causality and modern physics

Hegeler Institute

CAUSALITY AND MODERN PHYSICSAuthor(s): Henry MargenauSource: The Monist, Vol. 41, No. 1 (January, 1931), pp. 1-36Published by: Hegeler InstituteStable URL: http://www.jstor.org/stable/27901266 .

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VOL. XLI, No. 1. January, 1931

THE MONIST

CAUSALITY AND MODERN PHYSICS

THE thorough and rapid change in the attitude of

scientists toward causality is epitomized in the pro nouncements of two physicists whose brilliant discoveries formed the nuclei of the present scientific revolution. Max Planck stated, two years before the birth of quantum mechanics,1 that "the assumption of causality admitting no

exception, of a complete determinism, forms the presup position and the condition of scientific knowledge." Only four years later, Heisenberg wrote: "Because all experi

ments are subject to the laws of quantum mechanics the

invalidity of the causal law is definitely determined by means of quantum mechanics."2 Although the divergence of the two quotations marks to some extent the philo sophical taste of the two men, it is nevertheless character istic of a development which has gone on in the minds of

physicists and philosophers in general. But the removal of causality from the context of natural events calls for a new category; Planck's observation is felt to be correct in so far as it implies that the world cannot be left entirely to the reign of contingency. To succeed causality, prob ability has been chosen by the consent of most physicists and many philosophers. "We imagine a world in which all dependencies are of the same character by which the

appearance of one face of the die is connected with the 1 Lecture on Freedom delivered before the Academy of Science in 1923. *Zeitschr. f. Phys., 43, 1927.



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throw; each step in nature's happenings is a throw of

dice, and it is only the great probability of certain se

quences which has misled us to believe that there is a sure lawfulness concealed in them. . . . Such a world possesses in each of its elements a probability connection."3 The reason for this view is apparent from Reichenbach's fur ther statement: "It is the postulate of a minimum of

hypotheses which forces us to renounce strict causality." To investigate whether or not the substitution of prob ability for causality actually achieves a reduction in the number of hypotheses will be one of the purposes of the

present discussion.

Economy of hypotheses, however, was not the only consideration leading to the substitution in question. Phy sical theory, in the form of quantum mechanics, had abandoned the classical practice of formulating problems in terms of uniquely fixed and determinable quantities; it had introduced statistical uncertainty into its very pre

mises and thereby produced beautiful results. The prac tical successes of a method which operated with the

physical concept of probability seemed to suggest that the transcendental postulate of causality should be replaced by that of probability. In following this methodological clue little attention was given to the views of such philos ophers as Leibniz, Lotze, Poisson, E. V. Hartmann,

Brunschvicg and others who felt that probability has a

place only in a world governed by unique laws. It is at once clear that causality and probability, or

statistical determination, are not mutually exclusive. In fact even during an epoch when all physicists embraced the former principle statistical reasoning proved to be more powerful than the tracing of causal connections. The

8 Reichenbach, Kausalstruktur der Welt und der Unterschied von Ver gangenheit und Zukunft.



CAUSALITY AND MODERN PHYSICS 3

second law of thermodynamics, the most general "law" of

physics, could be established only on statistical grounds, yet it was admitted that elementary events were causally determined. The typical argument at the basis of the statistical theory of gases is the following : We are given a great number of similar systems, say molecules consti

tuting a gas. We suppose the principle of causality to be valid and assume in addition that we know the laws which

govern minutely the behavior of the individual systems. Utilizing these data an imaginary being like Laplace's demon, capable of knowing the complete state of the as

sembly at any time in terms of coordinates, momenta,

phase relations, etc. of the individual systems, could calcu late the state at any other time. But lacking such micro

scopic information we must have recourse to some other

method of determining the properties of the assembly. Mathematically speaking we are to solve a set of differen tial equations without knowing the boundary conditions of the problem-in this case, the values of all dependent variables at a certain time. Fortunately it can be shown

that, although under such conditions their exact individual values can never be known, they assume a certain de

terminable distribution after the laws have been in force for a very long time, and this distribution is independent of the initial state. Moreover the properties of what is called the "equilibrium state" of the gas can be calculated from this statistical distribution. In this connection it is

important to realize that, at least in the simpler applications of the statistical theory, not only was complete determin ism presupposed but also a knowledge of the laws of mo

tion and reaction of the individual systems. So far the various steps on the way from deterministic premises to a statistical conclusion were clearly seen and distinguished,



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and no apprehension was felt about the sudden appearance of a contradictory element in the realm of a causally de termined world, such as probability is now widely con

sidered to be. But differences began to develop when radioactivity

became a subject of theoretical investigation. It was found that the experimental discoveries all obeyed the

simple well-known formulae involving the decay constant of an element, or its mean life, both statistical concepts. These formulae are the immediate consequence of assign ing to each atom a probability of disruption which is con stant and independent of the time the atom has existed. The suggestion that these probabilities were merely results of causally working laws affecting the behavior of the constituents of an atom, in a sense outlined in connection with the statistical theory of gases, could by no means be

rejected. But the difficulty and the differentiating feature in the case of radioactivity was the physicist's complete ignorance of these laws. The task of discovering them seemed of little promise in view of the circumstance that the laws governing the more accessible phenomena in the

periphery of an atom were not even known with accuracy. Hence it was best to start reasoning by assuming prob abilities, not by explaining them. Of the relation : unique laws->probabilities-^experimental facts, only the latter

part was retained; the former, though it might exist, was not amenable to the treatment customary in physical prob lems. Consistently with this development arose the ques tion: If probability is at the basis of the physical argu

ment, could it not be considered the primary thing in nature? Why start with something which we do not know? The persuading force of such propositions is un

doubtedly strong. It is increased by the experience of the




success of further theories founded on probability axioms

(Einstein's probability coefficients for quantum transi tions, excitation probabilities4). But if the first part of the relation above be obliterated entirely, it must be shown to be intransitive; for, if probabilities always imply unique laws, then starting with probabilities would at most be a confession of ignorance.

