months. I re-entered Paris in a holiday mood and obtained, for five years, a position at half-salary at the Centre National de la Recherche Scientifique. I set myself the task of making clear what remained of rationalism when the Absolute was removed from its foundation. The history of Kantian interpretation showed the interpreters coming to grips with this question. The French philosophical world split at the time principally between the dogmas of Existentialism and Marxism: there an ontology of anguish and nausea, here an unscrupulous activism which, in Paris, continues to pervert thought even to the present day. In my doctoral thesis, I sought to examine

MAVIE

3

these dogmas critically, in order to take from them whatever finite reason was able to find worth discussing. I married and had two children, a girl and a boy. Maurice Merleau-Ponty, elected to the Sorbonne, proposed me as his successor at Lyon. I was not chosen. After a short stay at the Lycee, the Ministry named me, on condition that I reside there, to the University of Clermont-Ferrand, where I lived until 1962. I found old comrades there and made new friends. I was able to appoint assistants, all of whom have since made their way in the world. Up to this point, I had worked without method. I asked Descartes and Kant to reveal theirs to me. I noticed that their thought remained scarcely intelligible and sometimes impenetrable if one did not go back to the sciences which had inspired them or which they had created. Under the direction of Pierre Samuel, who taught in the science faculty, I deepened my mathematical knowledge a little. I was encouraged by Martial Gueroult, professor at the College de France, who honored me with his friendship. Gueroult, an historian of philosophy, insisted on the architectonic and proof methods by which philosophical systems characterize themselves. Around him we formed a kind of school - Ginette Dreyfus, Victor Goldschmidt, Louis Guillermit, and I - and worked together in concert. At the Sorbonne I had not received any education in logic. I learned it in the pages of Russell. Going from one idea to another, I discovered philosophy, or rather contemporary Anglo-Saxon philosophies, almost unknown in France at that time. In 1962, Gueroult sponsored me and brought about my election to the College de France, as successor to the suddenly deceased Merleau-Ponty. The retirement of Gueroult, my divorce, several visits abroad, in particular to Canada and to the United States, my remarriage, the explosion of 1968 at the College de France where I observed the revolutionary gesticulation, the death of Gueroult, of Goldschmidt, of

JULES VUILLEMIN

4

Guillermit, of Ginette Dreyfus, and of several others who are not inscribed in the philosophical register, a trip to Japan, such are the events which marked my life during this period. I had a difficult time at the beginning in Paris. Two auditor~ in my course on Russell, lost in an enormous room, permitted very few illusions. I did not give up. I continued to apply logical analysis, whose virtues I had just discovered, to the theologies of Aristotle and St. Anselm, to the constitution of the sensible world, from Russell and Whitehead to Carnap and Goodman. Making my way, however, a difference between me and the majority of Anglo-Saxon analysts emerged. There were those who, singlemindedly interested in chasing down grammatical errors in the talk of philosophers, forgot the existence of SCientific languages. But even those who applied the method of "rational reconstruction" to these latter more often imposed on them principles of their own choice. I resisted this violence done to history, and trusted in the sciences such as they are, and not such as they should be. Moreover, it is presumptuous to neglect the philosophical tradition. It therefore appeared to me necessary from the outset to put some order into this tradition by proposing a rational classification of possible types of philosophical systems. That done, it was necessary to ask whether contemporary science - and especially quantum mechanics - favors the selection of a given type of philosophical system or whether, taking the traditional classification apart, it suggests new concepts and new principles to philosophy. It is with the response to this question that I am now occupied. Jules Vuillemin

INDETERMINISM OR INSTABILITY, DOES IT MATTER?·

Patrick Suppes 1. Skepticism About Determinism

In my recent book, Probabilistic Metaphysics, I have argued at some length against determinism as a viable philosophical or scientific thesis. I want first to review those arguments and then go on to look at an alternative way of viewing phenomena. Instead of the dichotomy deterministic or indeterministic, perhaps the right one is stable or unstable. In expressing my skepticism about determinism I shall not linger over a technically precise definition. It seems to me that the intuitive notion that phenomena are deterministic when their past uniquely determines their future will serve quite adequately in the present context. The natural basis of skepticism is our remarkable inability to predict almost any complete phenomenon of interest, and even more, our inability to write down adequate difference or differential equations. Consider, for example, a gust of wind and its effect on

-The first draft of this paper was read at a symposium on indeterminism at the annual meeting of the Pacific Division of the American Philosophical Association, March 23, 1985. A later draft was the basis of a lecture at the College de France at the invitation of Jules Vuillemin on May 12, 1986. It is a pleasure to dedicate this paper to him. Our many conversations on a variety of philosophical topics have both enlightened and delighted me. 5

G. G. Brittan (ed.), Causality, Method, and Modality © Kluwer Academic Publishers 1991

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PATRICK SUPPES

leaves of grass, the branches of a tree, the particles of dust agitated in various ways. It seems utterly out of the question to predict these effects in any detail. Moreover, it seems hopeless even to think of writing down the equations, let alone solving them. It might be noted the particles of dust, at least, would be within the range of the phenomena of Brownian motion, and the hopelessness of actually predicting such motion has been recognized for a long time. Of course, this example of Brownian motion raises a problem that needs remarking. One standard view of classical physics is that all phenomena are deterministic,--it is just that we are unable to analyze some phenomena in adequate detail. But even here there is reason for skepticism. The standard result of the standard theory of Brownian motion is that because of the high incidence of collisions the path of a particle is continuous but differentiable almost nowhere (only on a set of measure zero). Given that the path is this kind of trajectory, it becomes obvious that determinism is out of the question just because of the many collisions. It is a familiar fact of classical mechanics that collisions in general cause great difficulty for deterministic theorems. The kind of result that we have in the case of Brownian motion is not just a matter of difficulty, it is a matter of principled hopelessness. So I take it that insofar as the phenomena I have just described fall within the purview of the theory of Brownian motion, determinism is ruled out. For many familiar human phenomena we do not even have the elements of schematic analysis given by the probabilistic theory of Brownian motion. Examples are easy to think of. A favorite of mine is the babble of speech. The idea of ever being able to determine the flow of talk even between just one set of persons, not to speak of a billion, given whatever knowledge you might hope to have seems ridiculous and absurd. There is no reason whatsoever to think we will ever have theories that lead to deterministic results. It is certainly true that in occasional high states of deliberation we formulate very

INDETERMINISM OR INSTABILITY

7

carefully the words we are going to utter, but this is not the standard condition of speech. Moreover, even in such states of high deliberation we do not and are not able consciously to control the prosodic contours of the utt~rance. In fact, as we descend from the abstract talk of grammarians and model theorists concerned with semantics to the intricate details of the actual sound-pressure waves emitted by speakers and received by listeners, the problem of having a deterministic theory of speech looms ever more hopeless. I have the same skepticism toward deterministic theories of vision. Such a theory for any serious level of detail seems out of the question. The reasons for thinking this are many in number. The long history of theories of vision and the difficulties we still have in giving detailed partial descriptions of what the visual system is sensing provide some evidence. Detailed physiological studies showing that the human eye is sensitive to even a single photon provide other kinds of evidence, as do quantitative studies of eye and head movements. The extraordinarily complicated nature of the transduction that takes place in the optical system in order to send messages to the central nervous systems is another case in point. Someone might want to claim that we could have a gross deterministic theory of vision, but such a theory would be superficial and uninteresting. The actual mechanisms seem intrinsically subtle and complex. Of course, there are some kinds of complex problems that we feel confident in tackling, but anyone who has taken a serious look at problems of vision will back away rather rapidly from optimistic claims about having within the framework of contemporary science, or science as we can foresee it to be in the future, a workable, detailed deterministic theory. What I have had to say about speech and vision applies also to the sense of smell. The evidence seems pretty good that this sense is sensitive down to the presence of a substance at the molecular level. Moreover, what theories there are of the activities of single

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recognition cells are probabilistic in character. As far as I know, no one has attempted to propose a serious deterministic theory of smell. These familiar phenomena I am using to buttress my reasons for skepticism about determinism are easily matched by a dozen others. Given the extraordinarily small number of phenomena about which we can have a deterministic theory, there is cause for psychological and philosophical speculation as to why the concept of determinism has ever achieved the importance it has in our thinking about the world around us. To adopt a broad deterministic view toward the world does require not quite the extreme faith of the early Christians, but at least that of such diverse eighteenth-century optimists as Kant and Laplace-. Surely one psychological root of the faith in determinism is its conflation with prediction. Hegel (1899, p. 278) reports that Napoleon in a conversation with Goethe remarked that the conceptual role of fate in the ancient world has been replaced by that of bureaucratic policy in modern times with the implication that uniform predictability of individual behavior subject to the bureaucracy is, in principle, what we can now have. The search for methods of prediction has ranged from zodiacs to chicken gizzards and is found in every land. The primitive urge to know the future has in no way been stilled by modern science, but only rechanneled into more austere forms. The new skepticism. so I am arguing, should be about the omnipresence of determinism, not the omniscience of God.

