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
nitely increasing stimulus expresses an exponential approach to the maximum. Suppose that the neuroelement leads to a region of the brain which contains a large number of neural circuits. Let N0 be the total number of circuits, excited and nonexcited, in the given region; let ε be the number of excited circuits which is initially null. Then the law of excitation is that the rate of change of number of excited circuits, ε, is proportional to intensity of excitation E and to the total number of unexcited circuits,

where A1 is a constant. If one assumes that the total number of circuits is large in comparison with the number of excited ones, and sets A1 N0 = A, one obtains

The foregoing result must be supplemented by including cessation of excitation. The decay of excitation in neural circuits conforms to a statistical law which is of the same form as that for radioactive atoms. A neural circuit is constituted by neurons which form a closed circuit; a neuron may synapse with another, and

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the latter in turn synapse with the first. Once excited, a circuit may remain in such state while the pulse of excitation continues to travel around the circuit. The excitation ceases through failure of sufficient numbers of terminal bulbs to maintain contact at a synapse. Cessation of excitation in an individual circuit is a matter of chance. For a large number of circuits, whose excitations cease by chance, the rate of "death" of excited circuits is proportional to the number thereof. If ε denotes the number of excited circuits and if a is a constant, the law -63-

of decay for the circuits is expressed by the differential equation

(2) We combine equations (1) and (2) and obtain

(3) A neuroelement may also inhibit circuits. If j is the number of inhibited circuits and B and b are appropriate constants, then

(4) Equations (3) and (4) provide a basis for a theory of excitation.

6. Microscopic Theory of Neural Circuits In the foregoing theory of neuroelements the statistical properties of large numbers of neurons were formulated as in macrophysics. From the microscopic point of view, the elementary processes of neurons are intrinsically discontinuous. An analysis of individual neural processes has been made by W. S. McCulloch and W. Pitts 58, 82 in terms of the operations of disjunction, conjunction, and negation of Boolean algebra. This analysis takes account of the discontinuous and all-or-none character of synaptic transmission of excitation. A number of simplifying assumptions are made. It is assumed that all synaptic delays are constant; the synaptic delay is used as a unit of time. The time required to conduct an excitation along a neural fiber is neglected. An example for analysis is the synapsis of a neuron designated N1 with a neuron

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designated N2. Let us assume that for excitation of N2 two terminal bulbs of N1 must -64-

make contact with the body of N2. A neuron is said to "fire" when it becomes excited. The proposition which expresses occurrence of excitation is: N2 fires at an instant t if, and only if, N1 fires at an instant earlier by the synaptic delay which is the unit of time. In symbols, N2 (t) ≡ N1 (t - 1). Let us next suppose that either of two neurons, N1 and N2, may cause a third neuron N3 to fire. Then action is expressed by the proposition: N3 fires at instant t if, and only if, either N1 or N2 fires at the earlier instant t - 1, N3 (t) ≡ N1 (t - 1) + N2 (t - 1). Another physical process is inhibition of one neuron by another. Then if N3 can be excited by N1 and inhibited by N2, action is expressed by: N3 fires at instant t if, and only if, N1 fires and N2 does not fire at the earlier instant t - 1.

In the preceding examples it was supposed that neurons N1 and N2 each had two terminal bulbs. Suppose that each of these neurons has only one terminal bulb, so that both neurons are required to excite N3. Action is expressed by: N3 fires at instant t if, and only if, neurons N1 and N2 both fire at the earlier instant t - 1. N3 (t) ≡ N1 (t - 1) · N2 (t - 1). Given a neural net of any complexity, its property always will be described by the proposition: Neurons N1, N2, . . . . fire at times t1, t2, . . . . respectively if, and only if, other neurons N1′, N2′, . . . . fire at times t1′, t2′, . . . . respectively and other neurons N1″, N2″, . . . . do not fire at times t1″, t2″, . . . . respectively. Thus a proposition about a neural net may be decomposed into a series of propositions which are connected by the operations of disjunction, conjunction, and negation of Boolean algebra. According to -65-

Rashevsky, a limitation of the foregoing analysis is that neurons are required to fire at intervals which are integral multiples of synaptic delay. Causal description of a phenomenon requires specification of states and

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expression of a functional relation between them. The state of a neuroelement at a given time is specified in terms of afferent stimulation and of activity of constituent neurons, each an "all-or-none" process. The functional relation is provided by specification of the neural net. From a given initial state one can infer the properties of a succeeding state. The occurrence of disjunctions in the descriptive analysis of neural processes prevents complete determination of the preceding state. The typical example of dynamical causality in classical physics renders possible inference to the past as well as to the future. In the realm of neural processes, however, causality is asymmetrical. Further, the activity of neural circuits is regenerative, which circumstance renders indefinite any reference to past time. Accordingly, McCulloch and Pitts conclude that knowledge of ourselves and of the world is incomplete with respect to space and indefinite with respect to time. This conclusion is confirmed, for example, by comparison of individual recollections with contemporaneous records.

7. Cybernetics In his pioneer work Cybernetics Norbert Wiener91 has sketched the development of physical devices which perform computations, predict the future, and serve as guides to physical operations. He points out analogies between the computing machine and neuroelements. The circuit of neurons has its analogue in the vacuum tube which is the unit of these machines. The synapse, which determines whether or not a combination of elements is sufficient to cause a next element to fire, has its analogue -66-

in the computing machine. The all-or-none character of excitation of neurons has its analogue in the choice made in determining the digit on the binary scale of a computing machine. Indeed, the employment of Boolean algebra for analysis of neural processes finds its analogue in the application of Boolean algebra to computing machines.

8. Order in Biology An organism like all material systems is constituted of molecules, which in turn are complex structures of simpler elements. Certain constituents of biological reality form groups which determine the characteristic nature of biological systems. The unit of life is the cell, within which are molecular structures that determine the traits of the organism. There are resemblances among organisms. Certain kinds of organisms are classified to form species; those species with certain common characteristics are united in larger groups. The systems of classification in biology indicate that an orderly arrangement exists among all forms of life. The biological basis of such relationship is common ancestry and evolution. Biological character manifests an

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essential degree of stability together with minor variability. The stability and variability of traits of organisms is explained by the theory of heredity. In this theory the gene is the causal agent which determines organic traits. A theory of genes is a biological atomism which serves to explain the continuing stability of kinds in terms of contiguous causality. Genes are subject to mutation through the causal action of external agents such as radiation. Thereby is explained the variability which characterizes organisms as well as stability. The genes are pictured as particular factors which are located at definite positions in the chromosomes. In prepa-67-

ration for cell division each chromosome splits lengthwise; when the cell divides each of the daughter cells carries with it a full complement of chromosomes. At times a series of cell divisions produces gametes or sex cells. In this process the chromosomes are distributed according to chance and in consequence all gametes do not carry the same genes. Free play for chance is further provided by the union of two gametes to form a fertilized egg, which then develops into a new organism. Researches in genetics have demonstrated that the transmission of genes in reproduction and evolution is subject to the laws of probability. Statistics is the basis of a theory of order in biology.

9. The Template The fact of order in living things has inspired attempts at scientific explanation. The basic problem is reproduction of those structures in which the protein molecule plays a fundamental role. A theory has been proposed that proteins are built to a pattern, or template, 6 which exists in a living cell. By hypothesis these templates are protein molecules, or parts thereof; a new protein molecule is fashioned on the template which serves as pattern. In view of the high degree of stability of kinds of organisms during evolution, it is further postulated that these templates have existed and have been reproduced for millions of years. The theory that protein molecules are patterned on a template employs the language of the foundry and the stamp mill. Mechanics, in the primitive sense of a theory of machines and which has been rejected for fine-scale, physical processes, now provides a model for biological phenomena. This recourse to pure mechanism has been described by Schroedinger86 in his book What is Life. He contrasts the basis of order in biology with that in physics. Order in macrophysical phenomena is founded -68-

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on the combination of an enormous number of constituent processes which individually act in a random manner. Except for minor fluctuations, the averages of resultants of microphysical processes conform to dynamical causal laws. Thus order arises out of disorder. By contrast, the genes of biology consist of a relatively small number of molecules, one thousand in order of magnitude. The size of statistical fluctuations for such a small number of systems renders impossible the reduction of biological order to disorder as in physics. Schroedinger offers as model for the order in vital processes the repetitive processes of the ideal clock of classical mechanism. To be sure, a real clock is subject to frictional and thermal influences. However, according to the third law of thermodynamics, as the absolute zero of temperature is approached, molecular disorder decreases. At ordinary temperatures a pendulum clock is to a good approximation free from disorderly influences. A clock is built of solids the form of which is maintained by forces that are not appreciably disturbed by random influences. Schroedinger suggests that the organism depends for its stability upon structures which have the properties of an aperiodic solid. The stability of solids is explained by quantum theory. -69-

VI

CAUSALITY AND RELATIVITY

1. Relativity in Classical Mechanics THE PROGRAM of classical physics was to represent physical processes as situated in space and time and as conformable to causal laws. Specification of position in space requires selection of a frame of reference and a standard of distance or length; specification of situation in time requires selection of an origin and a standard of time-span. The term system of reference will designate the frame of reference, standard rod, origin of time, and standard clock. Standard rods are equal in length if the rods are at rest in a specific frame of reference; standard clocks determine equal units of timespan if the clocks similarly are at rest. Each of a set of inertial systems of reference is permissible for classical mechanics. Among inertial systems is the Copernican astronomical one; further all systems that have a uniform velocity of translation with respect to the Copernican system. Classical mechanics satisfies a principle of a relativity of uniform motion: the same laws of mechanics hold with respect to all inertial systems of reference. The concept of causality is invariant with respect to transformations from one inertial system of reference to another inertial system. A given force, for example, causes the same acceleration with respect to all frames of reference. The detailed explanation is as follows: The same geometry of space and the same scale for reckoning time hold in all inertial systems of reference, so that the geometrical properties of bodies are the same in all systems.

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Force as a function of geometrical properties, such as distance and deformation, is the same in all inertial systems. Mass is postulated to be invariable in classical mechanics; and in view of the uniform velocity of inertial frames with respect to one another, the acceleration of a body is the same in all inertial systems. Hence the product of mass and acceleration is also the same in all inertial systems. It follows that force, and the product of mass and acceleration, are the same in all inertial systems. The concept of dynamical causality in classical mechanics conforms to the principle of relativity of uniform motion.