We saw in the discussion of the statistical theory of gases that statistical uncertainty was carried into the situa tion by an insufficiency in the determination of boundary conditions. Uncertainty, however, can also be introduced

by an ambiguity in the laws regulating the microscopic behavior of the molecules. Whether or not this uncer

tainty is statistical, and what is meant by the latter term, will be subject to further investigation. For the present we will assume that the uncertainty is of the same char acter as before. The word ambiguity calls for comment. Is a law still a law when it is not unique? Possibly not, in its strict and customary sense. In a perfectly chaotic

world any event may be followed by one of an unbounded

variety of events. In a perfectly lawful world the follow

ing event is uniquely determined. By an ambiguous law we shall mean a restrictive principle which will select out of the unbounded variety of the former instance a definite, determinable set of events. The admission of ambiguous laws, therefore, implies a denial of strict causality.

Up to this point, we have found no evidence for or

against causality; we have merely shown that on certain

assumptions-to be examined later-absence of strictly causal laws is compatible with our physical knowledge of the world. Surveying the situation as a whole from this

point of view one might observe that it is more cautious 4

Strictly speaking, excitation probabilities need not be considered as

axioms; they can be derived, if desired, by the statistical theory of gases.



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to uphold the validity of (unknown) unique laws, because, if support is given to the other alternative, our attitude toward these fundamental questions may have to be

changed if future discoveries reveal such laws. On the other hand, one dislikes to postulate things unknown. The

example of radioactivity, chosen here to illustrate the argu ment, is in no way exceptional. The quantum theory of emission and absorption of radiation, for instance, would have served the same purpose.

Physicists consider that the dubious status of causality established as a result of such theories has been relieved

by quantum mechanics. The following statement by Born, which puts the blame for the uncertainty definitely on the laws of nature, reflects a widely accepted belief: "The laws of nature themselves are of such character as to

prevent the exact determination of the momentary state."5

Philosophers would accept this view with utmost reserva tion. The discovery of new laws of such type, they would say, can never prove that all others are, and all those to

be discovered in the future will be, in accord with that statement. But the experiences which prompted Born to

make his contention are really superior to these objections. For, it is not a number of new laws which has led to this novel conclusion, but the discovery of a new, far-reaching, and very general principle from which laws of nature themselves may be derived.

It may be well to summarize the essential features of

quantum mechanics and to consider very briefly a few of its philosophical implications. We should be forcing mat ters if we treated the subject as a philosophical discipline which speculates about nature. Its chief concern is one of method. It has grown up around the question

* What

5 Vossiche Zeitung, April 12, 1928.




mathematical method best enables us to derive the facts of

experimental observation? The dominating perspective is mathematical. Hence we find in quantum mechanics per fect clearness of mathematical concepts, some dubiousness

in physical interpretation, and considerable uncertainty in its philosophical formulation. It will be recalled that

actually two methods, in principle equally successful, were

devised, (Schr dinger's and that of Heisenberg, Born, Jordan), Schr dinger's being more commonly known as wave mechanics. But the two methods were shown to be

equivalent in their results-a most striking and beautiful

coincidence, since their starting-points were completely diff rent (function theory on the one hand, matrix algebra on the other). When in the course of this discussion we

speak of quantum mechanics without further specification, we mean the method which is a result of the fusion of the

two, and which is used at present in most physical in

quiries concerning atomic or molecular phenomena. It

consists in writing down, to start with, the equation of conservation of energy in Hamilton's form, a relation

which derives its validity from classical, causal considera tions. This equation, however, is not the foundation of the new theory in the sense that it must be correct and

applicable to elementary events if the theory is to remain valid ; in fact, it is merely the formal carrier, the physical content of which is unimportant. The next step is to make certain mathematical changes and substitutions (multipli

h cation by substitution of-for pi), by which

2JI xi process one arrives at Schr dinger's equation. To be

sure, the latter may be derived in a variety of ways, but it is important to observe that the classical Hamiltonian is the indispensible means for obtaining it. The further



8 THE MONIST

work is then to solve Schroedinger's equation. This of course is a definite problem only if t|) is subject to certain conditions (continuity, single-valuedness, vanishing at in

finity) . Thus we are immediately confronted with the ques tion as to to the significance of the function AS long as we merely wish to calculate the stationary values of a physi cal problem, we are not forced to deal with this question at all ; we may even deny physical reality to ty and consider it as an auxiliary quantity introduced by the mechanism of calcu lation. But, if we desire information about processes in

volving non-stationary states, we are compelled to make

an interpretation. The mathematical apparatus, though

limiting the choice, does not associate definitely one con

cept with the symbol ty. TWO possibilities were suggested, the decision between which was largely a matter of pref erence with the individual scientists. Born's interpretation (ijn^dt is proportional to the probability that an electron

will be found at a given place at a given time) seems at

present to_be more generally upheld than that of Schr

dinger (i|n|)dt is proportional to the quantity of charge at a given place and time), and this not without good reason:

Heisenberg, in establishing his famous uncertainty rela

tion, showed that the exact simultaneous values of the

parameters of a quantum mechanical system cannot be

determined, that there must always be uncertainty. Hence

the emergence of probability in quantum mechanics, and of

uncertainty in every statement about nature which that

physical theory allows us to make. On these grounds it is usually claimed that the laws of nature are proved to be ambiguous; but let us be more cautious and say that our statement of the laws contains a necessary uncertainty. And what about causality? Quantum mechanics was characterized as a most potent system of mathematical




thought, capable of explaining in an amazing manner a

very large number of physical (experimental) facts but not of giving a detailed picture of the world in terms of exact instantaneous physical quantities. If we express the prin ciple of causality in a form which enjoys particular favor with scientists: "When the state of a closed system is

exactly known at any moment, then the laws of nature determine the state at any later time,"6 we must conclude

(again with Born) that causality is "empty," incapable of verification. The latter conclusion hinges, of course, on

the supposition that quantum mechanics is applicable to all phenomena and that it is the final form of physical reasoning.

Aside from this consideration it is quite evident that

quantum mechanics itself has not reached the final stage of its development, and it may justly be doubted that it will never change in its attitude towards the fundamental

questions of interpretation. One of its chief imperfec tions is felt to lie in the obligation of resorting to classical

physics for the material from which to construct Schr

dinger's equation. This circumstance is indeed somewhat

disturbing, for it involves the peculiar situation of one

theory utilizing results of another which (though the theories are non-contradictory) could not have been ob

tained strictly on its own premises. To physicists this state of affairs presents no serious difficulty. They have

met with it before in the apparent conflict between geo metrical and physical optics, where all contradictions van ished when the former was considered as a macroscopic approximation to the latter. The same remedy was here available. This procedure can also be justified from the

point of view of critical metaphysics. Every physical 6 Born, loe. cit



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theory has to make some transcendental postulates about nature. In classical physics these were statements of a

direct substantial character, such as, for instance: the

world consists of smallest, indivisible parts with certain definite properties. In quantum mechanics the postulates are more formal and less direct, and may even be called

methodological: the world is so constituted that we can ascertain its laws by means of a mathematical process pre scribed by the rules of quantum mechanics. Every rule is then a sort of recipe, and whether or not it is true would be a meaningless question. In this sense the use of class ical concepts in quantum mechanics is permissible.

