The kinds of mathematical quasi-empirism
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Abstract
The received view concerning mathematics is the one, that mathematics is a priori, and that mathematical knowledge develops via 'intelektuelle Anschauung' rather than by analyzing empirical data. Mathematical proofs seems to be immune to empirical refutation, and in particular the development of mathematics does not in any way resemble the development of e.g. physics. On the other hand, it is quite clear, that mathematics play a fundamental role in science, and it is often considered to be rather just a useful tool, which provides a language and a conceptual system allowing to express statements concerning empirical world. Such views stress the dependence of mathematics upon physics. In the article, the author presents two quite different aspects of this problem: the ontological and the methodological aspects. According to Quine, our argumentation in favor of mathematical realism should be based on the analysis of ontological commitment of empirical theories. There is no other compelling argument for mathematical realism. According to Lakatos, mathematical knowledge develops in a way similar to empirical science: it is fallible, and the proper model to describe it is the model of proofs and refutations. In the article the author describes and contrast these two points of view.
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Davis P.J., Hersh R., [1994] Świat matematyki, Warszawa, WNT.
Dieudonne J., [1970] “The Work of Nicholas Bourbaki”, American Mathematical Monthly 77, 134–145.
Gödel K., [1947/64] “What is Cantor’s Continuum Problem?”, American Mathematical Monthly 54, 515–525; [w rozszerzonej wersji przedrukowane w:] Philosophy of Mathematics, [eds.:] Benacerraf P., Putnam H., Prentice-Hall, Englewood Cliffs, New Jersey, 1964, 258–273; [a także w:] Collected Works, vol. 2, [eds.:] Feferman S. et al., Oxford University Press, 1990, 254–270.
Gödel K., [*1951] “Some Basic Theorems on the Foundations of Mathematics and Their Implications”, [in:] Collected Works, vol. 3, [eds.:] Feferman S. et al., Oxford University Press, 1995, 304–323.
Hammond A.L., [1983] „Matematyka — nasza niedostrzegalna kultura”, [w:] Matematyka współczesna. Dwanaście esejów, [red.:] Steen L.A., WNT, Warszawa, 26–48.
Hardy G.H., [1929] “Mathematical Proof”, Mind 38, 1–25.
Lakatos I., [1976] Proofs and Refutations. The Logic of Mathematical Discovery, Cambridge University Press, Cambridge; [pol. tłum. na podst. wyd. z 1999:] Dowody i refutacje. Logika odkrycia matematycznego, [red.:] Worral J., Zahar E., [tłum.:] Kozłowski M., Lipszyc K., Tikkun, Warszawa, 2005.
Lakatos I., [1978] “A Renaissance of Empiricism in the Recent Philosophy of Mathematics?”, [in:] Philosophical Papers, t. 2, Mathematics, Science and Epistemology, [eds.:] Worall J., Currie G., Cambridge University Press, 24–42; [pol. tłum.:] „Renesans empiryzmu we współczesnej filozofii matematyki?”, [w:] Współczesna filozofia matematyki, [red.:] Murawski R., PWN, Warszawa 2002, 215–243.
Mill J.St., [1962] System logiki dedukcyjnej i indukcyjnej, t. II, PWN, Warszawa.
Putnam H., [1975] “What is Mathematical Truth?”, [in:] Mathematics, Matter and Method: Philosophical Papers, t. 1, Cambridge University Press, 60–78; [tłum. pol.:] „Czym jest prawda matematyczna?”, [w:] Współczesna filozofia matematyki, [red.:] Murawski R., PWN, Warszawa 2002, 244–265.
Quine W.V.O., [1953a] “On What There Is”, [in:] From a Logical Point of View, Cambridge, Harvard University Press, 1–19; [tłum. pol.:] „O tym, co istnieje”, [w:] Z punktu widzenia logiki, [red.:] Stanosz B., PWN, Warszawa 1969, 9–34.
Quine W.V.O., [1953b] “Two Dogmas of Empiricism”, [in:] From a Logical Point of View, Cambridge, Harvard University Press, 20–46; [tłum. Pol.:] „Dwa dogmaty empiryzmu”, [w:] Z punktu widzenia logiki, PWN, Warszawa 1969, 35–70.
Quine W.V.O., [1981] “Things and Their Place in Theories”, [in:] Theories and Things, The Belknap Press of Harvard University Press, Cambridge, MA, 1–23; [tłum. pol.:] „Rzeczy i ich miejsca w teoriach”, [w:] Metafizyka w filozofii analitycznej, [red.:] Szubka T., Lublin, TN KUL, 1995, 31–52.
Quine W.V.O., [1984] “Review of Parsons C. Mathematics in Philosophy”, Journal of Philosophy 81, 783–794.
Wójtowicz K., [2007] „Filozofia matematyki Imre Lakatosa”, Roczniki Filozoficzne LV, (1), 229–247.