Causality in natural science (American lecture series, publication no. 213. American lectures in philosophy)

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CAUSALITY IN NATURAL SCIENCE By VICTOR F. LENZEN, PH.D. Professor of Physics University of California Berkeley, California

CHARLES C THOMAS · PUBLISHER Springfield · Illinois · U.S.A. -iii-

CHARLES C THOMAS · PUBLISHER BANNERSTONE HOUSE 301-327 East Lawrence Avenue, Springfield, Illinois, U.S.A. Published simultaneously in the British Commonwealth of Nations by BLACKWELL SCIENTIFIC PUBLICATIONS, LTD., OXFORD, ENGLAND Published simultaneously in Canada by THE RYERSON PRESS, TORONTO This monograph is protected by copyright. No part of it may be reproduced in any manner without written permission from the publisher.

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Copyright 1954, by CHARLES C THOMAS · PUBLISHER Library of Congress Catalog Card Number: 53-12610 Printed in the United States of America -iv-

CONTENTS

I. THE NATURE OF CAUSALITY 1. Introduction 2. Field of Causality 3. Causation Collision 4. Causality as Efficacy 5. The Criticism of Hume 6. Causality as Uniformity 7. Causality as Identity 8. Dynamical and Statistical Causality II. PRINCIPLE OF CAUSALITY 1. Example of Dynamical Causality 2. Applicability of Functional Relations 3. Recurrence of Causal Sequences 4. Principle of Causality 5. Cognitive Status of Principle 6. Extension of Principle of Causality III. COGNITION OF CAUSALITY 1. Causal Strands in Nature 2. Observational Methods 3. Mill's Canons of Induction 4. Experimentation 5. Frames of Space and Time 6. Nature of Experiment 7. Observation IV. CAUSALITY IN CLASSICAL PHYSICS 1. The Role of Mechanics

3 3 4 6 8 11 12 13 14 16 16 17 19 20 21 25 27 27 28 29 32 33 36 38 40 40

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2. 3. 4. 5. 6. 7. 8.

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Ancient and Mechanics Laws of Motion Force as Cause Differential Equation of Motion Laws of Conversation Reversible Motion Mechanics of Fields

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9. Classical Microphysics V. CAUSALITY IN BIOLOGY 1. Physical Explanation of Vital Processes 2. The Laws of Thermodynamics 3. Physical basis of Metabolism 4. Kinetic Theory of Diffusion 5. The Macrophysics of the Nervous System 6. Microscopic Theory of Neural Circuits 7. Cybernetics 8. Order in Biology 9. The template VI. CAUSALITY AND RELATIVITY 1. Relativity in Classical mechanics 2. Special Theory of Relativity 3. Structure of Space-Time 4. Temporal Order of Cause and Effect 5. Changes of Relative Quantities 6. General Theory of Relativity and Gravitation VII. CAUSALITY AND QUANTA 1. Problem of Quantum Theory 2. Origin of Quantum Theory 3. Dualism of Corpuscle and Wave 4. Unity of Quantum Theory

51 54 54 56 58 60 61 64 66 67 68 70 70 71 73 75 77 78 81 81 82 84 86

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5. Quantum Theory on a Corpuscular Basis 6. Principle of Indeterminacy 7. Quantum Theory on a Wave-Field Basis 8. Concept of State 9. Theory of Observation 10. Statistic in Quantum Theory 11. Casuality in Quantum Theory 12. Causality in Early and Present Quantum Theory 13. Complementarity 14. General Theory of Predictions 15. Logic of Complementarity Bibliography Index

87 92 94 94 95 98 99 101 103 105 107 111 117

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CAUSALITY IN NATURAL SCIENCE -1-

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I

THE NATURE OF CAUSALITY

1. Introduction THE FIELD of natural science is Nature: the realm of things, properties, and phenomena situated in space and time. Within nature exists the human individual who in diverse ways interacts with his environment. The natural environment provides man with sustenance, it stimulates him to respond, it inspires him to inquire into the constitution of things. In the effort to satisfy a native desire to understand nature, man has created science. Natural science is constituted of conceptual systems by which the mind of man orders, connects, and explains the properties and transformations of natural things in space and time. An essential element of science is expression of the connection between events through the concept of causality. Natural science originated and developed against a background of common experience. Primitive man perceived things in space, observed events in time, and made conjectures for the explanation of natural phenomena. He inferred from experience that the sun is the cause of light, that fire is the cause of smoke, that injury to his body is the cause of pain. Thus a concept of causality was an instrument of explanation in early stages of experience. The concept expresses causation: a process by which one phenomenon, the cause, gives rise to a succeeding phenomenon, the effect. The presuppositions of common experience are an in-3-

tegral constituent of the foundations of science. Natural science initially accepted the space-time world of nature and then refined, reconstructed, and elaborated the concepts required for the cognition thereof. This continuity between common experience and scientific method is exemplified by the concept of causality. The primitive concept of causality expressed efficacy of which the original basis was personal experience. Indeed, natural phenomena such as lightning and thunder once were explained as the acts of a wrathful god. Subsequently, efficient action

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was expressed by the physical concept of force. Critical analysis, however, generally has found regularity of sequence to be the essence of causality. This view is exemplified by the definition of John Stuart Mill:64"We may define the cause of a phenomenon to be the antecedent, or the concurrence of antecedents, on which it is invariably and unconditionally consequent." The subsequent analysis of the concept of causality in natural science will utilize the continuity between primitive experience and natural science.

2. Field of Causality I shall specify the field of causality in a manner to eliminate the problem of its ontological status. The work of natural science presupposes cognition of natural things by perception.56 A natural object enters the world of common experience with qualities that are manifested in sensation. The perceptual object is a frame of reference for sensory qualities. Non-sensuous properties also are ascribed to things by virtue of patterns of relations in which qualities of sensation stand. A natural quality such as whiteness is a universal which is grasped by thought through a concept. The terms of a language have meaning in the sense of connotation or in-4-

tension, and also have meaning in the sense of denotation or extension.57 I shall say that the connotation or intension of a term is the properties of the object to which the term applies. The signification of a term is the essential property of its intension. The quality whiteness is the signification of the term whiteness; the class of white things is the extension or denotation of the term. To understand a general term is to grasp through concepts the properties which are its intension. A physical record of cognition is constituted of written terms and sentences of a language which report the results of cognition. An object of thought is represented by the concepts of its properties. In daily life the conceptual object is clothed in sensory qualities and adequately substitutes for the perceptible thing. In science one ascribes properties to things which are not given by immediate experience. Thereby the conceptual object becomes distinguished from the perceptible thing. Science investigates the object which substitutes for the real thing, and places it within a wider conceptual scheme. The conceptual object may be called an image, a picture, a model. In the development of exact methods of description with the aid of mathematical concepts, the object of science acquires ideal status as a point, rigid body, ideal gas, perfect fluid. The object then becomes an ideal model which substitutes for the real world. In the most advanced stages of physical theory conceptual

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objects such as molecules, atoms, and electrons become completely dissociated from immediate sensory experience. The conceptual scheme of the physical world is a construction, the function of which is to order, connect, and explain experience. The physicist Max Planck76 has distinguished the "Weltbild" of physics from sensory data and from reality as well. The image or model which is constructed by scientific -5-

theory may be interpreted as an approximate conceptual substitute for reality, or it may be interpreted as a fiction which serves merely to correlate the data of perception. Causality is a relation within the realm of conceptual objects. The relation of cause and effect refers to conceptual events regardless of the relation of the latter to reality. In the prescientific stage of experience causality is attributed to an intuitively given world which confronts an observer. In the sophisticated stage of science causality must be attributed to a model which the scientist constructs out of concepts.

3. Causation in Collision The study of causality may be initiated by discussion of an example of causation in common experience: collision of two solid bodies. This example has provided a model for the explanation of complex physical processes; it is the basic physical process for an atomic theory. The collision of bodies has also provided an illustration for critical analysis of causality by the philosopher David Hume. Let us then suppose that a ball is set rolling on a horizontal plane surface. The rolling ball collides with a second ball, initially at rest, and sets the latter in motion. In this instance of causation does the term cause apply to the rolling ball, to its state of motion, or to the collision? At first sight the rolling ball may be viewed as the cause of an effect which is motion of the ball inititally at rest. Only by virtue of its motion, however, does the rolling ball produce motion in a second; hence the state of motion of the rolling ball appears to be the cause. But motion of the rolling ball causes the observed effect when the rolling ball strikes a second one. It would appear, then, that the cause is an event: impact of the rolling ball with the second one. If impact of one ball with the second one is specified as the -6-

cause, the contemporaneous acquisition of motion by the second ball is the effect. The present instance of causation exemplifies reciprocal action. While the second

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ball acquires motion during impact, it changes the motion of the rolling ball. The ball that suffers impact reacts on the rolling one, and this reaction is the cause of an effect, loss of motion of the rolling ball. Reciprocity of cause and effect during collision appears to demonstrate that effect is contemporaneous with cause. This appears to contradict the previously stated view that cause precedes effect in time. We may note, however, that the rolling ball plays the role of agent whose behavior gives rise to the event which produces new behavior in the second ball. If an experimenter were to repeat the collision, he would impress the same velocity upon the rolling ball. Accordingly, in the sequence of phenomena in which a rolling ball collides with a stationary one, at first sight the cause is the rolling ball as agent; on further analysis the cause is its state of motion; and finally the cause is impact of the rolling ball with the other. The events which in sequence pertain to the rolling ball constitute a total phenomenon which may be described as the antecedent cause; the subsequent events which in sequence pertain to the second ball likewise constitute a total phenomenon which may be described as the effect. Such loose use of the term cause occurs in common experience and frequently in natural science. However, even the analysis just presented lacks the precision which can be achieved by mathematical analysis based on quantitative description of physical processes. The lack of precision in the foregoing analysis of an example of causation derives from ambiguity in the concept of effect. A rolling ball on impact with a second one at -7-

rest sets the latter in motion. The communicated motion of the second ball starts from null and builds up to a final value. The effect of impact may be taken as the process of building up the final motion, or as the final state of motion. In dialectical discussions of causality the question is raised whether the effect is consequent to, or simultaneous with, the cause. The answer depends upon the definition of effect. As we shall see, in the mathematical description of motion we may describe the causal process by an integral law which expresses velocity or distance as a function of time; or we may describe causation as the building up of velocity in conformity to a differential law which expresses the time-rate of change of velocity as a function of state. Whether cause and effect are successive or simultaneous depends upon the employment of an integral or differential mode of description.