2. Special Theory of Relativity In the nineteenth century the theory of electric and magnetic fields led to the hypothesis that a substantial ether is the medium for the transmission of electric and magnetic action. In Lorentz's version of classical electromagnetic theory, the ether is stationary and constitutes a privileged frame of reference for electromagnetic phenomena. The principle of relativity of uniform motion did not hold for electromagnetism. Now light was interpreted to consist of vibrations of the electromagnetic field; its speed of propagation within the fixed ether was held to be constant and independent of the speed of the source of light. In view of the identity of schemes of reckoning space and time in all inertial systems, the speed of light with respect to a moving frame was inferred to depend on the speed of the frame with respect to the ether. Experiments, such as the Michelson-Morley experiment, were performed in order to measure the speed of the earth which at some stage must be in motion with respect to the ether. The results of experiments to detect the motion of the earth with respect to the ether were negative. It was inferred that electromagnetic phenomena also conform to a principle of relativity of uniform motion. But abandon-71-

ment of the principle of relativity appeared to be required by the principle of electromagnetism that the speed of light in the ether is independent of the speed of the source. Thus there arose a contradiction between the principle of relativity of uniform motion and the principle of the constancy of the speed of light. The contradiction presupposed, however, that the same schemes of reckoning space and time held for all inertial systems of reference. Einstein24 founded the special theory of relativity upon the principle of relativity and the principle of constancy of the speed of light. The joint use of these two postulates required resolution of the contradiction by reconstruction of the concepts of space and time; both space and time became relative to an inertial system of reference. In the special theory of relativity time is a relative order of events of which the structure is embodied in the periodic motion of a clock.

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Similarly, space is a relative order of bodies of which the structure is embodied in the positional relations of rigid bodies. Relative space and relative time were combined by Minkowski66 to form a four-dimensional space-time, in which events in the world of space-time can be specified in terms of three coordinates for space and an imaginary coordinate for time. There is an interval between two events which is constructed out of differences of their world coordinates and which is invariant with respect to transformations from one inertial system to another. For the graphical representation of motion of a particle along a straight line, one draws a horizontal line on which is marked a scale of coordinate distances from an origin. On a perpendicular to the space axis one marks a scale of imaginary coordinates for elapsed time. Between two points in the space-time diagram is the interval which is defined in terms of differences of space and time coordinates. If the imaginary coordinate is employed for time, the properties of the intervals -72-

between world events are analogous to distances in Euclidean space. The motion of a particle is constituted by a continuous series of events in space-time and is represented graphically by a world line in absolute space-time. The complete history of a body can be picturd by its world line. For the example of the one-dimensional particle: if the particle is at rest on the line, its world line is a straight line parallel to the time axis; if the particle moves with constant velocity, its world line is a straight line and inclined to the space axis.

3. Structure of Space-Time For the subsequent discussion I shall specify an event in the four-dimensional world of space-time by three spatial coordinates x, y, z and by a time coordinate ct. The law of the constancy of the speed of light with respect to a system of reference K is, if the light is emitted at the origin for space and time, x2 + y2 + z2 - c2t2 = 0. By the principle of relativity the same law holds with respect to a system of reference K′ which has a uniform velocity of translation v in the positive direction of the x axis with respect to K. Thus x′2 + y′2 - z′2 - c2t′2 = 0. From additional postulates that space and time have the same properties in all directions, one obtains equations which correlate coordinates of space and time in one system of reference with those in the other. The equations which express this Lorentz transformation are:

y' = y z' = z

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It is convenient to use units for space and time so that the speed of light is unity. The world line which represents the propagation of light then bisects the angles between the x and t axes. The world points which are attained by light emitted at the origin form the section of a vertical circular cone. By projecting the light lines backward one obtains a double cone.53 If an individual person is at the origin O, the world line of his body passes through the origin which divides his life line into past and future. The type of causal action which conforms to the theory of relativity imposes a specific structure on space-time. The transformations of space and time in classical mechanics presuppose that a signal can travel with infinite speed. In the graphical representation of space-time, the inclination of world lines of the signal to the space axis decreases as the speed of the signal increases. In the limit of infinite speed the world lines coincide with the positive and negative axes for space. On this presupposition of classical mechanics past and future are separated by the line t = 0. To the individual at the origin O, all events below t = 0 are past, all events above t = 0 are future. The finite speed of light imposes a new causal structure upon space-time. In the special theory of relativity, which entails that the finite speed of light is a maximum for physical processes, it is the cone with vertex at the origin that separates past and future. The events represented in the backward cone comprise the passive past of an observer represented at the origin O. These are events that he has witnessed himself or by which he has been influenced by processes in conformity with physical laws. The events represented in the forward cone comprise the observer's active future. These are events which can be influenced by his acts at O. -74-

The passive past and active future touch at O, but do not come into contact elsewhere. Between the backward and forward cones is a region with which the observer at O is connected neither passively nor actively.

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For the further discussion it is necessary to introduce the concept of space-time vector. A vector in ordinary space is the directed segment of a straight line and is described in terms of components with respect to Cartesian coordinates. A vector in four-dimensional space-time is specified in terms of three coordinates for space and one for time. Let x, y, z, and ct be the space-time coordinates of an event. A space-time vector runs from the origin to the point x, y, z, ct. Then s is the interval between the event represented by the origin and the event x, y, z, ct. The square of the absolute value of the invariants is defined by s2 = x2 + y2 + z2 - c2t2. The remainder of the discussion will be restricted to one spatial dimension, so that an event is specified by x and ct. I now introduce the distinction between space-like vector and time-like vector. A space-like vector joins the origin and points in the region between the cones. A timelike vector joins the origin and points within the cone.

4. Temporal Order of Cause and Effect Events at the origin and events in the intermediate region can be joined by a space-like vector OP. With respect to system K in which an event is specified by x, ct, event P in the intermediate region is later than O. But a system of reference Ko can be chosen so the O and P lie on the space axis and in this system P is simultaneous with O. Further, one can choose a system K′, in which an event is specified by x′, ct′, so that P is earlier than O.50 It might appear at first sight that if an event at O were cause and the event at P were effect, cause could precede, -75-

be simultaneous with, or succeed the effect. In view of this apparent ambiguity in temporal relations, it has been claimed that the traditional concept of causality is excluded by the special theory of relativity. The answer is that as described with respect to the frame K, a causal process from O to P would have to travel with a speed greater than that of light. The intermediate region, which is invariantly defined by the light cones, represents events which cannot be connected to the origin by a process for which the speed is less than that of light as prescribed by the special theory of relativity. Events at the origin and in the forward cone can be joined by a time-like vector OQ. With respect to the system K, event Q in the forward cone is later than an event at O. In no system K′ is the temporal order of events at O and Q changed. This result from geometry may be shown analytically as follows:68 In system K coordinates of the cause at the origin are 0, 0 and coordinates of the effect at Q are x, ct. The maximum speed of the process is c; hence T≧x/c. I take x positive, so that t is positive. In the system K the coordinates of the cause are also 0,0.

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The time of the effect at Q is

. Now x/c≦t0, and v/c≦1

. Hence. Thus

is positive or zero. Hence

is positive for the case in which the frame of K′ travels with a speed less than that of light. So if the cause at the origin precedes the effect at Q in one system of reference, it does so in all frames which travel with less than the speed of light. If the cause is -76-

simultaneous with the effect with respect to K, then x and ct are both null. Then t′ with respect to K′ will also be null. If two causally related events are simultaneous in one frame of reference, they will be simultaneous in all.

5. Changes of Relative Quantities The relativity of measures of space and time to a system of reference has raised questions concerning the causes of such changes. A load stretches a wire; heat expands a solid body. According to the special theory of relativity, a rod contracts when it is set in motion; for this contraction one has sought a cause.

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One explanation is that motion of the rod provides the cause for contraction of length which is a relation between rod and standard.75 Such an explanation lacks invariance, because the rod retains its length with respect to a system in which it remains at rest. A preferable explanation by Max Born11 is that the contraction of the rod upon motion does not fall under the concept of causality; the contraction is a consequence of change of standpoint. As shown on a graphical representation by Born, a persistent material rod is a space-time complex of events. If the rod is at rest in a system of reference, each point of the rod traces a world line parallel to the time axis. The graphical space-time representation of the one-dimensional rod is a strip of the (x, ct) plane. If the rod is in motion, the space-time strip is inclined to the timeaxis. According to classical mechanics, both strips would have the same width parallel to a fixed x axis. According to the theory of relativity, if a rod is at rest, first in one system of reference and then in another system which moves with respect to the first, the same rod will be represented by two different strips. The rod has the same length in the two systems in which it is at rest. Hence the two strips have the same width parallel to different x directions -77-

of relatively moving frames of reference. The contraction does not pertain to the strip, the complex of events which is the physical reality, but to a stretch intercepted on an x axis. Thus the contraction is not a causal process, but a consequence of change of system of reference.