Let us now return, for a moment, to the two quota tions at the beginning of this exposition. Heisenberg's statement can evidently not be upheld in its full rigor: causality can at most be pronounced empty, to use again Born's phrase. The error in Heisenberg's philosophical argument has been recognized clearly by Bergmann.7 The very instructive discussions of this philosopher also throw

light on the correctness of Planck's view that the assump tion of complete determinism is essential to scientific

knowledge. According to Bergmann, causality has to achieve a double task in ordering the data of sense per ception into the body of science : First, it must determine the relative position in time of different events; secondly, it must enable the scientist to make statements about the future through the correlation of his experiences. Then the question is asked whether or not these functions can be fulfilled if the understanding of causality is loosened to be in accord with the alleged consequences of quantum mechanics, i.e., to admit the existence of ambiguous laws. With regard to the first function this is to be answered in

1 Der Kampf um das Kausaigesets in der j ngsten Physik, p. 39.



CAUSALITY AND MODERN PHYSICS ll

the affirmative. "As far as the second function of the causal law is concerned, namely forecasting the future and

reconstructing the past, this would of course be impossible with reference to the single case," Bergmann finds.8 "Nevertheless this need not disturb us practically if only the infinite number of possibilities were, on their average, included in a definite range. For in this case, though we could not predict the definite effect or reconstruct the definite cause, we could predict with the certainty which the calculus of probability conveys that the effect will be

long to a definite range. Sommerfeld claims:9 'We must

postulate the exact prediction of what is to be observed under given circumstances as long as there is to be a natural science/ This postulate remains fulfilled also in the case of the statistical natural law." Although we can not spare this argument a slight criticism concerning the

vagueness of the postulate-the infinite number of possi bilities must, on their average, be included in a definite

range10-it seems convincing, and therefore calls for a

modification of Planck's statement which, as it stands, de mands too much. But before this verdict becomes final it will be necessary to examine the implications of "loosening up" causality in the manner advocated by Bergmann, and to assure ourselves that this can actually be done.

Up to this point our discussion has been somewhat in coherent and has lacked a central view. Its aim has been

merely to sketch the general philosophical trend and to point out certain difficulties with prevalent opinions. We shall now proceed more systematically in the search for a solu

8 Loe. cit., p. 52. 9 Zum gegenw rtigen Stande der Atomphysik," Phys. Zeitsch., 1927, p.

234. 10 Even if it be put in Schlick's form (Naturphilosophie, p. 457) : "That

which possesses the greater mathematical probability will occur in nature with

correspondingly greater frequency," as Bergmann does, this postulate does not

seem sufficiently clear.



12 THE MONIST

tion of our problem. First we shall attempt to fix the

place of causality in physics and consider if, and under what conditions, the philosophical content of this principle can be affected by physics. To this end we must deal somewhat with the various trends of physical thought and its limitations. Then it will be desirable to fix the mean

ing of causality by a definition most suitable to our needs.

Finally, after our tools have been sharpened by these pre liminary discussions, we shall have to inquire whether we are forced to abandon causality and to replace it by less

stringent hypotheses. This will be the more important part of the argument and involve primarily an analysis of the probability concept.

No matter in what particular form the principle of

causality is expressed, it must be taken to be a "syn thetisches Urteil a priori" in the Kantian sense. Ad herence to it constitutes the willingness to assume that any explanation which does not set down one phenomenon as

definitely determined by others is incomplete. Hence it is

impossible to disprove causality either by logical reasoning or by empirical evidence. This does not mean, however, that it is supreme and forever unchangeable, that it can not be reached by scientific argumentation, as Kant would have it, perhaps. Modern philosophy has left intact the transcendental character of a priori postulates, as formu lated by Kant, admitting in particular their importance as conditions for the possibility of experience, but it disputes their uniqueness and recognizes that under certain cir cumstances controlled by science one postulate may have to be replaced by another. That will be the case when a

postulate calls for too little uniformity in the course of natural events to be useful, or when it demands so much as to become burdensome to the progress of science. It




was for the former reason that teleology (everything hap pens to realize some inscrutable purpose of God) has been abandoned, and for the latter that physicists desire to have probability substituted for causality.

The purpose of physics is to collect and explain a cer tain type of nature's data. This statement means little unless we give the terms "nature" and "explanation" a

very definite significance beyond their colloquial meaning. To avoid lengthy discussions we wish to refer to a paper in which we dealt with these matters11 and use the terms in the sense there specified. Collecting data is chiefly the function of experimental physics, a discipline of explora tory and descriptive character. Its results, which, strictly speaking, never rise above the status of empirical evidence, do not touch causality, unless they are permeated with, and connected by, the fluidum of speculative interpretation. There have been scientists who claimed that the purpose of

physics exhausts itself in discovering and cataloguing facts about nature, in furnishing a complete photography of the

world; but we cannot convince ourselves that any sound

philosophy, whose function it is to assign to every category of thought its proper place within the context of human

affairs, could limit the purpose of physics to the extent of that supposition. To show why it is necessary to include in the apparatus of physics a certain symbolism whose elements do not have their origin in empirical data has been

attempted in the paper just mentioned.