*Historically we probably need to think of Kant as a cryptodeterminist.

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2. How to Save Determinism

Before making some direct comparisons with indeterminism in terms of instability there are some preliminary points to be made about unstable systems. The intuitive idea of instability in mechanics is this. Wide divergence i.n the behavior of two systems identical except for initial conditions is observed even when the initial conditions are extremely close. There are two aspects of unstable systems that make prediction of their behavior difficult, and therefore make difficult the realization of the deterministic program, even if the systems are, in fact, deterministic. One source of difficulty is that the initial conditions can be measured only approximately. If a system is not stable in the appropriate sense--I omit a technical definition here but it is straightforward to give one--, it will be impossible to predict its behavior for any but short intervals of time with any accuracy. In this case, we attributed predictive failures to a possibly small uncertainty in the initial conditions. We shall leave aside in the present discussion whether this uncertainty should be treated epistemologically or ontologically. Some later remarks will have something more to say about this issue. A second aspect of an unstable system can be that the solutions are not given in closed form, and calculations based on various methods of series expansion, etc., will not give accurate predictions. In other words, we cannot count on numerical methods to give us a detailed result for periods of prediction of any length. If the system is unstable, the accumulation of small errors in numerical methods of approximation, which may be the only ones available, can lead to unavoidable problems of accuracy. This last problem is especially true of systems that are governed by nonlinear differential equations. What I have said thus far applies to very simple systems of differential equations as well as complex ones. The solutions of the equations may be unstable but they do not seem to exhibit the kind

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of behavior we so directly associate with indeterministic or probabilistic behavior. It might be argued that the simplest systems of linear differential equations that are unstable do not represent something comparable to indeterminism. Yet it is true that for such unstable linear systems the accuracy of predictions will be poor, given, as is always the case in real situations, any errors in the measurement of initial conditions. In other words, unstable deterministic linear systems capture an important aspect of indeterminism, namely, our inability to predict future behavior on the basis of knowledge of present behavior. There is another aspect also of such linear systems that needs to be noted. In most applications, the linearity of the real system that is being modeled by the linear differential equations is only approximate. Almost always, deviations from linearity in the real system--the fact that the linear differential equations are only approximations--J, will make our ability to predict actual phenomena even more limited.

3.

Chaos and Symbolic Dynamics

We now get down to essentials. Those special unstable solutions of differential equations that exhibit chaotic behavior provide the intended alternative to indeterminism. It would have been more accurate in certain ways to entitle this paper 'Indeterminism or Chaos, Does it Matter?', but the meaning of chaos is too special, and so it is the central concept of instability that should be kept to the fore. So, what do we mean by chaos? A brief but not quite technically correct definition is the following. A solution of a deterministic system of differential equations is chaotic if and only if it exhibits some aspect of randornness--or, as an alternative, sufficient complexity. To some, this definition would seem to embody a contradiction, and therefore no solutions would satisfy it. On the left-

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hand side we refer to a deterministic system of equations and on the right-hand side to the random character of its solution. How can a deterministic system have a random solution? This is what chaos is all about, and the discovery of the new phenomena of chaos is certainly a watershed change in the history of determinism. Before turning to the recent discussions of chaos, it will be useful to go back over the earlier history of developing the theory of random processes within classical mechanics. The origin of the approach, usually called the method of arbitrary functions for a reason to be explained in a moment, originates with Poincare, but has been developed in detail by a number of mathematicians in the first half of this century. Already a rather short qualitative sketch of the ideas in very accessible form is given by Poincare in Science and Hypothesis (1913). (The history of developments since Poincare has been chronicled in some detail by von Plato (1983).) Here I shall just give a sketch of the analysis of coin flipping, one of the most natural cases to consider. To a large extent I shall follow the recent treatment due to Keller (1986), but as somewhat modified in Suppes (1987). Without going into details, we shall assume a circular coin that is symmetric in all the ways you would imagine; second, dissipating forces of friction are entirely neglected; third, it is assumed that the coin does not bounce but on its initial point of impact flattens out to a horizontal position. In other words, from the initial point of impact the face up does not change. With this idealized model, the physical analysis is simple. Newton's ordinary law of gravity governs the vertical motion of the particle--we assume there is no horizontal motion. Second, we assume that the rotational motion is that of constant angular velocity so there is no angular acceleration to the rotation. Now with this situation, if we knew the exact initial conditions, we could predict exactly how the coin would land, with either heads or tails face up. In fact, the classical analysis of this case assumes rightly enough that we do not know the exact value of the

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initial conditions. The method of arbitrary functions refers to the fact that we assume an arbitrary probability distribution of initial vertical velocity and initial rotational velocity. Then as the initial velocity tends to infinity, whatever the arbitrary distribution we begin with, the probability of a head will be one-half. In other words, the symmetry in the mechanical behavior of the system dominates completely as we approach the asymptotic solution. Of course, in real coin-flipping situations we are not imparting an arbitrarily large vertical velocity to the coin, but the variation in the way that we flip will lead to a very good approximation to one-half. The point is that in this typical analysis, the randomness enters only through the absence of knowledge of initial conditions. It is an important example of randomness in mechanical systems, one that has only recently begun to be recognized again as an important example, but it is not the kind of example on which I want to concentrate here. To show that the conventional philosophical dichotomy between determinism and randomness is mistaken, I consider two important and much discussed examples. The first is a special case of the three-body problem, certainly the most extensively studied problem in the history of mechanics. Our special case is this. There are two particles of equal mass moving according to Newton's inverse-square law of gravitation in an elliptic orbit relative to their common center of mass which is at rest. The third particle has a nearly negligible mass, so it does not affect the motion of the other two particles, but they affect its motion. This third particle is moving along a line perpendicular to the plane of motion of the first two particles and intersecting the plane at the center of their mass. From symmetry considerations, we can see that the third particle will not move off the line. The restricted problem is to describe the motion of the third particle. The analysis of this easily described situation is quite complicated and technical, but some of the results are simple to state in informal terms and directly

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relevant to my focus on determinism and randomness. Near the escape velocity for the third particle--the velocity at which it leaves and does not periodically return, the periodic motion is very irregular. In particular, the following remarkable theorem can be proved. Let t l' t2,... be the times at which the particle intersects the plane of motion of the other two particles. Let sk be the largest integer equal to or less than the difference between tk+1 and tk times a fixed constant-. Variation in the sks obviously measures the irregularity in the periodic motion. The theorem, due to the Russian mathematicians Sitnikov (1960) and Alekseev (1968a,b; 1969a,b), in the version given by Moser (1975) is this.

Given that the eccentricity of the elliptic orbit is positive but not too large, there exists an integer, say such that any infinite sequence of terms sk with sk::2 a corresponds to a solution of the deterministic differential equation governing the motion of the third particle··. mEOREM.

a

A corollary about random sequences immediately follows. Let s be any random sequence of heads and tails--for this purpose we can use any of the several variant definitions--Church, Kolmogorov, Martin-LOf, etc.. We pick two integers greater than to represent the random sequence--the lesser of the two representing heads, say, and the other tails. We then have:

a

-The constant is the reciprocal of the period of the motion of the two particles in the plane. **The correspondence between a solution of the differential equation and a sequence of integers is the source of the term symbolic dynamics. The idea of such a correspondence originated with G.D. Birkhoff in the 1930's.