6. General Theory of Relativity and Gravitation Newton's theory of gravitation admitted causality into classical mechanics as instantaneous action at a distance. This departure from contiguous causality disturbed contemporaries of Newton and the latter as well. The special theory of relativity prescribes that the finite speed of light be the maximum for physical processes. This required the creation of a theory of the gravitational field which is propagated with a finite speed. A solution of this problem was given by Einstein25 in the general theory of relativity. The general theory is based upon the concept of fourdimensional space-time with variable curvature. An intuitive analogy for such a concept is provided by the surface of a sphere. The geometry on the sphere may be interpreted as a non-Euclidean geometry for a curved space of two dimensions. Great circles on the sphere are geodesic lines which correspond to straight lines in an Euclidean plane. In triangles constructed of straight lines in an Euclidean plane, the sum of the angles is two right angles; in triangles constructed of geodesic lines on the sphere, the sum is greater than two right angles. In an Euclidean plane a cubical lattice can be constructed out of equal rods, and this constitutes a physical Cartesian coordinate system. On the surface of a sphere a Cartesian coordinate

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system can be introduced approximately for a limited region, but it cannot be extended over the whole surface. The different properties of configurations of rods in a plane and on a sphere can be expressed in terms of the curvature: the -78-

curvature is null for the plane and positive for the sphere. The geometrical theory of curved spaces was extended to space-time in which geodesic lines were also defined. In the Newtonian theory of gravitation the undisturbed motion of a body is inertial motion in a straight line with constant velocity. The sun is conceived to exert a gravitational force on a planet which is thereby deflected from its inertial motion. The concept of gravitational force as causal activity by matter is discarded in the theory of gravitation which is based on the general theory of relativity. The general theory of relativity is constructed with the concept of curved space-time. Events are described by Gaussian coordinates which are suited to systems of reference with varying states of motion. The geometry of curved space-time is specified by coefficients, the components of the metrical tensor, which also serve as the potentials of the gravitational field. The geometry of curved space-time is functionally related to the energy tensor of matter. Einstein discovered for the gravitational field a simple law which satisfies a general principle of relativity, that is, the law of gravitation is invariant with respect to transformations of Gaussian coordinates. The motion of a body in a gravitational field is along a geodesic line in curved space-time. Thus the action of gravitational force on the Newtonian theory is replaced by the action of curved spacetime in the general theory of relativity. The relativistic theory of gravitation is a field theory; modifications of field quantities are propagated through space by contiguity with a finite velocity. The principle of dynamical causality holds in the relativistic theory of gravitation and thus conforms to a general principle of relativity. The mathematical equations of a continuous field in the space-time of relativity may be viewed as the realiza-79-

tion of the classical ideal of dynamical causality, or regularity. The space-time of relativity presupposes unsymmetry between space and time variables. As we have seen, Minkowski created a geometry of space-time upon three real coordinates for space and an imaginary coordinate for time; one may also use three imaginary coordinates for space and a real coordinate for time. Pascual Jordan44 has utilized the theory of linear partial differential equations of the second order in order to expound the consequences of this space-time structure

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for causality. He states that if the four-dimensional world had four real dimensions, and if the differential equations of electromagnetism and of gravitation remained the same, there would be more than causality. From exact knowledge of a small region of the world, one could derive the physical state of regions arbitrarily distant in space and time. If the world had only two real dimensions and also two imaginary dimensions, there would no longer be causality. Motions could suddenly occur within a region without the intervention of a cause existing in the region or entering through the boundary. The space-time of relativity has three real and one imaginary dimension, or three imaginary and one real dimension. The principle of causality for relativistic field theory is a consequence of the mathematical theory of hyperbolic differential equations.34 -80-

VII

CAUSALITY AND QUANTA

1. Problem of Quantum Theory THE ANALYSIS of causality in classical physics included reference to the program of physical theory for the reduction of macrophysical, large-scale phenomena to microphysical, fine-scale processes. Microphysics originated in the ancient world as a metaphysics of atomism. It was founded by Leucippus and developed by Democritus, who taught that physical reality is constituted of atoms in the void. In the nineteenth century atomistic concepts were given quantitative form. The atomic theory was applied in chemistry; a kinetic-molecular theory of gases was created; and the theory of electromagnetism was founded on electrons. A macroscopic physical system was interpreted to consist of many microscopic systems. Macroscopic quantities were explained as the resultants of the microscopic quantities of the many constituent systems. The functional relations which express macrophysical laws were found to hold for the average values of the resultants of microphysical quantities. Since resultants of quantities for microphysical systems fluctuate about average values, macrophysical law is statistical. By the reduction of macrophysics to microphysics dynamical causality for observable phenomena was replaced by statistical regularity. Causality in the strict sense of dynamical regularity was transferred to fundamental microphysical theory. -81-

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An adequate microphysical. theory has required the development of a theory of quanta for microphysical processes. Atomism in microphysics expresses an element of discontinuity, or individuality, in physical events.8 The question then arises, what is the status of the concept of causality in the theory of quanta? As we shall see, quantum mechanics has made statistical regularity fundamental in physical theory. Under the leadership of Niels Bohr10 the concept of causality, in the sense of dynamical causality, has been generalized by the concept of complementarity.

2. Origin of Quantum Theory The element of discontinuity, or individuality, in microphysical processes has been discovered during the present century in the attempt to construct theories for the interaction of radiation and matter. Atomism in the theory of matter and electricity provided concepts for a theory that electrified corpuscles are the mechanism of emission and absorption of radiation by matter. In order to derive theoretically the law for the distribution of energy of thermal radiation in equilibrium with matter, Planck supposed that the mechanism of radiation is an oscillator which consists of an electrified particle that vibrates with simple harmonic motion like a body attached to the end of a spring. Planck78 found that derivation of a radiation law in agreement with experience required that he adopt the quantum hypothesis, according to which an oscillator can exist only in a discrete series of stationary states of energy. The energy of a state is given by the revised theory as E = (n + ½) hv, where v is the frequency, that is, number of vibrations per unit time, n is an integer, and h is the quantum of action. -82-

Subsequently, Einstein26 introduced the new hypothesis that radiation in empty space is constituted of corpuscles of energy E = hv and momentum ,

where v is the frequency of vibration which is assigned to the radiation on the basis of wave theory. The corpuscular aspect of radiation is exhibited by the exchange of energy in the photoelectric effect, the emission of electrons from a metal upon incidence of radiation. Later A. H. Compton16 found that experiments on the scattering of radiation by electrons could be interpreted in

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terms of exchange of energy and momentum by corpuscles. Radiation was thus found to have a dual nature: The propagation of radiation through space may be explained by a wave theory; the exchange of momentum and energy may be interpreted as collisions of corpuscles. A similar dualism for material particles was developed from a theory of matter waves which was initiated by Louis de Broglie.14 Further confirmation of the quantum theory was the successful application of the quantum hypothesis to the hydrogen atom by Bohr.7 The radiation emitted by energetically excited atomic hydrogen can be resolved by a prism or by a diffraction grating into components which severally produce lines on a photographic plate. From the positions of the lines one calculates wave lengths, and then frequencies. The Bohr theory of the atom postulates that the hydrogen atom consists of a positive nucleus and an electron which revolves around it in one of a discrete series of stationary states that are determined by a quantum condition. When an electron makes a transition from one stationary state to another, there is emission or absorption of a quantum of energy in accordance with the Bohr frequency principle, E1 - E2 = hv. -83-

The course of development from Bohr's initial hypothesis led to the discovery by Schroedinger86 of a differential equation for a function which represents the state of an atomic system. This equation provided a foundation for wave mechanics as a branch of quantum theory. The generally accepted statistical interpretation of the state function was originated by Max Born. Prior to the discovery of Schroedinger, however, Heisenberg had created quantum mechanics in the form based on the representation of physical quantities by operators. This original quantum mechanics and wave mechanics were then shown to be equivalent modes of representation. The present discussion of quantum mechanics is formulated in terms of the state function which satisfies Schroedinger's equation.

3. Dualism of Corpuscle and Wave The foregoing outline of the development of quantum theory shows that a dualism of corpuscular and wave aspects characterizes the microphysical constituents of physical reality. In the history of physics two points of view have been adopted for the interpretation of physical phenomena. Phenomena have been interpreted as manifestations of properties that have simple location, to use a term introduced by Whitehead, or as manifestations of properties of an extended field. On the one hand, physical phenomena have been explained as the action of corpuscles, or particles, which for mathematical theory may be idealized as

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physical points. The Newtonian corpuscular theory of light exemplifies the idea of simple location. On the corpuscular theory the propagation of light consists in the motion of corpuscles emitted from a luminous body. -84-

On the other hand, physical phenomena have been explained as the action of a field. Electric and magnetic fields as functions of position and time exemplify fields that are propagated through space during time by wave motion. Since corpuscle connotes simple location and field connotes extension, the spatial properties connoted by corpuscle and by field are logically incompatible. Thus the concept of corpuscular motion and that of motion of wavefields admit only applications that are mutually exclusive. The quantum theory is based on the existence of a dual aspect of physical reality. The dualism is exemplified by light. Experiments in which light passes through slits and forms patterns of bright and dark bands upon a screen, are interpreted to manifest the interference of waves which are propagated from the slits. Experiments in which light is incident upon a metal and causes the emission of an electron whose energy depends on the frequency, are interpreted to demonstrate that exchange of energy and momentum between matter and radiation occurs by collisions of corpuscles. There is no empirical contradiction between the dual aspects of physical reality, because wave and corpuscular properties are not manifested simultaneously. An experimental arrangement whereby a wave property of light is determined, for example, wave length, excludes the experimental determination of corpuscular properties, for example, localization of energy in collision with a corpuscle. The dual nature of elements of physical reality, such as electrons and radiation, demonstrates that a theory cannot attribute either corpuscular or wave properties to physical objects in every situation without restriction. However, both the classical corpuscle and wave-field provide the basis for a quantum theory. -85-

4. Unity of Quantum Theory Classical corpuscular and wave concepts each provide bases for a quantum theory which avoids the contradictions of a classical theory. An electron or radiation is a definite element of physical reality which may be described from different points of view. Two expressions of a language may differ in signification but have the same denotation, as for example, the expressions evening star and morning star. Similarly, radiation within an enclosure may be denoted by the phrase assembly of photons which obey the Bose-Einstein statistics, or by the phrase system of waves. In the former case, the radiation is conceived of as

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corpuscles and is described by a symmetrical function of the variables characterizing them. In the latter case, the radiation is conceived of as a system of waves, that is, simple harmonic oscillators, and is described by the numbers of oscillators in the various states of vibration. The transformation from one mode of description to the other has been demonstrated by Dirac.22 One may begin by thinking of radiation as an assembly of photons. The state of the assembly is represented by a function of variables which describe a photon; the function is symmetrical in the sense that it is unchanged if one interchanges variables that characterize two different photons. In view of the symmetry, the assembly is said to exemplify BoseEinstein statistics. Now the states of the assembly can be described equally well by numbers, the numbers of photons for which the characteristic variables have specific values. By a mathematical transformation the representatives of states of the assembly of photons may be expressed in terms of these numbers. The new variables and appropriate additional ones have the same formal properties as the variables which are used to describe the simple harmonic os-86-

cillator. Thus one is able to transform a description in terms of variables for corpuscles into one in terms of variables for simple harmonic oscillators. An assembly of photons is dynamically equivalent to a set of simple harmonic oscillators: to each independent state of a photon in the assembly corresponds an oscillator; to the number of photons in the state corresponds the quantum number of the oscillator. One may replace the set of simple harmonic oscillators by a train of waves, in which each Fourier component is dynamically equivalent to a simple harmonic oscillator. Hence the assembly of photons is equivalent to a system of waves. Although a classical corpuscular theory is inconsistent with a classical wave theory, the quantum theory of radiation is a unified theory.36 One physical reality, radiation, is described from two consistent and mathematically equivalent points of view. Two classical pictures, one of corpuscle and the other of wave, each provide a basis for a single quantum theory of radiation.