4. Causality as Efficacy In the light of causation in collision, we are prepared to determine the nature of causality. The language of causality includes terms such as production, efficacy, and necessity. These terms indicate that in the history of the subject causation has been described as production, as efficient action, and as necessary connection. The nature of an effect was held to be implicit and discoverable in

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the cause. An understanding of this aspect of causality will be assisted by historical considerations. The concept of causality exemplifies the law of development which was formulated by the positivist philosopher Auguste Comte.17 Comte's law of the three stages is that a concept originates in a theological state, passes through a metaphysical one, and finally arrives at the stage of positive science. In a theological stage man explains natural phenomena in terms derived from personal experience. -8-

The human individual has immediate experience that he is an active agent. He knows that he produces results by acts of will and bodily exertions. Specifically, I exert muscular force in order to lift a heavy body from the floor. The decision to act and the consequent muscular action are accompanied by inner experiences of diverse kinds. Contemporary psychologists may question the value of introspection as a mode of cognition about mind, but it is a plausible hypothesis that immediate experiences of personal activity furnished the basis for the conception of causation as production, efficacy, and necessity. In a theological stage of thought there was created an animistic conception of nature. Man initially interpreted natural phenomena as the manifestation of agents similar to himself. In a mythological era natural things and processes were personified as gods and goddesses. In the initial stages of science concepts of animism were interwoven with those of science. Thales was a pioneer in science, but he also declared that all things are full of gods. The lodestone, a natural magnet which orients itself with respect to the magnetic field of the earth, was interpreted to be the seat of a soul. In his theory of the heavens, Aristotle taught that the heavenly spheres revolve in perfect circular motions under the guidance of immaterial intelligences. Thus the efficient activity of the human individual was projected into natural processes for purposes of explanation. Causation in nature signified production and efficacy. The scope of the animistic conception of nature was gradually restricted through the progress of science and philosophy. A large realm of phenomena came to be interpreted as natural processes determined by inanimate forces which act in conformity to natural law. In the modern era there was created a mechanics for which force was -9-

conceived as an efficient cause that by necessity produces effects. The properties of force were formulated in Newton's three laws of motion, of which the second states that force is proportional to the rate of change of momentum with respect to time. Momentum depends upon mass and velocity jointly. Now

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mass is invariable and the time-rate of change of velocity is called acceleration. Hence force in classical mechanics is generally expressed as proportional to the product of mass and acceleration. In the seventeenth century, on foundations laid by Copernicus, Kepler, and Galileo, Newton created a system of the world upon the hypothesis that the sun exerts a force of gravitation on the earth and other planets which revolve around the sun in conformity with the laws of motion. The mechanical concept of force was also applied to the collision of two bodies of which the sequence of phenomena has already been described. The mechanical explanation of changes in the motions of the two balls upon collision is that during impact the colliding bodies undergo elastic deformations which call forth equal and opposite forces in conformity with the third law of motion. During the impact the elastic forces generate equal and opposite quantities of momentum. From the laws of motion that force is proportional to time-rate of change of momentum and that to every force there is an equal and opposite reacting force, one deduces that during the collision of two bodies there is conservation of momentum. Neglecting friction and other disturbing factors, the total momentum of the two balls after impact is equal to the momentum before impact. In the example of collision causation is an interaction which conforms to the laws of motion. The law of conservation of momentum also applies; this admits the interpretation that cause is equal to effect. The classical mechanical analysis of causation, as ex-10-

emplified by collision, represented causes as inanimate forces which act in conformity with natural law. From the standpoint of Comte's law of development the theological stage of causality had been succeeded by a metaphysical one. The term force signified exertion which was assigned a status in physical reality. A metaphysical necessity was attributed to the connection of cause and effect. The concept of force in classical mechanics was endowed at birth with the connotation of production, efficacy, and necessity.

5. The Criticism of Hume David Hume initiated a program of reflective criticism which has sought to eliminate the connotation of efficacy from the concept of causality. The basis of his criticism was the empiricist doctrine that every idea is copied from some preceding impression or sentiment. Hume cites our example of the collision of two bodies and declares, "Motion in one body is regarded upon impulse as the cause of motion in another. When we consider these objects with the utmost attention, we find only that the one body approaches the other; and that the motion of it precedes that of the other, but without any sensible interval." 39 Hume asserts that observation of the relation between cause and effect reveals no connection between them, but only conjunction. He says, "We are never able, in a single instance, to discover any power or necessary connection; any

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quality, which binds the effect to the cause, and renders the one an infallible consequence of the other. We only find that one does actually, in fact, follow the other."41 Hume contends that the idea of necessary connection between cause and effect arises from observation of a number of instances: "The first time a man saw the communication of motion by impulse, as by the shock of two billiard balls, he could not pronounce that the one event was con-11-

nected, but only that it was conjoined with the other. After he has observed several instances of this nature, he then pronounces them to be connected."42 The preceding discussion of Hume's analysis may be summarized by quoting his definition of cause: "We may define a cause to be 'an object precedent and contiguous to another, and where all the objects resembling the former are placed in like relations of precedency and contiguity to those objects that resemble the latter.'" 40

6. Causality as Uniformity The conclusion of Hume's criticism was that the essence of causality is not efficacy, but uniformity of sequence of phenomena. The definition of causality as uniformity brings the concept of causality to Comte's positive stage of development. In the nineteenth century force began to be eliminated from mechanics as an efficient activity. A pioneer in this development was the physicist Kirchhoff.49 In his work on mechanics he declared that its objective is not to discover the causes of motion; it is to describe completely and in the simplest manner the motions which occur in nature under specific conditions. When motion is described in quantitative terms, the dependence of motion upon conditions is expressed by functional relations which constitute the mathematical form of the laws of mechanics. A variable is said to be a function of another if the value of the first is determined by the value of the second. The physicist Ernst Mach59 declared that causality signifies functional relation between variables which characterize physical phenomena. The exposition of mechanics with explicit rejection of force as a constituent of physical reality is exemplified by the Traité de Mécanique Rationelle of Paul Appell.2 On his view the objective of mechanics is: From specification -12-

of motions which occur under given conditions, to predict what motions will occur under other conditions. The problem concerns only a body and its motion;

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it is not necessary to introduce force as third element. But for the sake of brevity, one adopts the following convention: if a given particle of a specified mass has a specified acceleration in the presence of other particles, the particle under consideration is acted on by a force which arises from the particles in its environment and which is determined by the product of mass and acceleration.

7. Causality as Identity We have concluded that causality is uniformity of sequence of phenomena; in more precise terms, causality is functional relation between variables which characterize phenomena. The expression of uniformity or functional relation constitutes natural law. That causality is lawfulness or conformity to law has been challenged by Emile Meyerson.63 He contends that causality is identity which is expressed by the equality of cause and effect. In the example of collision, when a rolling body collides with a second body, initially at rest, motion is produced in the second body. Equality of effect to cause appears to be exemplified, for the greater the antecedent speed of the rolling ball, the greater will be the consequent speed of the ball which is struck. The precise formulation of equality of cause and effect in the present example is in terms of momentum. Neglecting friction and other disturbing factors, the total momentum of the system constituted by the colliding bodies after impact is equal to the total momentum before impact. Causality as identity in the sense of Meyerson, therefore, is expressed by the law of conservation of momentum. However, the law of conservation of momentum for collision is derivable from the laws of classical me-13-

chanics to which collision conforms. Thus for this example, causality in the sense of identity is an aspect of causality as conformity to law. As we shall see, in classical mechanics conformity to law and equality and cause and effect are correlated aspects of causality.