Admitting, then, the legitimacy of a transcendental

symbolism of explanation, we must ask in what manner this latter is to be controlled by experimental physics. Is it necessary that, in constructing explanatory symbols, we take account of the limitations inherent in empirical meth

11 The Problem of Physical Explanation, Monist, July, 1929.



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ods ? The question is a very vital one. We shall illustrate it by two examples. The time of occurrence of a point event is a perfectly good physical concept. No logical con

straint would prevent us from constructing, out of it, the

concept of simultaneity. But because the time of events can only be established by means of signals, and these

signals require a finite time for their propagation, an em

pirical uncertainty is introduced into the concept of simul

taneity, and relativity prohibits the use of it. A similar situation is encountered in quantum mechanics. Momen

tum of an electron is a proper physical symbol, and we

may speak of the exact value of its momentum without

violating the principles of quantum mechanics. It has

sense, too, to designate the exact position in space of an electron. But Heisenberg's uncertainty principle states

that no two canonically conjugate quantities of a quantum mechanical system, such as momentum and position of an

electron at a given time, can be determined with perfect precision. The proof of the uncertainty relation is not

quite as transparent as the consideration which leads to

the rejection of simultaneity; however, it has its basis in the fact that quantum mechanics, in its very hypotheses, adapts itself more closely to the necessary imperfections of

experimental findings than classical physics does, and the

proof even appeals for part of its evidence to experimental facts. Thus we find again that, for reasons having to do with the empirical origin of knowledge, a combination of

concepts which is non-contradictory in itself is declared inadmissible. This state of affairs arises not from a mere shift of emphasis from theoretical to practical physical concern; it constitutes indeed an intrusion of empiricism into a field which had been reserved for pure reason. To decide whether or not it is justifiable appears to be a mat




ter of extreme importance. Clearly this attitude leads to what may be called forbidden concepts, logical constructs which are free from internal contradictions and discrep ancies, yet considered objectionable. And causality would

belong to the group of forbidden concepts since it involves

perfect precision of canonically conjugate quantities. Recognizing this, however, we do not wish to commit

the serious error of confusing "forbidden" with "meaning less." Causality does not become meaningless, even if no

physical measurement and no physical theory can ever

satisfy the conditions which it requires. This would be

necessary only if we attempted to prove the causal law which is far from our intentions and must be considered, after Kant's achievements, an undertaking about as useless

as the design of a perpetual motion machine.12 Now in what sense is the use of "forbidden" concepts

prohibited? In answering this question we must not lose

sight of the important and basic recognition regarding the character and natural limitations of all empirical knowledge. Scientific methods ought to take account of this fact; and if any method is capable of deliberately in

troducing the discreteness of natural data into its struc

ture, and of keeping all its consequences in conformity with the uncertain character of experimental statements, that method has a decided advantage over others. Quan

12 In his paper, "Kausalit t in der QuantenmechanikZS. f. Phys., 55, 1929, G. W. Kellner proposes to show that causality governs the phenomena of

quantum mechanics if op is a "Feldgrosse? i.e. if WdT is not a probability but a continuously distributed element of charge. He resolves the difficulty of indetermination by the peculiar device of waiting an infinitely long time. The

exposition points out that if a given form of quantum mechanics-Schr dinger 's -is used and the process of waiting infinitely long accepted as a valid means

for determining the initial state, then the complete state of an electron can be

calculated. We note that even if this were not the case such would not con

stitute a direct argument against causality.



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tum mechanics is such a method. But it is not a meta

physical doctrine. Physicists have of late rejected entirely all "forbidden" concepts and they have a perfect right to do so if they desire to limit arbitrarily their field of work and separate it sharply from metaphysics. This can not, however, do away with these concepts ; it merely transfers them to another discipline for consideration. Whether the

process is completely successful in purging physics from all metaphysical impurities-which we believe it is not need not concern us here.

It can not be denied that as long as the concepts of momentum and position separately have meaning, a state

can be imagined in which both are determined. If nature withholds from our knowledge the evidence of such states, and if, as quantum mechanics shows, we can get along without it in physics, that is quite another matter. One feels no logical necessity of abandoning "forbidden" con

cepts for that reason, and logic is the only discipline to

guide us here. The philosopher might almost be tempted to trace the repugnance against this concession to the fear that it implies some judgment as to "objective reality" or "realization in nature" of these concepts. But this fear is

wholly unfounded; philosophy has learned carefully to

guard against such confusion. Bridgman,13 following Einstein and others, goes so far as to suggest that all

physical concepts should be defined by operations. That may be a suggestion of great usefulness. Nevertheless it can not be tolerated as a general directive. For, in the first place, it would, if carried to its consequences, dissolve the world into an unmanageable variety of discrete con

cepts without logical coherence. There would be no way of telling, for instance, why a time interval read from a

13 In the Logic of Modern Physics.




clock is more closely related conceptually to a time interval measured by astronomical observations than to weight de termined by means of a balance. In the second place, such a proposal is likely to impede the progress of science by emphasizing too strongly its most rapidly changing ele ments: experimental methods. Finally, physical concepts are not by their own nature different from others and therefore do not require a mode of definition which is not

applicable to others. However, there are many useful con

cepts which can not be defined in terms of operations. As a consequence of these considerations, and with due

appreciation of the weight of the circumstances that have caused many physicists to taboo the forbidden concepts, we are compelled to pronounce them legitimate in a more

general sense. If we did not do so, our discussion would end here, for causality would lose its meaning. We are aware that, in the eyes of physicists, we are carrying our

problem from the domain of physics into metaphysics, where it properly belongs. Incidentally it may be recalled, however, that this particular branch of metaphysics has not always been thought foreign to science, as it now

generally seems to be. Kant, for instance, collected "the

a priori conditions for the possibility of experience," of which causality is one, under the name of "pure natural science."

Having now established the general relation of the

causality problem to philosophical and physical thought, we do well to fix its meaning with as much precision as is

possible. In the preceding pages the terms "causal law" and "causality principle" have occurred somewhat pro miscuously. A distinction between the two, such as Benno Kohn14and Brunschvicg15 have formulated, was not in

14 Cf. his Untersuchungen ber das Kausalproblem, 1881. 15 In his JJ exp rience humaine.



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tended. The use of "causality principle" seems preferable because the term law is usually reserved for more detailed statements about the performances of nature. The very common form of the principle previously referred to:

When the state of a system is exactly known, then every future state is determined, can now be dismissed without further comment, for this formulation makes the principle dependent on our exact knowledge of systems and there fore depresses it to the level of empirical relations, which, as we have seen, is improper. In addition to this fault it is contrary to the dominating tendency of physical science, which was seen by Planck and others in a pro gressive detachment of the physical interpretation of the world from the "individuality of the forming mind." Human knowledge must be regarded as accidental to the

validity of an a priori principle. Hence the latter should be modified to read : "The state of a system is determined

by that which exists before it," and this ought to be up held whether we know the previous state, or can know it, or not. The last mentioned form of the causality principle is approximately the one given by Brentano.16 Because the principle has so often been encumbered with empirical impedimenta its form was thought to be variable and sub ject to the progress of physical science. Such, for instance, seems to be the opinion of v. Mises who claims17 that causality is more or less a matter of habit, that prior to the discovery of the law of inertia motion without force

was held to be a violation of it. But it appears to us that these observations have nothing to do with the principle; they merely show that it was very frequently misunder stood and treated like an empirical datum.

