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2020 | 9(3/4) | 154--164
Tytuł artykułu

Anti-foundationalist Philosophy of Mathematics and Mathematical Proofs

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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The Euclidean ideal of mathematics as well as all the foundational schools in the philosophy of mathematics have been contested by the new approach, called the "maverick" trend in the philosophy of mathematics. Several points made by its main representatives are mentioned - from the revisability of actual proofs to the stress on real mathematical practice as opposed to its idealized reconstruction. Main features of real proofs are then mentioned; for example, whether they are convincing, understandable, and/or explanatory. Therefore, the new approach questions Hilbert's Thesis, according to which a correct mathematical proof is in principle reducible to a formal proof, based on explicit axioms and logic. (original abstract)
Słowa kluczowe
Czasopismo
Rocznik
Tom
Strony
154--164
Opis fizyczny
Twórcy
  • University of Warsaw, Poland
Bibliografia
  • Asprey, W., and P. Kitcher (eds.). History and philosophy of modern mathematics, Minneapolis: University of Minnesota Press, 1988.
  • Brown, J. R. Philosophy of Mathematics. A Contemporary Introduction to the World of Proofs and Pictures, New York and London: Routledge, 1999, 2008.
  • Byers, W. How Mathematicians Think; Using Ambiguity, Contradiction, and Paradox to Create Mathematics, Princeton: Princeton University Press, 2007.
  • Cellucci, C. Why Proof? What is a Proof? In G. Corsi and R. Lupacchini (eds.), Deduction, Computation, Experiment. Exploring the Effectiveness of Proof, Berlin: Springer-Verlag, 2008, pp. 1-27.
  • Cellucci, Carlo. Rethinking logic: Logic in relation to mathematics, evolution, and method, Dordrecht: Springer 2013.
  • Cellucci, C. Rethinking knowledge: The heuristic view, Dordrecht: Springer, 2017.
  • Davis, P. J., and R. Hersh. The Mathematical Experience, Boston, Basel, Berlin: Birkhauser, 1981, 19952.
  • Dehaene, S. The Number Sense: How the Mind Creates Mathematics, Oxford: Oxford University Press, 1997.
  • Enayat, A, and A. Visser. New constructions of satisfaction classes. Unifying the philosophy of truth, pp. 321-335, Logic, Epistemology, and the Unity of Science 36, Dordrecht: Springer, 2015.
  • Epstein, R. L. Five Ways of Saying "Therefore" Arguments, Proofs, Conditionals, Cause and Effect, Explanations, Belmont, CA: Wadsworth/Thomson Learning, 2002.
  • Ernest, P. Social Constructivism as a Philosophy of Mathematics, Albany, NY: SUNY Press, 1998.
  • Friend, M. Pluralism in Mathematics: A New Position in Philosophy of Mathematics, Dodrecht: Springer, 2014.
  • Giaquinto, M. Visualising in Mathematics, In P. Mancosu (ed.), The Philosophy of Mathematical Practice, Oxford: Oxford University Press, 2008, pp. 22-42.
  • Hardy, G. H. Mathematical Proof, Mind 38 (1928), pp. 1-25.
  • Hersh, R. Some proposals for reviving the philosophy of mathematics, Advances in Mathematics 31, 31-50, reprinted in T. Tymoczko (ed.), New Directions in the Philosophy of Mathematics, An Anlology, Basel: Birkhauser, 1986.
  • Hersh, R. Math Has a Front and a Back, Eureka 1988, and Synthese 88 (2), 1991.
  • Hersh, R. Proving is convincing and explaining, Educational Studies in Mathematics 24, 1993, pp. 389-399.
  • Hersh, R. What Is Mathematics, Really? Oxford: Oxford University Press, 1997.
  • Hersh, R. (ed.). 18 Unconventional Essays on the Nature of Mathematics, New York: Springer, 2006.
  • Hersh, R. Experiencing mathematics: What do we do, when we do mathematics? Providence: American Mathematical Society, 2014.