8. Dynamical and Statistical Causality In the analysis of causality thus far we have found that the term has been understood to signify 1) efficacy; 2) uniformity; and 3) identity. Contemporary critical analysis usually concludes that uniformity or regularity is the essence of causality. Meyerson has distinguished identity from légalité, but conservation laws which express identity can be derived from uniformity. The literature of the subject reveals, however, that efficacy in the sense of force or influence is claimed to constitute the essence of causality.43, 69 In the present work we recognize that efficacy is a constituent of causality as it is employed in scientific practice. Efficacy is exemplified when an acceleration, as effect, occurs simultaneously with a force, as cause. The intimate connection between cause and effect is represented in scientific experience through spatial

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contiguity of the terms. This contiguity represents causal efficacy which, insofar as it conforms to law, will be called dynamical causality. Classical mechanics offers a norm for dynamical causality. The present chapter on the nature of causality has expounded a classical concept of dynamical causality. The occurrence of effects in atomic phenomena which are distributed in conformity to a statistical law demonstrates the need for a concept of statistical causality. Planck77 cites two different kinds of causal connection of physical states: 1) absolute necessity of connection as expressed by dynamical regularity; and 2) probability of connection as -14-

expressed by statistical regularity. Planck has distinguished between dynamical regularity and statistical regularity. In order to recognize the element of efficacy in causality, I shall express the distinction as one between dynamical causality and statistical causality. -15-

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II PRINCIPLE OF CAUSALITY 1. Example of Dynamical Causality H ISTORICALLY, the principle of causality has been formulated for dynamical regularity, that is, causality as connection between particular events. The principle generally refers to the character of regularity which is expressed by functional relation in mathematical theory. For dynamical causality the functional relation correlates the variables of an individual process like the linear motion of a body. Now the significant problems for analysis pertain to a principle of causality which applies to the connection between particular events. In the present chapter the principal topic will be causality as functional relation between the variables for an individual process. As a basis of discussion I shall discuss a functional relation which was discovered by Galileo32 to hold for bodies that fall freely near the surface of the earth. Galileo verified the law of failing bodies from observations on the motion of a ball as it rolled down an inclined plane. He released a ball at the top of the plane and observed positions of the ball at successive instants of time. A typical set of observations is exemplified by the following table. If t is time in seconds and s is distance in centimeters measured from the origin, then: at t = 0, s = 0; at t = 1, s = 1; at t = 2, s = 4; at t = 3, s = 9; at t = 4, s = 16; etc. The distance traversed by the rolling ball was found to be directy proportional to the elapsed time: s = kt2, where in -16-

this example k = 1. Galileo inferred that a body falling freely in a vacuum would conform to the same law, but with a different factor of proportionality. In the standard notation the functional dependence of distance on time is expressed by s = ½gt2, where g is the acceleration of gravity. This functional relation may be represented graphically by a curve in a plane. On a vertical axis one specifies a scale for time, and on a horizontal axis a scale for distance. Instants of time and correlated distances of a body from the origin are coordinates of points which fall on a parabola with vertex at origin. In the example of free fall the element of force appears to be completely eliminated. Now the time-rate of change of distance is speed, and the time-rate of change of speed is acceleration, which turns out to be the factor g. The acceleration g is the effect of the force which the earth's gravitational field exerts upon the mass of the falling body. If one wishes to utilize the concept of force as a cause, the law of falling bodies, in the form acceleration a is equal to a constant g, can be interpreted as derived from the equation of motion that force is equal to the product of mass m and acceleration a. Since the active force on a freely falling body is its weight, which is proportional to mass and may be expressed mg, one obtains as the equation of motion, mg = ma, from which one derives g = a. Despite critical objections to force as efficient cause, the concept

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of force is a useful guide for the discovery and explanation of functional relations which constitute natural laws.

2. Applicability of Functional Relations The definition of causality as functional relation is subject to criticism in view of the circumstance that, in principle, it is always possible to find a functional relation which correlates any given sequence of data. This was clearly -17-

explained by Leibniz52 as follows: "Let us suppose for example that some one jots down a quantity of points upon a sheet of paper helter-skelter. Now I say that it is possible to find a geometrical line whose concept shall be uniform and constant, that is, in accordance with a certain formula, and which line at the same time shall pass through all of the points, and in the same order in which the hand jotted them down; also if a continuous line be traced, which is now straight, now circular, and now of any other description, it is possible to find a mental equivalent, a formula, or an equation, common to all the points of this line by virtue of which formula the changes in the direction of the line must occur. . . . When the formula is very complex, that which conforms to it passes for irregular. Thus we may say that in whatever manner God might have created the world, it would always have been regular and in a certain order." One can find functional relations by application of interpolation formulae of Newton and Lagrange. Hermann Weyl has restated the conclusion of Leibniz. Weyl90 declares that the concept of regularity loses its significance if there is no limitation upon the structural complexity of functional relations. He asserts that natural law is subject to the additional test of simplicity of structure. However, there are no generally accepted criteria for simplicity. It may be agreed that an equation of the first degree is simpler than one of higher degree. Degree of simplicity may also be defined as inversely proportional to the number of variables which occur in a functional relation. It was the judgment of Moritz Schlick85 that the criterion of simplicity is aesthetic and inadequate for the restriction of functional relations in a definition of causality. Schlick adopted as criterion of causality the capacity of a functional relation to serve for the prediction of results -18-

of experience. On his view the essence of causality is predictability. This presupposes that functional relations which are found to hold at one time can be found to hold at some future time. Predictability presupposes that causal

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sequences of phenomena are reproduced in nature or are reproducible by the art of experimentation. The experiment of Galileo with motion on the inclined plane has been reproduced countless times and has always yielded the same results. That functional relations are reproducible in nature or by experimentation is the basis of the value of the principle of causality.

3. Recurrence of Causal Sequences It is a generalization from experience that limited natural systems can be isolated to a high degree of approximation. The physical processes of such isolated systems are only slightly influenced by bodies external to them. Further, these relatively isolated systems periodically pass through similar sequences of states. The solar system constitutes a practically isolated system which is not influenced appreciably by the stars. The planets in their revolutions around the sun perform periodic motions which are described by Kepler's laws of planetary motion. The moon periodically revolves around the earth. Thus the celestial realm constitutes a laboratory in which nature periodically reproduces similar sequences of events. Reproducible sequences of phenomena can also be prepared on the surface of the earth. A pendulum consists of a solid body which is attached to a rod suspended from a fixed support. If the body is displaced from its position of equilibrium and released, it passes through the same series of states of motion in successive equal intervals of time. I have previously cited the often repeated experiment of Galileo in which a ball was released to roll down an in-19-

clined plane. The experiment on impact of bodies, with which our analysis of causality was initiated, is also reproducible. Observations on motions of the planets and experiments on rolling bodies, impact, and pendulums are reproducible to an approximation sufficient to have provided the basis for the laws of classical mechanics which exemplify causality as functional relation. The discussion demonstrates that causality as functional relation is limited in applicability to relatively isolated, limited systems.29 As applied to such systems causality is the basis of prediction. The concept of causality is not directly applicable to the universe as a whole. The universe exhibits an enormous variety of detail and complexity of processes. During the lifetime of science a particular state of the universe is not known to have recurred. Since past states of the universe appear not to recur, past sequences of states do not recur. The principle: If the same initial state is realized, the same sequence of states will also be realized, holds only vacuously since the condition is not satisfied.

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4. Principle of Causality A natural law which expresses a functional relation provides a basis for prediction by reason of the natural or prepared recurrences of instances of the functional relation. The concept of causality as functional relation is of value for natural science in view of the principle of causality. This principle may be expressed with varying degrees of precision. An uncritical formulation of the principle, but one of service to the experimentalist, is that the same cause always produces the same effect. A more adequate formulation is: If the same state of a system is realized, the -20-

same sequence of states will also be realized. The formulation of the principle must also admit the possibility that the same state may be realized at different places as well as at different times. Accordingly, I state the principle of causality in the manner of Paul Painlevé:74 If at two instants the same initial conditions of a system are realized, except for difference of positions in space and time, the same sequence of states will be realized after the two instants at corresponding positions in space and time. The principle of causality thus formulated implies that space and time are homogeneous, that is, space and time are not efficient causes of phenomena. For example, if a physical experiment is performed in a European laboratory, one infers that the same experiment and its results can be reproduced at some later time in an American laboratory.

5. Cognitive Status of Principle The principle of causality that the same initial conditions will be succeeded by the same sequences of phenomena is a general principle of natural science. The cognitive status of the principle has been the subject of philosophical inquiry. It is a fact of experience that sequences of phenomena which once have been realized also have recurred. To this extent the principle of causality expresses a generalization from experience. Verification of the principle by instances from past experience, however, does not constitute proof that the principle will hold in the future. That specific natural laws have been found to hold in the past is insufficient to prove that, if past initial conditions could be reestablished, the same laws would hold in the future. The philosopher Kant46 attributed to the principle of causality the status of an a priori proposition which is not dependent on, but constitutive of, experience. On his doc-21-

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trine the concept of causality is a condition of synthetical unity of phenomena in time and is the foundation of all experience. The concept of causality is a form of understanding through which empirical knowledge of nature becomes possible. On Kant's view the principle of causality, that all changes take place according to the connection between cause and effect, expresses a necessary mode of procedure for the construction of science. The principle of causality will hold for future experience in that the understanding will reject for science empirical data which cannot be connected by the category of causality. According to Kant, the principle of causality can neither be verified nor falsified by experience, because as founded on an a priori form of the understanding the principle is logically prior to, and therefore independent of, experience. The doctrine that the principle of causality is founded on an immutable form of thought is uncongenial to an era which is accustomed to relativity and change in the foundations of science. The significant contribution of Kant's doctrine perhaps is expressed by the interpretation of the principle of causality as a convention. That general principles of science are to be interpreted as conventions was expounded by Henri Poincaré. Philipp Frank has applied the doctrine of conventionalism to the principle of causality.30 He argued that the principle of causality is a definition of state which is laid down by convention. The interpretation of a principle or law of science as definition is founded on considerations which are embodied in a theory of postulates for a mathematical system. Things and properties of the real world stand in relations of all types. The conceptual significance of a term in a language, or the content of a concept, may be constituted by its relations with other terms. Significance resides, not in isolation but in context. A mathematical theory may be -22-

founded on postulates which implicitly define primitive concepts in terms of the relations between them. Similarly, a principle or law of natural science expresses a relation of concepts and thereby serves to define the constituent concepts.84 The principle of causality states that if the same initial state is realized, there follows the same sequence of states. The principle involves the basic concept of state of a system. If the state of a system is defined in terms of coordinates of position and components of velocity, as in classical mechanics, then it is true that the principle of causality has been verified in the past for mechanical systems. On the doctrine of conventionalism the principle may be interpreted as a definition of state of a system. If two apparently similar initial conditions are not succeeded by similar sequences of states, one may conclude from the principle of causality that the two initial states were not similar. A test of sameness of state is afforded by the nature of the consequent sequences of states. On the conventionalist interpretation of the principle of causality, the problem of natural science is to find specifications of states so that the principle of causality is exemplified by phenomena.