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Corollary. Any random sequence of heads and tails corresponds to a solution of the deterministic differential equation governing the motion of the third particle. In other words, for each random sequence there exists a set of initial conditions that determines the corresponding solution. Notice that in essential ways the motion of the particle is completely unpredictable even though deterministic. This is a consequence at once of the associated sequence being random. It is important to notice the difference from the earlier coin flipping case, for no distribution over initial conditions and thus no uncertainty about them is present in this tree-body problem. No single trajectory in the coin-flipping case exhibits in itself such random behavior. This example demonstrates the startling fact that the same phenomena can be both deterministic and random. The underlying explanation is the extraordinary instability of the deterministic phenomena. Before remarking further on the significance of this result, I turn to the second example which is an abstract discrete model of period doubling. Because the mathematics is more manageable it is a simple example of a type much studied now in the theory of chaos. The example also illustrates how a really simple case can still go a long way toward illustrating the basic ideas. Let f be the doubling function mapping the unit interval into itself. (1) xn+l = f(x n) = 2xn(mod 1), where mod 1 means taking away the integer part so that xn+llies in the unit interval. So if Xl = 2/3, Xz = 1/3, x3 = 2/3, x4 = 1/3 and so on periodically. The explicit solution of equation (1) is immediate: (2) xn+l + 2nx l (mod 1). With random sequences in mind, let us represent Xl in binary decimal notation, Le., as a sequence of l's and O's. Equation (1) now can be expressed as the rule: for each iteration from n to n+ 1 move the decimal point one position to the right, and drop whatever is to

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the left of the decimal point: .1011...-+ .011 .... We think of each xn as a point in the discrete trajectory of this apparently simple system. The remarks just made show immediately that the distance between successive discrete points of the trajectory cannot be predicted in general without complete knowledge of x l' If xl is a random number, i.e., a number between 0 and 1 whose binary decimal expansion is a random sequence, then such prediction will be out of the question unless xl is known. Moreover, any error in knowing xl spreads exponentially--the doubling system defined by equation (1) is highly unstable. Finally, it is a well-known result that almost all numbers are random numbers in the sense defined. Although the exact technical details are rather complicated for almost all chaotic systems, the first example of a restricted threebody problem was meant to illustrate orbital complexity and the second complexity of initial conditions. In any case, randomness can be an essential part of the behavior of what would seem to be quite simple deterministic systems.

4. The Troublesome Case of Quantum Mechanics. From what I have just said, the dements of a rejoinder to my earlier skepticism about determinism are apparent. The phenomena cited as examples of indeterminism are in fact just examples of highly complex, unstable deterministic systems whose future behavior cannot be predicted. The strongest argument against such view comes from quantum mechanics. Beginning in the 1930's there has been a series of proofs that deterministic theories are in principle inconsistent with quantum mechanics. The first proof of the impossibility of deterministic hidden variables was by von Neumann. The latest arguments have centered on the inequalities first formulated in 1964 by John Bell.

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Moreover, the associated experiments that have been performed have almost uniformly favored quantum mechanics over any deterministic theory satisfying the Bell inequalities. To those who accept the standard formulation of quantum mechanics, the various proofs about the nonexistence of hidden variables answer decisively the question in the title of this lecture. Indeterminism or instability, does it matter? For these folk the answer is affirmative. The negative results show chaotic unstable deterministic mechanical systems cannot be constructed to be consistent with standard quantum mechanics. The conclusion of this line of argument is that standard quantum mechanics is the most outstanding example of an intrinsically indeterministic theory. There is, however, a still live option for those of us who are not entirely happy with the orthodox theory of quantum mechanics and its many peculiar features. The option left open is to account for quantum phenomena in terms of something like the theory of Brownian motion, which is, of course, part of classical mechanics broadly construed. Nelson 0%7, 1985) has provided thus far the best defense of this approach. He has, for example, derived the Schroedinger equation, the most important equation of nonrelativistic quantum mechanics, from the assumptions of Newtonian mechanics. However, his recent analysis (985) ends up with Bell's theorem and the relevant experiments as a serious problem. The most feasible way out seems to be to develop a non-Markovian stochastic mechanics, which in itself represents a departure from classical nonlocality. The central problems of current physics are not much concerned with this alternative, but mathematicians and philosophers will continue to puzzle over the foundations of this century's most successful scientific theory. As long as the stochastic view in the sense of Brownian motion remains a viable option, the question posed in the title can be answered by a skeptical "Perhaps not". Consistent with this view, Laplace's concept of probability and thus

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17

of indeterminism also remains a viable option--probability is the expression of ignorance of deterministic causes.

5. Randomness as a Limiting Case of Unstable Determinism The existence of deep-seated randomness inside deterministic systems can be attributed to their great instability, and this suggests the road of rapprochement between determinism and randomness. A striking feature of randomness and instability is complexity. Moreover, recent definitions of randomness are in terms of complexity. The complexity of a sequence of finite symbols is measured by the length of a minimal computer program that will generate the sequence. (For asymptotic purposes, the particular computer or computer language does not matter.) A simple alternating sequence of l's and O's can be generated by a very short program. More intricate sequences require longer programs and are therefore more complex. Where this argument is going should be apparent. Random sequences are of maximal complexity. In fact, the programs required to generate them would have to be infinitely long. So what are random sequences? They are the limiting case of increasingly complex deterministic sequences. Randomness is just a feature of the most complex deterministic systems. And what of particular importance follows from this? The separation of determinism and predictability. The most complex deterministic systems are completely unpredictable in their behavior. Laplace's "higher intelligence" must be transfinite. He must be able to do arbitrarily complex computations arbitrarily fast. To give a modern ring to Laplace's basic idea, I propose this. Randomness is tbe expression of maximally complex deterministic causes.

PATRICK SUPPES

18

6. Does It Matter? Setting aside, for the moment, the -problem of hidden variables in quantum mechanics, we may argue that the philosophically most interesting conclusion to be drawn from the analysis outlined in this paper is that we cannot distinguish between determinism and indeterminism. The true-blue determinist can hold, without fear of contradiction, that all processes are determined. Confronted with the myriad examples of natural phenomena that cannot be predicted and that seem hopeless to try to predict, he can reply with serenity that even these processes are deterministic, but they are also unstable. The determinist can agree amiably enough that there are processes yet to be analyzed and that his belief that they too will turn out to be deterministic is only based on past experience. This last remark is meant to ring a Bayesian bell. Pure Bayesians are natural true-blue determinists. After all, de Finetti begins his two-volume treatise on probability by printing in capital letters: PROBABILIlY DOES NOT EXIST, a thesis Laplace would have heartily endorsed. The indeterminist, for his part, can just as firmly hold on to his beliefs, directly supported as they are by the phenomenological data in so many areas of experience. Moreover, with the possible exception of quantum mechaniCS, there seems to be no current possibility of giving a knockdown argument for either determinism or indeterminism. Under either theoretical view of the world, most natural phenomena cannot be analyzed in detail, and even less can be predicted. How drastic and serious these limitations are is not sufficiently appreciated. I gave a number of obvious examples in the first section, but even in that presumed citadel of mathematically developed science, classical mechaniCS, it is beyond our current capabilities to analyze a general

INDETERMINISM OR INSTABILITY

19

system of one particle having a potential with just two degrees of freedom·. Whichever philosophical view of the world is adopted, the impact on theoretical or experimental science will be slight. Probability has a fundamental role no matter what, and statistical practice is complacently consistent with either determinism or indeterminism. (The assumption of determinism plays no systematic role in Bayesian statistics, for example.) There remains the question of whether proofs of no hidden variables in quantum mechanics make a decisive argument against classical determinism. I have mentioned already some reasons for not accepting these results as the last word. I want to conclude with a more general argument. The essential point is the exceedingly thin probabilistic character of quantum mechanics. Roughly speaking, no correlations or other interactive measures can be computed in quantum mechanics. Perhaps most important, if we are examining the trajectory of a particle, no autocorrelations can be computed, Le., correlations of position at different times, but such a statistic is a most natural measure of probabilistic fluctuation in the temporal behavior of a particle. The probabilistic gruel dished out by the wave function of a quantum-mechanical system is too thin to nourish any really hearty indeterminist. Paradoxically enough, the reconstruction carried out so far of quantum phenomena within classical mechanics

•A system of one particle with two degrees of freedom is a system defined by the differential equations ··x - f(x), where x is a vector in the plane and f is a vector field on the plane. The system has a potential if there is a function U from the plane to the real numbers such that f - -dUJax.

