Is Criticism of Kolmogorov's Conditions of Using Mathematics Sound?

The article studies the foundations of criticism offered by the followers of the subjective interpretation of probability theory, namely Kolmogorov's condition about the proximity of the theoretical probability and empirical frequencies in the context of using mathematics. They consider this requirement to be redundant as it is deducible on the basis of Bernoulli's theorem being its conclusion. In my opinion, the question of redundancy of this condition has not only a historical value, but is also topical, because the inference about Kolmogorov's error is based on using Bernoulli's theorem and the problem of using mathematics does not receive adequate attention. To analyze the soundness of this criticism, I formulate the conditions of using Bernoulli's theorem in the frequency interpretation, because Kolmogorov remarked that regarding applications he followed Mises, the founder of the frequency interpretation. These conditions are: the correct use of the theorem presupposes estimation of the probability of success in research and verification of the independence of its results. I show that the verification of result independence is a hard problem, and, for this reason, the models of independent trials are usually accepted on informal foundations. I offer arguments in favor of the formal verification by means of Elyasberg's example. The example demonstrates that the dispersion of the sum of a thousand of independent random values differs from the dispersion of the sum of a thousand of weakly connected random values with a pair correlation coefficient 0.01 by more than one order. In contrast to the frequency interpretation, the requirements of using the subjective one are limited by accepting probability axiomatics only. Subjectivists believe that the probability of success in research and the independence of its results are known a priori. It appears that the using of the theorem by subjectivists does not guarantee objective results and, in fact, is not correct in the context of requirements to applications in the frequency interpretation. Thus, the criticism of Kolmogorov is not sound.

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