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Modification in the conception of state for the realization of causality may be illustrated by the concepts for mechanics. In ancient mechanics a body was characterized in terms of the physical property weight. A functional relation in which weight is a state variable is expressed by the principle of the lever: If bodies are attached to respective ends of a lever, the lever is in equilibrium when the product of weight and lever arm is the same for both. This principle of equilibrium is an element in the significance of weight as a physical quantity. For the formulation of the laws of mechanics in the modern era it was found necessary to ascribe to a material body another property, inertia, the -23-

quantitative determination of which is mass. The quantity mass enters as a factor in the laws of motion which may be interpreted as a definition of mass.53 Under conditions created in electrical experiments a body participates in new physical phenomena, and for this purpose electric charge is assigned. Every new state variable which is assigned to material bodies makes the state variable a term to some functional relation which constitutes a law. The functional relation constitutes an implicit definition of the state variable which signifies the property. A qualification of the conventionalist doctrine is that in practice the law as definition is supplemented by the creation of alternative methods for the determination of a physical property. If weight is defined by the law of the lever, the lever is apparatus for comparing weights. An alternative method of comparing weights of bodies is provided by a spring which indicates by its extension the weight of a body attached to it. Philipp Frank, who so clearly interpreted the principle of causality as a definition of state, subsequently explained that scientific practice requires an independent operational definition.31 In the history of natural science the principle of causality is set forth as a generalization from experience. Thus there is an empirical basis for the principle. It then provides a pattern for further knowledge. Under the guidance of the principle that the same initial state is followed by the same sequence of states, scientific investigation seeks definitions of states which satisfy the principle of causality. The principle plays the role of definition in a negative sense. Initial states which are not followed by the same sequences are by definition not the same. Adherence to the principle of causality stimulates investigation to find definitions of states which conform to the principle. A new state variable may be defined in terms of its functional -24-

relations to previously known variables. But practice requires that implicit definition be supplemented by alternative specialized methods of determination.

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The conclusion of the present discussion is that the principle of causality is a canon of procedure for the future. The work of natural science is to make observations and conduct experiments, in order to define concepts for the description of things and phenomena so that the course of natural phenomena will conform to a principle of causality.

6. Extension of Principle of Causality This chapter has been devoted to the principle of causality as formulated for dynamical regularity. But it has been stated that statistical laws also exist. The repetition of an experiment on a set of systems may yield results that are described by a distribution law. The same cause then does not produce the same effect; the effects exhibit a statistical distribution. This situation confronts us in atomic phenomena. An example of statistical regularity is the transmission of characters of an organism by heredity. An organism is characterized in terms of unit characters, such as tallness or shortness of a plant, whiteness or redness of a flower, and so forth. The distribution of a unit character among offspring confirms the hypothesis that all unit characters are transmitted with equal probability. For example, if a flowering plant, Mirabilis Jalapa ("four o'clock"), with pure white flower (W) is crossed with one having a red flower (R), the filial generation (F1) consists of intermediate pink flowers. If these are mated with one another the next generation (F2) consists of pure white, pink, and red in the proportions 1:2:1, respectively. The empirically discovered distribution of unit char-25-

acters in the generation (F2) is readily explained by the hypothesis that the original sex cells of the pure white variety are of form (W, W), and those of the red variety of form (R, R). On conjugation of male and female cells the fertilized ovum acquires a character from each parent. Thus the cells of the first filial generation (F1) are of the form (W, R). In the second filial generation (F2) the cells receive a W or an R with equal probability. The possibilities of combination are (W, W), (W, R), (R, W), (R, R), all of which have the same probability of occurrence. The analysis in terms of equal probabilities explains the observed distribution (1:2:1) in this special case. It is a law of Mendelian heredity that transmission of characters conforms to such statistical regularities. A physical basis for the law of heredity is provided by the theory that unit characters are determined by genes within the chromosomes of a cell. In classical physics it was assumed that a statistical regularity is the

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manifestation of more basic dynamical regularity. As we shall see, however, quantum theory has introduced statistical regularities which are not reducible, but fundamental. The principle of causality must then be generalized so as to cover statistical regularity as well as dynamical regularity. A formulation of a general principle of causality is: If a system is prepared so that it is in a state of maximum determinateness, and if the same experiment is performed on the system many times, then except for chance fluctuations the same frequencies of distribution of results will occur, at all instants of time and positions in space. If a system is subject to dynamical causality, the results of an experiment will be the same for all systems, at all instants and positions. -26-

III

COGNITION OF CAUSALITY

1. Causal Strands in Nature T HE ESSENCE of causality is connection between two phenomena: an antecedent phenomenon, the cause; a subsequent one, the effect. In dynamical causality the same cause is succeeded by the same effect; in statistical causality the same cause is followed by a distributed effect. Historically, the term causality has signified dynamical causality, or regularity. The existence of regularities of connection is a prerequisite of objective knowledge and therefore of science. Upon regularities rests the possibility of reproducible observations of natural things. Methods of statistical description presuppose dynamical causality, for control of conditions in a statistical experiment depends upon the preparation of macroscopic states of systems by application of dynamical laws. To the first approximation, at least, dynamical causality is exhibited in macroscopic phenomena. In this chapter the analysis will be directed to modes of cognition of dynamical causality, or regularity. In the present context it is not necessary to raise the question as to whether or not causation involves efficacy. It will suffice to recognize that dynamical causation is manifested in invariable sequence: If the same initial state is realized, the same sequence of states will also be realized. The precise formulation of causality as regularity of se-27-

quence is in terms of functional relation between variables which describe the state of a system. The scientific aim to discover regularities in natural phenomena is confronted by the obstacle that states of nature as a whole are not found to recur. The realm of nature, however, may be represented as a superposition of individual causal

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strands. Cognition of causality is achieved by isolating and observing particular strands of causation. For the scientific investigation of nature there are two procedures: observation in an initial stage, and experimentation in a final stage. Nature herself has provided examples of approximately isolated systems which exemplify causality. As we have previously noted, the revolution of the moon around the earth and the revolutions of the planets around the sun are examples of sequences of phenomena that exemplify a principle of causality to a high degree of approximation. By observations on these heavenly bodies the laws of classical mechanics were verified. On the surface of the earth many systems can be found which are sufficiently isolated to exhibit causal sequences. Experimentation, however, is the preferred method for the discovery of causal processes. Through experimental control initial states of a system can be prepared and varied at will; subsequent states and variations thereof can be observed; the influence of external agencies can be eliminated or controlled. Galileo's experiment with motion on an inclined plane illustrates the possibilities of precise cognition of a functional relation which constitutes causality.

2. Observational Methods The function of natural science is to describe and explain the things, properties, and phenomena of nature. Every science must begin with an observational stage in -28-

which one perceives natural things, describes their properties, and establishes sequences of phenomena by induction. Methods of procedure appropriate to observation were formulated by Sir Francis Bacon. Bacon expounded scientific method within the frame of Aristotle's classification of causes as material, formal, efficient, and final. This frame of concepts is not the basis of contemporary philosophy of science, but in the interest of historical background I shall outline briefly Bacon's method of induction by enumeration. His proposal was to construct tables which listed respectively positive, negative, and comparative instances of a phenomenon. By examination of the tables and by exclusion one would determine the Form or essence of the phenomenon. For example, Bacon listed as positive instances of heat:3 flames, rays of the sun, fiery meteors, friction; from these he concluded that the form of heat is motion. The method of enumeration was not adequate for induction from controlled experimentation under the guidance of hypotheses. A transitional stage in the formulation of scientific procedures is represented by the methods of induction of John Stuart Mill. In opposition to the rationalist doctrine of Kant that causality is imposed upon nature by thought, Mill took the empiricist position that causal relations are found by experience. He interpreted causation in the manner of Hume as invariable conjunction of antecedent and

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consequent. By his methods of induction Mill took account of both observational and experimental methods in science. For the sake of further historical background I shall briefly epitomize Mill's Canons of Induction.65

3. Mill's Canons of Induction By the method of agreement Mill recognized the basic role of observation of particular phenomena for natural -29-

science. The method of agreement is based on his first canon of induction which in abbreviated form states: If the instances of a phenomenon agree in one circumstance only, that circumstance is the cause (or effect) of the phenomenon. The advantages of experimental procedure were recognized by Mill in his method of difference. This method is based on the second canon: If instances of a phenomenon agree in all circumstances with instances in which it does not occur, except that one circumstance characterizes only the former, the circumstance in which alone the two sets of instances differ is the effect, or the cause, of the phenomenon. In Mill's judgment experimental procedure exemplifies the method of difference. Mill also applied the method of difference in order to improve the method of agreement. His combined method of agreement and difference is based on the third canon: If instances of a phenomenon agree in one circumstance only, while instances in which it does not occur agree only in the absence of the circumstance, the circumstance in which the two sets differ is the effect, or the cause, of the phenomenon. In the field of biology there are limitations to the control of conditions of an experiment. An organism is characterized by integration of functions, so that if an individual function is appreciably changed the object of experimentation may be injured. The biologist therefore may use two sets of organisms for an experiment: an experimental group by which the effect of a specific agent is to be determined, and a control group of similar organisms which are not subject to the specific agent. If the members of the experimental group develop characteristics which are absent from the control group, then Mill's joint method of agreement and difference justifies the inference that the agent -30-

applied to the experimental group is the cause of these characteristics as effect. As an example, the problem may be to investigate the biological action of a specific vitamin on growth. The vitamin is fed to an experimental group of animals and their characteristics are observed. Except for the vitamin, similar

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food is fed to a control group. If the members of the experimental group develop characteristics which are absent in the control group, then it can be inferred that the ingestion of the vitamin causes the observed characteristics of the experimental group. That natural phenomena generally are the resultants of many superposed processes was acknowledged by Mill with his method of residues. This method is based on the fourth canon: If one subtracts from any phenomenon such part as is known to be the effect of certain antecedents, the residue of the phenomenon is the effect of the remaining antecedents. By this method Mill acknowledged the importance of mathematical analysis in the explanation of phenomena. A classical illustration of the method of residues is the prediction of position of the planet Neptune on the hypothesis that a residue of the perturbations of Uranus was caused by an unknown planet. Finally, Mill acknowledged that in experimentation phenomena are investigated under variable conditions. His method of concomitant variations is based on the fifth canon: Whatever phenomenon varies whenever another phenomenon varies, is either a cause or effect of that phenomenon. For example, if a material rod is subject to a variable temperature the length of the rod varies concomitantly. Precise description of variation as functional relation requires quantitative description of temperature and length. By the method of concomitant variations Mill recognized causality as functional relation between variables which specify states of a system. In a broad sense -31-

the method of concomitant variations includes the others, for presence and absence of a property constitute values of a variable property.

