Causality in Contemporary Physics, from Physical Reality, ed. by Stephen Toulmin, 1970
1. INTRODUCTORY REMARKS The number of conceivable, logically possible, physical worlds is infinite; the human imagination, however, is surprisingly poor in conceiving and working out new possibilities. The power of imagination is so limited by the intuitive conditions of gross perceptual experience that it can hardly by itself progress a step beyond them. It is only by aid of the strict discipline of more refined scientific experience that our thought can transcend its habitual channels. The most colourful fairyland of the Thousand and One Nights is created by a slight rearrangement of the familiar material of everyday life. And upon reflection, when one examines them with more precision, one finds the same to be true of the boldest and most profound philosophical systems: If for the poet it was creation by aid of intuitive pictures, so for the philosopher it is construction by more abstract yet still familiar concepts, from which by apparently more transparent principles of combination new structures are formed. The physicist, too, at first proceeds in much the same way in the construction of hypotheses. This is particularly indicated by the tenacity of the belief, held for many centuries by the physicist, that to explain nature a copy of its processes in models perceptible to the senses is necessary. Thus, for instance, he repeatedly attributed characteristics of visible, tangible substances to the ether without the slightest reason for doing so. Only when observed facts either suggest or necessitate his use of the new systems of concepts does the physicist realise the new possibilities and break himself free from his former habits of thought; but then he readily and with the greatest ease makes the jump to, say, Riemannian space or to Einsteinian time, to concepts so daring and profound that neither the imagination of the poet nor the intellect of the philosopher could have been able to anticipate them. The turning point at which recent physics has arrived with respect to the question of causality could likewise not have been foreseen. Although there has been so much philosophising about determinism and indeterminism, about the content, validity, and mode of testing of the principle of causality, no one thought of precisely that possibility leading in quantum physics to the key that allows us a view of the real nature of the causal order. Only in retrospect do we realise how the new ideas differ from the old, and we are perhaps a little amazed that so far we have always missed the point. Now, however, after the significance of the concept of quantum theories has been demonstrated by the extraordinary results of its application, and we have had some years to accustom ourselves to the new ideas, it should not be premature to attempt to arrive at philosophical clarity as to the meaning and scope of the ideas that contemporary physics contributes to the problem of causality. 2. CAUSALITY AND THE PRINCIPLE OF CAUSALITY The observation that philosophical meditations did not foresee possibilities that were found later on, because of their close adherence to existing ideas, is true also of the ideas I propounded more than ten years ago.' Still it is probably not useless to return to some points of those earlier considerations, since in this way the progress accomplished in the meantime becomes so much clearer. First it is necessary to determine what the scientist actually means when he speaks of "causality." Where does he use this word? Obviously, wherever he supposes a "dependence" between certain events. (Nowadays it is self-evident that only events and not "things" come into question as elements of a causal relationship, since physics forms the four dimensional reality from events, and considers "things," three dimensional bodies, as mere abstractions.) But what does "dependence" mean? In science, in any case, it is always expressed by a law; causality is, accordingly, nothing but another word for the existence of a law. The content of the principle of causality then clearly lies in the assertion that everything in the world occurs according to laws; it is indifferent whether we affirm the validity of the principle of causality or of determinism. In order to formulate the principle of causality or the deterministic thesis, we must first have defined what is meant by a law of nature or by mutual "dependence" of natural events. For only when we know this are we able to understand the meaning of determinism, which states that every event is a member of a causal relation, that every process is wholly dependent upon other processes. (We shall not discuss whether the attempt to make a statement about "all" natural processes could lead to logical difficulties.) Thus in any event we distinguish the question of the meaning of the word "causality" or "natural law" from the question of the validity of the principle of causality or the law of causality, and we concern ourselves at the beginning with the first question only. The distinction we thus make coincides with that made by H. Reichenbach at the beginning of his essay "Die Kausalstruktur der Welt." He speaks there of the difference between two "forms of the hypothesis of causality." He calls the first the "implication form." It is given "when physics establishes laws, that is, makes statements of the form: 'if A then B.' " The second is "the deterministic form of the causal hypothesis;" it is identical with determinism, which states that the course of the world as a whole "remains unchangeable, that with a single cross section of the four dimensional world the past and future are fully determined." It seems to me simpler and more to the point to characterise the difference involved as the difference between the concept of causality and the principle of causality. The question then concerns the content of the concept of causality. When do we say that a process A "determines" another B, that B "depends" upon A, that B is related to A by law? What do the words "if—then," indicating causal relationship, mean in the statement "if A then B?" 3. LAW AND ORDER In the language of physics a natural process is represented as a sequence of values of definite physical magnitudes. We note already here that, of course, in a sequence only a finite number of values can be measured, that, therefore, experience affords only a discrete manifold of observed magnitudes, and further, that every value is conceived as subject to a certain inexactitude. Assuming that a large number of such observed magnitudes are given, we then ask quite generally: How does such a number of values have to be constituted so that we may say that it represents a law-like sequence, that there is a causal relationship between the observed magnitudes? We may, to begin with, presuppose that the data already possesses a natural order, namely the spatio-temporal one; that is to say, each quantitative value relates to a definite position in space and time. It is of course true that only with the aid of causal considerations are we able to indicate the position of events in physical space-time, by passing from phenomenal space-time, which represents the natural order of our experiences, to the physical world. But this complication may be excluded in our considerations, which limit themselves entirely to the realm of the physical world. Furthermore, our considerations are based upon a most fundamental assumption which I here mention in passing only, since it has already been discussed in a previous work (loc. cit. p. 463). It is the hypothesis that in nature there are certain "similarities," in the sense that different realms of nature are comparable to one another, so that we may say for instance: "the same" magnitude that in this place has value f1 has the value f, in another place. Comparability is then one of the presuppositions of measurability. It is not easy to give the real meaning of this assumption, but we need not concern ourselves with it since this last analysis is likewise irrelevant to our problem. According to these observations, our problem regarding the content of the concept of causality reduces itself to this: what characteristic must the spatio-temporally ordered group of values have so that it may be regarded as an expression of a "law of nature"? This characteristic can be nothing but an order, and indeed, since events extensive in space and time are already orderly, it must be a kind of intensive order. This order must be of the temporal sort, for, as is well known, we do not speak of causality in reference to spatial order (popularly expressed by "simultaneous," coexistent events) ; the concept of activity finds no application there. Spatial regularities, if such there be, would be called "coexistence laws." After limiting ourselves to the time dimension we must, I believe, now say: Every order of events in time, of whatever kind, is to be regarded as a causal relationship. Only complete chaos, complete irregularity, is to be designated as an acausal occurrence, as pure chance; every trace of an order would mean dependence, therefore causality. I believe that this use of the word "causal" is closer to its everyday sense than when confined, as seems to be done by many natural philosophers, to such an order as we could designate by "complete causality"—by which phrase it appears that something like "complete determination" of the event in question is meant (of course, we can express ourselves here in inexact terms only). If we should restrict the word to complete causality, we run the danger of finding no use at all for it in nature, while in some sense we do regard the existence of causality as a fact of experience. And there would be even less reason to place the boundary between law and chance at some other point. The only alternative that confronts us is thus: order or disorder? Causality and law are identical with order; irregularity and chance are identical with disorder. The result up to the present therefore seems to be: we call a natural process, described by a group of values, causal or regular if the values show any temporal order whatever. This definition becomes meaningful only when we know what is to be understood by "order," how it differs from chaos. A most puzzling problem! 