20

PATRICK SUPPES

is probabilistically much richer. It would be ironical indeed if the deepest probabilistic analyses of natural phenomena turn out to be within a deterministic rather than indeterministic framework. Stern Professor of Philosophy, Stanford University

INDETERMINISM OR INSTABILITY

21

References Alekseev, V. M. 096Ba), Quasirandom dynamical systems. I. Quasirandom diffeomorphisms. Mathematicheskie USSR Sbornik 5, 73-128. Alekseev, V. M. o 96Bb), Quasirandom dynamical systems. II. One-dimensional nonlinear oscillations in a field with periodic perturbation. Mathematicheskie USSR Sbornik 6, 505-560. Alekseev, V. M. 0969a), Quasirandom dynamical systems. III. Quasirandom oscillations of one-dimensional oscillators. Mathematicheskie USSR Sbornik 7, 1-43. Alekseev, V. M. o %9b) , Quasi-random dynamical systems. Doctoral Dissertation. Mathematicheskie Zametki 6(4), 489-498. Translation in Mathematical Notes, Academy of Sciences, USSR 6(4), 749-753. Bell,]. (964), On the Einstein Podolsky Rosen paradox. Physics 1, 195-200. de Finetti, B. (1984), Theory of Probability, Vol. 1, New York: Wiley. Hegel, G. W. F. (1899), The Philosophy of History 0. Sibree, trans.) Colonial Press. Reprinted by Dover Publications, New York, 1956. Keller,]. B. (986), The probability of heads. American Mathematical Monthly 93, 191-197. Moser,]. (1973), Stable and Random Motions in Dynamical Systems With Special Emphasis on Celestial Mechanics. Hermann Weyl Lectures, the Institute for Advance Study. Princeton, N.].: Princeton University Press. Nelson, E. 0%7), Dynamical Theories of Brownian Motion. Princeton, N.].: Princeton University Press. Nelson, E. (985), Quantum Fluctuations. Princeton, N.J.: Princeton University Press.

22

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Poincare H. (913), Science and Hypothesis. Lancaster, Pa.: The Science Press Sitnikov, K. 0%0), Existence of oscillating motions for the threebody problem. Doklady Akademii Nauk, USSR 133 (2), 303-306. Suppes, P. (984), Probabilistic Metaphysics, Oxford:Blackwell. Suppes, P. (984), Propensity representations of probability.

Erkenntnis, 26, 335-358. von Plato, J. (1983) The method of arbitrary functions. British Journal for the Philosophy of Science 34, 37-47.

COVARIANCE AND TIlE NON-PREFERENCE OF COORDINATE SYSTEMS

Erhard Scheibe I

In his famous paper of 1916 on the foundation of general relativity Einstein has formulated the folloWing prinCiple that he himself calls "the postulate of general covariance,,1: (C) The general laws of nature are to be expressed by equations that are valid for all coordinate systems, Le. that are covariant under arbitrary substitutions. This principle Einstein has viewed as a strengthening of what he in the same paper calls "the postulate of general relativity". This postulate is given the following formulation : (R) The laws of physics have to be such that they are valid for arbitrarily moved reference systems. Einstein argues that the relativity postulate follows from the covariance postulate by saying3: For among aD substitutions we at any rate find those which correspond to all relative motions of the (three-dimensional) coordinate systems. In arguing thus Einstein obviously assumes that the reference systems mentioned in the relativity postulate can be described by certain coordinate systems mentioned in the covariance postulate such that the relative motions of the former are described by certain transformations (= substitutions) of the latter. Under this assumption the relativity postulate says the same of every element of a certain set as the covariance postulate says of every element of a much larger set. Obviously, the former would then follow from the latter. 23

G. G. Brittan (ed.), Causality, Method, and Modality © Kluwer Academic Publishers 1991

24

ERHARD SCHEIBE

Only one year after Einstein's basic paper Kretschmann has objected against the above argumentation "that every physical theory, by a mere mathematical ... modification of the equations representing the theory and without changing its content, can be forced to satisfy even the most general relativity postulate" 4. In his rejoinder Einstein immediately admitted this objection and also the general opinion of the physicists - if there is such a thing-seems to have become "that in Einstein's words 5-, of necessity, every empirical law can be given a general covariant formulation". In spite of this giving in and in spite of this general tendency Einstein has continued to argue that physics is in need of a general principle of relativity and that this can be achieved by founding it on a principle of general covariance. Thus, to give but one example, in the widely known book "The Evolution of Physics" written by Infeld but authorized by Einstein, in passing from special to general relativity we read 6: "Can we formulate physical laws so that they are valid for all coordinate systems, not only those moving uniformly, but also those moving quite arbitrarily, relative to each other? If this can be done, our difficulties will be over." Now, whatever is meant by the difficulties to be overcome, it is suggested that we get rid of them merely by satisfying the principle of general covariance (which here is lumped together with the principle of general relativity). And the theory of general relativity then is introduced precisely by this step. In view of Einstein's adherence to his original standpoint in spite of this admission one can hardly escape the conclusion - believe it or not - that in formulating the principles quoted above Einstein did not succeed in fully expressing the idea that was really important to him. This idea must have been of a sort that would not allow admitting for it what Einstein did admit for its seeming equivalent. Already in the paper of 1916 it is spelled out with respect to both principles. In the sentence immediately preceding the quoted covariance postulate (C) Einstein summarizes a foregOing argument by saying7:

COVARIANCE AND NON-PREFERENCE

25

There is no alternative but viewing all possible coordinate systems as to be placed on the same footing in principal (prinzipiell gleichberechtigt). Similarly, with respect to the relativity principle (R) Einstein concludes a corresponding argument by sayingS: (R') Of all possible spaces in relative motion to each other no one must be preferred a priori. And immediately after this sentence there follows the above quoted postulate of general relativity (R). These facts about the text, I think, leave no doubt that in formulating (C) Einstein wanted to express and thought to ~ expressed what he had said more informally and provisionally by (C'), and Similarly with respect to (R) and (R') . But not only the actual wordings of (C) and (C'), as well as (R) and (R'), but also the arguments preceding these sentences show that something different from (C) and (R) was meant by (C') and (R') respectively. Newton's gravitational theory can easily be given a generally covariant (equivalent) formulation 9. But this does not do away with the fact that there are preferred (galilean) coordinate systems. Covariance with respect to a set of coordinate systems may be necessary in order that none of those systems is preferred. But, as the newtonian example clearly shows, it is by no means sufficient. Therefore, since Einstein wanted to avoid preferred coordinate systems (or, for that matter, reference systems) the question still was which was stronger, for the democracy among the systems as not only necessary but also sufficient principle distinguished the theory of general relativity from its predecessor Up to this point my argument concerned covariance as well as relativity. This was because the text of the paper in question shows that Einstein made his mistake with respect to both, and this may (C')

ERHARD SCHEIBE

26

make my claim that there ~ something wrong more plausible. From now on I shall concentrate on covariance, thereby deliberately leaving out the physically more interesting but, at the same time, much more difficult part of the matter. II

To obtain precise explications of the postulates (C) and (C') we have to get a clear idea of that part of a physical theory with respect to which those postulates can be given a precise reformulation. It turns out that the part usually called the "formalism" of a theory is sufficient for this purpose. A concept of formalism recently successfully applied to foundational studies is the concept of species of structures in the sense of Bourbaki lO . Roughly, the axioms of a theory then are statements (1)

L

[Xi 8]

about structures < XjS > where X is a finite system of principal base sets and s a finite system of elements of scale sets over the X, i.e. of sets generated by successively applying the operation of power set of a cartesian product, starting with the X. In this way the base sets X are structured by the typified sets s. It will be important for us that "concrete" sets such as the set of real numbers may, as so called auxiliary base sets, also take part in the process. Species of structures, well known from mathematics, are the algebraic ones, e.g. groups, rings, vector spaces etc., as well as the topological species of topological spaces, differentiable manifolds etc., but also mixed forms such as Hilbert spaces, Lie groups, etc. That also the formal part of a physical theory can be reconstructed as a species of structures will become plausible in particular through the species of geometrical structures that we are going to study.