5. Quantum Theory on a Corpuscular Basis The problem of causality in quantum theory is usually formulated in terms of the corpuscular model. It is therefore appropriate to begin with the quantum theory as founded upon the concept of corpuscle. The theory is known as quantum, or wave, mechanics, and is based on Schroedinger's equation for a function of state. The problem of causality in quantum theory arises from the effect of observation on the description of a microphysical object. I adopt a contextualist theory of physical concepts, according to which the significance of a concept is relative to

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the concrete situation to which the concept is applicable. The concepts of quantities are of measurable -87-

cillator. Thus one is able to transform a description in terms of variables for corpuscles into one in terms of variables for simple harmonic oscillators. An assembly of photons is dynamically equivalent to a set of simple harmonic oscillators: to each independent state of a photon in the assembly corresponds an oscillator; to the number of photons in the state corresponds the quantum number of the oscillator. One may replace the set of simple harmonic oscillators by a train of waves, in which each Fourier component is dynamically equivalent to a simple harmonic oscillator. Hence the assembly of photons is equivalent to a system of waves. Although a classical corpuscular theory is inconsistent with a classical wave theory, the quantum theory of radiation is a unified theory.36 One physical reality, radiation, is described from two consistent and mathematically equivalent points of view. Two classical pictures, one of corpuscle and the other of wave, each provide a basis for a single quantum theory of radiation.

5. Quantum Theory on a Corpuscular Basis The problem of causality in quantum theory is usually formulated in terms of the corpuscular model. It is therefore appropriate to begin with the quantum theory as founded upon the concept of corpuscle. The theory is known as quantum, or wave, mechanics, and is based on Schroedinger's equation for a function of state. The problem of causality in quantum theory arises from the effect of observation on the description of a microphysical object. I adopt a contextualist theory of physical concepts, according to which the significance of a concept is relative to the concrete situation to which the concept is applicable. The concepts of quantities are of measurable -87-

properties, so that a quantity is relative to a context in which an operation of measurement is performed. Physical properties are essentially dispositional attributes of objects which manifest themselves in interaction. In order to measure a dispositional attribute, the object is brought into interaction with a measuring instrument; through interpretation of the space-time processes of the instrument there is found the quantitative measure of the attribute. The interaction between object of investigation and apparatus requires that, in principle, one create a partition

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between them.37 The physical state of the apparatus on the observer's side of the partition is outside of consideration while the apparatus serves as instrument. According to classical physics, the place of the partition can be ignored; by unrestricted refinement of apparatus the reaction of apparatus on object can be diminished indefinitely. The concept of measurement is a classical one. An apparatus is a macroscopic system; the results of measurement are expressed in terms of classical concepts. The objects of microphysics, such as electrons, are constituents of physical reality which are not perceptible in the usual sense of the term. Observation of microphysical attributes is mediated by action of the microphysical object on an apparatus which reacts upon the object. According to quantum theory, there is a finite lower limit to the reaction of instrument upon the object during measurement. The effect of measurement upon the object is unpredictable and untraceable. The consequence is a limitation in the application of classical concepts for the description of microphysical reality. The disturbance of object by observation may be explained in terms of an example of Bohr.9 It is supposed that a particle is given an initial momentum and passes through -88-

a slit in a diaphragm. Preparation for measurement of position requires that the diaphragm be fixed rigidly to a support which defines the space frame of reference. During passage through the slit the particle exchanges momentum with the diaphragm; the momentum acquired by the diaphragm is absorbed by the supporting body of reference and therefore cannot be used to calculate the momentum acquired by the particle from conservation of momentum. If the momentum of the particle were precisely known before passage through the slit, that momentum would be less definite after the disturbance by measurement of position. Preparation for measurement of momentum requires that the diaphragm be left mobile. It is possible in principle to measure its momentum before and after the passage of the particle, and thus to calculate the momentum of the particle after it has passed through the slit. But determination of momentum of the diaphragm requires collision with a test body during which interaction there there is an untraceable displacement of the diaphragm. One loses knowledge of position of the particle when it passed through the slit. Bohr has stated that one must discriminate between essentially different experimental arrangements and procedures, which are suited either for an unambiguous use of the idea of space location, or for a legitimate application of the principle of conservation of momentum. With each of the mutually exclusive experimental arrangements we are confronted, not merely with ignorance of

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values of specific physical quantities, but with the impossibility of defining the concepts of these quantities in an unambiguous way. The foregoing example demonstrates that the description of an atomic object is dependent not only on the -89-

object, but also upon the mode of observation. The physical interaction between object and instrument alone does not fix the quantity that can be measured; the arrangement employed in cognition also plays a role. In the present example, if the diaphragm is mobile during passage of the particle through the slit, after passage of the particle one is left free to choose whether one wishes to find the momentum of the particle, or its initial position relative to the rest of the apparatus. One may either measure the momentum of the diaphragm and use it to calculate that of the particle, or one may attach the diaphragm to the support and fix position but renounce the possibility of determining momentum. The preceding analysis is confirmed by analysis of an example by C. F. von Weizsaecker.88 He considers the illumination of an electron and the possibilties of measuring position or momentum. It is supposed that the electron is in a given plane and is illuminated by light of low intensity, so that a single photon is scattered. To prepare for a measurement of position one would place a photographic plate in the appropriate image plane. If the light passes through a lens and is brought to a focus on the photographic plate, one could calculate from the position of the image the position of the electron at the time of collision with the photon. The theoretical basis of measurement is the wave theory of light, and one pictures a spherical wave which spreads out from the electron and by the converging action of the lens is brought to a focus to form an image. No definite direction characterizes the motion of the associated photon from its place of collision with the electron, so that the momentum conveyed to the electron by the photon cannot be traced. Precise determination of position is accompanied by an untraceable change in momentum. To prepare for a measurement of momentum one would -90-

place the photographic plate in the focal plane of the microscope. An image which results from the convergence of the light to a focus on the photographic plate is interpreted to demonstrate that the rays scattered by the electron all approach the microscope from the same direction. Suppose that the momentum of the photon were known before collision. From the direction of the light after collision with the electron, one could determine the change in momentum of the photon, and hence calculate the correlated change in momentum of the electron. The place in the object plane from which the light started is not known and so the position acquires indeterminacy. In both experimental arrangements of von

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Weizsaecker's example the same physical event, collision with a photon, occurs to an electron, but different quantities are determined with different experimental arrangements. After the photon collides with the electron and passes through the lens, one could in theory place the photographic plate either in the image or the focal plane, and thereby determine respectively position or momentum of the electron. The intellectual act of preparing a well defined arrangement for observation is an essential factor in the operation of measurement. In the example of von Weizsaecker it appears paradoxical that by appropriate setting of the plate one finds light scattered by the electron either as a spherical wave or as a plane wave with a definite direction. The electron, however, is not a corpuscle; it has wave properties which necessitate the use of an ensemble of systems in order to represent the results of measurement. One should imagine that there is an ensemble of many experiments in which light is scattered by a corpuscular electron in many ways.51, 55 One experimental arrangement will record one mode of interaction between light and electron, another -91-

will record another mode of interaction. By means of an ensemble of experiments one can give a picture of the results of measurement in terms of the corpuscular quantities position and momentum. The employment of ensembles explains a paradox which was offered by Einstein, Pololsky, and Rosen27 in support of an argument that quantum mechanical description is incomplete.

6. Principle of Indeterminacy The preceding examples demonstrate the limits of applicability of classical concepts for the description of atomic objects. Classical concepts are subject to a reciprocal relation, the principle of indeterminacy: The product of range of indeterminacy of momentum and range of indeterminacy of position is at least of the order of the quantum of action.35 A similar rule holds for energy and time. The origin of indeterminacy is the circumstance that an experimental arrangement which admits precise measurement of one quantity excludes similar precision for a correlated one. It follows that for quantum mechanics no sharp separation can be made between an independent behavior of objects and their interaction with measuring apparatus. It is necessary to distinguish between a physical quantity as measurable attribute of the physical world and the number which expresses the result of a measurement in terms of some unit. In classical theory a physical attribute is assigned to an object independently of the context in which it is manifested. The physical attribute and numerical value are thus interchangeable in thought. In quantum theory a physical attribute is relative to a context of observation and the result of a measurement in general cannot be predicted with certainty. Hence it is not appropriate to assume that the system possesses a particular value of an attribute out of

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its context of measurement. Dirac has introduced the word observable as a substitute for the classical term physical quantity. The purpose of measurement is to determine the value of an observable. The loose connection between a system and its attribute in quantum mechanics is symbolized by the fact that the operator represents an observable in the mathematical theory. In ordinary algebra the quantities on which algebraic operations are performed are subject to the commutative law of multiplication: ab = ba. In quantum theory the operators which represent physical quantities do not in general commute with one another. A quantum condition is expressed by the commutation law

An operator is exemplified by a linear transformation; if this operation is applied to a vector in a plane, it correlates to the vector a new vector which in general differs in length and direction from the initial one. It is possible to find mutually perpendicular directions, such that vectors in those directions are unchanged in direction by the linear transformation. If these vectors are chosen as the basis of a coordinate system, the coefficients of the linear transformation form a diagonal array, the terms of which are the characteristic values of the operator. One of the basic principles of quantum mechanics is that the results of measurement of an observable are the characteristic values of the corresponding operator. From the commutation law just cited one can derive the precise formulation of the principle of indeterminacy: The product of the standard deviations for measures of coordinate of position and component of momentum is equal to or greater than Planck's constant h divided by 4π.

