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2020 | 9(3/4) | 10--18
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Proof vs Truth in Mathematics

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EN
Abstrakty
EN
Two crucial concepts of the methodology and philosophy of mathematics are considered: proof and truth. We distinguish between informal proofs constructed by mathematicians in their research practice and formal proofs as defined in the foundations of mathematics (in metamathematics). Their role, features and interconnections are discussed. They are confronted with the concept of truth in mathematics. Relations between proofs and truth are analysed. (original abstract)
Słowa kluczowe
Czasopismo
Rocznik
Tom
Strony
10--18
Opis fizyczny
Twórcy
  • Adam Mickiewicz University in Poznań, Poland
Bibliografia
  • Aschbacher, M. Highly complex proofs and implications of such proofs. Philosophical Transactions of the Royal Society (A) 363, 2005, pp. 2401-2406.
  • Avigad, J. Mathematical method and proof, Synthese 153, 2006, pp. 105-159.
  • Barwise, J. An introduction to first-order logic, In J. Barwise (ed.), Handbook of Mathematical Logic, Amsterdam: North-Holland, 1977, pp. 5-46.
  • CadwalladerOlsker, T. What do we mean by mathematical proof? Journal of Humanistic Mathematics 1, 2011, pp. 33-60.
  • Detlefsen, M. Poincare against the logicians, Synthese 90 (3), 1992, pp. 349-378.
  • Detlefsen, M. (ed.). Proof, Logic and Formalization, London: Routledge, 1992,
  • De Villiers, M. D. Rethinking Proof with the Geometer's Sketchpad, Emeryville, CA: Key Curriculum Press, 1999.
  • Godel, K. Uber formal unentscheidbare Satze der Principia Mathematica und verwandter Systeme. I, Monatshefte fur Mathematik und Physik 38, 1931, pp. 173-198. Reprinted with English translation: On formally undecidable propositions of Principia Mathematica and related systems, In Godel K. Collected Works, vol. I: Publications 1929-1936, S. Feferman, J. W. Dawson, Jr., S. C. Kleene, G. H. Moore, R. M. Solovay and J. van Heijenoort eds.), New York: Oxford University Press, and Oxford: Clarendon Press, pp. 144-195.
  • Godel K. Collected Works, vol. I: Publications 1929-1936, S. Feferman, J. W. Dawson, Jr., S. C. Kleene, G. H. Moore, R. M. Solovay and J. van Heijenoort eds.), New York: Oxford University Press, and Oxford: Clarendon Press.
  • Hamami, Y. Mathematical inference and logical inference, The Review of Symbolic Logic 11 (4), 2019, pp. 665-704.
  • Kahle, R. Is there a "Hilbert thesis"? Studia Logica 107, 2019, pp. 145-165.
  • Kaye, R. Models of Peano Arithmetic, Oxford: Clarendon Press, 1991.
  • Kotlarski, H., and Z. Ratajczyk. Inductive full saisfaction classes, Annals of Pure and Applied Logic 47, 1990, pp. 199-223.
  • Kotlarski, H., and Z. Ratajczyk. More on induction in the language with a full satisfaction class, Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik 36, 1990, pp. 441-454.
  • Krajewski, S. Non-standard satisfaction classes, In W. Marek, M. Srebrny and A. Zarach (eds.), Set Theory and Hierarchy Theory, Proc. Bierutowice Conf. 1975, Lecture Notes in Mathematics 537, Berlin-Heidelberg-New York: Springer Verlag, 1976, pp. 121-144.
  • Kreisel, G. The formalist-positivist doctrine of mathematical precision in the light of experience, L 'Age de la Science 3, 1970, pp. 17-46.
  • Murawski R. Satisfaction classes - a survey, In R. Murawski and J. Pogonowski (eds.), Euphony and Logos, Amsterdam/Ątlanta, GA: Edition Rodopi, 1997, pp. 259-281.
  • Murawski, R. Recursive Functions and Metamathematics. Problems of Completeness and Decidability, Godel's Theorems, Dordrecht/Boston/London: Kluwer Academic Publishers, 1999.
  • Murawski, R. Truth vs. provability - philosophical and historical remarks. Logic and Logical Philosophy 10, 2002, pp. 93-117.
  • Murawski, R. On the distinction proof-truth in mathematics, In P. Gardenfors et al. (eds.), In the Scope of Logic, Methodology and Philosophy of Science, Dordrecht-Boston-London: Kluwer Academic, 2002, pp. 287-303.
  • Murawski, R. Troubles with (the concept of) truth in mathematics, Logic and Logical Philosophy 15, 2006, pp. 285-303. Reprinted in: R. Murawski, Lógos andMathema. Studies in the Philosophy of Mathematics and History of Logic, Frankfurt am Main: Peter Lang International Verlag der Wissenschaften, 2011, pp. 187-201.
  • Murawski, R. Some historical, philosophical and methodological remarks on proof in mathematics, In D. Probst and P. Schuster (Eds.), Concepts of Proof in Mathematics, Philosophy, and Computer Science, Ontos Mathematical Logic, Berlin: Walter de Gruyter, 2016, pp. 251-268.
  • Rota, G.-C. The phenomenology of mathematical proof, Synthese 111, 1997, pp. 183196.
  • Tarski, A. Pojęcie prawdy w językach nauk dedukcyjnych, Warszawa: Towarzystwo Naukowe Warszawskie, 1933, Wydział III Nauk Matematyczno-Fizycznych, vol. 34. Reprinted in: A. Tarski, Pisma logiczno-filozoficzne, vol. 1: Prawda, Warszawa: Wydawnictwo Naukowe PWN, 1995, pp. 131-172. English translation: The concept of truth in formalized languages, In A. Tarski, Logic, Semantics, Metamathematics. Papers from 1923 to 1938, Oxford: Clarendon Press, 1956, pp. 152-278 and in A. Tarski, Logic, Semantics, Metamathematics. Papers from 1923 to 1938, second edition edited and introduced by J. Corcoran, Indianapolis: Hackett Publishing Co., 1983, pp. 152-283.
  • Tarski, A. On the concept of following logically, History and Philosophy of Logic 23, 1936/2002, pp. 155-196.
  • Tarski, A. Logic, Semantics, Metamathematics. Papers from 1923 to 1938, Oxford: Clarendon Press, 1956.
  • Tarski, A. Truth and proof, Scientific American 220, No. 6, 1969, pp. 63-77.
  • Tarski, A. Logic, Semantics, Metamathematics. Papers from 1923 to 1938, second edition edited and introduced by J. Corcoran, Indianapolis: Hackett Publishing Co., 1983.
  • Wang, H. From Mathematics to Philosophy, London: Routledge and Kegan Paul, 1974.
  • Wang, H. Reflections on Kurt Godel, Cambridge, Mass: The MIT Press, 1987.
  • Woleński, J. Semantics and Truth, Logic, Epistemology and the Unity of Science 45, Berlin: Springer Verlag, 2019.
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Bibliografia
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