Mathematical objects, structures and proofs (introduction to the special issue)

Philosophy "Mathematical objects, structures and proofs" (thematic issue)

Authors

  • Lev D. Lamberov Ural Federal University named after the first President of Russia B.N. Yeltsin, 51, Lenin av., Ekaterinburg, 620002, Russia

DOI:

https://doi.org/10.17072/2078-7898/2022-3-361-367

Keywords:

mathematical objects, structures, proofs, subject of mathematics, philosophy of mathematics

Abstract

The paper serves as an introduction to the issues discussed in the following articles. It raises the problem (challenge) of integration, according to which an adequate solution of a philosophical problem should simultaneously be an answer to both ontological and epistemological questions. This problem is described speculatively and by referring to P. Benacerraf’s dilemma. In addition, the problem is illustrated by comparing classical and intuitionistic mathematics and also through interpretation of the concept of computer proof. The paper demonstrates that adequate philosophy of mathematics must simultaneously take into account the ontological and epistemological aspects of mathematics and mathematical practice.

Author Biography

Lev D. Lamberov, Ural Federal University named after the first President of Russia B.N. Yeltsin, 51, Lenin av., Ekaterinburg, 620002, Russia

Candidate of Philosophy, Docent,Associate Professor of the Departmentof Ontology and Theory of Knowledge

References

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References

Appel, K. and Haken, W. (1977). Every planar map is four colorable. Part I: Discharging. Illinois Journal of Mathematics. Vol. 21, iss. 3, pp. 429–490. DOI: https://doi.org/10.1215/ijm/1256049011

Appel, K., Haken, W. and Koch, J. (1977). Every planar map is four colorable. Part II: Reducibility. Il linois Journal of Mathematics. Vol. 21, iss. 3, pp. 491–567. DOI: https://doi.org/10.1215/ ijm/1256049012

Atten, M. van (2004). On Brouwer. Belmont, CA: Wadsworth/Thomson Learning Publ., 96 p.

Atten, M. van (2007). Brouwer meets Husserl: On the phenomenology of choice sequences. Dordrecht, NL: Springer Publ., 219 p. DOI: https://doi.org/ 10.1007/978-1-4020-5087-9

Benacerraf, P. (1973). Mathematical truth. The Journal of Philosophy. Vol. 70, iss. 19, pp. 661–679. DOI: https://doi.org/10.2307/2025075

Brouwer, L.E.J. (auth.), D. van Dalen (ed.) (1981). Brouwer’s Cambridge lectures on intuition ism. Cambridge, UK: Cambridge University Press, 122 p.

Field, H. (1989). Realism, mathematics and mo dality. Oxford, UK: Basil Blackwell Publ., 304 p.

Goldman, A.I. (1967). A causal theory of know ing. The Journal of Philosophy. Vol. 64, iss. 12, pp. 357–372. DOI: https://doi.org/10.2307/2024268

Goldman, A.I. (1975). Innate knowledge. S.P. Stich (ed.) Innate ideas. Berkeley, CA: Universi ty of California Press, pp. 111–120.

Heyting, A. (1956). Intuitionism. An introduction. Amsterdam, NL: North-Holland Publishing Compa ny, 147 p.

Linnebo, Ø. (2006). Epistemological challenges to mathematical Platonism. Philosophical Studies. Vol. 129, iss. 3, pp. 545–574. DOI: https://doi.org/10.1007/s11098-004-3388-1

Markov, A.A. (1954). Teoriya algorifmov [The theory of algorithms]. Moscow, Leningrad: AS USSR Publ., 376 p.

Markov, A.A. (1972). O logike konstruktivnoy matematiki [On the logic of constructive mathemat ics]. Moscow: Znanie Publ., 47 p.

Peacocke, C. (1999). Being known. New York: Oxford University Press, 368 p. DOI: https://doi.org/10.1093/0198238606.001.0001

Quine, W.V.O. (1969). Epistemology naturalized. Ontological relativity and other essays. New York: Columbia University Press, pp. 69–90. DOI: https://doi.org/10.7312/quin92204-004

Shanker, S. (1987). Wittgenstein and the turning point in the philosophy of mathematics. London, UK: Croom Helm Publ., 370 p. DOI: https://doi.org/ 10.4324/9781315823492 Tarski, A. (1935). [The concept of truth in formal ized languages]. Studia Philosophica. Vol. 1, pp. 261–405.

Tarski, A. (1944). The semantical concept of truth and the foundations of semantics. Philosophy and Phenomenological Research. Vol. 4, no. 3, pp. 341– 375. DOI: https://doi.org/10.2307/2102968

Teller, P. (1980). Computer proof. The Journal of Philosophy. Vol. 77, iss. 12, pp. 797–803. DOI: https://doi.org/10.2307/2025805

Tymoczko, T. (1979). The four-color theorem and its philosophical significance. The Journal of Philos ophy. Vol. 76, iss. 2, pp. 57–83. DOI: https://doi.org/10.2307/2025976

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2022-09-29

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