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The values of an observable are in general dispersed about a mean value. The results of a repeated measurement, which in classical theory could be idealized

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as identical, now constitute a collective. Thus one arrives at the statistical interpretation of quantum theory.

7. Quantum Theory on a Wave-Field Basis The preceding discussion has been based upon the corpuscular picture as the basis for quantum theory. For the sake of completeness I interpolate a brief discussion of the quantum theory of wave-fields. The example is the electromagnetic field which is described classically by Maxwell's equations. The potentials which describe the field are transformed for quantum theory into operators that obey commutation relations. The application of these commutation relations imposes quantization on the field. By this process, which is called second quantization, one obtains results which are interpreted to represent numbers of particles. Thus the application of quantum conditions in the form of commutation relations to operators which represent the classical field quantities, imposes corpuscular properties upon the field. Relativistic quantum theories of wave-fields are a principal topic in contemporary theoretical physics.61 This type of theory has provided a method of approach to the problem of nuclear forces. The meson of contemporary physics is a corpuscle associated with the fields of the fundamental constituents of the nucleus.

8. Concept of State The theoretical formulation of quantum theory requires a new conception of state for microphysical systems. The classical mechanical definition of state of a system is expressed in terms of coordinates of position and components -94-

of momentum. Analysis of measurements shows that it is not possible simultaneously to assign precise values to all these quantities for atomic systems, so the classical concept of state cannot be exemplified. For quantum theory it is postulated that a system can be prepared so that a maximum number of its observables have definite values. The system is then said to be in a determinate state, a state which has been called a pure case by Weyl.89 A particle for which the three coordinates of position are determinate is in a well defined state; the conjugate components of momentum are completely indeterminate. The determinate state is represented in the mathematical theory by a function of the values of a maximum number of observables; on the corpuscular picture they are usually coordinates of position. This state function is often called a wave function, because in some examples the function represents wave motion.

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The states of a system can be represented by vectors, better, directions, in a linear vector space of infinitely many dimensions. This function space is spanned by mutually perpendicular characteristic functions which serve as base vectors to define axes for the space. The state function may be represented by a point, the terminus of a unit vector, in the function space which is named after Hilbert.

9. Theory of Observation In the general theory of observation a system to be observed is prepared so that it is in a determinate state. For example, to prepare plane polarized light, ordinary light is permitted to fall upon a Nicol prism from which emerges light polarized in a specific plane. That is, the vibrations which constitute light take place in a specific direction with respect to a plane, the plane of polarization. If one -95-

conceives of light as constituted by photons, the plane polarized light will consist of photons polarized in a given direction, and may be represented by an appropriate state function.20 Preparation for an experiment on plane polarized light is to permit a single plane polarized photon to pass through a second Nicol prism. By action of the second Nicol on the photon the original beam is resolved into two beams which are polarized at right angles to each other. The original beam is then distributed over two component beams, each of which is represented by its characteristic state function. The state function of the photon may be expressed as a superposition of two functions, each of which characterizes a photon polarized in one of two mutually perpendicular directions. Observation on the photon which emerges from the second Nicol has not yet been completed, for the two component beams are still able to interfere with each other. If one component beam activates a sensitive substance on a photographic plate, the total energy of the photon is concentrated there. The photon has made a discontinuous transition to the state which characterizes that beam. This is the decisive stage in the observation. There is a probability that upon observation the photon will be found in one beam, and another probability that it will be found in the other. The results of observation conform to a statistical law. The principle of dynamical causality has been abandoned, but a principle of statistical causality does hold for the behavior of the photon. A diffraction grating, a set of slits between lines ruled on the surface, merely extends the pattern exemplified by the Nicol prism. If a beam of light passes through a grating, the beam is resolved into component beams and is thereby prepared for the decisive step in observation. To employ -96-

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terms introduced by Eddington, the grating is a sieve which resolves a physical process into component ones. The preparation of a system for observation makes it appropriate to represent the initial determinate state as a superposition of states. The decisive stage in observation throws the system from the initial prepared state into a component state. This process is represented by the operation of normal projection of the original state vector upon an axis, or more generally, a subspace. The square of the absolute value of the normal projection of a unit state vector upon a subspace is the probability of realization of the state which falls in the subspace. By the observation a pure case is transformed into a mixture. The decisive step in observation produces registration by a record, such as activation of a grain on a photographic plate. Registration is physical; subsequent thereto is perception of the record. We have seen that after passage of light through a grating an additional factor is needed to cause the discontinuous change of the system from its original state to a component one. And here we come to a debatable question: Is the factor that produces the decision an objective or subjective one? Pascual Jordan45 interprets J. von Neumann to hold that the conversion from pure case to mixture is a mental process of the observer: If the observer forgets those relations between component wave functions which make them able to interfere with each other, then in the mind of the observer the pure case is turned into a mixture. According to the view ascribed to von Neumann, the state function expresses the actual knowledge of the observer and not his potential knowledge. Jordan and Henry Margenau adopt an objective view: The decision between the various possibilities is made by -97-

a physical process such as the absorption of a photon by a photographic plate. The decison is registered by a record which is a macrophysical factor and an ordinary object of perception. The process of decision has been illustrated by polarization, an example of Dirac, who appears to hold the physical interpretation.

10. Statistics in Quantum Theory We have seen that the result of a measurement of an observable for a microphysical system can in general be predicted only with a probability. If a series of measurements of the same observable is performed on a system which is each time prepared to be in the same initial state, one obtains a set of values which will be distributed in accordance with a reproductible law of probability. Thus a set of observations on an atomic system satisfies a principle of statistical

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causality. The quantum mechanical system in a determinate state is properly represented by an ensemble of systems every one of which is in the same state. The ensemble for such a pure case is called a unitary ensemble by von Neumann. In a unitary ensemble every system can be predicted with certainty to yield definite values for a specific maximum set of observables. The results of measurement of conjugate observables cannot be predicted with certainty; the values of a conjugate observable will be dispersed about an average value in accordance with a law of probability. After performance of a particular measurement on each system of a given unitary ensemble, the latter will be found to be decomposable into sub-ensembles, in each of which the observable has been found to have a specific value. Each sub-ensemble is then unitary with respect to the observable that was measured. The original unitary ensemble then becomes a mixture of sub-ensembles. A mixture repre-98-

sents a system for which there is known only the probabilities of occurrence of the several results, each of which characterizes every system of a sub-ensemble of the total ensemble.

11. Causality in Quantum Theory According to quantum mechanics, as expounded by J. von Neumann,71 the changes that may occur to a system are of two types: First, there is the continuous change of a state function in conformity to the differential equation discovered by Schroedinger. According to this equation the time-rate of change of the state function depends on the energy operator applied to the state function:

The change of state function is a causal process which satisfies a principle of dynamical causality. From the solution of the equation and an initial value, the state function can be calculated for any earlier or later time. Second, there is the discontinuous change of the state function upon observation. If a system is in a determinate state, measurement on an

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observable generally will yield a result which is one of a spectrum of values, each of which will have a calculable probability of occurrence. A special case is one in which the immediate repetition of a measurement will yield the same result as the first. By the first measurement of an observable a system is thrown discontinuously into a state in which the observable has a characteristic value. The system is then in a determinate state as defined with respect to values of the observable; one can predict with certainty that a repetition of the measurement will yield the same result. If the system initially is -99-

in a determinate state, that state is re-established by a measurement on the observable of which the value characterizes the state. In general, however, a measurement throws a system into a new characteristic state by a discontinuous, statistically causal process. We have previously seen that the indeterminism of quantum mechanics arises from an untraceable disturbance of the system upon observation. The dependence of properties upon the experimental situation requires that one establish a partition, or "cut," between object of observation and measuring apparatus. Heisenberg37 has expounded the view that the cut is the seat of the indeterminacy. On the object side of the cut, states of physical systems are transformed in conformity to dynamical causal laws. On the observer's side of the cut, physical effects proceed through the apparatus and register perceptible results. The indeterminable disturbance of the object by the apparatus provides the free play at the cut which is necessary to provide compatibility of predictions of quantum mechanics with the indications of classical measuring apparatus. The measuring apparatus and its classical laws of operation are presupposed in a given investigation. We may, however, look upon the situation from a meta-theoretical point of view. To study Bohr's example of the passage of a particle through a slit in a diaphragm, one may select a frame with respect to which the interaction between particle and diaphragm is an object to be described by the methods of quantum mechanics. If one determines spatial positions of particle and diaphragm with respect to the new frame, one must renounce determinability of momentum of particle and diaphragm. The cut between object of observation and measuring apparatus can be displaced arbitrarily with a corresponding change in the object of study. This fact is confirmed by a -100-

theory of measurement in quantum mechanics, which shows that the results of observation can be explained as the correlation of properties of object and apparatus by measurement.72

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12. Causality in Early and Present Quantum Theory The role of causality in quantum theory has changed during the development of the theory. The original quantum theory of Planck was founded on the hypothesis that an atomic system can exist in one of a discrete set of stationary states of constant energy. In the Bohr theory of the hydrogen atom the stationary states were constituted by the revolutions of an electron in circular orbits about a positively charged nucleus. If an atom was in a stationary state of higher energy E 1 , it could jump discontinuously to a state of lower energy E 2 with the emission of radiation. In terms of concepts of corpuscular theory one would say that a photon is emitted during the quantum jump from one stationary state to another. If the atom is in interaction with a field of radiation, it can absorb as well as emit radiations. The probabilities of absorption, of induced emission, and of spontaneous emission of radiation by the atom were determined by correspondence with the results of classical theory. Causal regularity was replaced by statistical regularity; indeed, complete abandonment of causality was implied by spontaneous emission. The Bohr theory of the atom proved incapable of generalization. After the creation of a new quantum mechanics, Dirac21 constructed with it a theory of the interaction of atom and radiation. In his theory atom and radiation are treated as a single system which evolves in accordance with the fundamental law of quantum mechanics. Initially atom and radiation are conceived to be in states of determinate energy respectively. The total sys-101-

tem is formed by the union of the two elements and is represented by a state function which evolves continuously in time. The total system passes through a continuous series of states all of which are determinate; but after the initial time neither the atom nor the radiation by itself will be in a state of determinate energy. If the interaction were to cease, each constituent of the total system would be expressible as a superposition of component states of energy. Disturbance of a system by observation overcomes this indeterminateness, for observation of the energy of atom or radiation causes a discontinuous transition into a state of determinate energy.60 If the system at the instant of observation happens to be in a state of definite energy, observation in this exceptional case will not cause discontinuous transition. In general, however, if the atom is initially in the stationary state characterized by the quantum number n, it might later be found upon observation to be in the state m. The theory enables one to calculate the probability of such a change of state. Probabilities of absorption and induced emission depend on the intensity of the radiation. There is also a probability for spontaneous emission. From the standpoint of the earlier quantum theory one would say that quantum jumps occur in an independent physical reality. According to present theory, the statistical element does not

