The Notion of Mathematical Proof and the Problem of Explanation in Mathematics

Main Article Content

Krzysztof Wójtowicz

Abstract

In the article, I present two possible points of view concerning mathematical proofs: (a) the formal view (according to which the formalized versions of mathematical proofs reveal their “essence”); (b) the semantic view (according to which mathematical proofs are sequences of intellectual acts, and a form of intuitive “grasp” is crucial). The problem of formalizability of mathematical proofs is discussed, as well as the problem of explanation in mathematics – in particular the problem of explanatory versus non-explanatory character of mathematical proofs. I argue, that this problem can be analyzed in a fruitful way only from the semantic point of view.

Article Details

How to Cite
Wójtowicz, K. (2015). The Notion of Mathematical Proof and the Problem of Explanation in Mathematics. Philosophical Problems in Science (Zagadnienia Filozoficzne W Nauce), (58), 89–114. Retrieved from https://zfn.edu.pl/index.php/zfn/article/view/9
Section
Articles

References

Azzouni J., The derivation-indicator view of mathematical practice, „Philosophia Mathematica” 2004, 3 (12), s. 81–105.

Barwise J., Mathematical proofs of computer system correctness, „Notices of the American Mathematical Society” 1989, 36, s. 844– 851.

Detlefsen M., Formalism, [w:] The Oxford Handbook of Philosophy of Mathematics and Logic, (red.) S. Shapiro S., Oxford University Press, Oxford 2005, s. 236–317.

Gödel K., Russell’s Mathematical Logic, [w:] The philosophy of Bertrand Russell. Library of Living philosophers, vol. 5, (red.) P.A. Schlipp, Open Court Publishing Company, La Salle, Ill. 1944, s. 123–153. Polskie tłumaczenie [w:] Współczesna filozofia matematyki. Wybór tekstów, (red.) R. Murawski, Wydawnictwa Naukowe PWN, Warszawa 2002, s. 77–102.

Hahn H., Empiricism, Logic and Mathematics, D. Reidel, Dordrecht – London – Boston 1980.

Hilbert D., Über das Unendliche, „Mathematische Annalen” 1926, 95, s. 161–190. Tłumaczenie polskie [w:] Filozofia matematyki. Antologia tekstów klasycznych, (red.) R. Murawski, Wydawnictwa UAM, Poznań, s. 288–307.

Hilbert D., Die Grundlagen der Mathematik, „Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität” 1928, 6, s. 65–85. Angielskie tłumaczenie [w:] From Frege to Gödel: A Sourcebook in Mathematical Logic, 1879–1931, (red.) J. Van Heijenoort, Harvard University Press, Cambridge, Mass. 2002, s. 464–479.

Kartezjusz, [1958] Prawidła kierowania umysłem; poszukiwanie prawdy poprzez światło przyrodzone rozumu, tłum. L. Chmaj, PWN, Warszawa 1958.

Mancosu P., Mathematical explanation: problems and prospects, „Topoi” 2001, 20, s. 97–117.
Pasch M., Vorlesungen über neuere Geometrie, Teubner, Leipzig 1882.
Pringsheim A., Vorlesungen über Zahlen- und Funktionenlehre, Zweiter Band, Erste Abteilung: Grundlagen der Theorie der analytischen Funktionen einer komplexen Veränderlichen, B.G. Teubner, Leipzig – Berlin 1925.

Rav Y., Why do we prove theorems?, „Philosophia Mathematica” 1999, 7, s. 5–41.
Rav Y., A critique of a formalist-mechanist version of the justification of arguments in mathematicians’ proof practices, „Philosophia Mathematica” 2007, 15, s. 291–320.

Resnik M.D., Kushner D., Explanation, independence and realism in mathematics, „British Journal for the Philosophy of Science” 1987, 38 (2), s. 141–158.

Russell B.A.W., Logical Atomism, [w:] Contemporary British Philosophy, (red.) J. M. Muirhead, George Allen & Unwin, London 1924, s. 357–383. Przedrukowane [w:] Logic and Knowledge, (red.) R.C. Marsh, London, George Allen & Unwin, s. 323–343.

Russell B.A.W., Essays in Analysis, (red.) D. Lackey, George Allen & Unwin, London 1973.

Steiner M., Mathematics, explanation and scientific knowledge, „Nous” 1978, 12, s. 17–28.

Wójtowicz K., O pojęciu dowodu w matematyce, seria Monografie Fundacji Na Rzecz Nauki Polskiej, Wydawnictwo Naukowe UMK, Toruń 2012, s. 250.