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Nikolai Shanin

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N.A.Shanin, 2010

Nikolai Aleksandrovich Shanin (Russian: Николай Александрович Шанин), a Soviet and Russian mathematician and the founder of a big scientific school in Leningrad/Saint Petersburg, was born on May 25, 1919, in Pskov, Russia, in a doctor’s family and passed away on September 17, 2011, in Saint Petersburg, Russia.[1] His father Alexander Protasyevitch Shanin (Russian: Александр Протасьевич Шанин, 1886–1973[2]), was a known specialist in skin cancer.

In 1935 N. A. Shanin entered the Faculty of Mathematics and Mechanics of Leningrad State University, and in 1939 he began his PhD studies there. Andrey Andreyevich Markov, Jr. became his supervisor, his second superviser was Pavel Sergeyevich Alexandrov. Markov’s ideas and personality had a decisive influence on the formation of Shanin’s research interests. In 1942 he defended his PhD dissertation “On extension of topological spaces” , and in 1946 his D.Sc. dissertation “On the product of topological spaces”. From 1941 to 1945, during the war between the USSR and Germany, Shanin served in the Red Army. In October 1945 he became a senior research fellow at Steklov Mathematical Institute of the USSR Academy of Sciences in Leningrad (later Saint Petersburg) division (LOMI/POMI),[3] where he worked until the end of his life. While working in the Academy of Sciences, he also taught for many years at Leningrad/Saint Petersburg State University at the Faculty of Mathematics and Mechanics (he became a professor at this faculty in 1957) and at that of Philosophy.

Shanin’s research activity can be divided into two periods: topological and logical–constructivist.[4] The first period continued up to the end of the 1940s. His results in general topology[5] were influential for a long time.[6] The second period was much longer and in addition to numerous scientific results it gave a big Leningrad school of mathematical logic and proof theory with ramifications to computability (e.g., Yuri Matiyasevich[7]), algorithmics and computational complexity,[8] applications of computers to research in mathematics.[9][10]

For A. A. Markov Jr., and then N. A. Shanin, ineffectiveness of purely existential theorems was a source of ‘discomfort’ for the foundations of mathematics, and thus, the ideas of intuitionism were attractive.

N. A. Shanin started with generalizations of A. N. Kolmogorov[11] and K.Gödel[12] approach to imbedding operations that transform a formula F of classical logic into a formula Fœ ' of intuitionistic (constructive) logic in such a way that Fœ ' is deducible in intuitionistic logic iff F is deducible in classical one. Besides that, such a transformation should preserve the syntax of F ‘as much as possible’. N. A. Shanin constructed a series of subtle and general operations, in particular, described classes of classical formulas with ∃ and ∨ that are deducible in intuitionistic logic without changes[13]. This paper[13] was one of the first papers on intuitionistic (the term "constructive" was often used instead, partially for political reasons) logic in the USSR, and it influenced the research in this field. Later N. A. Shanin applied his ideas to other formal systems.

The next area of research of N. A. Shanin concerned constructive semantics and was also influenced by intuitionism. However, the semantics of intuitionists is vague.The first rigorous semantics of intuitionistic logic was S. C. Kleene’s realizability.[14] According to S. C. Kleene a formula ∀x∃yA(x,y) is true if there is an algorithm that for each x constructs y such that A(x,y). In S. C. Kleene’s realizability the reduced formula is not simpler than the initial one. N. A. Shanin’s procedure (algorithm) of `elicitation of constructive problem'[15] reduces the initial formula to a formula of the form ∃x1...∃xkF , where F contains neither ∃ nor ∨, and due to embedding operations it suffices to prove this formula F in classical logic. This N. A. Shanin’s procedure facilitated communications between the Russian constructivist school and constructivists in the West, in particular, intuitionists. S. C. Kleene observed[16] that in purely logical terms this algorithm is a consequence of just two principles: Markov's principle and a variant of Church's thesis. Further development of these ideas gave a `finitary' approach (in the sense of Hilbert) to constructive mathematics.[17]

Based on the constructive semantics, N. A. Shanin started, from the middle of the 50ies, a revision of classical mathematics, in fact of calculus and functional analysis, from constructivist viewpoint.[18][19] A priori it is not evident what notion of computable real number is productive. N. A. Shanin defined constructive real number as 'duplex' where rational approximations and the rate of convergence are algorithms, and showed that it works well. Similar algorithmic approaches to reals were developed in the West[20] (see computable number).

In 1961 N. A. Shanin organized a group of logic (in Russian of ‘mathematical logic’) in Leningrad Department of Steklov Mathematical Institute of the Academy of Sciences of USSR. The first goal was to develop and implement an algorithm of automatic theorem proving, and first of all, for classical propositional calculus. The three first members of the group were: Gennady Davydov (1939–2016), Sergey Maslov (1939–1982), Grigory Mints (1939–2014). More people were recruited next years (including V.P.Orevkov [1], A.O.Slissenko, Yu.V.Matiyasevich). These years one could see a worldwide enthusiasm concerning automatic theorem proving, in particular, in pure logic. Starting from Gentzen sequent calculus N.A.Shanin developed a proof search algorithm that outputs a `natural' human friendly proof. He placed emphasis on the use of heuristics and on obtaining a result in the form of a natural deduction. The algorithm was implemented[21] and showed a very good performance.