4. ATTEMPTS AT DEFINITION OF REGULARITY It is certain that in our daily life as well as in science we differentiate rather clearly between order and disorder, regularity and irregularity. How are we to understand this? At first glance the answer does not seem very difficult. It appears that we need only make sure how physics actually represents laws of nature, in which form it describes the dependence of events. Now, this form is the mathematical function. The dependence of one event upon another is expressed by the fact that the values of apart of the magnitudes are represented as functions of the others. Every order of numbers is mathematically represented by a function; and so it appears as if the desired criterion of order, which differentiates it from disorder, is expressibility by a function. But as soon as the idea of the identity of function and law is expressed, we see that it cannot possibly be correct. For, as is well-known, whatever be the distribution of given magnitudes, functions may always be found that represent just that distribution with any degree of accuracy; and this means that every possible distribution of magnitudes, every conceivable series of values is to be considered as an order. There would be no chaos. Thus we do not in this way successfully distinguish causality from chance, order from disorder, or succeed in defining rule and law in this manner. As was shown also in our previous considerations, there seems to remain only the alternative of imposing certain requirements on the functions that describe the observed series of values, and by means of them to determine the concept of order. We should then have to say: if the functions that describe the distribution of values of the magnitudes have such and such a definite structure, the represented sequence is in accordance with law, otherwise it is unordered. Thereby we find ourselves in a rather hopeless situation, for it is clear that in this way arbitrariness is given free rein, and a distinction between law and chance resting upon such an arbitrary basis could never be satisfactory. It could be so only if a fundamental and sharp distinction in the structure of the functions could be ascertained, which at the same time possessed such definite empirical possibilities of application that everyone would immediately recognise them as the correct formulation of the concepts of regularity and irregularity as they are applied in science. Here there appear simultaneously two ways, both of which men have tried to adopt. The first was already used by Maxwell to define causality. It consists of the following stipulation: the space and time co-ordinates are not to appear explicitly in the equations that describe the sequence in question. This requirement is equivalent to the notion popularly expressed in the phrase: similar causes, similar effects. In fact it means that a process that takes place anywhere and any time in a definite manner will take place in exactly the same manner in every other place and time, under the same circumstances. In other words, the rule states the universal validity of the relationship represented. Universal validity, however, is, as has been generally acknowledged, exactly that which in laws of nature has been designated by the ambiguous term "necessity," so that it appears as if the essential nature of the causal relation had been correctly hit upon by this stipulation. Concerning Maxwell's definition of law, which I myself previously defended (in the passage cited), we may say the following: The concept of law in physics is undoubtedly such that this requirement is always fulfilled. Actually no investigator thinks of formulating laws of nature that refer explicitly to definite positions and moments in the universe. If space and time occurred explicitly in the physical equations, they would have quite a different significance from the one they actually have in our world. The relativity of space and time, fundamental to our world-view, would be denied, and time and space could no longer assume the peculiar role of "forms" of occurrence which they have in our cosmos. We should therefore be free to maintain the Maxwellian condition of causality — would it, however, be a necessary condition? We shall hardly be allowed to say that, for surely a world is conceivable in which all events would have to be expressed in formulas in which space and time appear explicitly, without our denying that these formulas represent true laws and that this world is completely orderly. So far as I can see, it would be conceivable, for instance, that uniform measurements of the elementary quantum of electricity (electric charge) would give values for these magnitudes that would fluctuate about 5 per cent in seven hours and then again in seven hours, and then ten hours, without our being able to find the slightest "reason" for it. And besides that, perhaps still another variation might appear for which we would make an absolute change of position of the earth in space responsible. In this case then the Maxwellian condition would not be fulfilled, but we would surely not find the world to be disorderly and we would formulate its regularity and be able to make predictions by means of it. We shall therefore be inclined to the view that the Maxwellian definition is too limited, and we shall ask ourselves what the criterion of law would be in the hypothetical case we have discussed. Now, the decisive factor of the hypothetical case seems to be that we could so easily consider the influence of space and time, that they enter into the formulas in such a simple manner. If, in our example, the electric charge were to behave differently every week and every hour, or form a completely "irregular curve," we could of course afterwards represent its dependence upon time by a function, but this function would be very complicated. We would then say that no law exists but that the variations of magnitudes are governed by "chance." We do not have to invent cases of this kind since, as is well known, the new physics accepts them as commonplace. The discontinuous events in the atom that Bohr's theory, interpreted as jumps of an electron from one orbit into another are regarded as purely accidental, as "acausal," although we may subsequently think of their occurrence as a function of time. But this function would be very complicated, not periodic, not readily grasped, and it is for this reason only that we say that no regularity exists. But as soon as any simple statement regarding the jumps is formulable — if, for instance, the time intervals become increasingly larger — this would at once appear to us as a regularity, even though time would explicitly enter into the formula. Accordingly, it appears as if we speak of order, law, causality when the course of events is described by functions of simple form; while complexity of the formula is the criterion for disorder, lawlessness, chance. And so one very easily arrives at the point of defining causality by the simplicity of the descriptive functions. Simplicity is, however, a half pragmatic, half aesthetic concept. We may therefore call this definition aesthetic. Also, without being able to state what is here meant by "simplicity," we must yet affirm the fact that every investigator who has succeeded in representing a series of observations in a simple formula (for instance, linear, quadratic, exponential function) is quite sure of having discovered a law, and so the aesthetic definition, as well as the Maxwellian one, obviously discloses a characteristic of causality that is considered a decisive criterion. Which of the two attempts at formulating the concept of law shall we accept? Or shall we formulate a new definition by combining both? 5. INADEQUACY OF THE ATTEMPTS AT DEFINITION To sum up: the Maxwellian definition has in its favour the fact that all known laws of nature satisfy it, and that it may be considered to be an adequate expression of the proposition, "similar causes, similar effects." Against this definition is the fact that cases are conceivable in which we should certainly acknowledge regularity without the fulfillment of the criterion. The "aesthetic" definition has in its favour that it is also applicable to the above considered cases, to which the other one is not, and that also undoubtedly in the prosecution of science "simplicity" of functions is used as a criterion of order and law. Against it, however, is the fact that simplicity is clearly a relative and indefinite concept, so that a strict definition of causality is not obtained, and law and chance cannot be sufficiently distinguished. It might indeed be possible that we have to take this last idea into account, and that a "law of nature" is actually not something so precisely conceivable as one might at first think; however, such a point of view will be accepted only when one is sure that no other possibility remains. It is certain that the concept of simplicity can only be fixed by a convention that must always remain arbitrary. We should probably be inclined to consider a function of the first degree simpler than a function of the second degree; however, even the latter undoubtedly represents an unexceptionable law when it describes the data of observation with great accuracy. The Newtonian formula of gravitation, in which the square of the distance occurs, is generally still regarded as a paradigm case of a simple natural law. One may, for example, agree further that of all continuous curves that pass through a given number of points with sufficient proximity, we may consider as simplest the one that everywhere on the average has the greatest radius of curvature. (There is an unpublished work on this by Marcel Natkin.) However, such artifices seem unnatural, and the fact alone that there are degrees of simplicity makes the definition of causality based upon it unsatisfactory. This state of affairs is made even worse by the fact that, as we know, it is not at all a matter of the simplicity of an isolated law, but of the simplicity of the system of all natural laws. And so, for instance, the true equation of the law of gases has by no means the simple form given to it by Boyle-Mariotte; yet we know that its complicated form may be explained by a particularly simple set of elementary laws. In principle it should be much more difficult to find rules for the simplicity of a system of formulas. They would always remain provisional, so that apparent order could with progressive knowledge turn out to be disorder. And so neither the Maxwellian nor the aesthetic criterion seems to give a really satisfactory answer to the question as to what causality actually is. The first seems too narrow, the second too vague. No progress in principle is made by a combination of both attempts, and one may readily see that the shortcomings cannot be removed by improvements along the same lines. The shortcomings observed are clearly of a fundamental nature, and that gives us the idea of revising the present point of departure and of considering whether we are in general on the right path. 6. PREDICTION AS CRITERION OF CAUSALITY Until now we have assumed that a definite distribution of values is given, and asked: when does it represent a regular and when a chance sequence? It may be that this question cannot be answered at all by mere consideration of the distribution of values, but that it is necessary to go beyond this domain. Let us for a moment consider the consequences that the statements made about the concept of causality have for the principle