4. Experimentation Adequate cognition of causality in nature requires experimentation. By experiment a causal strand is isolated and the law of its process can be determined. The possibility of experimental isolation of a particular natural process rests on the circumstance that influences external to a system decrease rapidly with distance and that environmental conditions can be controlled. In experimentation initial states of a system are prepared, controlled, and varied and subsequent states are determined in the sense of being traced by description. An experiment may yield only qualitative results; thus constituents of a compound are recognized by the methods of qualitative analysis in chemistry. The goal of experimentation, however, is quantitative control of conditions with a view to numerical description of results. Representation of states in terms of results of measurement renders possible discovery of functional relations between the state variables of a system. Controlled conditions of experiment, and quantitative description especially, are characteristic of physical science. We

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have already examined the classical experiment of Galileo by which he verified the law of motion of a ball on an inclined plane. The constant incline of the plane provided constant modification of conditions of fall under gravity. The experimenter controlled initial time, initial position, and initial speed of the body. At successive instants after its release, the positions of the rolling ball were marked on the incline. The distance s of positions from the origin was found to be related to the corresponding instant t by the functional relation s = kt2. The conditions of Gali-32-

leo's original experiment were not as precisely controlled as would be possible today. Too great precision, however, would have obscured the simple law which holds for the ideal case of motion in a vacuum. From the law of motion under approximately controlled conditions on the plane, Galileo inferred the law of freely falling bodies for a vacuum. Exact method is introduced into the various fields of natural science by description of natural phenomena in physical terms. Indeed, a program for the unity of science has been founded on the principle that the terms of every natural science shall be defined or reduced to the terms of physical science. Controlled investigation in all fields of natural science is patterned after the methods of physical experimentation.15 The presuppositions and procedures of experimentation, therefore, are to be investigated by analysis of a physical experiment.

5. Frames of Space and Time The basic presupposition of physical experimentation is that physical things have position in space and that physical events occur in time. Space may be represented to consist of points, or positions, which determine the relations in which coexistent things stand to one another. Time may be represented to consist of instants which determine relations in which successive events stand to one another. Space and time are schemes of order for objects of study in physical science. The space of physical objects is homogeneous; no point of space is distinguished from the rest. The time of physical events likewise is homogeneous; no instant of time is distinguished from the rest. The homogeneity of space and time are presuppositions of a principle of causality. To prepare for an experiment the initial step is to introduce a scheme of reckoning for space and time. For space -33-

one must select a body of reference with respect to which position may be described. In the history of science the initial frame of reference for space was

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the surface of the earth or some structure of solid bodies with was attached to the earth. In a laboratory reproduction of Galileo's experiment with rolling bodies on an inclined plane, the material body which provides the incline also constitutes a frame of reference for space. But this frame rests upon a floor which in turn rests upon supports which are anchored in the earth. Measurements more precise than those of Galileo demonstrate that the earth is not an adequate frame of reference for the description of motion by classical mechanics. An adequate frame, and practically the one adopted by Copernicus for the description of the motions of planets, is a frame which is imagined to have the origin at the center of mass of the solar system and axes oriented with respect to the average positions of the fixed stars. For metrical description of position relative to the chosen frame of reference a standard of distance, or length, must be selected or constructed. The international standard is specified on a specially constructed solid body under specified conditions. Solid bodies are practically rigid: If two bodies are placed adjacent, so that separated points of one coincide respectively with corresponding points of the other, the coincidences are preserved during motion, and if interrupted by motion can be restored. A separated pair of points on a solid body determines a stretch. Two stretches are congruent if the endpoints of one can be made to coincide with the corresponding endpoints of the other. Distance is a relation between the endpoints of a stretch. The standard of distance is the relation between the endpoints of a specific stretch on the standard body; a unit of distance can be defined in terms of the standard of distance. The measure of distance between two points, or the length -34-

of the line joining them, is the number of unit stretches which will fill the line from one end to the other. By the determination of a standard of distance a metrical structure is imposed upon space. The metrical structure of the space of experiment is described by the propositions of Euclidean geometry. For description of events in time one must select an origin with respect to which the date of an event may be determined. The metrical description of events relative to an origin further requires that a standard of time-span be selected or constructed. Now there exist in nature physical processes which are repeated in space in durations of time that are declared equal by convention. Thus the earth rotates about its axes once every twenty-four hours; a pendulum performs a vibration in a constant period; a body under no forces traverses equal distances in equal spans of time. A clock is an apparatus which performs equal motions in space during equal spans of time. In a particular experiment, a specific instant as indicated by the clock may be chosen as origin of time. The instant of a subsequent event may be determined by perception of simultaneity of the event with a particular indication of the clock. Distance and span of time occupy a special status among physical properties. In a sense both are intuitively exhibited properties. A distance is measured by

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superposing upon it an equal multiple of the standard of distance. The measurement of time-span similarly requires superposition. The establishment of superposition for the cognition of equality is presupposed not to involve interaction between object of measurement and instrument. The perception of superposition is a macrophysical process and its idealization for theoretical purposes abstracts from gravitational or other action between superposed bodies. The presuppositions concerning space and time are -35-

basic for the principle of causality. The specification of states requires that one have measuring rods to measure distances in space and clocks to measure spans of time. It is a postulate of measurement that these instruments of measurement preserve their self-identity when transported in space and time. The distance between the endpoints of a standard measuring rod is invariant; the period of one specified motion of the clock is invariant. As Painlevé has emphasized, it is a fact of experience that with ordinary measuring rods and clocks one measures distances and spans of time so that, if the same state of a system is realized at different places and at different times, the same sequence of states will be realized.

6. Nature of Experiment As we have seen, measurement of spatial and temporal quantities occurs by superposition; thereby one determines identity of quantity measured with that of a standard. Physical properties otherwise are dispositional attributes that are manifested in interaction. Indeed, measurement of physical quantities is a form of experiment. I shall proceed directly to further analysis of experimentation. The essential character of an experiment is that some object is subjected by an apparatus to an interaction; from the space-time indications of the apparatus one determines the properties of the object. The object of investigation may be a field. The experiment by which Galileo demonstrated the law of falling bodies can be interpreted as one on the strength of the earth's gravitational field. The apparatus in Galileo's experiment consisted of a ball, a test body, and an inclined plane which reduced the motion and facilitated control of the initial conditions. Galileo found that distance traversed by the ball is proportional to the square of the elapsed time. -36-

From this law for the body and the laws of motion one can infer that the earth's gravitational field is uniform over a small region near the surface of the earth.

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The object of investigation may be the properties of a body, for example, an electrified particle. In this case the apparatus may include a magnetic field which deflects an electrically charged body in motion through the field. The path of the particle may be detected by projecting it through a chamber filled with saturated water vapor; upon expansion the water vapor condenses on ions that are created by the particle on collision. From the strength of the field and the radius of curvature of the condensation track, one infers by physical laws a functional relation between charge, mass, and speed of the particle. Interpretation of results of observation presupposes that space-time indications of apparatus are correlated by physical laws with properties of the object. In Galileo's experiment, positions of the rolling body at successive instants indicate constant acceleration, from which one infers the strength of the field by the laws of motion. In the experiment on an electrified particle, the condensation track indicates the curvature of path in an imposed magnetic field, from which one infers properties of the particle by the laws of motion and of the field. The illustrations which have been used to describe the nature of experiment constitute measurements of a physical quantity. We may cite further examples to show that an apparatus which is employed for measurement embodies some physical law. The spring balance is used to measure force by virtue of the law that extension of spring is proportional to load. The ammeter is used to measure electric current by virtue of the law that magnetic field is proportional to strength of current. A thermometer is used to -37-

measure temperature by virtue of the law that volume of a liquid is proportional to temperature. A physical system may be subjected to an experiment in which a number of quantities are measured at intervals during a span of time. Each measurement of a quantity constitutes a subexperiment. Now measurement requires that the object act on an apparatus; the apparatus then reacts on the object. This reaction of apparatus upon object should be made as small as possible, in order that the law for the object of the main experiment may be determined as accurately as possible. In classical physics it was assumed that the effect of apparatus in measurement could be indefinitely diminished. Classical physics operated with the concept of measure which precisely represents the state of a system at the instant of measurement.

7. Observation The completion of an experiment requires observation of the results. For example, one may read the position of a pointer on a scale. Observation may also consist in determining positions of permanent effects, such as lines on a film. The results provide data for theoretical interpretation.