N. A. Shanin was a dynamic and vigorous professor who, in particular, very well explained the most basic concepts of logic that do not have mathematical definitions via simpler notions (like integer – recall Leopolde Kronecker's joke about integers that can be translated in English as "God made the integers, all else is the work of man"). His analysis of various semantical issues had a considerable influence on philosophers.[22]

Further reading

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  • Vsemirnov, M A (2001), "Nikolai Aleksandrovich Shanin (on his 80th birthday)", Russian Math. Surveys, 56 (3): 601–605, Bibcode:2001RuMaS..56..601V, doi:10.1070/RM2001v056n03ABEH000412
  • Vsemirnov, M A; et al. (2013), "Nikolai Aleksandrovich Shanin (obituary)", Russ. Math. Surv., 68 (4): 763–767, Bibcode:2013RuMaS..68..763V, doi:10.1070/RM2013v068n04ABEH004852

References

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  1. ^ Vsemirnov, M. A.; Hirsch, È. A.; Grigor'Ev, D. Yu; Davydov, G. V.; Dantsin, E. Ya; Zaslavskii, I. D.; Karavaev, E. F.; Konev, B. Yu; Kossovskii, N. K.; Lifschitz, V. A.; Margenstern, M.; Matiyasevich, Yu V.; Mints, G. E.; Orevkov, V. P.; Pliuškevičius, R.; Slisenko, A. O.; Solov'Ev, S. V.; Chernov, V. P. (2013). "Nikolai Aleksandrovich Shanin (obituary)". Russian Math. Surveys. 68 (4): 763–767. Bibcode:2013RuMaS..68..763V. doi:10.1070/RM2013v068n04ABEH004852.
  2. ^ "К 95-летию НМИЦ онкологии им. Н.Н. Петрова: их имена звучат по-особому". НМИЦ онкологии им. Н.Н. Петрова (in Russian). Retrieved 2025-01-28.
  3. ^ https://www.pdmi.ras.ru/pdmi/history
  4. ^ "Shanin N. A. * Publication List *". logic.pdmi.ras.ru. Retrieved 2025-01-28.
  5. ^ Shanin, Nikolai. "On the product of topological spaces (in Russian)". Trudy Mat. Inst. Steklov, Academy of Sciencies of the USSR. 24 (1948): 1–112.
  6. ^ Malykhin, V.; Ponomarev, V. "General topology (set-theoretic trend)". Journal of Soviet Mathematics, 7:4 (1977). 7 (1977) (4): 587–629.
  7. ^ Matiyasevich, Yuri (1993). Hilbert's 10th Problem. The MIT Press. pp. 288 pp. ISBN 9780262132954.
  8. ^ A. Slissenko. St.Petersburg/Leningrad (1961-1998): From Logic to Complexity and Further, In: "People and Ideas In Theoretical Computer Science", Springer Verlag, 1998, 274-313. ISBN 981-4021-13-X
  9. ^ Matiyasevich, Yuri. "Computational rediscovery of Ramanujan's tau numbers". Integers. Electronic Journal of Combinatorial Number Theory. 18 (2018) (A): 1–8.
  10. ^ Matiyasevich, Yuri. "In search of approximate polynomial dependencies among the derivatives of the alternating zeta function. To appear". Journal of Experimental Mathematics.
  11. ^ Kolmogoroff, A. "Zur Deutung der intuitionistischen Logik". Math. Z. 35 (1932): 58–65.
  12. ^ Gödel, K. "Eine Interpretation des intuitionischen Aussagenkalküls". Ergebnisse Math. Colloq. 4 (1933): 39–40.
  13. ^ a b Shanin, N.A. "On some logical problems of arithmetics. (in Russian)". Trudy Mat. Inst. Steklov, Academy of Sciencies of the USSR. 43(1955): 1–112.
  14. ^ Kleene, S.C. "On the interpretation of intuitionistic number theory". J. Symbolic Logic. 10 (1945) (4): 109–124.
  15. ^ Shanin, N. A. "On constructive understanding of mathematical judgements. (in Russian)". Trudy Mat. Inst. Steklov, Academy of Sciencies of the USSR. 52 (1958): 226–311.
  16. ^ Kleene, S. C. "Realizability and Shanin's algorithm for the constructive deciphering of mathematical sentences". Logique et Analyse (Nouvelle Serie). 3 (1960): 154–165.
  17. ^ Shanin, N. A. "On a hierarchy of ways of understanding judgments in constructive mathematics (In Russian). English translation appeared in 1976". Proc. Steklov Inst. Math. 129 (1973): 209–271.
  18. ^ Shanin, N. A. "Constructive real numbers and constructive function spaces. (in Russian)". Tr. Mat. Inst. Steklov. 67 (1962): 15–294.
  19. ^ Shanin, N. A. (1968). Constructive real numbers and constructive function spaces. Translations of Mathematical Monographs. 21. Amer. Math. Soc. pp. 325 p.p.
  20. ^ Bishop, E.; Bridges, Douglas S. (1985). Constructive Analysis. Springer-Verlag.
  21. ^ N. Shanin, G. Davydov, S. Maslov, G. Mints, V. Orevkov, and A. Slisenko (Slissenko). An algorithm for machine search of a natural logical deduction in a propositional calculus. In J.Siekmann and G.Wrightson, editors, The Automation of Reasoning I. Classical Papers on Computational Logic 1957–1966, pages 424–483. Springer-Verlag, 1983 (Russian original was published by Publishing House Nauka, Leningrad, 1965, 39p.).
  22. ^ Николай Александрович Шанин. Некролог. Логико-философские штудии. 9:1 (2011). ISSN: 2223-3954.
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