7. Probability and Statistics Soviet Essays - Sheynin, Oscar

to be determined experimentally, but rather in the limiting, asymptotic frequencies ofnumbers of some class, regularly distributed in these sequences. These limiting frequenciespresent some similarities with mathematical probabilities heuristically very valuable for thetheory, but the confusion of these two concepts is an inadmissible misunderstanding.[4] Analogous paradoxes occur also in some geometrical problems of probability theory.An investigation of this subject from the viewpoint of the theory’s unity and possibility of anempirical checking of its deductions leads to the conclusion that the theory of probability isfar from being able to consider all abstract sets. However, under appropriate restrictions andtaking some precautions, on which I cannot dwell here, an arithmetization of infinite sets, i.e.,the determination of the probabilities of all their sensible propositions, is possible withoutcontradiction. I shall only remark that the main source of the paradoxes was that thearithmetization of infinite sets had been carried out more or less intuitively rather than bydistinctly formulating which of the two principles, continuity or discontinuity, was chosen asits foundation. The former corresponds to the assumption that it is senseless to state that twomagnitudes are equal one to another because in an experiment equalities can only be realizedwith some error and not absolutely precisely. On the contrary, the latter is applicable tomagnitudes whose precise equality admits of actual checking; their totality is alwayscalculable and they cannot be equally probable (if the totality is infinite). In each particularcase, only experiment can and should decide which of the two hypotheses is true.Thus, the theory of radiation of a blackbody, issuing from the principle of continuity,arrives at the Rayleigh law of distribution of radiant energy. However, since this law did notconform to experiment, Planck was compelled to assume the principle of discontinuity, or ofstep-wise changes of energy, and to create his celebrated quantum theory that was entirelycorroborated not only by all the phenomena of radiation, but also by the properties of heatcapacity of bodies at all possible temperatures. Classical mechanics, whose laws werederived by observing the movement of bodies of finite dimensions, demands essentialchanges in order to interpret phenomena connected with radiation. A deeper reason for thisconsists in that this discipline leads to a uniform distribution of energy among an infinitenumber of degrees of freedom, i.e., in essence, to the contradictory assumption of an infiniteset of equally possible incompatible cases, see above.Quantum mechanics encounters serious difficulties as well; these, however, are not in theplane of probability theory but in the sphere of our mechanical and geometric ideas which wedesire to apply to such elements as the electron, never observed all by themselves. In anycase, from the standpoint to be developed below, it is not necessary to seek for a definitemechanical-geometric model of the atom; we may be satisfied by a physical theoryconstructed on a pattern of a harmless play between all the electrons of a given body with itsrules being dictated by observed macroscopic phenomena.[5] Returning to the principles of probability theory, it may be thought, as it seems to me,that the main formal logical difficulties of its construction have by now been surmounted 3 . Itis not enough, however, to acknowledge that, when keeping to the well-known rules ofcalculation accepted by the theory, we may without logical contradictions attribute definiteprobabilities to various facts. Is there any physical sense in saying that two different factspossess equal probabilities? Do not we sin against the law of causality when stating that intwo trials, when throwing two identical dice, the probability of a six is the same in both caseswhereas the actual outcomes were a six and a five?Much attention, both previously and recently, was given to this question important boththeoretically and practically. If we are now closer to some like-mindedness with respect to itssolution than we were several decades ago, we owe it not to deeper philosophical reasoningas compared with the deliberations expressed by Laplace or Cournot, but to the experimental