of causality. We imagine that for as many internal and boundary points of a physical system as possible we attempt to determine the value of the state variables by precise observation. Now one is in the habit of saying that the principle of causality is valid if from the state of the system during a very short time and from the boundary conditions all other states of the system may be deduced. Such a deduction is, however, possible under all circumstances, for according to what has been said, functions may always be found that represent all observed values with any desired accuracy. And as soon as we have such functions we may by means of them compute all states already observed, whether earlier or later, from any state of the system. For functions have been chosen in such a way that they represent everything observed in the system. In other words: the principle of causality would be satisfied under all circumstances. A statement, however, that is applicable to any system whatever, no matter what its properties, says nothing at all about this system, is an empty statement, a mere tautology, and it is futile to construct it. Hence if the causal law is actually to mean something, if it has content, the formulation from which we began must be false, for the law has turned out to be tautological. If, however, we make the qualification that the equations used are not to contain the space and time co-ordinates explicitly, or that they are to be very "simple," the principle acquires, to be sure, a real content; but, in the first case the reflection is valid that we have formulated too limited a concept of causality, and in the second the sole characteristic would be that the computation would be easier. However, we should certainly not want to formulate the difference between chaos and order in such manner that we say the first is comprehensible only to an excellent mathematician, the second to an average one. We must therefore begin anew and attempt to formulate the meaning of the causal law differently. Our error until now was that we did not conform with enough precision to the actual procedure by which, in science, one actually tests whether processes are or are not dependent upon one another, whether or not a law, a causal sequence, exists. Until now we only investigated how a law is constructed. To learn its real meaning, however, one must observe how it is tested. It is always the case that the significance of a statement is revealed only by the manner of its verification. How then is the test made? After we have succeeded in finding a function that satisfactorily connects a group of observational data, we are by no means satisfied, even when the function found has a very simple structure, for now comes the main thing, which our considerations hitherto have not touched: We observe whether the formula obtained also represents correctly those observations that were not used in achieving the formula. For the physicist, as an investigator of reality, the only important, decisive, and essential thing is that equations derived from certain data be applicable to other, new data. Only when this is true does he consider his formula to be a law. In other words, the true criterion of law, the essential sign of causality, is the success of prediction. By success of prediction is to be understood, according to what has been said, nothing but the confirmation of a formula for such data as have not been used in its construction. Whether these data have already been observed or are only subsequently determined is in this connection of no consequence whatever. This observation is of great importance: past and future data are altogether on the same footing in this respect, the future is not of special significance; the criterion of causality is not confirmation in the future but confirmation in general. It is self-evident that the test of a law can occur only after its formulation, but this gives no special distinction to the future. What is essential is that it is indifferent whether the verifying data are in the past or the future; it is incidental when they become known or are used for verification. The confirmation remains the same whether the data were known before the formulation of a theory, as in the case of the anomaly of Mercury's movement, or whether it was prophesied by the theory, as in the case of the red-shift of the spectral lines. Only for the application of science, for technique, is it of fundamental importance that natural laws allow prediction of something new, observed by no one as yet. And so earlier philosophers, Bacon, Hume, Comte, have long known that knowledge of reality coincides with the possibility, of prediction. Thus fundamentally they correctly formulated the essence of causality. 7. ELUCIDATION OF THE RESULT If we accept the success of prediction as the true criterion of a causal relationship—and, with an important limitation to be mentioned presently, we shall have to do so—we thereby admit as well that the previous attempts at definition no longer enter into consideration. In fact, if we can really predict new observations, it does not matter how the formulas that enabled us to do so were constructed, whether they seem simple or complicated, whether time and space enter explicitly or not. As soon as someone can calculate the new observation data from the old, we shall admit that he has grasped the law governing the processes; prediction is therefore a sufficient characteristic of causality. We easily realize that confirmation is also a necessary characteristic, and that the Maxwellian and aesthetic criteria do not suffice, when we imagine we have found a formula of -great simplicity that describes a definite natural process with great precision but at once ceases to work when applied to the further phases of the process, to new observations. Obviously we should then say that the distribution of magnitudes occurring once has simulated a dependence of natural events which in reality does not exist, that it was a matter of mere chance the particular sequence could be described by simple formulas. That there was no natural law is proven by the fact that our formula can stand no test, for in the attempt to repeat the observations the sequence occurs quite differently; the formula is no longer applicable. A second alternative seems, of course, to be that one may say the law was valid at the time of the observation but no longer holds. It is clear, however, that this is only another way of expressing the absence of law, the universality of the law being denied. The "regularity" observed for the single sequence was not true regularity, but merely chance. The confirmation of prediction is therefore the only criterion of causality. Through it alone does reality speak to us; the construction of laws and formulas is simply the work of man. Here I must include two observations that go together and are of basic importance. First, I said previously that we may recognise the "verification" of a regularity as the adequate characteristic of causality only subject to a limitation. This limitation consists in the fact that the confirmation of a prediction never actually proves the existence of causality but always only makes it probable. Further observations may indeed show the supposed law to be always incorrect, and then we should have to say that "it expressed the sequence only by chance." A final verification is therefore, so to say, impossible in principle. We deduce therefrom that a causal statement logically does not at all have the characteristic of a proposition, for a genuine proposition must in the end allow itself to be verified. We shall return to this shortly without, however, being able to explain this apparent paradox fully here, where we are not concerned with logic. The second observation concerns the fact that between the criterion of confirmation and the two rejected attempts at definition a remarkable relationship exists nevertheless. It lies simply in the fact that actually the different characteristics go hand in hand. We certainly expect with great assurance that precisely those formulas satisfying the Maxwellian criterion and distinguished by aesthetic simplicity will be confirmed, and that the propositions made by their help will be true. And even if we should sometimes be disappointed in this expectation, the fact remains that the laws that have really proved to be valid were always of a profound simplicity, and always fulfilled the Maxwellian definition. But what the significance of this "simplicity" is, is difficult to say, and much erroneous thinking has been done in this connection; we do not wish to put too much stress upon it. It is certain that we may imagine much "simpler" worlds than our own. There is also a "simplicity" that is merely a matter of representation, that is, pertains to symbolism by means of which we express facts. Its consideration leads to the question of "conventionalism" and does not interest us in this connection. At any rate we see that if a formula corresponds to both of the earlier and inadequate criteria we consider it probable that it is really the expression of a law, of an actually existing order, that it will therefore be confirmed. If it has been confirmed, we think it again probable that it will continue to be so. (And indeed, it is understood, without bringing in new hypotheses, for in general physical laws are so constructed that they may always be maintained by new hypotheses brought in ad hoc. But if these become too complicated, one says that the law nevertheless does not hold, the right order has not been found.) The word probability, which we use here, moreover designates something completely different from the concept treated in the calculus of probability and occurring in statistical physics. For the sake of logical clarity (for philosophers this is the prime concern) it is of greatest importance to realise the situation precisely. We have seen that, basically, causality is not at all definable in the sense that for an already given sequence one could answer the question: Was it causal or not? Only in reference to the single case, to the single verification, can one say: It behaves as causality demands. For advancement in knowledge of nature (and this is the main concern of the physicist) this is fortunately quite sufficient. If a few verifications — under some circumstances only one — are successful, we build practically on the verified law, with the unqualified reliance with which we trust our life to a motor constructed according to natural laws. It has indeed been frequently observed that one cannot actually speak of an absolute verification of a law, since we always, so to say, silently reserve the right to modify it on the basis of further experience. If I may in passing add a few words about the logical situation, the circumstance mentioned means that basically a natural law does not have the logical character of a "proposition" but represents "a direction for the formulation of propositions." (I owe these ideas and terms to Ludwig Wittgenstein.) We have already indicated this, above, regarding causal statements, and actually a causal statement is identical with a law. The statement, "The principle of energy holds," for instance, says no more nor less about nature than the principle of energy itself says. As is well known, only the individual propositions derived from a natural law are testable, and these always have the form: "Under such and such circumstances this indicator will point to that mark on the scale," "Under such and such circumstances there occurs a darkening on this point of the photographic plate," and the like. The verifiable propositions are of this nature and of this nature is every verification. Verification in general, the success of a prediction, confirmation in experience, is therefore the criterion of causality, simply; and indeed in the practical sense in which alone we may speak of the test of a law. In this sense, however, the question regarding the existence of causality is testable. That confirmation in experience, the success of a prediction, is something final, not subject to further analysis, cannot be over-emphasised. No number of propositions can state when it must occur, but we must simply await whether it occur or not. 8. CAUSALITY AND QUANTUM THEORY In the previous considerations nothing was said except what, in my opinion, may be read out of the procedure of the scientist. No concept of causality was constructed; only the role it actually plays in physics was determined. Now the attitude of most physicists towards certain results of quantum theory shows that they see the essence of causality just where the foregoing considerations also found it, namely, in the possibility of prediction. When the physicists say that complete validity of the causal principle is not compatible with the quantum theory, the basis, indeed the meaning, of this assertion, lies simply in the fact that the theory makes precise predictions impossible. We must try to make this really clear to ourselves. In contemporary physics it is possible to say, in a manner of speaking, that, with certain limitations to be mentioned, each physical system is to be considered as a system of protons and electrons, and that its state is completely determined by the position and momentum of its particles being known at every moment. Now, as is well known, a certain formula is derived from the quantum theory, the so-called "uncertainty principle" of Heisenberg, which teaches that it is impossible to indicate for a particle both determinants, place and velocity, with any desired precision, and that the more precise the value of one co-ordinate the greater inexactness we must expect for the other. If we know say that the place coordinate lies within a small interval Δp, the velocity co-ordinate q may be indicated only with such precision that its value remains undetermined in the interval Δq, and indeed so that the product ΔpΔq is of the order of magnitude of Planck's quantum effect h. In principle then the one co-ordinate could be determined with any degree of precision, but absolutely precise observation of it will have as a consequence that we can say nothing more about the other co-ordinate. This principle of indeterminacy has been so frequently illustrated, even in popular form, that we need not describe the situation any more closely; our task must be to understand exactly its real meaning. When we ask for the meaning of a statement this always means (not only in physics) : by which particular experiences do we test its truth? When thus, for example, we conceive the place of an electron to be determined by observation with an inexactness Δp, what does it mean when I say, for instance, the direction of the velocity of this electron may be indicated only with an inexactness ΔΘ? How do I determine whether this statement is true or false? Now, that a particle has gone in a definite direction may be tested only by its arrival at a definite point. To give the velocity of a particle signifies absolutely nothing more than to predict that in a certain time it will arrive at a certain point. "The inexactness of direction amounts to ΔΘ" means: in a certain experiment I shall find the electron within the angle ΔΘ; however, I do not know exactly where therein. And if I repeat "the same" experiment I shall find the electron at various points within the angle, and I never know beforehand at which point in it. If the position of the particle is observed with absolute precision the result would be that in principle we could not know at all in which direction the electron would be found after a short time. Only further observation could subsequently tell us this, and with very frequent repetition of "the same" experiment it must appear that on the average no one direction predominates. The fact that both position and velocity of an electron cannot be precisely measured is usually interpreted as saying: it would be impossible to describe fully the state of a system at a definite point in time, and therefore the principle of causality becomes inapplicable. Since the principle asserts that the future states of the system are determined by its initial state, since, thus, it presupposes that the initial state may be described in principle exactly, the principle of causality collapses, for this presupposition has not been fulfilled. I should not like to call this idea false, but it seems useless to me, because it does not express the essential point clearly. What is essential is that one realise that the indeterminacy that the Heisenberg-relation expresses is in truth an indeterminacy of prediction. In principle nothing interferes (this is also emphasised by Eddington in a similar thought context) with our determining the position of an electron twice at any two closely adjacent points in time, and with our considering these measurements equivalent to position and velocity measurements. But the vital point is that with data about a state obtained in such a manner we are never in a position to predict a future state with precision. If, that is, we should define the velocity of the electron in the usual manner (distance divided by time) by means of the observed places and times, the velocity would nevertheless be different in the next moment, for, as we know, it must be assumed that its course is disturbed by observation in a quite uncontrollable manner. This alone is the true significance of the statement that a momentary state is not precisely determinable; that is, the impossibility of prediction alone is the actual reason why the physicist deems necessary the denial of causality. There is no doubt, therefore, that quantum physics finds the criterion of causality precisely where we too have discovered it, and speaks of the failure of the principle of causality only because it has become impossible to make predictions with any desired degree of accuracy. I cite M. Born, Naturwiss. 17 (1929):
The impossibility of measuring exactly all the data of a state prevents the predetermination of its further course of development. Because of this principle of causality in its usual formulation loses all significance. For when it is impossible in principle to know all the conditions (causes) of an event, it is empty talk to say that every event has a cause.Causality as such, the existence of laws, is however not denied. There are still valid predictions, but they do not consist in the expression of exact magnitude values, but are of the form: the A- magnitude X will lie in the interval between a and Δa. What is new in the contribution of the most recent physics to the problem of causality does not consist in the fact that the validity of the causal principle is contested at all, nor that, say, the microstructure of nature is described by statistical rather than causal regularities, nor in the fact that the realisation of the merely probable validity of natural laws has displaced belief in their absolute validity. All these ideas have, in part, long since been expressed. The novelty rather consists in the hitherto unsuspected discovery that through natural laws themselves a limit is set in principle to precision in predictions. That is something quite different from the rather obvious idea that actually and practically there is a limit to precision in observations and that the assumption of absolutely precise natural laws is in every case unnecessary if one wants to give an account of every experience. Previously it must always have seemed as if the question of determinism had to remain undecided in principle. The kind of decision now available, namely, by means of a natural law itself (the Heisenberg relation), was not foreseen. In any case, one who today speaks of a decidability and holds the question to be answered unfavorably for determinism must assume that law of nature as actually existing and raised beyond all doubt. That we are absolutely sure of this, or ever could be, a careful investigator will hesitate to state. But the principle of indeterminacy is an integral part of the structure of the quantum theory, and we must trust its correctness so long as new experiments and new observations do not force us to revise the quantum theory. (In fact it is daily better confirmed.) But to have shown that a theory of such structure is at all possible in the description of nature is in itself a great accomplishment of modern physics. It signifies an important philosophical clarification of the basic concepts of natural science. The progress in principle is clear. One may now speak of an empirical test of the principle of causality in the same sense as of the test of some special law of nature. And that we may in some sense justifiably speak of it is proven simply by the existence of science. 