  • Kitcher, P. The Nature of Mathematical Knowledge, Oxford: Oxford University Press, 1985.
  • Kotlarski, H., S. Krajewski, and A. H. Lachlan. Construction of satisfaction classes for nonstandard models, Canadian Math. Bulletin 24, 1981, pp. 283-293.
  • Krajewski, S. Remarks on Mathematical Explanation, In B. Brozek, M. Heller and M. Hohol (eds.), The Concept of Explanation, Kraków: Copernicus Center Press, 2015, pp. 89-104.
  • Krajewski, S. On Suprasubjective Existence in Mathematics, Studia semiotyczne XXXII (No 2), 2018, pp. 75-86.
  • Lakatos, I. Proofs and refutations; The logic of mathematical discovery, edited by J. Worral and E. Zahar, Cambridge: Cambridge University Press 1976, 20152. (Original papers in British Journal for the Philosophy of Science 1963-64.)
  • Lakatos, I. A renaissance of empiricism in the recent philosophy of mathematics? In I. Lakatos, Philosophical Papers, vol. 2, edited by J. Worrall and G. Curie, Cambridge: Cambridge University Press, 1978, pp. 24-42.
  • Lakoff, G., and R. E. Nunez. Where Mathematics Comes From. How the Embodied Mind Brings Mathematics into Being, New York: Basic Books, 2000.
  • Mancosu, P (ed.). The Philosophy of Mathematical Practice, Oxford: Oxford University Press, 2008.
  • Polya, G. Mathematics and Plausible Reasoning, vol I, Princeton: Princeton University Press, 1973.
  • Putnam, H. What is Mathematical Truth? In Philosophical Papers, vol. 1, Mathematics, Matter and Method, Cambridge: Cambridge University Press, 1975, pp. 60-78.
  • Rav, Y. Philosophical Problems of Mathematics in the Light of Evolutionary Epistemology, Philosophica 43, No. 1, 1989, pp. 49-78; also Chapter 5 in Hersh, R (ed.). 18 Unconventional Essays on the Nature of Mathematics, New York: Springer, 2006.
  • Rav, Y. Why Do We Prove Theorems? Philosophia Mathematica (3) 7, 1999, pp. 5-41.
  • Rav, Y. The Axiomatic Method in Theory and in Practice, Logique & Analyse 202, 2008, pp. 125-147.
  • Rotman, B. Toward a Semiotics of Mathematics, Semiotica 72, 1988, pp. 1- 35.
  • Rotman, B. Ad Infinitum ... the Ghost ln Turing's Machine: Taking God Out Of Mathematics And Putting the Body Back In, Stanford, CA: Stanford University Press, 1993.
  • Rotman, B. Mathematics as Sign: Writing, Imagining, Counting, Stanford, CA: Stanford University Press, 2000.
  • Sriraman, B. (ed.). Humanizing Mathematics and its Philosophy. Essays Celebrating the 90th Birthday of Reuben Hersh, Basel: Birkhauser 2017.
  • Schmerl, J. Kernels, Truth and Satisfaction, to be published.
  • Tymoczko, T. The four-color problem and its philosophical significance, The Journal of Philosophy 76, 1979, pp. 57-83.
  • Tymoczko, T. (ed.), New Directions in the Philosophy of Mathematics, An Antology, Basel: Birkhauser, 1986.
  • Wagner, R. Making and Breaking Mathematical Sense: Histories and Philosophies of Mathematical Practice, Princeton: Princeton University Press, 2017.
  • White, L. The Locus of Mathematical Reality: An Anthropological Footnote, Philosophy of Science 14, 1947, pp. 289-303, reprinted in Hersh, R (ed.). 18 Unconventional Essays on the Nature of Mathematics, New York: Springer, 2006, pp. 304-319.
  • Wilder, R. L. The Cultural Basis of Mathematics, in Proceedings of the International Congress of Mathematicians, AMS 1952, pp. 258-271.
  • Wilder, R. L. Mathematics as a cultural system, Oxford: Pergamon Press, 1981.
Typ dokumentu
Bibliografia
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