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reside in the unfolding of states; it enters through observation by which a system is thrown discontinuously into a state which is made possible by continuous interaction of atom and radiation. To sum up the role of causality in quantum theory: A system, which may be composed of partial systems, is represented by a state function that evolves continuously in time in conformity to a dynamical causal law. Upon observation there occurs a discontinuous change of state which conforms to statistical regularity. Observation is -102-

described by von Neumann as a consequence of interaction with measuring apparatus; therefore I deem it appropriate to describe the action of apparatus on system as statistical causality. Quantum theory satisfies a general principle of causality: dynamical causality holds for the evolution of states; the more general statistical causality holds for the results of observation. Both aspects of determinism and indeterminism occur in quantum theory. Margenau62 emphasizes dynamical causality in quantum theory. The state function does satisfy Schroedinger's equation. On Margenau's view the concept of physical state is molded so as to satisfy a principle of causality in the classical sense. Edwin C. Kemble,48 on the other hand, emphasizes the indeterministic aspect of quantum theory. He declares that the state function is a tool for the calculation of probabilities; it is subjective and cannot be projected into the external world.

13. Complementarity The statistical interpretation of quantum theory, which is now before us, has been placed in a wider setting from the standpoint of complementarity. The classical description of physical phenomena was in terms of a space-time process which exemplifies dynamical causality. According to Bohr,10 quantum theoretical description exemplifies complementary. As expounded by Bohr, the requirement of dynamical causality is fulfilled by description of an atomic system in terms of momentum and energy. However, observation of space-time position requires experimental arrangements which exclude simultaneous measurements of the conjugate quantities momentum and energy. In view of the finiteness of the quantum of action, such conjugate quantities cannot -103-

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be assigned simultaneously with perfect precision. The two modes of description, which can be combined in classical theory, are complementary for quantum theory. The term complementary is frequently applied to the corpuscular and wave properties of matter and radiation. Max Born12, 13 has pointed out, however, that such application is not intended by Bohr. The concepts of corpuscle and wave express dual rather than complementary aspects of matter and radiation. The fulfillment of dynamical causality in interaction between radiation and matter is expressed by conservation of momentum and energy. This causal representation involves renunciation of description of the processes in space during time. Classical kinematical and dynamical concepts are both required for physical description. But these classical concepts must be used in a complementary manner for the representation of a microphysical process. Bohr declares that the concept of complementarity constitutes a rational generalization of the ideal of dynamical causality. The concept of complementarity has been extended by Bohr to biological phenomena.8 As has been seen, in experimental biology properties of living systems are described in physical terms. Now the process of observation involves an intrusion into the processes of the organism; such action disturbs vital processes and may even produce death. The strict application of concepts which are suited to the description of inanimate nature stands in an exclusive relation to the discernment of regularities for vital processes. Bohr has further applied the concept of complementarity to the relation between the psychical realm and the physical processes which underly them. He states that the experience of free will, which characterizes the life of the spirit, stands in a complementary relation to the causal con-104-

nection of the accompanying physiological processes. As in the atomic realm, every observation of physiological processes is accompanied by an untraceable change in them. Hence an attempt to observe the physical basis of mental experiences will change the accompanying feeling of volition. According to Bohr, the basic problem of science is to communicate experiences and ideas to others by means of language, and in the performance of this function the practical use of every word stands in a complementary relation to attempts at its strict definition.

14. General Theory of Predictions A general theoretical frame for complementarity has been provided by investigations of J. L. Destouches.18 He has constructed a general theory of

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predictions for physical systems and has generalized the procedures of wave mechanics. A basic concept in the general theory of predictions is that of state quantity, from which all others can be derived. If there exists a state quantity for a system, all other quantities which pertain thereto are simultaneously measurable in principle. If there exists no state quantity for a system, there exist pairs of quantities that are not simultaneously measurable. M. Destouches divides physical theories into two classes: 1) Theories for which there exists a state quantity and which are objectivist, for example, classical theories; 2) Theories for which there exists no state quantity and which are subjectivist, for example, wave mechanics. In an objectivist theory the results of a measurement appear as intrinsic properties of observed physical systems. In a subjectivist theory the results of measurements cannot be considered intrinsic properties of the observed system, but only -105-

as properties of the complex of apparatus-system. Objectivist theories are characterized by causality, subjectivist theories by complementarity. An objectivist theory is deterministic, a subjectivist theory is indeterministic. M. Destouches finds that the character of subjectivity is a consequence of the circumstance that cognition in microphysics requires employment of a macroscopic apparatus which cannot be eliminated in principle. A consequence is that a theory of atomism necessarily is a subjectivist theory. A macroscopic theory is objectivist in that intervention of an apparatus of measurement is eliminable in principle. M. Destouches presents the view that complementarity also appears under a dialectical aspect. Two opposing theories, such as a wave theory and a corpuscular theory, are analogous to thesis and antithesis. Complementarity enables one to conciliate the opposing points of view by a synthesis. General philosophical implications of complementarity have been expounded by Ferdinand Gonseth.33 He declares that progress in science occurs through unveilment of successive horizons of reality: for physical science the natural horizon or personal world, the classical horizon, and the quantum horizon. Within successive horizons one may distinguish an apparent horizon and a profound horizon. An event of the profound horizon is known experimentally only by phenomenal traces in the apparent one. A profound event may have two complementary traces which are irreducible to one another within the apparent horizon. Such traces, if of two kinds, are complementary. The idea of complementarity achieves the reconciliation of two horizons, of which one plays the role of apparent horizon and the other the role of profound horizon. -106-

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15. Logic of Complementarity Complementarity has been introduced into physical theory as a consequence of the relativity of physical attributes to the context of observation. Experimental arrangements for the measurement of conjugate quantities are mutually exclusive; hence there are pairs of observations that cannot be made simultaneously. The restriction upon simultaneous observability entails that one cannot freely apply the logical operation conjunction to propositions which state experimental results. Conjunction in classical logic conforms to the rule that the logical product can be asserted as true if the factors thereof are asserted to be true. Quantum theory, however, requires that one classify propositions into combinable and incombinable pairs, which respectively correspond to measurements that can and cannot be performed simultaneously. This limitation on the classical operation of conjunction has inspired the creation of logics suited to quantum theory. A contribution to a logic of complementarity has been made by Paulette Destouches-Février.28 She states that the logic suited to the early quantum theory as supplemented by complementarity is a logic of three values. The values true (T) and false (F) occur in classical logic. A new third value (A) is absolutely false, or excluded, which applies to propositions which cannot be obtained experimentally and to the logical product of incombinable propositions. A three-valued logic for quantum mechanics has been constructed by Hans Reichenbach.83 He attributes to Bohr and Heisenberg a restrictive interpretation of quantum mechanics which is based on the rule: Only statements about measured entities are admissible; statements about unmeasured entities, or interphenomena, are meaningless. -107-

Reichenbach proposes an alternative restrictive interpretation, according to which statements about unmeasured entities are indeterminate (I) and have a status in a threevalued logic. He constructs truth tables for the operations of three-valued logic by a method analogous to the construction of tables for two-valued logic of classical physics. Quantum mechanics, as a formalism which is founded on Schroedinger's equation and which is interpreted in statistical terms, introduces complementarity automatically. A logic of quantum mechanics has been constructed by Garrett Birkhoff and John von Neumann.5 In classical mechanics a point in the phase-space of a system represents coordinates of position and components of momentum. The experimental propositions concerning any system in classical mechanics correspond to a "field" of subsets of its phase-space; the propositions of classical mechanics

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form a Boolean algebra. In quantum mechanics the motion of a system is represented by a wave function, that is, by a point in a function space. To the statement of a result of measurement there corresponds a subset of initial wave functions. Thus the phase-space of quantum mechanics is Hilbert space; the mathematical representative of any experimental proposition is a closed linear subspace of Hilbert space. To each logical operation on experimental propositions corresponds an operation on sets. To the logical product corresponds the intersection of associated sets, to the relation of implication corresponds inclusion of subsets. Thus there is homomorphism between the calculus of propositions and a calculus of sets. The rules of operation for the logic of quantum mechanics are similar to those of projective geometry. The logic of Birkhoff and von Neumann preserves the relation of implication. They conclude that the distributive law is the one which fails in quantum mechanics, and that this -108-

failure is the adequate formal description of possible nonsimultaneous observability. Mme. Destouches-Février19 has generalized the logic of Birkhoff and von Neumann on the basis of a general theory of predictions by a logic of complementarity and subjectivity. She has also investigated the applicability to physics of an intuitionist mathematics of G. F. C. Griss, the logic of which prohibits reasoning that involves negation and is also a logic of complementarity. Mme. Destouches-Février has further considered the bearing of complementarity on the most general characteristics of thought about reality. Classical science employs a logic with the principle of excluded middle which involves negation and contradiction. In contrast to the dialectic of contradiction, that of complementarity appears as a positive dialectic. According to Mme. DestouchesFévrier, the appearance of contradiction in atomic physics arises from the simultaneous play of thought on two planes which are difficult to separate. There is a concrete, pre-physical plane that is bound to sensory experience; and also an abstract, homogeneous plane of mathematical physics in which occurs complementarity. In the view of Bohr, who has long guided the development of quantum theory, the generalization of causality by complementarity for the description of atomic processes expresses the circumstance that man is both actor and spectator in the drama of existence. -109-

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BIBLIOGRAPHY 1.