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Observation may be analyzed into two factors: registration and perception. Registration is exemplified by the adjustment of a pointer to a specific position on a scale; also by the production of a permanent record upon a screen or other apparatus. Perception is exemplified by ordinary perception of a line on a photographic plate. Registration of effect on apparatus involves interaction between object of investigation and apparatus. According to theories of classical physics, the magnitude of an interaction could be made indefinitely small. The procedure of registration itself falls within the province of scientific theory. Registration involves correlation between state of -38-

the object and state of apparatus. Thus a condition of registration is that the principle of causality holds for the process; if this were not the case, a science of reproducible results would be impossible. Scientific theory is applicable also to the several stages of perception. According to the ordinary scientific account this process is as follows: Light is scattered by the record, travels through space, and is brought to a focus on the retina of the eye of the observer; nervous impulses are initiated in the retina, travel along the optic nerve, and finally give rise to processes in brain tissue. To the process in the brain is correlated the content of perception. The relation between perceptual content and brain process is a subject on which philosophers disagree. Regardless of theory, however, cognition of natural things presupposes that the content of perception stands in one-one correspondence through mediate physical processes with the record. J. von Neumann70 has expressed the situation by the statement that the principle of psycho-physical parallelism is a basic presupposition of science. -39-

IV CAUSALITY IN CLASSICAL PHYSICS 1. The Role of Mechanics T HE CONCEPT of causality was first given precise definition through the laws of classical mechanics. The field of mechanics is the motions of natural bodies which occur in space during time. Galileo contributed a theory of motion of bodies subject to gravity near the earth; Huygens discovered the mechanical properties of physical pendulums; Newton constructed a comprehensive theory upon laws of motion and supplemented it with a law of gravitation. The earlier adoption of a heliocentric frame of reference by Copernicus for the description of

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motions of the planets, and Kepler's discovery of the laws of planetary motion, prepared the way for the creation of a system of the world by Newton, a system which Whitehead has called the synthesis of the seventeenth century. Mechanics was extended by d'Alembert in the eighteenth century, and then was given generalized form by Lagrange in his Mécanique Analytique. Mechanics subsequently provided the basis for theories of other physical phenomena. The phenomena of light were attributed to vibrations in an ether endowed with mechanical properties. A mechanical theory of heat reduced heat to the energy of disordered motion of molecules and atoms which constitute a material body. Electric and magnetic actions were explained through the stresses and motions of -40-

mechanical models. In the nineteenth century it became the ideal to reduce all physical phenomena to those of motion. Causality in mechanics was the model for classical physics, indeed, for natural science in general. A metaphysical theory of materialism sought to explain life by the interactions of material corpuscles.

2. Ancient and Modern Mechanics The field of mechanics is the motion of material bodies. A body moves in space during time under material conditions and in conformity with laws of motion. The classical laws of motion satisfy the principle of causality: If the same initial state is realized at any time in any place, the same sequence of states occurs. Space and time are presupposed to be homogeneous: the laws do not depend explicitly on position and date of motion. The form of causality in mechanics depends on the concept of state of a moving body. Our understanding of modern mechanics will be facilitated if we compare it with ancient mechanics, as represented by Aristotle and expounded by Scholastic Aristotelians.67 Ancient and modern mechanics differ in their conceptions of state of a moving body. According to the theory of Aristotle, the state of a terrestrial body depends on its position. Thus force is required to maintain the velocity of a body. If forces ceased to act on a moving body, the body would stop. According to modern classical mechanics, the state of a body is determined by its position and velocity. Force changes the state of a body by changing its velocity as well as its position. Thus force is required to maintain the acceleration of a body. If forces ceased to act on a moving body, the body would continue to move uniformly with the same velocity as at the instant of cessation of force. The characteristic of modern mechanics which essen-41-

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tially distinguishes it from the ancient is the principle of inertia. This principle was expressed in complete generality by Newton in the first law of motion: A body continues in its state of rest or of uniform motion in a straight line, except insofar as it is compelled to change that state by impressed forces. Newton's first law characterizes force as that which changes the velocity of a body. Force, which represents the influence of surrounding bodies upon a given body, is characterized as cause of change of position in ancient mechanics, as cause of change of velocity in modern mechanics.

3. Laws of Motion Newton based classical mechanics on three laws of motion. The first law is the principle of inertia; the law ascribes to material bodies the property inertia, the disposition to continue in a state of rest or of uniform motion. The physical quantity mass is inertia specified as measurable. One introduces the physical quantity momentum which depends jointly on mass and velocity. The dependence of motion on force is then expressed by the second law: Timerate of change of momentum is proportional to the force acting. In classical mechanics mass is independent of velocity, so that rate of change of momentum is the product of mass and rate of change of velocity, or acceleration. The second law can then be stated as: The product of mass and acceleration is proportional to the force acting. By suitable choice of units one obtains as the fundamental equation of mechanics ma

=

F.

The third law is: To every force there is an equal and opposite reacting force. Thus the reciprocity which is exemplified by the collision of two balls is recognized to hold generally for causality. -42-

4. Force as Cause It is a fact of experience that the acceleration of a given body is subject to the presence of external bodies which are sufficiently near to it. The second law appears to state that the external bodies exert forces which cause the acceleration of the given body. The critical standpoint represented by Appell discards force as cause and defines the term force as a symbol for the product of mass and acceleration. In the present work I shall interpret the force on a given body, neither as an activity nor as a mere symbol, but as a characteristic of the external bodies in the environment of the given body. The equation ma = F provides a blank form for F, which takes on different forms under different conditions. The common property of force resides in the circumstance that the product of mass and acceleration is proportional to it.

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If a body of mass m is attached to an extended spring and the position of the body is specified by coordinate x, the concrete equation of motion is ma = -kx. Appell defines force as ma; I define force as -kx. F = -kx designates the action of the spring upon the body. This force involves two factors, the elastic constant of the spring k, and the coordinate of position x which also represents the extension of the spring. The expression -kx represents the force as a linear field which originates in the spring. The significance of the equation ma = F resides in the circumstance that in physically interesting cases, force can be expressed as a simple function of factors in the environment of the body. In the present example, the coordinate x which specifies extension of spring also specifies position of the body. One can therefore interpret the equation of motion to express the acceleration of the body as a function -43-

of its position. In this form the concepts of force and of cause in an uncritical sense do not enter. Causality becomes functional relation.

5. Differential Equation of Motion An equation of motion in mechanics states that the instantaneous acceleration of a body is a function of its state, as specified in terms of position and velocity, and possibly of the time. Since acceleration is time-rate of change of velocity, which in turn is time-rate of change of position, the equation of motion is expressible as a differential equation. If x is positional coordinate which depends on time t as independent variable, the differential equation states that the second derivative of x with respect to time is a function of time, position, and velocity:

The general solution of the differential equation expresses the coordinate x as a function of the time and two arbitrary constants which may be determined from initial values of coordinate x and its rate of change x + ̇. In the example of a body falling under gravity, the acceleration is a constant. A first integration yields an expression for velocity as a linear function of the time. A second integration yields the coordinate of position x as a function of the square of the time. This is the functional relation, which in the integral form s = ½gt2, was demonstrated to hold by Galileo's experiment with motion on an inclined plane.

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In classical mechanics causality is functional relation which is expressed by natural laws in the form of differential equations and solutions thereof. A differential equation expresses the law of motion in the form: rates of change of state variables are functions of the state. The -44-

solution expresses the law in the form: state variables are functions of the time and of arbitrary initial conditions. At first sight it appears that all reference to force has been eliminated. However, the functional relation involves empirical factors which depend on conditions of the motion. In the differential equation for falling bodies the second derivative of coordinate with respect to time is equal to the constant g. This constant is the acceleration of gravity, but it is also intensity of gravitational field which depends on the mass of the earth and its radius. In uncritical terms, the cause of the second derivative of coordinate for a body of mass m is the force mg which is exerted upon it by the earth. Force may also be expressed in terms of potential energy. The work of a force is defined as the product of force and component of displacement in the direction of the force. The potential energy of a body at a given position in a field of force, and with respect to a standard position, is the work which can be done by the field as the body is displaced from the given to the standard position. The force exerted by a field upon a body can be expressed as the negative gradient of the potential energy, that is, as the negative rate of change of potential energy with distance in the direction for which it is a maximum. The kinetic energy of a body is its capacity to do work in virtue of its motion. In generalized mechanics the state of a system is described in terms of generalized coordinates of position and their rates of change. It is possible to express the kinetic energy as a function of the generalized state variables, and to form a function, the Lagrangian function, which is the kinetic energy minus the potential energy. In terms of the Lagrangian function one formulates differential equations of motion for the generalized coordinates as functions of the time. -45-

It is further possible to define generalized components of momentum which are correlated with generalized coordinates; generalized coordinates and conjugate components of momentum then become the state variables. One may then form a Hamiltonian function H which is usually the sum of kinetic energy and potential energy expressed as functions of coordinates and conjugate momenta. In terms of the Hamiltonian function one formulates the differential equations of motion for the coordinates and momenta as functions of the time.