COVARIANCE AND NON-PREFERENCE

27

Before doing so we have to introduce a second concept on the general level: equivalence of species of structures. For already Einstein's statement, connected with (C), "that every empirical law can be given a generally covariant formulation" signalizes that general covariance is not the invariable concomitant of a theory. The most we can expect in general is the existence of a covariant version of a theory which then, of course, must be equivalent to the original one. Similarly, the idea of a class of preferred coordinate systems, related to (CO), essentially means that we could have a version of our theory, equivalent to the original one, in which we can get along with that restricted class. In both cases equivalence includes the actual change of the internal concepts of the theory in question, i.e. a change of the typified sets s. We therefore have to provide for equivalence transformations (2)

t

= q[X, 81

, 8 = q- 1 [X, t1

which have to satisfy certain obvious conditions. A familiar equivalence of this type is, for instance, the one between boolean algebras and boolean lattices. In our further investigations we shall be concerned with certain species of geometrical structures only . But it was important to define the concept of equivalence on a most general level. Approaching geometrical theories we begin with the rather special but fundamental case of coordinate geometries. A coordinate geometry is about a space X structured by a set F of (local) coordinate systems and by nothing else. The coordinate systems, relating subsets of X to subsets of Rn , are transformed by (local) coordinate transformations of a given pseudo-group G of Rn 12. The best known coordinate geometry is the species of C00-differentiable manifolds. In this case G is a very large pseudo-group Goo' But we can choose also a relatively small group as, for instance, the Galileo group or the Lorentz group

28

ERHARD SCHEIBE

or the euclidean group (of Rn). With them are associated well known classical geometries underlying various physical theories. It is true that these geometries usually are not formulated as coordinate geometries 13. But they are equivalent (in the sense of (2)) to coordinate geometries. The matter is closely related to Felix Klein's Erlangen program14, and we may therefore call species of structures equivalent to coordinate geometries Klein geometries. Evidently not every geometry in the traditional sense, e.g. riemannian geometry, is a Klein geometry, - let alone physical theories in general. Let us, therefore, generalize the concept of coordinate geometry by the concept of analytical geometry. The structures that are the subject of these geometries are species structured not only by a set F of coordinate systems but also by further typified sets s, - geometrical objects as they are sometimes called. Following the somewhat old-fashioned way of presenting geometry15 we assume the s not only to have but just to be coordinate representations: they are partial (!) functions

(3a)

8 :

Fl.

~ D.

assigning to every coordinate system cP of a subset F of F an element s ( ) in the representation space Os of s. Os is a subset of a scale seconstructed from R, and the underlying pseudogroup G is represented on Os such that in the intersection of any two y, cP , 'II sF,

(3b)

v. (W . cP- 1) . 8(cp) = 8(W)

where rs is the representation of G. Well known examples of geometrical objects in the sense of (3) are fields. For them

(4)

Fl.

= F , D. ~ Pow (Rn

X

R(n H1))

COVARIANCE AND NON-PREFERENCE

29

and the elements of Ds in fact are partial functions on Rn with values in

R(nk+l).

Riemannian metrics, Lorentzian metrics, affine

connections are special cases of fields. But also curves and submanifolds can be viewed as geometrical objects covered by (3). It is, of course, true that such objects as well as fields can also be described by an intrinsic method not directly making use of coordinate systems. In fact, the intrinsic representation of geometrical objects has become fashionable of late 16. However, although this method may be more adequate in some cases, it has to be emphasized that the problems connected with covariance and the non-preference of coordinate systems would become meaningless if we were to follow that method throughout. This may be evident already from the wordings of the principles (C) and (C'): the term "coordinate system" explicitly occurs in them, and it is not to be seen how it could be eliminated without depriving the principles of their essential content. Part of this content is already contained in an assumption that we have to make if we now complete this outline of the concept of analytical geometry. Up to this point we were concerned mainly with the typification of the sets s. If it now comes to the axioms proper it is evident that a condition of invariance has to be imposed on them: Since our geometrical objects are represented by coordinate representations and since these representations are in general different from coordinate system to coordinate system, what we want to say about those objects s in terms of their representations seq»~ must be invariant under coordinate transformations. If, for instance, in differential geometry (G - G ) we were to establish the relation between a tensor field g ~ ~v and an affine connection rlCA.6 that the covariant derivation of the former with respect to the latter vanishes we would express this as usual by

ERHARD SCHEIBE

30

(5)

using any coordinate system. Now the essential thing about a statement like (5) is that it is invariant under coordinate transformations (3b): As a consequence of the special transformation rules for the 8"tv and rA.~, if (5) holds in one coordinate system it holds in any other. Obviously, this is what we have to require in general: It is on pain of inconsistency that we have to require that something said about s( belonging to I:I from the corresponding set F of < Xi F,S> belonging to I:aff . The best known examples of such equivalences are the differential geometric reformulations of the classical geometries. Let us, for instance, define (locally) affine geometry I: as the coordinate geometry belonging to the pseudo-group Gaff of all (locally) affine transformations of Rn. Then I:aff is equivalent to the differential geometry, i.e. G I - Goo ' I:aff of a flat affine connection. Clearly (6a) holds for this case, and (6b) is a consequence of the natural definition of F I as being just the set of coordinate systems on X generated by F and Goo' Similar situations occur by reformulating (locally) euclidean geometry as a special case of riemannian geometry and (locally) minkowskian geometry as the species of flat lorentzian manifolds. It is worthwhile to pause for a moment and ask how the phenomenon described by (6) is possible. In the examples given so far one of the geometries was supposed to be a coordinate geometry. A case more typical for the general situation is the following. We consider a field theory governed by the simple relativistic wave equation (7a)

based on minkowskian geometry as a coordinate geometry. The amazing thing about a differential geometric formulation of this

ERHARD SCHEmE

32

theory is that the equation (7 a) , though invariant under the Lorentz group, simply is not invariant under arbitrary coordinate transformations of G 00 . On the other hand, the wanted formulation certainly has to include an equation that is invariant under the transformations of G00 . How does this come about? The answer is that what is at work in (7a) not only is the wave function f but also the minkowskian metric g which, however, is disguised since it enters the stage only through special coordinate systems for which

for k = 1,2,3 for IL f. v

(Sa)

Thus in fact our dramatis personae are f and g, and there is the wave equation

(7b)

v IA

(glA"!.L) ax"

= 0

evlJ. the covariant differentiation with respect to g) relating f

and an arbitrary lorentzian metric g and invariant under Goo in precisely the same sense as (7a) is invariant under the Lorentz group. Of course, (7b) is still much too general. But if we require g to be flat by the equation (Bb)

RIA "IA>'

=0

(R IJ.vd.' the curvature tensor), likewise invariant under Goo we are led back to the original equation (7a) through the existence of special coordinate systems with (8a).

COVARIANCE AND NON-PREFERENCE

33

The study of such examples does, of course, mean little with respect to the question of general theorems related to our phenomenon. As regards theorems, the principles from which we started come to mind 17 . First, we have seen Einstein suggesting that the laws of nature should be expressed by equations covariant with respect to arbitrary coordinate transformations. Translated into the terminology developed so far this would mean that those laws have to be formulated as axioms of a differential geometry. One reaction to this proposal is that, since we do not yet know the laws of nature, only the future development of physics will tell us whether Einstein was right. But then there came the objection that the postulate might be vacuous after all, - that we can always satisfy it whatever the final laws of nature may be. In contrast to the intention that Einstein may have had with his original proposal, the intention connected with its analytical version can hardly be anything but to bring about a prooJof this version. If, however, we want to prove something we must give it a fairly precise formulation, replacing such expressions as "the laws of nature" by some well defined concept of physical theory. Let us take as such a concept the concept of geometry developed in II. Then the logico-analytical version of the principle of general covariance becomes (el) Every analytical geometry (with G :s; Goo) is equivalent to an (analytical) differential geometry, i.e. an analytical geometry having the pseudo-group Goo' Is this provable? For a proof we could proceed as follows. Let

(9a)

L

[X; F, s]

be the given analytical geometry with pseudo-group G. Then the conjunction

(9b)

CG[X; Foo ]

"L [X; F, s] "F ~ Foo

ERHARD SCHEmE

34

where the first member is the coordinate geometry belonging to G00 evidently is equivalent to (930) with (6) being satisfied. To establish the equivalence we only have to define F00 as being the set of coordinate systems generated by F and Goo. But (9b) is not yet an analytical geometry with respect to Goo. We would still have to bring about the situation described in II, especially by (3). It is far from clear whether this can be done in each and every case, and we will not go any further into this matter 18. It must suffice to make the reader feel that as soon as we try to be a bit more precise in this business as is usual we find ourselves in a situation not easy to control. IV