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

ADAMS G. P., LOEWENBERG J., AND PEPPER S. C., editors: Causality, University of California Publications in Philosophy, Vol. 15, 1932. APPELL P.: Traité de Mécanique Rationelle, 4th ed., Paris, GauthierVillars, 1919, 1:94. BACON FRANCIS: The Philosophical Works of Francis Bacon, edited by J. M. Robertson, London, Routledge, 1905, Aphorisms: Book 2, Aphorism 11. BERNARD CLAUDE: Les Phénomènes de la Vie, 2nd ed, Paris, Baillière, 1885, 1:51. BIRKHOFF G., AND NEUMANN J. VON: "The Logic of Quantum Mechanics", Annals of Mathematics, 37:823, 1936. BLUM H. F.: Time's Arrow and Evolution, Princeton, University Press, 1951. BOHR N.: "On the Constitution of Atoms and Molecules", Philosophical Magazine, series 6, 26:1, 1913. -----: Atomic Theory and the Description of Nature, Cambridge, University Press, 1934. -----: "Can quantum-mechanical description of physical reality be considered complete?" Physical Review, 48:696, 1935. -----: "On the Notions of Causality and Complementarity", Dialectica, 2:312, 1948. BORN M.: Die Relativitaetstheorie Einstein, Berlin, Springer, 1922, p. 1 91). English translation by H. L. BROSE, London, Methuen, 1924. -----: Natural Philosophy of Cause and Chance, Oxford, Clarendon, 1949. -----: The Restless Universe, 2nd ed., New York, Dover, 1951, p. 283. BROGLIE L. DE: "Recherches sur la Théorie des Quanta", Annales de Physique, series 10, 3:22, 1925. BRUNSWIK E.: "The Conceptual Framework of Psychology", International Encyclopedia of Unified Science, Chicago, University Press, 1952, Vol. 1, No. 10. COMPTON A. H.: "A Quantum Theory of the Scattering of X-Rays by Light Elements", Physical Review, 21:483, 1923. COMTE A.: Philosophie Positive, 2nd ed., Paris Baillière, 1864, Vol. 1. -111-

18. DESTOUCHES J. L.: "Quelques aspects théoriques de la notion de complémentarité", Dialectica, 2:351, 1948. 19. DESTOUCHES-FÉVRIER P.: "Manifestations et sens de la notion de complémentarité", Dialectica, 2:383, 1948. 20. DIRAC P. A. M.: Principles of Quantum Mechanics, 1st ed., Oxford, Clarendon, 1930, p. 2.

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21. Ibid., chap. 10. 22. Ibid., chap. 12. 23. DRIESCH H.: Science and Philosophy of the Organism, 2nd ed., London, Black, 1929. 24. EINSTEIN A.: "Zur Elektrodynamik bewegter Koerper", Annalen der Physik, 17:891, 1905. Reprinted in LORENTZ H. A., EINSTEIN A., AND MINKOWSKI H.: Das Relativitaetsprinzip, 4th ed, LeipzigBerlin, Teubner, 1922. English translation by W. PERRETT AND G. B. JEFFERY ( 1923), New York, Dover, 1952. 25. -----: "Die Grundlage der allgemeinen Relativitaetstheorie", Annalen der Physik, 49:769, 1916. Reprinted in Das Relativitaetsprinzip, op. cit. 26. -----: "Ueber einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt", Annalen der Physik, 17: 132, 1905. 27. EINSTEIN A., PODOLSKY B., AND ROSEN N.: "Can quantum-mechanical description of physical reality be considered complete?" Physical Review, 47:777, 1935. 28. FÉVRIER P.: "Les relations d'incertitude de Heisenberg et la logique", Comptes Rendus de l'Academie des Sciences, 204:481 and 959, 1937. 29. FRANK P.: Das Kausalgesetz und seine Grenzen, Wien, Springer, 1932. 30. -----: "The Law of Causality and Experience" ( 1908), Between Physics and Philosophy, Cambridge, Harvard, 1941. 31. -----: "Foundations of Physics", International Encyclopedia of Unified Science, Chicago, University Press, 1946, Vol. 1, No. 7. 32. GALILEI GALILEO: Dialogues Concerning Two New Sciences ( 1638), translated by HENRY CREW AND A. DESALVIO, New York, Macmillan, 1914. 33. GONSETH F.: "Remarque sur l'idee de complémentarité", Dialectica, 2:413, 1948. 34. HADAMARD J.: Lectures on Cauchy's Problem, New Haven, Yale University, 1923. 35. HEISENBERG W.: "Ueber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik", Zeitschrift fuer Physik, 43:172, 1927. -112-

-----: The Physical Principles: of the Quantrum Theory, Chicago, University Press, 1930, p. 181. -----: Wandlungen in den Grundlagen der Naturwissenschaften, 8th ed., Zuerich, Hirzel, 1948, p. 11. HERMANN GRETE: "Die naturphilosophischen Grundlagen der Quantenmechanik", Abhandlungen der Fries'schen Schule, 6:75, 1935. HUME., D.: Treatise of Human Nature ( 1739), Selby-Bigge ed., Oxford, Clarendon, 1896, p. 76. Ibid., p. 170. -----: An Enquiry Concerning Human Understanding ( 1748), Selby-Bigge ed., Oxford, Clarendon 1894, p. 63. Ibid., p. 75.

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JONAS H.: Causality and Perception, Journal of Philosophy, 47:319, 1950. JORDAN P.: "Kausalitaet und Statistik in der modernen Physik", Die Naturuwissenschaften, 15:105, 1927. -----: "On the Process of Measurement in Quantum Mechanics", Philosophy of Science, 16:269, 1949. KANT I.: Critique of Pure Reason ( 1781), translated by MAX MUELLER , New York, Macmillan, 1896. -----: Critique of Teleological Judgment ( 1790), translated by J. C. MEREDITH , Oxford, Clarendon, 1928, p. 54. KEMBLE E. C.: "Reality, Measurement, and the State of the System in Quantum Mechanics", Philosophy of Science, 18:273, 1951. KIRCHHOFF G.: Vorlesungen ueber Mathematische Physik, 2nd ed., Leipzig, Teubner, 1877, Vol. 1, preface. KOPFF A.: Grundzuege der Einsteinschen Relativitaetstheorie, 2nd ed., Leipzig, Hirzel, 1923, fig. 1. English translation by H. LEVY, London, Methuen, 1923. KRATZER A.: "Wissenschaftstheoretische Betrachtungen zur Atomphysik", Abhandlungen der Fries'schen Schule, 6:291, 1937. LEIBNIZ G. W.: Discourse on Metaphysics, translated by GEORGE R. MONTGOMERY , Chicago, Open Court, 1902, sec. 6. LENZEN V. F.: The Nature of Physical Theory, New York, Wiley, 1931. -----: "Procedures of Empirical Science", International Encyclopedia of Unified Science, Chicago, University Press, 1938, Vol. 1, No. 5. -----: "Philosophical Problems of the Statistical Interpretation of Quantum Mechanics", Second Berkeley Symposium on Mathematical Statistics and Probability, Berkeley and Los Angeles, University of California, 1951, p. 567. -----: "Verification in Science", Revue internationale de Philosophie, 5:323, 1951. -113-

57. LEWIS C. I.: An Analysis of Knowledge and Valuation, LaSalle, Open Court, 1946, p. 39. 58. McCULLOCH W. S, AND PITTS W.: "A Logical Calculus of the Ideas Immanent in Nervous Activity", Bulletin of Mathematical Biophysics, 5:115, 1943. 59. MACH E.: Science of Mechanics, translated by T. J. McCORMACK, 4th ed, Chicago, Open Court, 1919, p. 579. 60. MARCH A.: Die Grundlagen der Quantenmechanik, Leipzig, Barth, 1931, p. 189. 61. -----: Quantum Mechanics of Particles and Wave Fields, New York, Wiley, 1951. 62. MARGENAU H.: The Nature of Physical Reality, New York, McGraw-Hill, 1950. 63. MEYERSON E.: Identity and Reality ( 1908), translated by K. LOEWENBERG , London, Allen and Unwin, 1930. 64. MILL J. S.: System of Logic, 10th ed, London, Longmans Green, 1879, Vol. 1, p. 392.

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65. Ibid., chap. 8. 66. MINKOWSKI H.: "Raum und Zeit" ( 1908). Reprinted in Das Relativitaetsprinzip, op. cit., 24. 67. MOODY E. A.: "Galileo and Avempace", Journal of the History of Ideas, 12:163 and 375, 1951. 68. MUNICK R. J.: "Causality, Relativity and Language", American Journal of Physics, 19:438, 1951. 69. NELSON E.: "A Defense of Substance", Philosophical Review, 56:492, 1947. 70. NEUMANN J. VON: Mathematische Grurdlagen der Quantemnechanik, Berlin, Springer, 1932, p. 223. 71. Ibid., chap. 5. 72. Ibid., chap. 6. 73. NEWTON I.: Mathematical Principles of Natural Philosophy ( 1687), translated by M. MOTTE, edited by F. CAJORI, Berkeley, University of California, 1934. 74. PAINLEVÉ P.: Les Axiomes de la Mécanique, Paris, Gauthier-Villars, 1922. 75. PAULI W., JUN.: Relativitaetstheorie, Leipzig-Berlin, Teubner, 1921, p. 560. 76. PLANCK M.: "Die Einheit des physikalischen Weltbildes" ( 1908), Physikalische Rundblicke, Leipzig, Hirzel, 1922. 77. Ibid., Dynamische und statistische Gesetzmaessigkeit. 78. -----: "Zur Theorie des Gesetzes der Energieverteilung im Normalspektrum", Verhandlungen der Deutschen Physikalische Gesellschaft, 2:237, 1900. -114-

79. RASHEVSKY N.: Mathematical Biophysics, 2nd ed., Chicago, University Press, 1948. 80. Ibid., chap. 8. 81. Ibid., chap. 25. 82. Ibid., chap. 46. 83. REICHENBACH H.: Philosophic Foundations of Quantum Mechanics, Berkeley and Los Angeles, University of California, 1944. 84. SCHLICK M.: Algemeine Erkenntnislehre, 2nd ed., Berlin, Springer, 1925, p. 29. 85. -----: "Die Kausalitaet in der gegenwaertigen Physik", Die Naturwissenschaften, 19:145, 1931. 86. SCHROEDINGER E.: "Quantisierung als Eigenwertproblem", Annalen der Physik, 79:361, 1926. Also in Collected Papers on Wave Mechanics, translated by J. F. SHEARER AND W. M. DEANS, London and Glasgow, Blackie, 1928. 87. -----: What is Life? Gambridge, University Press, 1946. 88. WEIZSAECKER C. F. VON: "Zur Deutung der Quanteanmechanik", Zeitschrift fuer Physik, 118:489, 1941. 89. WEYL H.: The Theory of Groups and Quantum Mechanics, translated by H. P. ROBERTSON, London, Methuen, 1931, p. 77. 90. -----: Philosophy of Mathematics and Natural Science, rev. ed., Princeton, University Press, 1949, p. 191.