9. IS THE PRINCIPLE OF CAUSALITY IN THE QUANTUM THEORY FALSE OR EMPTY? In order to understand the situation it is necessary to compare two formulations that the criticism of the causal principle assumes in physics. Some say that the quantum theory has shown (of course presupposing that it is correct in its present form) that the principle is not valid in nature. The others say that it is empty. The former believe thus that it makes a definite assertion about reality that experience has proven false; the others believe that the proposition in which it is apparently expressed is not at all a genuine assertion but is a meaningless succession of words. As evidence for the first point of view, Heisenberg's oft cited article (in Z. Physik, 1927, 43) is quoted, which says: "Because all experiments are subject to the laws of quantum mechanics, the invalidity of the causal law is definitively determined by quantum mechanics." Born is commonly named as representing the second point of view (cf. the passage cited above). Hugo Bergmann ("Der Kampf um das Kausalgesetz in der jungsten Physik," Braunschweig, 1929) and Thilo Vogel ("Zur Erkenntnistheorie der quantentheoretischen Grundbegriffe" Diss., Giessen, 1929) have concerned themselves with the philosophical aspects of this dilemma. Both of these authors correctly assume that those physicists who reject the causal principle are none the less of the same opinion basically, even if they say different things, and that the apparent difference is to be attributed to the inexact language of the one party. Both are of the opinion that it is Heisenberg who is guilty of the inexactness, and that therefore it should not be said that the quantum theory has proven the principle to be false. Both emphasize the fact that the causal law can neither be affirmed nor rejected by experience. Shall we consider this interpretation as correct? First we must affirm that we consider the bases upon which H. Bergmann forms his opinion to be quite incorrect. For him the causal law is to be neither discredited nor affirmed, because he considers it to be a synthetic judgment a priori in the Kantian sense. On the one hand, as is well known, such a judgment is supposed to express a genuine cognition (this is conveyed by the word "synthetic") , on the other hand it must be incapable of test by experience because "the possibility of experience" rests upon it (that is conveyed by the words a priori). We know today that these two requirements contradict each other; there are no synthetic judgments a priori. If a proposition says anything at all about reality (and only when it does so does it contain knowledge) its truth or falsity must be determinable by observation of reality. If there is no possibility of such a test in principle, if the proposition is compatible with every experience, it must be empty and cannot contain any knowledge of nature. If, on the assumption of the falsity of the proposition, something in the world of experience were different from what it would be if the proposition were true, then of course it could be tested. Consequently not-testable-throughexperience means that the way the world appears to us is quite independent of the truth or falsity of the proposition, hence it says nothing about the world. Kant, of course, believed that the principle of causality says a great deal about the empirical world, even determines its essential nature. Therefore one does no favour to Kantianism or a-priorism when one affirms that the principle cannot be tested. With this we have rejected H. Bergmann's point of view (the same would hold true of Th. Vogel's opinion in so far as he inclines towards a moderated, a priorism; however his formulations at the end of the treatise cited are not quite clear to me) and we must therefore consider a new phase of the question. Does the falsity of the principle of causality really follow from results of quantum mechanics? Or does it follow rather that the proposition is without content? A sequence of words may be meaningless in two ways: either it is tautological (empty) or it is not a proposition at all, not an assertion in the logical sense. It would appear at first sight as if the latter possibility did not enter into our considerations here, for if the words that are to express the causal principle do not represent a real proposition, they must simply be a meaningless, absurd succession of words. One must, however, bear in mind that there are sequences of words that are not propositions and express no facts and yet fulfil very significant functions in life; so-called question and command sentences. And even if the causal principle is expressed in the grammatical form of a declarative sentence, we know from modern logic that one can hardly judge the logical content of a sentence by its form. And thus it is quite possible that beneath the categorical form of the causal principle a kind of command, a demand exists—thus, approximately, what Kant calls a "regulative principle." A similar opinion regarding this principle is indeed held by those philosophers who see in it merely the expression of a postulate or of a "decision"5 never to give up the quest for laws and causes. This point of view therefore must be carefully considered. Accordingly, we must decide between the following three possibilities: I. The principle of causality is a tautology. In this case it would always be true but without content. II. It is an empirical proposition. In this case it could be either true or false, either knowledge or error. III. It represents a postulate, an injunction to continue to seek causes. In this case it cannot be either true or false but is at most either appropriate or inappropriate. I. We shall soon become clear regarding the first possibility, especially as we have already mentioned it above (§6). We found that the causal principle as expressed in the form, "All events occur according to law," is certainly tautological if by lawfulness is meant, "representable by some formula or other." From this, however, we inferred that this could not be the true content of the principle, and we looked fora new formulation. In fact, science in principle has no interest in a tautological proposition. If the causal principle were of this nature, determinism would be self-evident, but empty. And indeterminism, its opposite, would be self-contradictory, for the negation of a tautology is a contradiction. The ,question as to which one of the two is correct could not be raised at all. Therefore if modern physics not only formulates the question but believes it to be definitely answered by experience, what physics means by determinism and the principle of causality surely cannot be a tautology. In order to know whether a proposition is tautological or not one obviously needs no experience at all, one need only realise its meaning. If one should say that physics has demonstrated the tautological nature of the causal principle, it would be as senseless as to say that astronomy has shown that 2 times 2 equals 4. Since the time of Poincar6, we have learned to note that apparently certain general statements enter into a description of nature which are not subject to confirmation or disproval, namely the "conventions." The genuine conventions, which are actually a type of definition, must in fact be formulated as tautologies. Here, however, it is not necessary to go further into this matter. We conclude only that since we have already admitted that modern physics at any rate teaches us something about the validity of the principle of causality, it cannot be an empty proposition, a tautology, a convention; but it must be of such a nature that in some way it is subject to the judgment of experience. II. Is the principle of causality simply a proposition the truth or falsity of which may be determined by observation of nature? Our previous considerations seem to support this interpretation. If it is correct we shall have to side with Heisenberg, therefore, in opposition to H. Bergmann and Th. Vogel in the above mentioned apparent opposition between the formulations of Heisenberg and Born, in which these investigators express the results of the quantum theory. I call that opposition apparent, for while Heisenberg speaks of the invalidity and Born of the senselessness of the causal principle, Born yet adds "in its usual formulation." Therefore it may well be that the usual formulation gives rise to nothing but a tautology, but that the real meaning of the principle could be formulated in a genuine statement which could be proven false by quantum experiments. In order to determine this we must again consider which formulation of the causal principle we found ourselves driven to accept. According to our former statements the content of the principle may be expressed thus: "All events are in principle predictable." If this statement were a genuine proposition it would be verifiable—and not only this, but we would be able to say that the verification has been attempted and has so far given a negative result. But what is the case with our principle? Can the meaning of the word "predictable" be clearly indicated? We called an event "predicted" when it was deduced by the help of a formula that was constructed on the basis of a series of observations of other events. Mathematically expressed, prediction is an extrapolation. The denial of exact predictability, as the quantum theory teaches, would mean then that it is impossible to derive from a series of observations a formula that will also represent, exactly, new observation data. But what again does this "impossible" mean? One may, as we saw, subsequently always find a function that includes the new as connects the previous data with the new data and makes both appear derivative from the same natural regularity. That impossibility is therefore not a logical one; it does not mean that there is no formula with the desired properties. Strictly speaking, however, it is also not a real impossibility; for it is possible that someone should by mere chance, by pure guess, always get the correct formula. No natural law prevents correct guesses regarding the future. No, that impossibility means that it is impossible to seek that formula, that is, there is no rule for obtaining such a formula. This, however, cannot be expressed in a legitimate proposition. Our efforts to find a testable proposition equivalent to the causal principle have therefore failed. Our attempts at formulations have led only to pseudo-propositions. This result is not entirely unexpected, however, for we have already said that the causal principle may be tested for its correctness in the same sense that a natural law may be tested. We also noted, however, that natural laws, strictly analysed, are not propositions that are true or false, but are, rather, x "directions" for the construction of such propositions. If this holds also of the causal principle, we find ourselves referred to the third possibility: III. The principle of causality does not directly express a fact to us, say, about the regularity of the world, but it constitutes an imperative, a precept to seek regularity, to describe events by laws. Such a direction is not true or false but is good or bad, useful or-,, useless. And what quantum physics teaches us is just this: that the principle is bad, useless, impracticable within the limits precisely laid down by the principle of indeterminacy. Within those limits it is impossible to seek for causes. Quantum mechanics actually teaches us this, and thus gives us a guiding thread to the activity that is called investigation of nature, an opposing rule against the causal principle. Here one sees again how much the situation created by physics is different from the possibilities that have been thought out by philosophy. The causal principle is no postulate in the sense in which this concept occurs in earlier philosophers, for there it means a rule to which we must adhere under all circumstances. Experience, however, decides upon the