With respect to the equivalences of analytical geometries satisfying (6) there is complete symmetry between the two following questions (A) Given an analytical geometry 1:. with pseudo-group G. Is there an analytical geometry 1:.1 having a larger pseudo-group G1,Le. satisfying (6a), but still equivalent to 1:. in the sense of (6b)? (A') Given an analytical geometry 1:.1 with pseudo-group G1. Is there an analytical geometry 1:. having a smaller pseudo-group G, Le. satisfying (6a), but still equivalent to 1:.1 in the sense of (6b)? In the previous section we have discussed (A) for the extreme case that G 1 - Goo. A far reaching positive answer to (A) in this case was (e+). But we raised doubts as to its validity. The corresponding positive answer to (A') certainly is wrong: There is no logico-analytical version (e' +) of (e') as there may be one for (e). Rather we have (e~ There are differential geometries that are not equivalent in the sense of (6b) to any analytical geometry having a

COVARIANCE AND NON-PREFERENCE

35

smaller pseudo-group, cf. (6a) with G 1 = Goo' An uninteresting instance of (C') would be the coordinate geometry with pseudo-group Goo' i.e. the theory of infinitely open differentiable manifolds. But also Einstein's theory of general relativity, if it is given a suitable formulation, seems to be a candidate for (C') although a proof is still missing. However, pointing out (C') I do not pretend to have found an adequate explication of Einstein's original (C'). Taken literally it in fact is an explication. But it grants a theory its virtue of not distinguishing special coordinate systems simply by letting its axioms being sufficiently weak. And thiS, in turn, does not seem to be a virtue of a (metatheoretical) principle. It is here where our decision to concentrate on covariance and the nonpreference of coordinate systems leads to consequences showing that that viewpoint may be a bit too narrow. Nonetheless I shall conclude this paper by discussing some variations of the idea of non-preference of coordinate systems. To this end let me introduce two concepts related to the one in question. The essential concept entering (C') was: (B) The analytical geometry ~l with pseudo-group G 1 is not equivalent in the sense of (6b) to any analytical geometry ~l having a smaller pseudo-group G in the sense of (6a). Consider now the following concept (B 1) For any relevant condition on a coordinate system, if it can be proven from ~1 that there are coordinate systems satisfying that condition then it can also be proven that every coordinate system satisfies the condition. In other words: There is no condition for which it could be proven that some but not all coordinate systems satisfy it. This is perhaps the most direct explication of the idea that in the geometry ~1 no coordinate systems are preferred to others: In the field theory defined by (7b) and (Bb) there are privileged coordinate systems

ERHARD SCHEIBE

36

precisely in the sense that we can prove that in some coordinate systems (7a) (or (8a)) holds whereas in others it does not. The new concept (B 1) is stronger than (B). For by virtue of (6b) any reduction of the pseudo-group of 1:1 immediately leads to a condition distinguishing certain coordinate systems. On the other hand, (B l) would not hold for general. relativity because for this theory there are conditions distinguishing certain coordinate systems without reducing G00 . The condition on a coordinate system adapting it to the light cones at every point of its domain is a case in point. Besides (B 1) there is another concept (B 2) related to (B) but presumably weaker than it. This concept was suggested by]. Anderson 20 and made more precise by M. Friedman 21 . In the following I give my own version of the matter. Let 1:1 be an analytical geometry whose pseudo-group G 1 of coordinate transformations is a group acting on Rn22. It may then happen that G 1 is categorical in the following restricted sense: With respect to the arguments "Xl" "F l " and "sl" in

(10) any two models < Xl; F1". sl'" > and < Xl; F1,,· sl'" > are isomorphic. If this happens and < Xl; F1". sl'" > is a model of 1:1 ' then s 1 is called an absolute object in that structure. There are absolute objects occurring of necessity: Any two models of 1:1 necessarily are isomorphic with respect to their sets of coordinate systems F 1 and F 1'- If, therefore, s1 is definable in terms of F 1 then it will be an absolute object. Such is the case, for instance, if the coordinate geometry on which 1:1 based has the Lorentz group as its group of coordinate transformations and sl is the usual metric definable on this ground. But there are cases of absolute objects not definable in the coordinate geometry. If 1:1 is the differential geometric formulation of euclidean geometry we have categoricity

COVARIANCE AND NON-PREFERENCE

37

without the possibility of defining the metric in the coordinate geometry which, in this case, is the species of Goo - differentiable manifolds. The case of non-definable absolute objects leads to a stronger version of categoricity: 1:1 is strongly categorical with respect to "sl" if it is categorical and "sl" is not definable in terms of "F 1". Our third concept of irreducibility then is (B 2) The analytical geometry 1: 1, is a coordinate geometry or it is not strongly categorical with respect to any of its arguments "sl". One can easily see that (B2) follows from (B). For if (B2) does not hold 1:1 is not a coordinate geometry. Moreover, it is strongly categorical with respect to at least one of the arguments, say "sl". Given a model < Xl; F I'" S I'" > of 1: 1 we define a set of preferred coordinate systems F < F1 as follows: Because of the categoricity the model is isomorphic to a standard model < Rn; G 1". Sl'" > of 1: 1 , The isomorphism is effected by a coordinate system in Fl' The set of coordinate systems thus distinguished is smaller than F 1 because 1:1 was assumed to be strongly categorical. In this way equivalence to a theory with a smaller group G and the absolute object sl being eliminated can be shown. Of course, the general concept (B 2) does not do away with absolute objects altogether: If G is one of the classical groups we still are where we ever were. Consequently, just as in the case of general covariance the interesting case is the differential geometric one. Anderson wanted to avoid absolute objects under all circumstances - whether they are definable or not. In order to avoid the definable cases we have to make the group G as large as possible. And this nicely fits into the bunch of ideas originally introduced by Einstein. Professor of Philosophy. University of Heidelberg

ERHARD SCHEmE

38

Notes 1

A. Einstein: Die Grondlage der allgemeinen Reltitivitatstheorie. Ann. d. Phys.49 (1916) 769-822. Quoted from § 3.

2

ibid. § 2.

3

ibid. § 3.

4

E.

5

A. Einstein: Prinzipie//es zur a//gemeinen Relativittitstheorie. Ann. d. Phys. 55 (1918) 241-4. Quoted from p . 242 .

6

A. Einstein and L. Infeld: tbe Evolution of Physics. Cambridge 1938 . Quoted from p. 212 . As I learned from Don Howard the book was written entirely by Infeld,and Einstein only gave his name to fasten the sale.

7

A. Einstein: Die Grondlage der allgemeinen Reltitivitatstheorie. Ann. d. Phys, 49 (1916) 769-822. Quoted from § 2.

8

ibid. § 3.

9

E. Cartan: Sur Ies varietes a connexion affine et /a thoorie de la relativite generalisee. Ann. sci. Ecole Normale Super. 40 (1923) 326-412 and 41 (924) 1-25.

Kretschmann: Ober den physikalischen Sinn der Relativittitspostulate. A. Einsteins neue und seine ursprllngltche Relativitatstheorie. Ann. d. Phys. 53 (1917) 575-614. Quoted from p . 576.

COVARIANCE AND NON-PREFERENCE

39

10

N. Bourbaki: Elements of Mathematics. Theory of Sets. Reading Mass., 1%8 . CH. IV. For physical application see G. Ludwig: Die Grundstrukturen einer physikalischen Theone. Berlin 1978.

11

For the following see E. Scheibe: Invariance and Covariance. In: Scientific Philosophy Today, Essays in Honor of Mario Bunge. Ed. by]. Agassi and R. S. Cohen. Dordrecht 1982 . 311 - 31 .

12

For details see the Encyclopedic Dictionary of Mathematics. Ed. by S. Iyanaga and Y. Kawada. Cambridge, Mass., 1977.92 D and (a narrower concept) 108 Z.

13

A recent exception is W. G. Dixon: Special Relativity. CUP 1978. pp. 42 ff.

14

F. Klein: Elementarmathematik vom h6heren Standpunkt aus. Vol. II: Geometrie. Berlin 1925.

15

The standard monograph is]. A. Schouten: Ricci-Calculus. Berlin 1954.

