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Andrei markov and mathematical constructivism

1995; Elsevier BV; Linguagem: Inglês

10.1016/s0049-237x(06)80057-1

2542-6702

N. M. Nagorny,

History and Theory of Mathematics

This chapter discusses the contribution of Andrei Markov in mathematical constructivism. His research in the theory of algorithms, mathematical logic, and constructive mathematics that spanned 30 years of his life was the height of his scientific career. On Markov's initiative, the trend of mathematics he inaugurated is called “constructivist.” Markovian constructivism holds a legitimate place in mathematics. Markov's switch to constructivism and his first steps in this new field coincided with one of the unhappiest periods in Soviet history. Markov's program for mathematics suggests that constructive mathematics should deal exclusively with constructive objects. These objects should be treated only within the framework of potential feasibility abstraction, the abstraction of actual infinity being banned. The term “algorithm” should be understood in a precise way. Because of the nature of the allowed objects and abstractions, the understanding of mathematical statements should be based upon specially conceived constructive logical principles. The used apparatus of logical deduction should be based upon a constructive logic that would exclude the provability of pure existence theorems.