The constructive mathematics of A. A. Markov: some reflections

Abstract

A. A. Markov's constructive mathematics: some reflections. A review of constructive mathematics in the Markovian sense is presented, with particular attention to constructive analysis. An account is given of the basic methodological principles of constructive mathematics and of expectations connected with them. A number of results of constructive mathematical analysis, including basic facts of constructive continuum theory and functions on the constructive continuum are presented. Additional questions of constructive analysis, including problems of topology and functional analysis, are considered as well. It is pointed out that although the initial revolutionary intentions are, perhaps, behind us now, the constructive direction still provides a valuable experience of development of mathematics on a purely syntactic and effectivist basis. The last two circumstances are of special importance in the light of recent developments in computer science.