16

See Misner, Ch. W., Thorne, K .S., and]. A. Wheeler: Gravitation. San Francisco 1973. What I am emphasizing is that , although the definition of, say, the concept of a vector field need not refer to coordinate systems, the definition is based on the concept of a differentiable manifold and this concept usually is defined by using coordinate systems.

17

A different analysis of the principle of general covariance can be

ERHARD SCHEIBE

40

found in S. Weinberg: Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York 1972. pp. 91 ff. 18

For some further thoughts on the matter see the paper mentioned in n.11.

19

There are, of course, explications different from (C+). One possibility is to restrict the whole question to field theories in the sense of (4). Yet the problem of proving (C+) thus modified again is a matter not too easily settled.

20

J. L. Anderson: Principles of Relativity Physics. New York 1967. J. L. Anderson: Covariance, Invariance, and Equivalence: a Viewpoint. Gen. Rei. Grav. 2 (1971) 161-72.

21

M. Friedman: Relativity Principles, Absolute Objects, and Symmetry Groups. In: Space, Time, and Geometry. Ed. by P. Suppes. Dordrecht 1973. 2%-320.

22

This assumption simplifies the concept formation and the argument. But it seems not essential for the matter.

KANT'S "PLATONIC" ARGUMENT IN BEHALF OF TIlE A PRIORI CHARACTER OF TIlE REPRESENTATION OF SPACE

Pierre Laberge The transcendental exposition of the concept of space has perhaps received more attention than the metaphysical exposition. 1 This is undoubtedly because of the threat which non-Euclidean geometries pose to it and because of the interest which for this reason philosophies of mathematics sometimes have in it. Gottfried Martin has given a little more spice to the arguments of the metaphysical exposition in behalf of the a priori character of the representation of space by qualifying the first as "platonic" and the second as "aristotelian." 2 In what sense does the first deserve such dignity? On the other hand, are there any problems with it? The first argument (A23/B38) of the metaphysical exposition establishes the a priori character of the representation of space, as Martin summarizes it 3, by making the following case: "in order that it be possible to represent to myself certain things (etwas) as juxtaposed (als nebeneinander), it is necessary that the representation of the juxtaposition (des Nebeneinanders) be already presupposed." So, he continues, it has to do with the "Grundargument' 4 that Plato regularly uses to establish the Theory of Ideas and thanks to which we can prove, if proof there be, the a priori character of all representations a priori. Plato employs this argument, for example, in the Pbaedo (72e - 71) in order to prove the a priori character of the concept of equality, that is to say, to establish that equality is an Idea. "In order that I can recognize two things as equal," Martin again summarizes,5 ".. .1 must already have available the concept of 41

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equality." Only the reminiscence of the Idea of equality, of the "Equal in itself" (74a) makes possible recognition of the equality of these sticks of wood or of these stones (74b). The first argument of the metaphysical exposition is thus nothing but the celebrated reminiscence argument, demythologized and applied to the particular case of the representation of space. Martin's interpretation assumes that the reminiscence argument is intended to establish the Theory of Ideas. But it seems to me rather that the reminiscence argument presupposes the Theory of Ideas. Recognition of the equality of these sticks of wood or of these stones is reminiscence of the Idea of equality because it can only be reminiscence. It can only be reminiscence because (a) we must claim (and isn't this just to take the Theory of Ideas as given?) that the "Equal in itself.. .is something" (74a and b) and (b) much more is needed than the equality of sticks of wood or of stones "to match that which is Equal" (74d). Socrates intends to establish that the knowledge of this "Equal in itself," of which the recognition of the equality of sticks of wood or of stones can only be a reminiscence, must have been acquired before birth. To suppose that the reminiscence argument is intended to establish the Theory of Ideas renders it circular: there are Ideas; therefore we recall them when we perceive; since we recall them when we perceive, there are Ideas! To call the first Kantian argument "platonic" in this sense is equivalent to lending to Plato and to Kant a circular argument, circular in the same way, perhaps, as that imagined by D.P. Dryer: "In order to recognize objects as red, one must already have the concept of redness, "6 hence the concept of redness is not empirical. The conclusion would hold only on condition that one agreed from the outset either that the

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concept of redness is a Priori or that one cannot already possess the concept of redness without it being a Priori. If it corresponded to the argument imagined by Dryer, Kant's first argument would be no more successful: in order to recognize objects as spatial, one must already possess the concept of space, hence the concept of space is not empirical. That is perhaps the interpretation of P F. Strawson, for whom Kant's first argument, a banal tautology 7, does not even attain the majesty of a circle. If we have been able to loan Kant a platonic argument after having loaned Plato a circular argument, we might have been able to loan Kant a circular argument directly. Has this been done? VaihingerB put me on the track of Ueberweg. This latter criticizes Kant's argument for being a Zirkelschluss. After summarizing it as follows "Space is not an empirical concept drawn from external experiences; for the representation of space must already be at the foundation (schon zum Grunde liegen) of all concrete localization (Lokalisierung)," he adds in a footnote: "Was freilich ein Zirkelschluss ist." 9 Thus he defines a" Cirkelbewei1' in his System der Logik: 'when A is proved by Band B in its turn by A.10" In the Zirkelschluss for which Kant is criticized, the B can only be "already at the foundation," and the A the non-empirical or a Priori character of the representation of space. Kant would set out to prove the a priori character by the foundational character, but he would succeed in doing so only by tacitly basing the second on the first. It is in this way, moreover, that Hermann Cohen 11 interprets Ueberweg's objection since he tries to reply: if Kant intends to prove that the representation of space is not empirical, he does not undertake to prove at the same time that it is a priori. Whence Kant's merely

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negative formulation: space is not an empirical concept. Everything would transpire as if Kant had chosen this negative formulation in order to escape the circularity objection. For how could there be a circle if Kant is not trying to prove A? Is Kant's argument circular or "platonic" in Martin's sense? We can agree with Vaihinger 12 and Martin 13 in seeing in its first sentence the thesis to be established: "Space is not an empirical concept which has been derived from outer experiencesj" in the second sentence which begins with "Denn," the argument properly socalled: "for in order that certain sensations be referred to something outside me (that is, to something in another region of space from that in which I find myself), and Similarly in order that I may be able to represent them as outside and alongside one another, and accordingly not only as different but as in different places, the representation of space must be presupposedj" in the third sentence which begins with •Demnach," the restatement of the theSiS, whose truth now appears coupled with the soundness of the argument. The third sentence thus paraphrases the first two sentences in linking them: "The representation of space cannot, therefore, be empirically obtained from the relations of outer appearance. on the contrary, this outer experience is itself possible at all only through that representation." We may therefore forget it after having taken note of its antileibniziano-wolffian accent (A40, 856-57). To tell the truth, the argument divides into two sub-arguments: on the one hand, the representation of space, it is declared, is presupposed in order that I can refer impressions to something outside me (ausser mich), on the other hand, it is also presupposed in order that I can represent to myself things as outside and next to each

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other (ausser und nebeneinander). The first sub-argument is difficult to interpret. A tautology powerless to shake the empiricist? In fact, what empiricist would not subscribe to the impossibility of representing things outside of him without the representation of space? Unless the "ausser" in "ausser micb" is to be understood in a nonspatial sense, in the sense of praeter rather than extra? Whence the attempt of H. E. Allison to exonerate Kant of the tautology with which Straws on finds fault. Allison goes so far as to suggest that the second sub-argument must also be sheltered from Strawson's criticism. It would be tautological to assert that space must be presupposed for one to represent "things as outside and along side one another, and accordingly not only as different but as in different places." It would therefore be necessary to interpret the second sub-argument in the following manner: "in order to be aware of things as numerically distinct from one another, it is necessary to be aware, not only of their qualitative differences, but also of the fact that they are located in different places." 14 But, even if sound, 15 the second sub-argument thus understood would be advanced by Kant without proofl Moreover, the parallel with the argument in behalf of the a priori character of the concept of time: "For neither coexistence nor succession would ever come within our perception, if the representation of time were not presupposed as underlying them a priori' (A30/B46) begs for a literal interpretation: the representation of the coexistent or of the successive there, of the ausser und nebeneinander here, presupposes there the representation of time, here the representation of space. As regards the "ausser micb" of the first sub-argument, it is difficult to interpret it in a non-spatial way given Kant's parenthetical