causal principle, of course, not upon its truth or falsity—that would be senseless—but upon its utility. And natural laws themselves decide the limits of utility. In this lies the novelty of the situation. There are no postulates in the sense of the older philosophy. Each postulate may be limited by an opposing rule taken from experience, that is, may be recognised as inappropriate and thus nullified. One might perhaps believe that this point of view would lead to a type of pragmatism, since the validity of natural laws and of causality depends only on their confirmation and on nothing else. But here there is a big difference that must be sharply emphasised. The statement of pragmatism that the truth of propositions consists entirely in their confirmation, in their usefulness, must from our standpoint be rejected. Truth and confirmation are not identical for us. On the contrary, since in the case of the causal principle we may test only its confirmation, only the usefulness of its -precept, we may not speak of its "truth," and we deny to it the nature of a genuine statement. Of course, pragmatism may be understood psychologically and its teaching may, as it were, be excused, by saying that it is really difficult and requires thorough reflection to see the difference between a true proposition and useful rule, between a false proposition and useless rule. For "directions" of this type occur grammatically in the form of ordinary propositions. While for a real assertion it is essential that it be in principle verifiable or falsifiable, the usefulness of a direction can never be absolutely proven because later observations may still prove it to be inappropriate. The Heisenberg relation itself expresses a natural law, and as such has the nature of a direction. On this basis alone the rejection of determinism arising therefrom cannot be considered proof of the falsity of a definite proposition, but may be considered only an indication of the inadequacy of a rule. The hope therefore always remains that with further knowledge the causal principle will again triumph. The expert will observe that by considerations such as the above the so-called problem of "induction," too, ceases to have application, and is thus solved in the way in which Hume solved it. For the problem of induction consists in the question of the logical justification of general propositions regarding reality, which are always extrapolations from individual observations. We recognise, as Hume does, that there is no logical justification for them; there cannot be one because they are not real propositions. Natural laws are not (in the logician's language) "general implications," because they cannot be verified for all cases, but are rules, instructions, aiding the investigator to find his way about in reality, to discover true propositions, to expect certain events. This expectation, this practical attitude, is what Hume expresses by the word "belief." We must not forget that observation and experimentation are actions by which we come into direct contact with nature. The relations between reality and us are sometimes expressed in sentences that have the grammatical form of propositions, but whose real meaning lies in their being directives for possible action. To sum up: The rejection of determinism by modern physics means neither the falsity nor the emptiness of a definite proposition about nature, but the uselessness of the rule that, as the "causal principle," points the way to every induction and every natural law. And in fact the inapplicability of the rule is asserted only for a definitely circumscribed realm; there, however, with all the certainty that pertains to investigation in the exact physical sciences. 10. ORDER, DISORDER, AND "STATISTICAL REGULARITY" After the peculiar nature of the causal principle has become clear to us, we may now also understand the role actually played by the previously discussed, but then rejected, criterion of simplicity. It had to be rejected only in so far as it does not accord with the concept of cause. However we noticed that de facto it coincides with the true criterion, that of confirmation. For it clearly represents the special precept fruitful in our world, by which the general injunction of the causal principle, to seek regularity, is supplemented and perfected. The causal principle directs us to construct, from given observations, functions that lead to prediction of new ones. The principle of simplicity gives us the practical method by which we follow this direction, by saying: Connect the observation data by the "simplest" curve—which will then represent the function sought. The causal principle could remain valid even if the rule, leading to success, were quite different. Therefore this rule does not suffice to determine the causal concept, but merely represents a narrower, more special application. As a matter of fact it often does not suffice to attain the correct extrapolation. If we thus recognise the purely practical nature of the principle of simplicity, it becomes clear that "simplicity" is not to be strictly defined. Here, however, the vagueness does not matter. If, say, we should draw the simplest curve through the points representing data of the quantum processes in some experiment (for instance electronic jumps in the atom) it will be of no use in making predictions. And since we know of no other rule by which this aim would be realised, we say that the processes follow no law but are accidental. De facto, however, there does exist a marked concordance between simplicity and lawfulness, between chance and complexity. This leads us to an important consideration. It is conceivable that extrapolation with the help of the simplest curve would almost always lead to the correct result; that, however, with no ascertainable reason now and then some single observation will not match in the prediction. In order to make the idea firm let us imagine the following simple case: By means of a very long series of observations in nature we determine that in 99 per cent of the cases, an event A is followed by an event B; but not in the remaining (irregulary distributed) 1 per cent — without it being possible to find the slightest "cause" for the exception. We would say of such a world that it is still quite orderly, since our prophecies would be fulfilled on the average of 99 per cent (therefore much better than at present in meteorology or in many phases of medicine). We should therefore ascribe to this world causality, though of an "imperfect" sort. Every time that A occurs we shall expect the occurrence of B with great confidence; we shall rely upon it and get along not at all badly. Let us assume that the world otherwise is quite intelligible. If, then, with the best methods and the greatest efforts, science cannot account for the average 1 per cent deviation we shall finally rest content with that, and shall explain the world as orderly within limits. In such a case we have a "statistical law" before us. It is important to observe that a law of this kind, wherever we encounter it in science, is as it were a result of two components, in that the incomplete or statistical causality is divided into strict law and chance, which are superimposed upon each other. In the above example we should say that it is a strict law that on the average B follows A, in 99 out of 100 cases and that the distribution of the 1 per cent of deviating cases over the whole is completely a matter of chance. An example from physics: in the kinetic theory of gases the laws according to which each particle moves are accepted as perfectly strict; but the distribution of single particles and their states are, however, assumed to be completely "random" at any given moment. From a combination of both hypotheses, then, the macroscopic laws of gases result (for instance Van der Waal's law of gases) as well as the imperfect regularity of the Brownian movement. Thus in a scientific description of the process we separate a purely causal from a purely accidental part. For the former we construct a strict theory, and the latter we view in the statistical manner, that is, using "laws" of probability, which, however, are not actually laws but (as will be shown) represent the definition of the "accidental." In other words, we are not satisfied with a statistical law of the above form, but conceive it as a mixture of strict regularity and complete irregularity. Another example obviously occurs in the Schr6dinger quantum mechanics (in the interpretation of Born). There the description of the processes is also split into two parts: into the strictly regular diffusion of the ψ waves, and in the occurrence of a particle or a quantum, which is simply accidental, within the limits of "probability" determined by the ψ value at the point concerned. (That is, the value of ψ tells us, for instance, that at a definite point on the average 1,000 quanta per second enter. These 1,000, however, show quite an irregular distribution in themselves.) What does "simply accidental" or "chance" or "completely unordered" mean here? From the previous case, of the regular occurrence of A and B together in an average of 99 per cent of the observations, which does not represent complete order, we may by gradual transitions pass to disorder. Let us assume, say, that observation shows that on the average the process B follows the process A in 50 per cent of the cases, the process C follows A in 40 per cent, and D follows A in the remaining 10 per cent. We should still speak of a definite regularity, of statistical causality, but we should then judge a much smaller degree of order to be present than in the first case. (A metaphysician would perhaps say that the process A has a certain "tendency" to call forth the process B, a slighter tendency toward the process C, etc.) When would we state that there is no kind of regularity, that therefore the events, A, B, C, D are completely independent on one another (in which case the metaphysician would say that there is no inherent tendency in A to produce its consequent)? Obviously only when, after a very long sequence of observations every series formed out of the different events by permutation (with repetition) occurs on the average with the same frequency (where the series would have to be small with respect to the whole sequence of observations). We should then say that nature has no predilection for a definite succession of processes, that the succession therefore takes place quite irregularly. Such a distribution of the events has usually been called a distribution "according to the rules of probability." Where such a distribution exists we speak of complete independence of the events in question; we say they are not causally connected with one another. And according to what has been said, this manner of speech does not signify merely an indication of the lack of regularity but is identical with it by definition. The