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Causality in classical mechanics is functional relation which may be expressed by a differential equation or by its solution. As previously noted, Meyerson interpreted causality as identity which he distinguished from legality, or conformity to law. Identity, however, can be subsumed under lawfulness. During the motion of a system which conforms to the differential equations of motion, specific functions of the state variables remain constant. From the equations of motion one deduces laws of conservation which express identity, that is, equality of cause and effect. We have previously noted that in a collision of two bodies there is conservation of momentum. It follows quite generally from the second and third laws of motion that the total momentum of an isolated system is a constant of motion. If a system is subject to forces which are expressible in terms of the negative derivatives of a potential energy, total energy is also a constant of the motion. An example of conservation of energy is the motion of a body which is projected upwards from the surface of the earth. Let s be the distance of the body from the earth, v the speed at any instant, and a = -g the acceleration, which is negative because the upward direction for s is positive. -46-

Neglecting the resistance of the air, the sufficiently accurate equation of motion is

,(1) which states that the product of mass and time-rate of change of velocity is equal to the weight of the body. The weight is directed downwards and therefore is represented as negative. One can transform rate of change of velocity

, so that the differential equation becomes

. The solution of this equation is

(2) If the initial conditions are: At s = s o , v = v o , the constant is determined to be

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. Substituting this constant in the solution (2), and rearranging terms, one finally obtains

. (3) From the differential equation which expresses the law of motion we have derived a function of the state variables,

, which at any stage of the motion is equal to the initial value. Now, energy is defined as the capacity to do work, i.e., to exert a force in a displacement. The quantity ½mv2 is called kinetic energy and expresses capacity to do work by virtue of motion. The quantity mgs is called potential energy and expresses capacity to do work -47-

by virtue of position above the surface of the earth. If a body is raised above the earth, work must be done by a force to overcome the weight of the body; this work is regained from the weight on return displacement to the surface. Total energy is the sum of kinetic energy and potential energy. The equation (3), then, may be interpreted to state that the total energy of the projected body, in the state described by values of variables v and s, is equal to the energy for an initial state described by values v o and s o . Thus a solution of the equations of motion, and definitions of kinetic and potential energy, yield the law of conservation of energy for motion of a body under gravity near the earth. Equality of cause and effect, which is expressed by conservation of energy, is thus demonstrated to be a consequence of conformity to the law of motion which is expressed by a differential equation. The correlation between causality as identity and as functional relationship may be expressed in general terms as follows: The properties of a mechanical system can be expressed by functions constructed from kinetic energy and potential energy. One formulates the differential equations of motion for generalized coordinates by a Lagrangian function which is the kinetic energy minus the potential energy. One further formulates equations for coordinates and momenta

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by a Hamiltonian function which is usually the total energy. The concept of force as expression of causality is not exhibited explicitly when a system is described in terms of a Lagrangian or Hamiltonian function. Causality is functional relationship which is expressed by the differential equations or their solutions. The differential equations have integrals, each of which is a function of the variables of state and is equal to a constant during the motion. If the time does not occur explicitly in the basic functions for the system, there are integrals which do not -48-

involve the time. The case of motion under the action of gravity exemplifies conservation of energy as an integral of the equations of motion which does not involve the time. As a further contribution to the theory of integrals I cite the problem of an isolated system of particles which exert gravitational forces upon one another. The system of differential equations of motion for this case admits ten classical integrals. In view of three independent modes of motion in space, there are ten integrals: integral of energy, three integrals of components of momentum, three integrals of components of moment of momentum, and three integrals which express constancy of components of velocity of the center of mass. These integrals express conservation during motion and thus express identity in time in the sense of Meyerson. Causality as identity is an aspect of causality as lawfulness.

7. Reversible Motion The motion of a mechanical system is reversible, if it is conservative, i.e., if the forces are derivable from a potential. If at any instant the instantaneous velocities are reversed in direction, the system will pass with reversed velocities through configurations previously occupied in the original motion. For example, if a body moves along a straight line under no forces its velocity will be constant. The motion will be reversed by reversal of velocity at any time. If in the equations of motion for a conservative system the time as independent variable is replaced by its negative, the equations remain unchanged. The symmetry of motion with respect to time is a further expression of identity. The motions of observable natural systems are subject to frictional resistance which depends upon velocity. The motions of such isolated non-conservative systems are irre-49-

versible. In the previous example of the body which moves along a straight line, if it is subject to frictional resistance it will gradually slow down. If the velocity is reversed at any instant, the body will move in the reverse direction with

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decreasing speed. The motion which is initiated by reversal of velocity does not retrace the states of the original one in an opposite direction.

8. Mechanics of Fields Thus far we have considered causality in classical mechanics, for which the states of individual particles and systems are described in terms of positions and velocities, or momenta. The dependent variables, coordinates and momenta, may be called individual functions of the time as independent variables. For the description of a continuous material medium in space it is advantageous to employ field variables. The space occupied by the medium is called a field which is described by quantities, such as displacement, velocity, density, and stress which are expressed as functions of position and time. These field functions are employed in the theory of continuous media in preference to the individual functions of the mechanics of particles. The state of stress, which is the force per unit area which one part of a continuous medium exerts upon a contiguous part, is expressed as a field function of coordinates and time. The field functions satisfy partial differential equations in which coordinates of position and time are independent variables. Solutions of such equations for a region and a span of time are determined by the field throughout the region at an initial time and by conditions on the boundary during a corresponding span of time. The equations of motion admit solutions which represent the propagation of -50-

the field variables through space during time by wave motion with a finite velocity. The partial differential equations for field variables of a continuous medium express causation as contiguous action in space and time. Thereby we present the most adequate formulation of dynamical causality in classical physics.

9. Classical Microphysics The physical laws which have been cited as examples of causality refer to macrophysical, that is, large-scale phenomena. Macrophysical laws are exemplified by Newton's laws of motion, Maxwell's equations of electromagnetism, and the laws of thermodynamics. The physical scientist also undertakes to reduce large-scale to fine-scale phenomena, macrophysical to microphysical processes. Such reduction is the aim of atomistic theories in the general sense of the term. The program of microphysical theory is to postulate laws for microphysical processes and then to derive laws for macrophysical phenomena. The hypothetical laws of microphysics initially were obtained by applying classical macrophysical laws to microphysical objects.

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Thus a kinetic-molecular theory for a gas was based upon the constructive hypothesis that a gas consists of a large number of molecules which move and interact in conformity to the laws of classical mechanics. In particular, the laws of conservation of momentum and energy were postulated to hold during a collision between molecules. The molecules of a gas are required for theory to be so large in number and small in size that they are not directly perceptible. It is neither possible to solve the equations of motion for a single molecule nor to observe its state at an instant. However, it is averages of resultants of molecular quantities which determine the observable properties of a gas. Such averages can be calculated with -51-

the aid of statistical distribution laws which can be derived by considerations of probability. The inference from microphysical averages to macrophysical quantities requires postulates of correlation. For a gas two postulates of correlation are: 1) pressure of a gas in a state of equilibrium is the average time-rate of transfer of momentum per unit area; 2) temperature of a gas is proportional to the average kinetic energy of the molecules per degree of freedom. From postulates for microphysical processes and postulates of correlation it is possible to derive classical macrophysical laws, for example, the general gas law that the product of pressure and volume of a gas is proportional to the temperature as measured on an absolute scale. The reduction of the macrophysical to the microphysical was successful only with qualification. Macrophysical laws which had been deemed to be dynamical regularities were transformed into statistical regularities. A macroscopic state is a resultant of molecular processes which fluctuates about an average value. Hence macrophysical functional relations hold only on the average for observable phenomena. In view of the large number of molecules in a material system, the fluctuations are negligible for most practical purposes. Strong fluctuations are improbable but possible. A dynamical regularity of macrophysics is explained as a statistical one on a microphysical foundation. The second law of thermodynamics is an example of transformation of dynamical into statistical regularity. The second law states: Natural processes within an isolated system tends to go irreversibly in a unique direction. Irreversible processes are exemplified by the flow of heat from a region of higher to one of lower temperature, by diffusion, and by the conversion of mechanical energy into heat through friction. -52-

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A macroscopic state for thermodynamics is determined by the distribution of molecular quantities. For microscopic theory there is assigned a probability to a distribution. This probability is expressed in terms of the number of ways in which molecules can be assigned molecular properties so as to realize the distribution. Entropy in thermodynamics is proportional to the logarithm of the probability of distribution which determines the macroscopic state. The probability of a state is a measure of its disorder, or randomness with respect to molecular properties. The statistical mechanical interpretation of the second law is that the state of an isolated system probably will change from less to greater probability, from order to disorder, from less to greater entropy. The state of equilibrium is one of maximum probability, of maximum disorder, and of maximum entropy. Thus the second law of thermodynamics becomes a statistical law. It is only probable that natural processes within an isolated system will go irreversibly in the direction of degradation of energy. Classical atomistic theories have been illustrated by the kinetic-molecular theory of matter. Maxwell's equations of the electromagnetic field were derived by Lorentz from the hypotheses of a theory of electrons and thus offer another example of the reduction of macrophysical phenomena to microphysical processes. The hypotheses for microphysical processes were obtained by extrapolating the laws for macrophysical phenomena to the microscopic realm. Classical microphysics exemplified causality as functional relation which expresses dynamical regularity for microphysical elements of physical reality. In the subsequent discussion of the theory of quanta it will be shown that statistical causality is required in the foundations of microphysics. -53-

V

CAUSALITY IN BIOLOGY

1. Physical Explanation of Vital Processes IN THE INTRODUCTORY discussion of causality the hypothesis was advanced that man initially became aware of causation through efforts of his own. Primitive thought interpreted nature as constituted of living things akin to man. This animistic conception of nature was given its most picturesque formulation in the personification of natural forces. Thus the original model for explanation of natural phenomena was the activity of life. In the modern era, however, scientific method as exemplified by physical science has worked to purge the concept of causality of its animistic element. Causality has become functional relationship which is most adequately expressed by the differential equation.