16 Cf. his Versuch ber die Erkenntnis.

17 In his essay Ueber kausale und statistische Gesetzm ssigkeit in der Physik, Naturwissenschaften, 1930, Heft 7.




We desire to reduce the principle of causality to as

simple a formula as we can find. For this reason we pro

pose a form which is not altogether customary but enjoys the advantages of clearness and generality. In the paper already refered to,18 mention was, made of a postulate called the principle of consistency of nature. It stated

essentially (and roughly) : Under the same conditions, nature behaves alike. It involves an assertion that there

are unique laws. We feel that this (though inadequately phrased at present) characterizes the state of affairs which the principle of causality is designed to express, and there fore use it in place of the latter. Consistency of nature, in this sense, postulates nothing that is not implied in the usual conception of causality as a transcendental principle, and it can easily be shown to result in a perfect determin ism such as strict causality calls for. Let us first, how ever, improve its precision by referring it, not to nature as a whole, but to closed systems. We may then state it thus :

When a closed system undergoes changes, one state, if realized more than once, is always followed by the same other state. Or, if a process in a closed system occurs once as a consequence of a certain condition, it will occur

in the same manner every time that condition returns.

Reference to closed systems is necessary, of course, if the

formulation is to have physical sense; for they are the

only objects with regard to which physical statements have

meaning at all. This point is perhaps of some interest to those who endeavor to found all physical concepts on oper ations. A closed system, a very necessary and useful con

cept, is certainly an idealization of experience; it can at most be called a speculative extrapolation upon an ever

approximating set of operations. From our point of view, such concepts have nothing that is obnoxious.

i* Monist, July, 1929.



20 THE MONIST

Starting with the principle of consistency of nature, by what manner of reasoning is one led to the assumption of determinism? If nature were periodical in time, de terminism would be a consequence of consistency. It was to take account of the possibility of a non-periodical nature that the principle of consistency had to be postulated with

regard to isolated or closed systems. Each of these is a miniature world, susceptible of comparison to any other in the light of the principle. If the changes in any system are periodical, the unique future behavior of that system is immediately established, for it must continue to pass through the cycles which are fixed by its own past. If the changes are not periodical and we encounter a new

state, we must only suppose that a similar system has been in the same state at an earlier time. Then, the flow of time being unique and its direction fixed, there lies in the

past some definite state which is the consequent of the one under consideration. Thus we arrive at a determinism for all individual closed systems, and hence for all natural oc currences. This latter generalization, which may appear

unwarranted, is certainly permissible if nature is separable into a number of closed systems. For, by the definition of a closed system, its effects, if compounded with those of other closed systems, are additive, and the course of nature must be uniquely determined as long as that of all single systems is determined. But nature can not be resolved into independent (closed) systems; the usual process of

doing so is only an approximation. Is the generalization inherent in the postulate of determinism for nature as a

whole still to be maintained on the basis of this state of affairs? The answer is in the affirmative. Whenever several systems whose laws are known when they are con

sidered closed are compounded in such a way as to take




account of interactions between them, there exists always some mathematical process (either direct, or a method of

perturbations) by means of which the laws of the composite system can be obtained. Whether or not the details of the calculation can actually be carried out is unimportant in this connection. The essential feature is the possibility of

deriving unique laws if mathematical difficulties can be IO overcome.

The symmetry of the causal relation has occasionally been disputed. Reichenbach20 claims the existence of "nodal points" in the chain of natural events, points at which the causal nexus becomes ambiguous since one effect

may have more than one cause or vice versa. In his

Philosophie der Raum-Zeit-Lehre he argues: "If Ex is the cause of E2, small variations in Ex will be connected with small variations in E2. However, small variations in

E2 are not connected with variations in Ei." For a more

detailed discussion of these matters we wish to refer to

Hugo Bergmann,21 who maintains that Reichenbach's as

sertion results from a confusion of total and partial causes. Bergmann's reasoning appears to us to be con

vincing. But whatever may be the evidence for the other

point of view, asymmetry of causal relations is incom

patible with the principle of consistency of nature which we regard as the basis of the causality postulate. It fixes the future as definitely as the past and admits of no "in transitive forks," to use Reichenbach's terminology. We shall therefore exclude asymmetry as foreign to the

thought of strict causality, without saying anything, at 19

Quantum mechanics, of course, does not arrive at unique laws by any method. This is due to the usual supposition of indeterminism in the indi vidual systems.

TLoc. cit

^Loc. cit., pp. 16 ff.



22 THE MONIST

this point, about the right with which it may be postulated as an a priori hypothesis.

Before discussing why, as a result of modern physical discoveries, determinism should, or should not be, aban

doned, let us investigate on what fact belief in determinism, in classical physics, was founded. Aside from ever present encumbrances due to habits of thought, this basis was an

unfailing correspondence between causal forecasts of physi cal events and their actual occurrence. One might think of a na ve form of determinism in which a calculation would fix with utmost precision the outcome of an experiment. This, however, was by no means the case in classical physics. The correspondence was known to be of a more indirect character : calculations yielded a value which was identical with a certain one included within a range of values de termined experimentally. The theory of errors, which is a branch of the probability calculus, was called upon to

specify which value, selected from the experimental range, should be compared with the results of calculations. Strict

ly speaking, nature was not expected to respond exactly to causal reasoning. Nevertheless, determinism was

thought to be intact. It seems of great importance to realize that measurements do not, by themselves, fix a

physical quantity or verify a law. They furnish a sequence of values, in fact whole numbers, which are combined ac

cording to probability rules to give a certain number which

is, in general, not among the ones obtained by measure ment. But the application of the probability calculus is

possible only if certain conditions, to be imposed on the set of numbers in question, are satisfied. And these con

ditions, in turn, involve new postulates. The set of num




bers must represent a "probability aggregate." By this term we wish to denote what R. v. Mises called a Kol

lektiv."22 There are two properties which make a set of

events a probability aggregate : first, the existence of limits for the relative frequencies; secondly, invariance of these limits against a systematic choice of events {Prinzip vom

ausgeschlossenen Spielsystem). We will illustrate these

requirements with reference to the game of dice. Suppose that we throw one die n times and that a six appears xii