so-called "probability distribution" is simply the definition of complete disorder, pure chance. It seems to be generally admitted that to speak of "laws of chance" is a very poor way of expressing the matter (since chance means the exact opposite of law). One too easily tends to ask the meaningless question (the so-called "problem of application" relates to this) how it happens that even chance is subject to law. I cannot therefore accept Reichenbach's point of view when he speaks of a "principle of distribution according to the law of probability" as a presupposition of all the natural sciences; which principle, along with the principle of causality, is to form the basis of all physical knowledge. That principle, he thinks, consists in the assumption that the irrelevant factors in a causal relationship, the "remaining factors," "exert their influence in accordance with the law of probability." It seems to me that these "laws of probability" are nothing more than the definition of causal independence. Of course we must here include an observation that although without practical significance is yet of great importance, logically as well as in principle. The above definition of absolute disorder (equi-frequent average occurrence of all possible sequences of events) would be correct only in the case of an infinite number of observations. For it must be valid for series of any magnitude and every one of these must, according to the previous observation, be regardable as small in comparison with the total number of cases, that is, the total number of cases must surpass all limits. Since in reality this, of course, is impossible, we cannot, strictly speaking, decide whether disorder conclusively exists in any case. That this must be so follows, moreover, from our previous result, that for an already given sequence we cannot decide whether it is "orderly" or not. The same difficulty of principle exists here that makes it impossible to define the probability of any event in nature by the relative frequency of its occurrence. To arrive at correct estimates such as are required for mathematical computation (probability calculations) we should have to pass to the limit for an infinite number of cases—naturally a senseless demand for empiricism. This is often not sufficiently considered. The only useful method of defining probabilities is the one of Spielraume, logical range (Bolzano, V. Kries, Wittgenstein, Waismann; see the above cited article of Waismann). This, however, does not belong to our theme. We now proceed to derive some consequences from the above considerations and to criticise others that are drawn here and there in this connection. 11. WHAT DOES "DETERMINED" MEAN? Since, generally, we speak of causality by saying that one process determines another, that the future is determined by the present, we want once more to clarify for ourselves the true significance of this unhappy word "determine." That a certain state determines another later one can in the first place not mean that there is a hidden connection between them called causality, which could somehow be found or must be thought. For such naive ways of thinking are, for us, surely no longer possible, 200 years after Hume. Now we have already given the positive answer, at the beginning of our deliberations: "A determines B" cannot mean anything but: B may be calculated from A. And this again means: there is a general formula which describes the state B as soon as certain values of the "initial state" A are put into it, and as soon as a certain value is given to certain variables, for instance that of time, t. That the formula is "general" means, again, that besides A and B there are any number of other states connected with one another by the same formula and in the same manner. Indeed a large part of our efforts was directed to answering the question of when one may say there is such a formula (called "natural law"). And the answer was that the criterion lay in nothing but the actual observation of the B computed from A. Only when we can indicate a formula that is used successfully in prediction can we say that there is a formula (order is present). The word "determined" therefore means exactly the same as "predictable" or "computable in advance." This simple viewpoint alone is needed to resolve a well-known paradox, important for the problem of causality, which perplexed Aristotle and is even today a source of confusion. It is the paradox of so-called "logical determinism." It states that the principles of contradiction and of excluded middle would not be valid for propositions regarding future facts if determinism is not valid. In fact, so Aristotle argued, if indeterminism is correct, if the future is not already determined, it seems that the proposition, "the event E will take place the day after tomorrow," can be today neither true nor false. For if, for instance, it were true the event would have to occur, it would already be determined, contrary to the indeterministic assumption. Even in our day this argument is sometimes held to be conclusive, and is even said to be the basis of a new logic. There must of course be an error here, for logical propositions, which are only rules of our symbolism, cannot depend as regards their validity upon the existence of causality in the world; every proposition must have truth or falsity as a timeless characteristic. The correct interpretation of determinism removes the difficulty at once and leaves to the logical principles their validity. The proposition "the event E occurs on such and such a day" is timeless—therefore even now either true or false, and only one of the two, quite independently of whether determinism or indeterminism holds in the world. For the latter by no means asserts that today the proposition about the future E is not unambiguously true or false, but only that the truth or falsity of that proposition cannot be calculated on the basis of propositions about present events. This means, then, that we cannot know whether the proposition is true before the corresponding point in time has passed, but this has nothing to do with its being true or with the basic laws of logic. 12. DETERMINATION OF THE PAST If physics today, speaking indeterministically, says that the future is (within certain limits) undetermined, it means nothing more or less than: it is impossible to find a formula by which we may calculate the future from the present. (More correctly it would mean: it is impossible to seek such a formula, there is no rule for its discovery; it could be guessed only by pure chance.) It is perhaps comforting to observe that in quite the same sense (and I cannot imagine any other meaning of the word "undetermined") we must say of the past that in a certain respect it, too, is undetermined. Let us assume for example that the velocity of an electron has been precisely measured and then its location observed: in this case the equations of the quantum theory also enable us to compute, exactly, previous positions of the electrons However, actually, this indication of position is physically meaningless, for its correctness cannot in principle be tested, since it is impossible to verify subsequently whether the electron appeared in the computed place at the given time. If, however, one had observed it in this computed place, it would certainly not have reached those places later noted, since its course is known to be disturbed by observation in an incalculable manner. Heisenberg says (p. 15): "Whether or not one attributes a physical reality to the calculation of the past of the electron is merely a matter of taste." However, I should prefer to express myself even more strongly, in complete agreement with what I believe to be the fundamental point of view of Bohr and Heisenberg himself. If an assertion regarding the position of an electron is not verifiable in atomic dimensions, then we cannot attribute any meaning to it; it becomes impossible to speak of the "path" of a particle between two points where it has been observed. (This of course is not true of bodies of molar dimensions. If a bullet is now here and a second later at a distance of 10 metres, then it must have passed the points in between during that second, even if no one has perceived it; for in principle it is possible to verify subsequently that it has been at the intervening points.) One may treat this as the sharpened formulation of a proposition of the general theory of relativity: just as transformations that leave all point coincidences — intersection points of world lines — unchanged have no physical meaning, so we may say here, there is no sense at all in attributing physical reality to the segments of world-lines between the points of intersection. The most concise description of the state of affairs we have discussed is perhaps to say (as the most important investigators of the quantum problems do) that the validity of the usual spatiotemporal concepts is limited to that which is macroscopically observable; they are not applicable to atomic dimensions. None the less let us spend another moment with the results just arrived at concerning determination of the past. We sometimes find it asserted in current literature that contemporary physics has reestablished the ancient Aristotelian concept of "final cause," in the form of that which is earlier being determined by the later, but not vice versa. This idea occurs in the interpretation of the formulas of atomic radiation which, according to the theory of Bohr, is supposed to take place so that the atom sends out a light quantum every time an electron jumps from a higher to a lower orbit. The frequency of the light quantum depends upon the initial orbit and the final orbit of the electron (it is proportional to the difference in energy values of the two orbits) ; it is therefore obviously determined by a future event (the entrance of the electron into the final orbit). Let us test the meaning of this idea. Aside from the fact that the concept of final cause must have had a different content for Aristotle, this idea, according to our analysis of "determine," states that in certain cases it is impossible to compute a future event Z from the data of past events V, but that, on the other hand, V may be derived from the known Z. Good, let us imagine that the formula for this is given and that a V has been computed therefrom. How do we test the correctness of the formula? Only by comparing that which is computed with the observed V. V, however, is already in the past (it existed before Z, which has also already occurred and had to be known in order to be insertable into the formula) ; it cannot be observed post factum. If then we have not previously ascertained it, the proposition that the computed V occurred is not verifiable in principle and is therefore meaningless. If however V has already been observed, we have a formula that connects events already observed. There is