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remark: "(that is, to something in another region of space from that in which I find myselO." In brief, eiJher the first argument deserves Strawson's criticism, or it is to be interpreted, although with difficulty, in Allison's way (in which case Kant advances it as gratuitously as the second sub-argument as interpreted by Allison), or it is simply a special case of the second sub argument. As well in consequence to discard it as at best superfluous, which leaves us with the single second sub-argument completely parallel to the argument (without sub-argument) in behalf of the a Priori character of the representation of time. It remains to ask if these two parallel arguments do not prove to be tautological or at bast circular. They would be tautological if they led back to the argument already mentioned in behalf of the a priori character of the concepts of redness and of space: "in order to recognize objects as red, one must already have the concept of redness;" in order to recognize objects as spatial, one must already haye the concept of space. But is this really the case? For Kant does not write that it is necessary to already have the concept of space to be able to recognize XiS as spatial, but that it is necessary to already have it in order to recognize them as ausser und nebeneinander. (One can, as Martin, blast away the "ausser' to save only the "neben".)16 In the same way, as regards time, Kant does not write that it is necessary to already possess the concept of time in order to recognize XiS as temporal, but that it is necessary to already have it in order to recognize them as co-existent or successive. But, it will be objected, don't we escape the tautology only to sink into circularity? How to prove that one cannot already possess concepts of space and time without them being a Priori and this

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without tacitly taking for granted either that these concepts are .a priori or that one cannot already possess them without their being .a priori? Perhaps the Dissertation of 1770 provides us with the response in paragraph 14 on time. I7 If it is necessary to already possess the concept of time, if the concept of time presupposed, it cannot be empirical. For if it is presupposed, one cannot define it therefore through that which presupposes it. One cannot define it by the order of succession I8 or "per seriem actualium post se invicem exsistentium." For quid significiet vocula post, non intelligo, nisi praevio jam temporis conceptu" (2, 399). This is to say that it cannot be defined empirically, since experience does not provide us with anything but this "series." Kant alludes to this difficulty when in paragraph 14.5, he castigates the circulus vitiosus into which "Leibnizius et asseclae' (2, 400-400 fall. Without doubt, he has in mind an analogous difficulty concerning space when he warns again against the danger of circularity in paragraph 15.d (2, 404). Nam, he could write, quid significent vocula "extra se invicem" non intelligo, nisi praevio jam spatii conceptu. The first argument in behalf of the a priori character of the representation of space thus escapes circularity, if it does escape it, to the extent that the definition of space to which its opponents would be driven presupposes that which it wishes to define and would thus itself be circular. 19 Professor of Philosophy. University of Ottawa

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Notes

lQuotations of the Critique of Pure Reason are from the translation by Norman Kemp Smith (Macmillan and Co., Ltd., 1933), following the A and/or B editions. All other texts of Kant's follow the Akademie Ausgabe. 2Gottfried Martin, Immanuel Kant (Berlin: Walter de Gruyter, 1%9), pp. 34-39. In a doctoral thesis defended in 1987 at the Universite de Provence, Aix-Marseille I, and which deserves publication, FrancoisXavier Chenet traces back to Johann Georg Feder the affirmation of a tie between the first argument of the metaphysical exposition and Plato. See Francois-Xavier Chenet, L' Esthetique transcendentale de la Critique de la raison pure. Etude analytique et critique~ p. 150. Martin, for his part, traces back the rediscovery of this tie to Natorp (Platos Ideenlehre, zweite Auflage, Leipzig, Felix Meiner, 1921, pp. 34ft). I have, however, not found a trace of this rediscovery in the passage indicated or elsewhere in Platos Ideenlehre.

3Ibid., p. 34. 4Ibid, p. 34. 5Ibid., p. 35. 60. p. Dryer, Kant's Solution for Verification in Metaphysics (London:

George Allen and Unwin, 1%6), p. 173.

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7p.F. Strawson, The Bounds 19(6), p. 58.

0/ Sense

(London: Methuen and Co.,

BHans Vaihinger, Kommentar zu Kants Kritik der reinen VernunJt, Band 2, zweite Auflage (Stuttgart: Union Deutsche Verlagsgesellschaft, 1922), p. 179 9F. Ueberweg, Grundriss der Geschichte der Philosophie~ Dritter Teil (Berlin: Ernst Siegfried Mittler und Sohn, 1907), p. 306. 1°F. Ueberweg, System der Logik, dritte Auflage (Bonn: Adolph Marcus, 1968), pp. 403-404. 11 Hermann Cohen, Kants Theorie der Er/ahrung, zweite Auflage (Berlin: Dumrnler, 1885),pp. 96-97. 120p. cit., pp. 156-157. BOp. cit., P. 33.

14Henry E. Allison, Kant's Transcendental Idealism (New Haven: Yale University Press, 19832, p. 83. 15Compare to Metaphysik Mrongovius, 29.1.2, p.830. 160p . cit., p. 35. Jules Vuillemin underlines the importance of attaching the "neben" to the "ausser" in the second edition of the

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Critique. See Jules Vuillemin, La philosophie de I'algebre. tome 1 (Paris: P.U.F., 1967), p. 457, note 3. 17Compare to 1) Metaphysik L',;, "Raum is die Ordnung der Dinge, sofern sie ausserhalb oder neben einander zugleich sind, Zeit ist die Ordnung der Dinge, sofern sie nach einander sind. Diese Definition ist aber tautologisch; will man sie verstehen, so muss schon der Begriff von Zeit und Raum vorhergehen. Denn nebeneinander bedeutet in verschiedenen Orten sein, also liegen die Begriffe von Zeit und Raum der Definition schon zum Grunde" (28.1, pp. 177-178). 2) Metaphysik Mrongovius: "Der autor erklaert Raum durch Ordo extra se positorum. Extra se positorum sind Dinge in verschiedenen Orten. Der Begriff des Orts sezt den Begriff des Raurns voraus, und der Begriff wird schon als bekannt angenommen: ordo plurium, quatenus post se existunt est tempus; nach einander seyn ist zu verschiedenen zeiten seyn, also ist idem per idem erklaert" (29.1.2, p.831).

18Baumgarten defines space and time as follows in paragraph 239 of his Metaphysica: "Ordo simultaneorum extra se invicem positorum est spatium, successivorim, tempus" (17, p. 79), Wolff, according to Kant, defines space as "ein bloss der empirischen Anschauung (Wahrnehmung) gegebenes Nebeneinandersein des Mannigfaltigen ausser einander" (6, p. 208). 19J3ut why, it will be asked, does Kant no longer invoke in the Critique the objection of the Dissertation? I have no response to this

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question. According to Vaihinger, op. cit., p. 369, Kant would have discovered between 1770 and 1781 that the defect attributed to the Leibniziano-wolffians was not a .. circulus vitiosus' but rather a "Cirkel in der genetischen Ableitung." But was this a reason to abandon the objection? Would it not have sufficed to label it differently?

TIlE SENSE OF TIlE A PRIORI MEmOD

IN LEIBNIZ'S DYNAMICS Fran~ois

Duchesneau

In the opuscule De primae philosophiae emendatione notione substantiae (1694), Leibniz mentions that he has devoted a special science, dynamics, to the notion of force, and that this methodological elaboration clarifies the metaphysics of substance. I To a certain extent this assertion causes problems. The scientific papers published by Leibniz up to that time do not support this claim to have founded a science of dynamics. Certainly, since the Brevis demonstratio erroris memorabibis cartesii (1686), Leibniz had been working to invalidate demonstratively the Cartesian principle of the conservation of quantity of motion. He proposes as an alternative a new measure of the motive force which would be conserved in mechanical exchanges, a measure based on the estimate of absolute force (potentia absoJuta) or living force (vis viva), expressed by the product mv 2. Since then, a stormy quarrel with the Cartesians concerning the possibility of deducing the new principle as foundation of the system of the laws of nature had broken out 2 . But on what basis was Leibniz able to claim that he had founded a science of dynamics? One factor in the answer to this question emerges from a more careful examination of the appeals to the a priori method of demonstration which appear progressively in his unpublished work and in certain exchanges of letters, in particular with De Voider, Johann Bernouilli, and Christian Wolff. Our ambition here will be limited to presenting some reasons which militate in favor of such a re-examination, then to initiate the process of analysis through a study of the first part of the Dynamica de potentia (1689-90) where recourse to the a priori method is first sketched. 53

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FRAN