Vii

times. Then the fraction - is called the relative fre n

quency of the appearance of a six, and the first require ni

ment demands that lim - has a definite value for any n^00 n

sequence of throws. This implies the possibility, if not the actual carrying out, of an infinite number of throws. The

ni limiting values of the relative frequencies

- , as n ap n

proaches infinity, are called probabilities, and their assign ment to the various events (appearance of 1, 2, ... 6), which constitutes the "distribution" of the probability ag gregate is made according to principles not included in the

probability calculus. For instance, if it is agreed that the

probability of throwing a six, or any other number, with 1

one die is - , this is not derived by probability considera 6

tions. Probability starts with a given distribution and

permits the calculation of the distribution in a modified

aggregate (for instance one which results if two dice 22 In this connection reference ought to be made to v. Mises* excellent

discussion of the foundation of statistics in his article: JJ eher kausale und

statistische Gesetzm ssigkeit in der Physik, Naturwissenschaften, 18 Jahrg, Heft 7, 1930.



24 THE MONIST

are thrown at once, etc.). The second requirement is ni

that lim - must remain unchanged when certain n"^00 n

throws are selected from the infinite sequence according to any preassigned rule (say, every 2d throw). In com

puting the "most probable value" from a set of measure ments we expect that nature, in presenting its data, has

supplied us with a probability aggregate, which is equiv alent to assuming the truth of the following propositions : Each single measurement is forever repeatable; in any set

consisting of a great number of measurements the various

results are so distributed that each occurs with a constant relative frequency; a similar set of measurements derived from another by means of selection rules respecting only the order of values in the primary set has the same dis tribution of relative frequencies.

To say that the probability of throwing a six or any other smaller number with one die equals 1/6, reflects the physical condition that all six sides of a die are equivalent. Loading a die does not alter the nature of a sequence of throws as a probability aggregate, but changes its distri bution by shifting the values of the various probabilities. In a set of measurements there is no physical considera tion by which the distribution may be known ; we are here in need of a special hypothesis. The so-called error law of Gauss, which relates the probability of a certain meas ured value to its numerical magnitude (the "error" is simply a measured value from which the arithmetical mean has been subtracted) is commonly assumed to determine the distribution. Analytically it amounts to, and may be derived from, either of the following two suppositions: ( 1 ) The most probable value of a set of measurements is the arithmetical mean computed from the direct results of




observation; (2) The most probable value is the one for which the sum of the squares of the deviations is a mini mum. The second may be shown to follow from the first.

Clearly, there is no compulsion about making such an hy pothesis. The only foundation for the choice is its sim

plicity, which is apparent from the first of the suppositions. The arithmetical mean is here arbitrarily specified, yet it is not difficult to think of examples where the mean of a

higher power than the first has greater physical signifi cance than the arithmetical mean. While the error law can not be proved by a priori arguments except with the aid of special assumptions, it may, in a certain sense, be said to be derivable from experience. For, if in the great

majority of instances we find the calculated value of a

physical quantity to coincide more nearly with the arith metical mean of measurements than with any other, we feel safe in upholding the validity of the Gaussian distribu tion. But it is to be remembered that such coincidence re

quires the feasibility of calculations, which is strongly impaired, if not destroyed, as soon as determinism is abandoned. Causality thus offers a chance of justifying empirically the error law; but this is not of sufficient

weight to demand adherence to the principle of consistency excluding other considerations.

Physical measurements, the elements of empirical knowledge, have been shown to be merely individual events of a series forming a probability aggregate. The simplest operations of the experimental physicist have meaning only if placed in a larger statistical scheme. Nevertheless this situation, as viewed by scientists twenty years ago, was not subversive to the general belief in causality. We have here the same state of affairs as we found in con nection with the statistical theory of gases: nature is



26 THE MONIST

thought to be consistent, to follow unique laws; uncer

tainty is introduced by our ignorance regarding boundary conditions. But whereas in the case of the gas theory no

attention is paid to the initial state, we make in all physical measurements a very deliberate and systematic attempt to

determine the experimental conditions. In measuring the

length of a table we are very careful to have the edge of the yardstick flush with the edge of the table. Now the

imperfection of our senses and our limited experimental skill do not permit us to realize this condition with arbi

trary precision, nor are we capable of reading exactly the

fraction of a scale division coincident with the other end.

Moreover, the yardstick is not perfectly accurate, for the

person who made it was subject to the same limitations. This example shows that the uncertainty of measurements, or their statistical character, results, not from an inherent

property of nature, but from cricumstances concerning

physiological and mental imperfections of the measuring individual. The assumption that the mean of a set of

measurements is the "true" value of a physical quantity, therefore, involves a proposition regarding the organiza tion of the subject in quest of knowledge rather than the

object which is to be known. It is possible, and it is a per fectly legitimate procedure, of course, to forego this an

alysis deliberately and to stop all inquiry with the sum

mary statement that nature presents us with data forming a probability aggregate whose mean is the "true" value. But this statement, which is much in the manner of modern physical thought, voices a skepticism that appears less satisfactory from a philosophical point of view than the explicit attitude of classical physics, which explained




the statistical character of measurements by the postulate of consistency of nature combined with assumptions about the qualities of the measuring individual.23

It has been pointed out that there is no cogent a priori reason for attaching greater weight to the mean of a set of measurements than to any single measurement. Hence

it is not entirely out of place to ask the question : Could it not be that the choice of the mean produces a distorted

picture of the world? Is another choice more suitable? We easily convince ourselves that a different choice would cause all laws of nature to be different and would therefore

change the aspect of the world. But the only criterion for the suitability of a choice is in the simplicity of the laws, and this would indeed not be preserved if any value essen

tially different from the mean were selected. While experimental physics has dealt extensively with

statistical concepts, theoretical physicists has until recently 23 This more explicit attitude has one minor disadvantage. Let us sup

pose that in the various measurements made on a physical quantity the ob

served boundary conditions are such that their arithmetical mean is the "true" value. Then as long as the measurement itself involves only linear relations

(as for instance the measurement of length by means of a yardstick) the results of the operation are such that their mean is again the "true" value

of the measurement, that is, the one obtained if the "true" value of the boundary conditions had effected the result of the measurement. In general, however, measurements may be more complex, and one can no longer expect the mean

value of the measured results to be true in the same sense. Suppose, for ex

ample, the time in which a car travels the distance between two milestones is to be measured by means of a stop watch, and that several people in the car are carrying on the observations. For simplicity we will assume that the

milestones are accurately spaced and that the observers make no error at all in pressing the watch as the car passes the first stone. The errors made in

stopping the watch are so distributed that the average of the readings is the

"true" reading. Suppose now that the speed of the car is to be determined

by the same method. It would obviously be the reciprocal of the "true" reading thus obtained. If, however, an instrument recording velocities instead of

times (by automatically taking reciprocals) were used in place of the stop

watch, the average of the readings would not be identical with the "true" one.