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The biological sciences have advanced towards the status of exact natural science by the gradual explanation of vital processes in terms of the laws of chemistry and physics, ultimately in terms of the laws of physics. Living things have a material basis which is constituted of complex molecules of the element carbon. An organism is the seat of physico-chemical processes which require nutrients; such as air, water, salts, and constituents of plants and animals. The organism exchanges energy with its environment and thereby is subject to the laws of thermodynamics. The phenomena of heredity are explained in terms of the causal action of specific factors, the genes, -54-

which are analogous to the atoms of the chemist. Biological phenomena thus offer examples of causal processes like those of physical science. It remains an open question whether or not life can be completely reduced to physical processes. Aristotle, in his doctrine of the four causes, assigned the basic role for the explanation of biological phenomena to the final cause. It was the τέλς, or end, that determines development from potentiality to actuality. Windelband has described the system of the world which was based on the mechanics of Newton as a mechanistic despiritualization of nature. Nevertheless, Kant,47 who may be described as the philosopher of causality in classical mechanics, declared that the organism cannot be reduced to mechanical principles. The eminent physiologist Claude Bernard4 stated, "The vital force directs phenomena which it does not produce; the physical agents produce phenomena which they do not direct." Recently Hans Driesch23 has expounded a philosophy of the organism. In the effort to explain characteristic vital processes of self-regeneration Driesch proposed the hypothesis of an entelechy which controls the processes of life. Vital force and entelechy have not been amenable to experimental control, and therefore fall outside the scope of causality as it is exemplified in natural science. It is justifiable to assert that a materialism, such as was held in the nineteenth century and which explained life in terms of mechanical interactions of simple, indestructible atoms, is no longer available as a possible basis for biology. Physical science itself has completely abandoned such simple ideas. Contemporary physics offers theories about the structure of elementary particles which have wave as well as corpuscular properties. The objects of microphysical study have properties and conform to laws which are far removed from the mechanisms of an earlier era. -55-

Whether or not these new concepts and laws will serve to explain the laws of biology in terms of physics is undecided. In view of the continuing development

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of physical concepts the outcome cannot be predicted. In the following I undertake only to present some examples of causality in the biological field.

2. The Laws of Thermodynamics A living thing, a biological system, is first of all a portion of matter that is enclosed within a boundary which separates it from the rest of nature. The organism interacts with its environment: it receives new matter which is employed to form new tissues, and produces waste which is eliminated. The organism acts on other things through the agency of parts of its body; thus a human being applies forces through his arms and legs. Interaction between living thing and natural habitat involves application and transformation of energy. The general transformations of work and energy are the subject of thermodynamics. Accordingly, the energetic processes of living things conform to the laws of thermodynamics. An organism receives heat and other forms of energy from external things, and also expends energy in the performance of mechanical work. The transformations of energy in which biological systems participate therefore fall under the first law of thermodynamics which expresses the principle of conservation of energy. In order to formulate the first law, a thermodynamic system is characterized by an internal energy which is a function of its state, and also by external forces which the system exerts upon natural things external to it. The first law, which presupposes the mechanical theory of heat, states that the gain in internal energy of a system is equal -56-

to the mechanical equivalent of the heat added plus the work done on the system. The green plant stores energy in carbon compounds which it manufactures from carbon-dioxide and water by photosynthesis. From the reactants carbondioxide, water, and radiant energy are produced carbon compounds, oxygen, and heat through the agency of chlorophyll of the green plant. Conversely, an organism which feeds upon plants expends energy by oxidation. Thus the carbon compounds and oxygen react with the consequent production of carbondioxide, water, heat, and work. The second law of thermodynamics describes the direction in which natural processes in an isolated system tend to go. In its status as a law of thermodynamics the second law is a macrophysical one. It states that natural processes are irreversible; the characteristic irreversible process is the flow of heat from a place of higher to one of lower temperature. In its status as a theorem of statistical mechanics the second law states that a system of microphysical elements probably will go in the direction of increasing disorder. A

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difference in temperature of two bodies manifests a type of order which permits the transformation of heat into useful work. If two bodies come to temperature equilibrium, their heat in the form of molecular energy is no longer directly available for the performance of mechanical work. At first sight the processes of an organism appear to violate the second law. The organism receives materials and energy from external sources and creates orderly structures. The processes of life appear to reverse the trend towards disorder. An organism has been said to receive streams of negative entropy. Phenomena in an organism can be brought into conformity with the second law by extending the system. A -57-

living thing which builds order exchanges materials and energy with its environment. The creation of ordered stores of energy within the organism requires that it receive radiant energy from the sun. In order to study an organism from the thermodynamic point of view the system under consideration must include the sun and other environmental influences. It is scientific opinion, as represented by the physiologist Harold F. Blum,6 that if order increases in one part of the total system there is a compensating decrease in the order of the whole. The system which consists of organism and its environment exhibits irreversible change in conformity to the second law.

3. Physical Basis of Metabolism The unit of living things is the cell. Processes of life are constituted by the activity, growth, and multiplication of cells. The basic process of a cell is metabolism: this process is constituted of physico-chemical reactions, in the maintenance of which the cell receives nutrient materials from its environment and eliminates waste materials thereto. The results of mathematical biophysics which have been obtained by N. Rashevsky79 and others demonstrate that the processes of cells conform to physical and chemical laws. For a theoretical discussion the cell is assigned simplified properties. This procedure corresponds to the kinetic theory of gases for which molecules have been conceived as smooth, elastic spheres. As in kinetic theory, representation of biological reality by an idealized model makes possible formulation of functional relations which express causality for the processes of life. The absorption and elimination of materials by a cell requires transport of materials. Nutrient materials for a cell consist of air, water, the organic constituents of plants and animals. Transport of materials occurs by diffusion, -58-

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which as a large-scale process conforms to a differential equation. The equation expresses the law of diffusion and exemplifies contiguous causality in space and time. The law of diffusion involves the concept of concentration which is represented by c and is defined as the mass of material per unit volume. Material of a given kind diffuses from a place of higher concentration to one of lower concentration. The gradient of the concentration is the rate of change of concentration with respect to distance in the direction in which the rate is a maximum. As a first approximation, the equation of diffusion is based on the hypothesis that the rate of transport of material in mass per unit time per unit area is proportional to the negative gradient of the concentration. That is, if K is the coefficient of diffusion, the rate of transport in the direction of an axis for x is expressed by

. The negative sign indicates that transport occurs in the direction of decreasing concentration. The equation of diffusion is derived by equating two expressions of increase in material in an elementary volume. In order to sketch the derivation, let us suppose that diffusion proceeds in an x direction and calculate the rate of increase of material in a rectangular element of volume of length dx and area A. Across the boundary with coordinate x, at which the gradient is

, the rate of transport of mass is

. Across the boundary with coordinate x + dx, at which the gradient is

, the rate of transport of mass is

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. The net rate -59-

of transport of mass across the boundaries into the volume element is

The mass within the element of volume is the product of concentration c and the volume dx A. Hence the time-rate of change of mass within the element is

. By the law of conservation of matter, the net mass of the material transported across the boundary is equal to the increase in mass within the volume, so that by equating the above expressions one obtains for the one-dimensional case the diffusion equation.

4. Kinetic Theory of Diffusion Non-uniform concentration of nutrient materials and waste products is the condition of transport of materials required for the metabolism of the cell. The mathematical biophysicist Rashevsky reports a kinetic theory of diffusion which has been proposed by H. D. Landahl.80 The problem is to explain the forces which act on each element of volume of a system in consequence of diffusion. It is postulated that the metabolic material exists as a dilute solution, so that the molecules of the solute behave like those of a gas. On account of uncertainties of a kinetic theory of liquids, the solvent is treated as a dense gas. A molecule of a solute which varies in concentration is subject to unsymmetrical bombardment from neighboring molecules. The changes in velocity of the molecules are interpreted as the action of force. For theoretical discussion a cell may be assigned a spherical form and placed in -60-

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an infinitely extended liquid in medium. On account of surface tension a droplet of liquid tends to assume a spherical form. The forces created by non-uniformities of concentration may cause a liquid system to assume a non-spherical form. As the concentration of a material changes while the cell produces or consumes a material, forces are generated which act outwards on elements of the cell. These forces tend to disrupt the cell; when a critical size is reached the disruptive forces produce a spontaneous division of the cell into two halves. The size at which cell division occurs has been calculated and found to be of the order of magnitude of actual living cells. The condition for the spontaneous transition from single cell to two half cells is that during division the work done by the forces be positive. Positive work implies that the energy of the system decreases, a result which conforms to the principle of mechanics that the condition of stable equilibrium is one of the minimum potential energy. The preceding discussion shows that metabolism provides a sufficient basis for cellular growth and multiplication. Other factors in the process are change of surface tension and electric charge.

5. The Macrophysics of the Nervous System Contemporary psychology explains conscious activity in terms of functions of the central nervous system. A problem in mathematical biophysics is the formulation of laws for processes of the nervous system. The unit of the nervous system is the neuron; it consists of a cell body to which are attached filaments. Among these filaments is the axon which has branches that terminate in small bulbs. Through the terminal bulbs of its axon a neuron makes contact with the body of another neuron; -61-

such contact between neurons is called a synapse. A group of neurons is called a neuroelement, the excitation of which determines the gross behavior of an organism. Physiology provides elementary laws which govern the interaction of neurons. Through the action of a stimulus a neuron may be excited or "fired." The elementary process of excitation is of short duration and is a localized phenomenon. By contiguous causality the excitation travels along the axon and upon reaching a synapse may excite a neuron of higher order. To produce excitation a stimulus must exceed a threshold. The "all-or-none" law is that once the threshold is exceeded the intensity of the consequent excitation is independent of the stimulus. Excitation is accompanied by some metabolic change and is measured in physical terms by an electric action potential.

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It is an objective of mathematical biophysics to determine the laws for the interaction of neuroelements from the laws for the interaction of neurons. Rashevsky points out that the laws for neuroelements, which contain a large number of neurons, are statistical laws like those for macrophysical phenomena which have been reduced to microphysical processes. We may illustrate causality in the nervous system by a theory of excitation of neuroelements as expounded by Rashevsky.81 Let S be intensity of stimulus, L the threshold, and E intensity of excitation which is defined as the average number of excitation impulses in a neuroelement per unit time. Then for moderate stimuli a first approximation to a law is that intensity of excitation is directly proportional to the excess of stimulus over threshold, E ∝ (S - L); for S

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