It is in mathematics that Russia and the Soviet Union have made the greatest contributions. Today the Soviet Union is a world power in mathematics. Indeed, Moscow probably has the greatest concentration of talent of any city. The main competitor is no doubt Paris, since mathematicians in the United States, another leader in mathematics in the last generation, are more widely distributed geographically.
This great tradition in mathematics dates back to Leonhard Euler and Jakob Bernouilli in the early eighteenth century, both of whom did important work while living in Russia. N.I. Lobachevskii, M.V. Ostrogradskii, and P.L. Chebyshev in the nineteenth century solidified the reputation of Russia in mathematics. By the early twentieth century Russian mathematicians were working at the leading edge of mathematics in many areas: Chebyshev and A. A. Markov in the theory of numbers and probability; V. A. Steklov and A. N. Krylov in differential equations; D.F. Egorov, K.A. Andreev, and A. K. Vlasov in geometry; D.A. Grave, S. O. Shatunovskii, and F. E. Molin in algebra; N. N. Luzin in theory of functions; and many others. In the later Soviet period outstanding mathematicians are far too numerous to name, but they include L.M. Gerfand, A. N. Kolmogorov, A. Ia. Khinchin, S.N. Bernshtein, N.N. Bogoliubov, L.V. Kantorovich, L.S. Pontriagin, L.R. Shafarevich, S. L. Sobolev, and I. M. Vinogradov.
Unfortunately, the importance of the history of Russian and Soviet mathematics is poorly reflected in English-language sources. Not even Lobachevskii, the creator of non-Euclidean geometry, is the subject of a full biography in English. V.F. KaganÕs N. Lobachevsky and His Contribution to Science (Moscow: Foreign Languages Publishing House, 1957) is perhaps the source most often cited, but it is clearly inadequate. Alexander Vucinich has explored some of the nontechnical aspects of Lobachevskii’s life in his “Nikolai Ivanovich Lobachevskii: The Man Behind the First Non-Euclidean Geometry,” Isis, 1962, 53:465-481. The best source on the circumstances of the creation of Lobachevskii geometry is a senior thesis by Gregory Crowe, “The Life and Work of Nikolai Ivanovich Lobachevsky: A Study of the Factors Leading to the Discovery and Acceptance of the First Non-Euclidean Geometry” (Harvard Univ., 1986).
A happy exception to the dearth of English-language materials on the history of Russian and Soviet mathematics is Anne Hibner Koblitz’s biography of the first significant woman mathematician of modern times, A Convergence of Lives: Sofia Kovalevskaia: Scientist, Writer, Revolutionary (Boston: BirkhBuser, 1983). Biographical material is also available on Nikolai Luzin, a founder of the twentieth-century “Moscow School” of mathematics, in two articles: Esther Luzin R. Phillips, “Nicolai Nicolaevich and the Moscow School of the Theory of Functions,” Historia Mathematica 1978, 5:275-30; and Allen Shields, ÒYears Ago: Luzin and Egorov,” The Mathematical lntelligencer, 1987, 9(4):24-27.