The twentieth century science paradoxes

The isolation of hypothetical theories from the realities of living matter has caused mysticism to penetrate scientific theories. With mystical thinking, the idea of using an analytical method to solve cognitive problems does not occur. Dialectical logic, in contrast to mysticism, states the opposite: any problematic tasks of cognizing the vital processes and phenomena of the universe are solvable exclusively in an analytic way, with the only method. The author created a universal and formal theory of solving intellectual (i.e., having no previously known algorithms for solving) problems associated with the knowledge of the vital functions of natural and man-made processes in any phenomena of the universe - the Kondratenko method of axiomatic modeling, the effectiveness of which is achieved by correctly setting the problem and solving it purely formal method. The correctness of the statement of the problem means, first of all, the recognition of the failure of all hypothetical (not confirmed by the results of full-scale experimentation with the subject of knowledge) theories. This requirement, in particular, to the mathematical tools used to solve problems of cognition, it revealed paradoxes in the foundations of mathematics, which are discussed in the article. At present, in the natural and applied sciences in most publications, i.e. more than 90% associated with the construction of formal theories in these sciences, the proof of theorems is carried out: firstly, in a meaningful way, which contradicts the urgent requirement of philosophers of science to use exclusively formal evidence, which is a criterion for assessing the correctness and reliability of evidence; secondly, in substantive evidence in 95% of cases, an exclusively standard list of tautologies is used, which by definition is incorrect for the purpose of proving theorems on phenomena and processes of the universe based on exclusively true axioms obtained as a result of full-scale experimentation with these phenomena and processes. The article deals with the paradox in the classical approach to proving theorems, which consists in the inappropriateness of generally accepted stereotypical tautologies of classical mathematics for proving theorems.