Abstract
AbstractThis article contains an extended exposition of my talk “Yu. I. Neimark and mathematical rigor” at the special session of the conference MMCT-2020. It discusses the role of mathematical rigor in applied mathematics. The discussion focuses on the question of relationship between continuous processes and their computer modeling. The question of mathematically rigorous formalization of physical theories, which goes back to Hilbert’s 6th problem and the widespread the point of view that continuous mathematics is an approximation of the discrete one, and not vice versa are also discussed. A new axiomatic of set theory is introduced that includes vague definitions and concepts at the same level of rigor as in modern classical mathematics, which operates only with well-defined objects. This allows to consider some non-rigorous arguments of applied mathematics as rigorous ones and, thus, to be sure that they are consistent.KeywordsNonstandard analysisInfinitely large numbersHyperfinite set
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