no reason why such a formula should not be reversible. (For in practice one-many functions do not occur in physics.) If by means of it V may be calculated from Z, it must be equally possible to determine Z by means of it when V is given. We therefore encounter a contradiction when we say the past may be calculated from the present but not vice versa. Logically both are the same. Note well: the essence of this argument is that the data of the events V and Z enter into the natural law with entirely equal right; they must all already have been observed if the formula is to be verifiable. For the rest, here too, all the obscurities are basically due to the lack of clear distinction between that which may be formulated as a contribution of thought and that which has really been observed. Here again we see the great advantage of Heisenberg's point of view, which would offer a purely mathematical and not an apparently intuitive model of the atom; with it the temptation to introduce so-called "final causes" falls to the ground. It seems to me that the mere elucidation of the meaning of the word "determine" shows that it is under all circumstances impermissible to assume (quite independently of the question of determinism) that a later event determines an earlier one, but that the reverse is not true. 13. TOWARD DIFFERENTIATION OF PAST AND FUTURE The last considerations seem to teach that an inference regarding past events has precisely the same nature logically as one regarding future events. In so far as, and to the extent that, causality holds at all, we may say with equal justice that the earlier determines the later and the later determines the earlier. In accordance with this all attempts to differentiate conceptually between the temporal direction from the past to the future and from future to past fail. This I believe is true also of H. Reichenbach's attempt (in the treatise cited in the Bayrischen Sitzungsberichten) to demonstrate the asymmetry of the causal relationship, and by its help to ascertain conceptually the positive temporal direction and thereby to be able to define even the time of the present, the now. He believes that the causal structure in the direction of the future differs topologically from that of the reverse direction. The arguments he gives for this belief I consider incorrect. However, I do not want to dwell upon this (compare for instance the critique, in need of some further elaboration, of Reichenbach's ideas by H. Bergmann in "Der Kampf um das Kausalgesetz in der jungsten Physik") but merely to mention that the demand fora definition of the now is logically meaningless. The difference between earlier and later in physics may be described objectively, and in fact, as far as I can see, only by aid of the principle of entropy. But in this way the direction past-future is only differentiated from the opposite. However, that real events proceed in the first direction and not in the reverse cannot be said at all, and no natural law can express it. Eddington (The Nature of the Physical World) describes this in an intuitive way, in claiming that a positive temporal direction (time's arrow) may be defined physically, but that it is not possible to formulate conceptually the passage from the past to the future (becoming). H. Bergmann rightly sees, in opposition to H. Reichenbach, that physics has no means whatever of distinguishing the now, of defining the concept of the present. He seems, however, to assume falsely tha'. by means of "psychological categories" this may not be impossible. In truth the meaning of the word "now" may only be shown, just as we may only show and not define what we understand by "blue" or by "happiness." That the causal relation is asymmetrical, unidirectional (as Reichenbach, loc. cit. believes) is falsely suggested by facts connected with the principle of entropy. It is only due to this law that in everyday life the earlier may be more readily derived from the later than vice versa. The calculation of the later is of course not by itself identical with an inference to the future, and neither is the calculation of earlier itself identical with an inference to the past. This is the case only when the temporal point from which we make the inference is the present. Reichenbach believes (loc. cit., p. 155) that the latter case is actually distinguished by the fact that the past is objectively determined whereas the future is objectively undetermined. Brief analyses show that all that is meant by "objectively determined" is "inferable from a partial effect." The future is "objectively undetermined" because it cannot be inferred from a partial cause, for the totality of all partial causes can not be defined in the absence of determinism. All sorts of things may be said against the concepts of partial cause and partial effect, and we have already indicated that the apparently easier process of inference is falsely suggested by facts involved in the principle of entropy. But even if the argument contained no error it would again only characterise the difference between the earlier and later, not the difference between past and future. 14. INDETERMINATENESS OF NATURE AND FREEDOM OF THE WILL The psychological reason for the sort of ideas last mentioned (and that is why I referred to them) seems to me to lie in the circumstance that, in addition to the simple meaning our analysis found for the word "undetermined," implicitly a sort of metaphysical, related meaning is attributed to it; namely, as if one could attribute determinateness or indeterminateness to a process in itself. That, however, is meaningless. Since "determined" means calculable by means of certain data, to speak of determinism makes sense only when we add: By what? Each real process whether it belongs to the past or to the future is as it is; being undetermined cannot belong to its characteristics. Regarding the natural processes themselves one cannot sensibly assert a "vagueness" of "indefiniteness." Only in reference to our thoughts may we speak of such (namely, when we do not know definitely which propositions are true, which representations are correct.) Sommerfeld evidently means just this when he says:10 "It is not the experimentally that are indeterminate. With sufficient attention to experimental conditions, these may be precisely treated. Indeterminism applies only to our ideational forms which accompany physical facts." One must not believe therefore that modern physics has any place for the misconception of natural processes "undetermined in themselves." If, for instance, in an experiment it is not possible to give an electron a precise location, and if the same is true of its momentum, this means nothing more than that position and momentum values of a punctiform electron are not suitable means for the description of the process that takes place in nature. The modern formulations of the quantum theory recognise this and take it into account. Just as little as the present situation in modern physics allows the formulation of a metaphysical concept of indeterminism does it allow speculations about the so-called "problem of freedom of the will" which is connected with it. This must be sharply emphasised, for not only philosophers but also men of science have not been able to withstand the temptation to utter thoughts such as the following: Science shows us that the physical universe is not fully determined; it follows, (1) that indeterminism is in the right and that physics therefore does not contradict the assertion of freedom of the will; (2) that nature, since strict causality does not prevail in it, provides room for spiritual or mental factors. In answer to (1) we may say: the real problem regarding freedom of the will as it occurs in ethics has been confused with the question of indeterminism only because of crude errors which, since Hume, have long since been corrected. The moral freedom that the concept of responsibility presupposes does not stand in opposition to causality but would be entirely destroyed without it. To (2) we may say: the statement implies a dualism, the juxtaposition of a spiritual and physical world between which there may be an interaction because of the imperfect causality of the latter. In my opinion no philosopher has succeeded in elucidating the real meaning of such a proposition, that is, no one has shown which experiences would enable us to confirm its truth and which experiences would disclose its falsity. Quite the contrary, logical analysis (for which of course there is no place here) leads to the conclusion that in the data of experience there is no legitimate ground for that dualism. It is therefore a meaningless, untestable metaphysical proposition. It seems to be believed that the possibility of "psychical" factors entering through possible loopholes of "physical" causality has consequences relating to our world outlook that satisfy certain emotional needs. However this is an illusion (since the purely theoretical interpretation of the world has no relationship to emotional needs correctly understood) . If the tiny gaps in causality could in some way be filled in, it would only mean that the above mentioned, practically insignificant traces of indeterminism existent in the modern world-picture would again be partly wiped out. In this realm the metaphysics of earlier times was guilty of certain errors which sometimes occur even now where metaphysical motives are completely absent. Thus we read in Reichenbach (p. 141) : "If determinism is correct nothing can justify our undertaking an action for tomorrow but not for yesterday. It is clear that we then have no possibility at all even to abstain from the plan for the morrow's action and from the belief in freedom — certainly not, but in that case our action does not make sense." It seems to me that the exact opposite is the case: our actions and plans obviously make sense only in so far as the future is determined by them. Here there is simply a confusion of determinism with fatalism, which has so often been criticised in the literature that we need not dwell upon it. Moreover, he who still represents the opinion criticised above would not be helped at all by the indeterminism of modern physics. For with utmost consideration of all relevant facts, happenings are still so precisely calculable beforehand in it, the remaining indeterminateness is so slight, that the significance our actions would have in this world of ours would still be vanishingly small. Precisely the last considerations teach us again how different the contributions of modern physics to the question of causality are from those of earlier philosophical thinking; and how correct we were in saying at the very beginning that human imagination was in no position to foresee the structure of the world as revealed to us by patient investigation. For it is even difficult for it to progress in the steps science has already shown to be possible.