This difficulty is of course absent if the distinguished status of the mean is postulated with regard to the results of measurements instead of the obser

vations on boundary conditions, but it is of no consequence in the above case

or any similar one, for the discrepancy there encountered is of higher order

than the error and vanishes when the errors become small.



28 THE MONIST

operated with fixed quantities. Quantum mchanics, rec

ognizing the role of probability in experimental methods, has now enlarged the field of application of statistical con

cepts b yincluding probability in its formalism. Differen tial equations no longer determine at what point in space an element of electric charge is to be found but give the

probability of its being there. What was previously con sidered a charged region of space has become a probability aggregate with a definite distribution. The degree of determination may range from absolute certainty to equal probability for every point of space, and is dependent on the exactness with which the conjugate quantity (here the

momentum of the charge) is known. In realizing this we must not fail to recall the requirements which a probability aggregate must satisfy, and not deceive ourselves as to the amount and severeness of the postulates involved. It can not be denied that there is considerable satisfaction in this new development, for it establishes a parallelism between the manner in which nature's data are gathered and the

formulation of laws. Instead of producing one sharply defined value to be compared wtih a series of experimental ones, it sets up a probability aggregate on the side of theoretical physics to correspond to another one on the side of experimental physics. The working material in both fields has thus become more nearly of the same grain.

This correspondence we desire to examine more closely. There are two ideal ways in which a coordination between the results of experimental observation and the predictions of theoretical physics might exist. First, nature supplies definite, sharply defined data; calculations yield values of equal sharpness ; and the two should be identical. Sec

ondly, nature presents a range of uncertainty with definite limits within which a physical quantity should lie; calcu




lations determine a similar range; and the two should agree. The latter supposition would be in accordance with the existence of ambiguous laws (implying the abandon ment of the principle of consistency of nature) if the

range of uncertainty is one of ignorance, that is, one with out a probability distribution. The first supposition is

evidently impossible, for it has been shown that experi mental data are not unique, but form probability aggre gates. The second is thought by many to be the one which

quantum mechanics proves to be correct. But this, un

fortunately, must be considered a fallacy. It is here of

importance to distinguish between ignorance and statistical

uncertainty. The former implies nothing at all, but the latter involves everything that pertains to probability ag gregates. Quantum theory produces statements about statistical uncertainty, not about ignorance. It is true

that the uncertainty relation makes all space the range in which an electron with a fixed velocity is moving, but it states that it may be found with equal probability at every point. There is a great deal of difference between saying : an electron is somewhere within space, but I do not, or can

not, know where; and: each point is an equally probable place for the electron. For, the last statement contains in

addition to the former the assertion that if I perform a

very great number of observations the results will be distributed equally over all space.

Heisenberg's famous imaginary experiment in which the exact place of an electron is to be determined by means of a Y-ray microscope seems to leave us in utter ignorance as to the momentum of the electron. The emphasis which has so frequently been placed on it has done much to foster the idea that, in physical measurements of this kind, we

really do not obtain a probability aggregate, but arrive at



30 THE MONIST

perfect uncertainty. This circumstance would then be

contradictory to our earlier considerations. Is Heisen

berg's experiment, as a physical operation, essentially dif ferent from observations dealing with large scale phe nomena ? To be sure, the fact that measuring instruments disturb the state to be measured has long been recognized and has never called for more than casual comment. The

noteworthy feature in Heisenberg's argument is the focus

sing of attention upon one single measurement. We

are compelled to remark that this is not in line with good experimental practice. If the experiment is to mean any

thing at all, it should have been performed by placing a

great number of y-ray microscopes around the electron

and making observations through all of them as nearly as

possible at the same time. In that case the operations would have resulted in a series of values forming a prob ability aggregate in the same sense that all other measure ments do. The errors would certainly be large and the range of uncertainty wide, but it does not differ in an essential way from those encountered in connection with

other experiments. In view of this situation we are forced to conclude that

the correspondence which we set out to investigate is not characterized by one of the two simple suppositions made at the beginning of this inquiry; that it consists in the correlation of two probability aggregates. Moreover, since statistical uncertainty may be a consequence of the action of unique laws, we have found no basis for assuming indeterminism in nature.

After these excursions we are prepared to return to the central part of the causality problem. It is quite generally proposed to replace the postulate of consistency of nature

by an axiom concerning the application of the probability




calculus to nature. This would mean that we start reason

ing by assuming, as the elements of physical knowledge, the two types of probability aggregates of which we have

spoken, one experimental and one theoretical, and that we

suppress all further questions as to why they should be such aggregates. And it seems to be commonly supposed that the assumptions inherent in this proposition are less severe. Typical, perhaps, is Bergmann's attitude, who

says :24 "It is an open question whether we must, in order

to assure statistical laws of nature, go as far as does the

causal law, namely to presuppose full, unambiguous neces

sity for single occurrences, or whether we may satisfy ourselves by assuming as the most basic postulate the fol

lowing: That which possesses the greater mathematical

probability will occur in nature with correspondingly greater frequency/ This we wish to propose here. This postluate makes fewer assumptions than the law of

causality itself and has for this reason an advantage over it." However, we feel that this must be denied. Demand

ing that the probability calculus be applicable to nature sounds very modest, but the foregoing analysis has de composed this seemingly unimposing postulate into a spec trum of axioms of different appearance. It is not suffi

ciently simple to lie at the basis of physical thought. Con

sistency of nature, on the other hand, is a definite, unin volved, and clear-cut postulate, and therefore formally

preferable to the other. Whether or not it is more re strictive can by no means be settled by cursory considera tions. The weight of these formal arguments is definitely in favor of consistency. But even if we agree to give up adherence to this principle, claiming probabilities to be the ultima of physical reasoning, we should be aware that this

change does not amount to accepting indeterminism. 24 Loe. cit., p. 49.



