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(a) Probability and Experience The theory of probability had its origin in an empirical question: Chevalier de M&r&'s gambling problem. Equally, the present-day physicist finds no difficulty in empirically testing probabilities which have been theoretically predicted, by measuring the relative frequencies of the occurrence of certain events. On the other hand, the epistemological discussion on the meaning of the application of the so-called mathematical concept of probability to empirical reality is by no means settled. The battle is still raging between 'objectivist', 'subjectivist', and even other interpretations of probability. Probability is one of the outstanding examples of the 'epistemological paradox' that we can successfully use our basic concepts without actually understanding them. In many apparent paradoxes which are connected with fundamental philosophical problems the first step towards their solution consists in accepting the seemingly paradoxical situation as a phenomenon, and in this sense as a fact. Thus we will have to understand that it is the very nature of basic concepts to be practically useful without, or at least before, being analytically clarified. This clarification will have to use other concepts in an unanalysed manner. It may mean a step forward in such an analysis to see whether a hierarchy exists in the practical use of basic concepts, and which concepts then practically depend on the availability of which other concepts, and also to see where concepts

A striking characteristic of contemporary science is its emphasis upon probability. This is especially notable in quantum mechanics.There is a respect in which probability is the same for all scientific theories. Verifiability requires that any theory predict certain numbers which can be compared with the numbers gained by actual operations of measuring. In actual practice these numbers, which we shall term theoretical measurables and operative measurables respectively, never correspond. It becomes necessary, therefore, for the scientist to specify when the deviation between them is such that verification occurs. These specifications are defined by the theory of errors in which the concept of probability has an essential place and a specific meaning. But the theory of errors is the same for all applications; hence, probability as defined by the theory of errors is the same for all scientific theories.

In the ongoing investigation of the conceptual basis of quantum physics a certain tradition has developed concerning the use of probabilistic concepts. While the necessity for these concepts is generally recognized, the tradition in question holds that the actual probabilistic concepts employed in quantum physics differ in certain essential respects from the probabilistic notions used in other areas of physics and in ordinary statistical contexts. This line of thought apparently originates with BORN himself (see Jammer [5], p. 285). In some writings it seems that BOHR (for example, see [1], p. 4) has taken a similar point of view. And more recently the tradition has found a convincing spokesman in the work of Henry Margenau (see [8] and references there).

The view I attempt to support in this paper by various sorts of arguments is that if we examine the structure of quantum mechanics for paradoxical or surprising implications, it is not the interpretation of the uncertainty principle as showing the untenability of determinism that is most important. I take it that many would now agree that strict determinism was a fragile flower nurtured for a few decades in the special climate of classical mechanics, particularly as formulated by Laplace. The once broad claims for determinism have in many ways been replaced by universal claims for the theory of probability and the methodology of statistics, although I do not wish to suggest for a moment that determinism as a thesis and probability theory as a methodological cornerstone of science are on the same conceptual footing. Rather, my point is to try to show that the really radical intellectual thing about quantum mechanics is not its challenge to determinism as a philosophical thesis, but its challenge to probability theory and modern mathematical statistics as the universal methodology of all empirical science.

After a brief discussion of the reasons for the complete failure of a deterministic interpretation of quantum mechanics (§ 1)Niels Bohr's ideas on quantum mechanics are exposed. The importance of Bohr's idea on the necessity of combining the quantum-mechanical description of atomic objects with a classical description of the instruments is stressed (§ 2).It is pointed out, however, that the terminology used by Bohr often gives rise to misunderstandings and to a positivistic interpretation of his ideas. This, in turn, creates an opposition to the new ideas (de Eroglie school) (§ 3).In the following the concept of relativity with respect to observation means is introduced and it is stressed that relativity does not exclude objectivity (§ 4).The meaning of the term “instrument” as used in quantum mechanics is explained (§ 5).It is pointed out that wave-corpuscular duality is connected with the inherent possibility of the electron to manifest itself, according to the external conditions prevailing, as a wave, or as a corpuscle, or in an intermediate way. A more literal interpretation of duality based on some model is inadmissible (§ 6).The probabilistic description of the interaction of an atomic object with an instrument, resulting in a probability distribution for the measured quantity, is discussed and the primary character of probability in quantum mechanics is stressed (§ 7).If a subdivision of the experimental arrangement in a preparatory part, a working part and a registration part is possible, one can vary the last stage of experiment and obtain probability distributions referring to the same initial state. Since these are parametrically expressed in terms of one and the same wave function, this function is independent of the last stage of experiment. By making the corresponding abstraction one is led to the quantum-mechanical concept of state as described by a wave function (§ 8).The necessity of distinguishing in quantum mechanics the potentially possible from the accomplished is compared with the concepts of classical physics and with the notions of everyday life (§ 9).The nature of the statistical collective to which the quantum-mechanical probabilities refer is discussed and it is shown that the statistical character of quantum mechanics does not contradict the fact that this theory describes the properties of an individual object (§ 10).The role and position of the causality principle in quantum mechanics is discussed and the nature of the so-called “reduction of a wave-packet” explained (§ 11).The necessity of incorporating the new ideas in the materialistics philosophy and of a further development of dialectic materialism is stressed (§ 12).

The notion and methods of calculating probabilities in quantum mechanics is strikingly different from classical mechanics and standard probability theory. Quantum mechanics manipulates mysteriously, from a classical probabilistic point of view, complex functions and operators to nevertheless end up with observable quantities which are interpreted as if they were derived from a standard type of probability theory. Whether the strange methods of obtaining the predictions of the theory are essential, in the sense that given the type of results quantum mechanics predicts implies a quantum mechanical type of mathematical formalism, is not totally clear. Certainly many of the paradoxes seem to be based on the mode of calculating probabilities. For example, the paradox of the double slit experiment arises from the fact that the probability of the particle being at a certain point on the screen is not the same probability that it went through one slit or the other. Since standard probability theory arose out of simplistic considerations it may be that it is restricted and that its future development will generalize it to a degree where quantum mechanics will fit in naturally. Weiner (1963) made some attempts along these lines by showing some examples from ordinary probability theory that yield results similar in mathematical form as in quantum mechanics. Another approach to the understanding of the probabilistic methods is to attempt to formulate quantum mechanics as a traditional probability theory. We shall describe this attempt, and although we shall show that this is strictly speaking impossible, the impossibility will illuminate certain aspects of the theory. In particular we will emphasize the question of joint distributions of non-commuting variables and show the one-to-one relationship between so-called correspondence rules and the distribution functions. As we shall see, the question as to whether there can exist a consistent rule for obtaining quantum mechanical operators (a correspondence rule) from their classical counterpart and the question whether there exists a legitimate joint distribution function are one and the same.