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91. WIENER N.: Cybernetics, Cambridge, Technology Press, and New York, Wiley, 1948. -115-

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A

B

C

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Acceleration, definition of, 10 of gravity, 17 , 45 Action, at a distance, 78 quantum of, 82 , 92 Adams, G. P., 110 Animism, in science, 9 , 54 Appell, P., 12 , 43 , 110 Aristotle, doctrine of four causes, 29 , 55 theory of the heavens, 9 theory of mechanics, 41 Atom, Bohr theory of, 83 , 101 , 110 Atomism, theory of, 81 Bacon, Francis, method of induction, 29 , 110 Bernard, Claude, concept of vital phenomena, 55 , 110 Biophysics, mathematical, 58 ff Birkhoff, G., 108 , 110 Blum, H. F., 58 , 110 Bohr, N., analysis of observation, 88 , 100 , 103 , 110 concept of complementarity, 82 , 103 , 110 interpretation of quantum mechanics, 107 theory of atom, 83 , 101 , 110 Boolean algebra, 64 , 67 , 108 Born, M., explanation of complementarity, 104 , 110 explanation of contraction in relativity, 77 , 110 statistical interpretation of quantum mechanics, 84 , 110 Bose-Einstein statistics, 86 Broglie, L. de, wave mechanics, 83 , 110 Brunswik, E., 110 Causality, according to: Aristotle, 29 ; Bohr, 82 ; Frank, 22 ; Hume, 12 ; Kant, 21 ; Mach, 12 ; Meyerson, 13 , 46 ; Mill, 4 , 29 ; Planck, 14 ; Poincaré, 22 as: contiguity, 12 , 14 , 51 , 62 , 79 ; efficacy, 4 , 8 , 14 ; functional relation, 12 , 17 ; identity, 13 , 46 ; necessity, 8 , 11 ; predictability, 19 ; relation between conceptual objects, 6 ; uniformity, 4 , 12 asymmetry in neural processes, 66

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dynamical and statistical, 14 , 26 , 52 , 81 , 96 , 103 in classical mechanics, 44 , 71 in quantum theory, 99 ff principle of, 16 ff reciprocity of 7 , 42 symmetry of, 49 Classical mechanics, 40 ff, 70 , 108 Collision of bodies, 6 Commutation law, 93 Complementarity, in physics, 82 , 103 -117-

in biology, 104 logic of, 107 Compton, A, H., 83 , 110 Compton effect, 83 Comte, A., law of development, 8 , 110 Concept, 4 Conservation, laws of, 10 , 13 , 46 , 51 , 60 , 89 Convention, principle of causality as, 22 Copernicus, 10 , 40 Corpuscle, 83 , 84 Cybernetics, 66

D

D'Alembert, J., 40 Definition, implicit, 23 operatioual, 24 Democritus, 81 Destouches, J. L., 105 , 111 Destouches-Février, P., 107 , 109 , 111 Development, Comte's law of, 8 , 110 Differential law, 8 , 44 Diffusion, equation of, 59 kinetic theory of, 60 Dirac, P. A. M., concept of observable, 93 , 111 theory of radiation, 86 , 101 , 111 Driesch, H., 55 , 111 Dualism, of corpuscle and wave, 83

E

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Eddington, A. S., 97 Einstein, A., special theory of relativity, 72 ff, 111 general theory of relativity, 78 , 111 theory of radiation, 83 , 111

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on quantum mechanics, 92 , 111 Equation, differential, 44 , 50 , 59 , 80 , 99 Energy, definition of, 47 conservation of, 46 kinetic, 45 , 47 potential, 45 , 47 , 61 of radiation, 83 Entelechy, Driesch's concept of, 55 Entropy, 53 , 57 Ether, 71 Event, space-time, 72 Excitation, of neurons, 62 Experimentation, 32 ff

F

Falling bodies, Law of, 16 , 33 Février, P., 111 Field theory, in classical physics, 50 , 84 in quantum theory, 94 Force, in classical mechanics, 10 , 42 , 71 , 79 as cause, 17 , 43 , 79 Frame of reference, 34 , 70 , 89 Frank, P., 22 , 24 , 111

G

Galileo, 10 , 16 , 19 , 28 , 32 , 36 , 37 , 40 , 44 , 111 Gonseth, F., 106 , 111 Gravitntion, Newton's theory of, 10 , 40 , 79 Einstein's theory of, 78 Griss, G. F. C., 109

H

Hadamard, J., 111 Hamiltonian function, 46 , 48 Heisenberg, W., 84 , 100 , 107 , 111 , 112 Heredity, law of Mendelian, 25 , 67 Hermann, G., 112 Hilbert D. 95 , 108 -118-

Horizons of reality, 106 Hume, D., 6 , 11 , 112 Huygens, C., 40

I

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Identity, Meyerson's concept of, 46

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Indeterminacy, principle of, 92 Induction, Bacon's methods, 29 Mills canons, 29 ff Integral, 48 Integral, law, 8 Interval, 72

J K

L

Jonas, H., 112 Jordan, P., 80 , 97 , 112 Kant, I., 21 , 29 , 55 , 112 Kemble, E. C., 103 , 112 Kepler, J., 10 , 40 Kirchhoff, G., 12 , 112 Kopff, A., 112 Kratzer, A., 112 Lagrange, J. L., 18 , 40 Lagrangian function, 45 , 48 Landahl, H. D., 60 Laws, of motion, 10 , 42 Leibniz, G., 18 , 112 Lenzen, V. F., 112 Leucippus, 81 Lewis, C. I., 113 Light, corpuscular theory of, 84 wave theory of, 83 Loewenberg, J., 110 Logic, of quantum mechanics, 108 of three values, 107 Lorentz, H. A., 53 , 71 Lorentz transformations, 73

M

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McCulloch, W. S., 64 , 66 , 113 Mach, E., 12 , 113 Macrophysics, 51 , 61 March, A., 113 Margenau, H., 97 , 103 , 113 Mass, 10 , 71 Measurement, 33 , 88 Mechanics, ancient, 23 , 41 classical, 20 , 28 , 40 ff, 70 , 108 quantum, 84 ff Metabolism, physical basis of, 58

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Meyerson, E., 13 , 49 , 113 Michelson-Morley experiment, 71 Microphysics, 51 , 81 Mill, J. S., 4 , 29 , 113 Minkowski, H., 72 , 80 , 113 Model, 5 , 58 Momentum, 10 , 42 , 83 , 88 Moody, E. A., 113 Motion, on inclined plane, 16 of falling bodies, 17 laws of, 42 Munick, R. J., 113

N

Nelson, E., 113 Neuroelement, law of excitation of, 62 microscopic theory of, 64 Neumann, J. von, 39 , 97 , 98 , 103 , 108 , 113 Newton, I., 10 , 18 , 40 , 42 , 55 , 78 , 113

O

Observation, in quantum mechanics, 87 , 93 , 95 , 102 methods of, 28 , 38 Operator, in quantum mechanics, 84 , 93 Order, in biology, 67 -119-

P

Q

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Painlevé, P., 21 , 36 , 113 Partition between object and subject, 88 , 100 Pauli, W. Jun., 113 Pepper, S. C., 110 Perception, 4 , 39 Photo-electric effect, 83 Pitts, W., 64 , 66 , 112 Planck, Max, 5 , 14 , 82 , 101 , 113 Podolsky, B., 92 Poincaré, H., 22 Predictions theory of, 105 Probability, 26 , 53 , 97 , 101 Psycho-physical parallelism, 39 Pure case, 95 Quantum theory, 81 ff

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unity of, 86

R

S

Rashevsky, N., 58 , 60 , 62 , 66 , 114 Reichenbach, H., 107 , 114 Relativity, 70 ff changes in, 77 general theory of, 78 principle of, 70 special theory of, 72 temporal order of cause and effect in, 75 Rosen, N., 92 Schlick, M., 18 , 114 Schroedinger, E., 68 , 84 , 87 , 99 , 114 Schroedinger's equation, 84 , 99 , 108 , 114 Second quantization, 94 Simplicity, criteria of, 18 Space, 33 , 72 Space, Hilbert, 95 , 108 Space-Time of relativity, 72 , 78 causal structure of, 74 State of system, macroscopic state of, 27 , 53 in ancient mechanics, 41 in classical mechanics, 23 , 41 , 45 in quantum mechanics, 84 , 94 Statistics, in classical theory, 51 in quantum mechanics, 84 , 94 , 98 Superposition, principle of, 96

T

Template, 68 Term, connotation or intention of, 4 denotation or extension of, 5 Thales, 9 Theories, macroscopic, 51 f microscopic, 51 f objectivist, 105 subjectivist, 105 Thermodynamics, laws of, 51 , 56 first law of, 56 second law of, 52 third law of, 69 Time, 33 , 72

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Transformation, of coordinates in classical mechanics, 70 Lorentz, 73 in quantum mechanics, 93

U V W

Unitary ensemble, 98 Unity of Science, 33 Vector, 75 , 93 , 95 Wave-field, quantum theory of, 94 Wave function, 95 Weight, 17 , 23 Weizsaecker, C. F. von, 90 , 114 -120-

Weyl, H., 18 , 95 , 114 Whitehead, A. N., 40 , 84 Wiener, N., 66 , 114 Windelband, W., 55 Work, 45 World, space-time, 72 line, 73 -121-

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