32 THE MONIST

It has been attempted to show, once in connection with the statistical theory of gases and radioactivity, and later in discussing the nature of physical measurements, that the principle of consistency of nature is compatible with, and, if combined with the uncertainty of boundary condi

tions, leads to, the use of probabilities. It will now be

necessary to investigate under what conditions nature may be governed by probability relations but not obey the prin ciple of consistency. This is indeed the state of affairs advocated by some physicists. Its truth is mostly claimed

directly for small scale events, and the argument is then rounded out by observing that, if probabilities exist for

elementary processes, and many of these processes com

bine to form a large scale phenomenon, the latter must show regularity. Hence the apparent lawfulness of the universe. In answering the question as to the existence of

probability aggregates in a fundamentally inconsistent

nature, it is immaterial whether we refer to small scale or

large scale phenomena. Let us consider the question in

connection with a very simple example. Imagine a small,

perfectly spherical ball to be held in a position vertically above an equilateral wedge whose edge is toward the ball. If allowed to drop the ball will roll down either on the left or on the right face of the wedge. We know that if we

attempt to release the ball from a point lying in a plumb line through the edge, the results of dropping it will be a

probability aggregate with a simple distribution: out of a great number of observations the ball will roll down to the left as often as to the right.

In case the principle of consistency is assumed to be

valid, this is to be understood as follows. Whenever the ball is incident on the left face, it will roll down that face. This circumstance, in turn, is controlled by the boundary




condition: when the ball is released from a point slightly to the left of the plumb-line it will be incident on the left face. The distribution of the various initial positions of the ball is responsible for the fact that the results form a

probability aggregate. We know of our tendency to pro duce symmetrical errors, that is, to place the ball initially just as often to the left as to the right of the plumb-line. This does not amount to an additional postulate about nature in the sense that consistency is a postulate, as was

previously pointed out. Moreover, we can convince our

selves of this subjective tendency by performing another

experiment. One might, for instance, try to place a stick across a line so that its center coincides with the line. Per

forming this a number of times and measuring the error each time would prove the statement correct. Next we will suppose that a draft tends to displace the ball during its flight from left to right. We no longer observe equal probabilities for the two events, although the distribution of the boundary conditions remains unchanged. The dis tribution within the aggregate has been changed in an

intelligible and determinable manner.25 How can these matters be explained if the principle of

consistency does not hold? Now the ball, if incident on the left face of the wedge, may be permitted to jump over to the right and roll down that face. Or it may not fall

along the plumb-line at all. We know here as well as before that the boundary conditions are distributed in a certain way over a long sequence of experiments, but this fact is in no definite manner linked with the results of our observations. Let these form a probability aggregate. The details of all individual happenings remain here en

tirely obscure. To account for the collective properties of 25 In quantum mechanics, too, the intoduction of additional physical agencies,

such as forces, changes the \f/ -function, i.e. the distribution of the probability

aggregate.



34 THE MONIST

the observations as a whole it seems necessary to assume

that nature counts the single results in some mysterious fashion and thereby effects statistical regularity of the entire set. The change in the distribution of the aggre gate produced by a draft is not intelligible in this case. Axiomatic and non-physical hypotheses must be appealed to in order to understand this change.

A doctrine of this character is not contradictory in itself and is therefore beyond logical refutation. It is less

explicit but not for that reason less restrictive. For, al

though it denies adherence to the principle of consistency, it substitutes in place of it another one, namely that by

means of which the entire set of observations is a prob ability aggregate, and which we characterized by saying that nature counts. The two postulates are of different

types and can hardly be compared as to their restrictive ness, except by their effects, which are the same inasmuch as both supply probability properties for an aggregate of observations. However, we can not refrain from voicing with respect to the latter point of view some objections to which the one embracing the principle of consistency is immune. The first has already been touched upon: it is the fact that the dependence of the distribution of an ag gregate on physical agencies (in our example, the draft) becomes unintelligible. Another lies in the difficulty of

accounting for the invariance with arbitrary selection rules of a distribution. If nature, unguided by causal bonds, counts results to achieve the final aim of a statistical dis

tribution, it is not clearly to be seen why human intel

ligence should not be able to upset this effect. The assump tion of a natural intelligence conflicting with human design would most likely be pronounced unreasonable by scientists. There then remains only the alternative of supposing that




each single event is dependent on the way in which it is to be combined with others, which involves a dependence of the past on the future. Such an hypothesis is, of course, not impossible per se, but it is well known that it can stand only on strictly deterministic ground. For, if the future is undetermined, it is, in the present moment, non-existent, and to postulate its reaction upon the perfect

ly definite past is only a play on words. We see, there

fore, that this alternative is contradictory to the assump tion which it was designed to support.

The admission of determinism in the manner just con sidered is occasionally said to have an important bearing on the problem of freedom. It is at once apparent that it can at most concern freedom of action, i.e., the effect of

an already completed volition upon the physical world. This volition could now be the only decisive factor in de

termining the course of natural processes, while according to the deterministic view it was merely an extraneous com

ponent added to a number of causal determinants. Nothing is changed with regard to the status of the problem by this new turn. Furthermore the above conclusion is not even correct in a strict sense. It is only for phenomena on an atomic scale that there is no complete causal determination.

Large scale processes, the type subject to human control, exhibit perfect determinism in spite of contingency acting in elementary events. We do not feel, therefore, that the situation in question has an essential influence upon the

problem of freedom. The foregoing discussions were intended to show that

modern developments of physical thought, in particular, quantum mechanics, do not require the abandonment of the causality postulate. Moreover they lead to the ad

vocacy of the view that a substitution of probability rela



36 THE MONIST

tions for deterministic uniqueness of elementary events offers no philosophical advantages. They contain an an

alysis of probability concepts and attempt to demonstrate the futility of simplifying basic assumptions by introduc

ing terms like statistical uncertainty, perfect irregularity, which, though they sound unassuming, imply very definite and far-reaching axioms about nature. On the whole,

causality appears to be a valuable postulate which it is

profitable to retain.

HENRY MARGENAU.

YALE UNIVERSITY.



causality and modern physics

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