The Constructive Mathematics of A. A. Markov

Abstract

Andrei Andreevich Markov Jr. (born September 22, 1903, in St. Petersburg; died Oc tober 11, 1979, in Moscow) was the late and only child of the great Russian mathe matician Andrei Andreevich Markov Sr. (born June 14, 1856, in Ryazan7; died July 20, 1922, in Petrograd), universally recognized, in particular, for his contributions to the theory of probability (e.g., Markov chains and Markov processes). At his father's suggestion, young Andrei entered the chemistry section of the School of Physics and Mathematics at the University of Petrograd (formerly St. Petersburg, then Leningrad, and now again St. Petersburg). The young man was fascinated by chemistry and al ready by 1920 had taken part in chemical research. The results of his and his coau thor's work were published in 1924. Thus Markov's first paper dealt with chemistry. In his sophomore year he became interested in theoretical physics, and he graduated in 1924 with a physics degree. Markov's publications in chemistry were followed by a series of papers on the three body problem and dynamical systems (1926-1937), and a paper on Schr?dinger's quantum mechanics. The latter was one of the first papers on quantum mechanics published in the U.S.S.R., appearing less than a year after Schr?dinger's own ground breaking series of publications. In this connection it should be noted that it was Markov who, in 1931, introduced the concept of an abstract (topological) dynamical system. In 1932 Markov published an intriguing paper (in German) on relativity titled Deriving a World Metric from the Relation 'Earlier Than.' His interest in abstract mathematics is represented by series of papers on topology, algebra, analysis, and geometry. After World War II Markov's interests turned to axiomatic set theory, mathematical logic, and the foundations of mathematics. He founded the Russian school of con structive mathematics in the late 1940s and early 1950s. But in private conversations Markov often said that he had nurtured constructivist convictions for a very long time, in fact, long before the war. The Moscow mathematical school had been interested in constructivism, especially intuitionism, since its inception in the 1920s. It is enough to mention the 1925 work of Kolmogorov on intuitionistic logic [10]. It may be that this interest was due, at