Philosophical and Methodological Problem of Consistency of Mathematical Theories

Increased abstraction of modern mathematical theories has revived interest in traditional philosophical and methodological problem of internally consistent system of axioms where the contradicting each other statements can’t be deduced. If we are talking about axioms describing a well-known area of mathematical objects from the standpoint of local consistency this problem does not appear to be as relevant. But these problems are associated with the various attempts of formalists to explain the mathematical existence through consistency. But, for example, with regard to the problem of establishing of consistency of mathematical analysis the solution of which would clarify the fate of Hilbert's proof theory it has not solved yet so as the problem of the consistency of axiomatic set theory. Therefore it can be assumed that the criterion of consistency despite its essential role in axiomatic systems both formal and substantive nature is the same auxiliary logical criterion as well as mathematical provability. An adequate solution of the problem of consistency of mathematics can be achieved in the area of methodological and substantive arguments revealing the mechanism of appearance of contradictions in the mathematical theory. The paper shows that from a systemic point of view in the context of philosophical and methodological synthesis of various directions of justification of modern mathematics it can’t insist on only the rationale for consistency of mathematical theories.