Proof vs Truth in Mathematics

Roman Murawski

doi:10.2478/sh-2020-0025

.blurhash-client-img { display: none !important; }

Proof vs Truth in Mathematics

Studia Humana

Volume 9 (2020): Issue 3-4 (October 2020)

By: Roman Murawski

Open Access

|Nov 2020

Abstract

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.

References

1. Aschbacher, M. Highly complex proofs and implications of such proofs. Philosophical Transactions of the Royal Society (A) 363, 2005, pp. 2401–2406.10.1098/rsta.2005.1655
Search in Google Scholar Back to article
2. Avigad, J. Mathematical method and proof, Synthese 153, 2006, pp. 105–159.10.1007/s11229-005-4064-5
Search in Google Scholar Back to article
3. Barwise, J. An introduction to first-order logic, In J. Barwise (ed.), Handbook of Mathematical Logic, Amsterdam: North-Holland, 1977, pp. 5–46.10.1016/S0049-237X(08)71097-8
Search in Google Scholar Back to article
4. CadwalladerOlsker, T. What do we mean by mathematical proof? Journal of Humanistic Mathematics 1, 2011, pp. 33–60.10.5642/jhummath.201101.04
Search in Google Scholar Back to article
5. Detlefsen, M. Poincaré against the logicians, Synthese 90 (3), 1992, pp. 349–378.10.1007/BF00500033
Search in Google Scholar Back to article
6. Detlefsen, M. (ed.). Proof, Logic and Formalization, London: Routledge, 1992,
Search in Google Scholar Back to article
7. De Villiers, M. D. Rethinking Proof with the Geometer’s Sketchpad, Emeryville, CA: Key Curriculum Press, 1999.
Search in Google Scholar Back to article
8. Gödel, K. Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. I, Monatshefte für Mathematik und Physik 38, 1931, pp. 173-198. Reprinted with English translation: On formally undecidable propositions of Principia Mathematica and related systems, In Gödel 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. 144–195.10.1007/BF01700692
Search in Google Scholar Back to article
9. Gödel 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.
Search in Google Scholar Back to article
10. Hamami, Y. Mathematical inference and logical inference, The Review of Symbolic Logic 11 (4), 2019, pp. 665–704.10.1017/S1755020317000326
Search in Google Scholar Back to article
11. Kahle, R. Is there a “Hilbert thesis”? Studia Logica 107, 2019, pp. 145–165.10.1007/s11225-017-9776-2
Search in Google Scholar Back to article
12. Kaye, R. Models of Peano Arithmetic, Oxford: Clarendon Press, 1991.10.1093/oso/9780198532132.001.0001
Search in Google Scholar Back to article
13. Kotlarski, H., and Z. Ratajczyk. Inductive full saisfaction classes, Annals of Pure and Applied Logic 47, 1990, pp. 199-223.10.1016/0168-0072(90)90035-Z
Search in Google Scholar Back to article
14. Kotlarski, H., and Z. Ratajczyk. More on induction in the language with a full satisfaction class, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 36, 1990, pp. 441-454.10.1002/malq.19900360509
Search in Google Scholar Back to article
15. 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.
Search in Google Scholar Back to article
16. Kreisel, G. The formalist-positivist doctrine of mathematical precision in the light of experience, L’Âge de la Science 3, 1970, pp. 17–46.
Search in Google Scholar Back to article
17. Murawski R. Satisfaction classes – a survey, In R. Murawski and J. Pogonowski (eds.), Euphony and Logos, Amsterdam/Ątlanta, GA: Edition Rodopi, 1997, pp. 259–281.10.1163/9789004457560
Search in Google Scholar Back to article
18. Murawski, R. Recursive Functions and Metamathematics. Problems of Completeness and Decidability, Gödel’s Theorems, Dordrecht/Boston/London: Kluwer Academic Publishers, 1999.10.1007/978-94-017-2866-9_4
Search in Google Scholar Back to article
19. Murawski, R. Truth vs. provability – philosophical and historical remarks. Logic and Logical Philosophy 10, 2002, pp. 93–117.10.12775/LLP.2002.006
Search in Google Scholar Back to article
20. Murawski, R. On the distinction proof-truth in mathematics, In P. Gärdenfors et al. (eds.), In the Scope of Logic, Methodology and Philosophy of Science, Dordrecht–Boston–London: Kluwer Academic, 2002, pp. 287–303.
Search in Google Scholar Back to article
21. Murawski, R. Troubles with (the concept of) truth in mathematics, Logic and Logical Philosophy 15, 2006, pp. 285–303. Reprinted in: R. Murawski, Lógos and Máthēma. Studies in the Philosophy of Mathematics and History of Logic, Frankfurt am Main: Peter Lang International Verlag der Wissenschaften, 2011, pp. 187–201.10.12775/LLP.2006.017
Search in Google Scholar Back to article
22. 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. 251–268.10.1515/9781501502620-015
Search in Google Scholar Back to article
23. Rota, G.–C. The phenomenology of mathematical proof, Synthese 111, 1997, pp. 183–196.10.1023/A:1004974521326
Search in Google Scholar Back to article
24. 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.
Search in Google Scholar Back to article
25. Tarski, A. On the concept of following logically, History and Philosophy of Logic 23, 1936/2002, pp. 155–196.10.1080/0144534021000036683
Search in Google Scholar Back to article
26. Tarski, A. Logic, Semantics, Metamathematics. Papers from 1923 to 1938, Oxford: Clarendon Press, 1956.
Search in Google Scholar Back to article
27. Tarski, A. Truth and proof, Scientific American 220, No. 6, 1969, pp. 63–77.10.1038/scientificamerican0669-635769563
Search in Google Scholar Back to article
28. Tarski, A. Logic, Semantics, Metamathematics. Papers from 1923 to 1938, second edition edited and introduced by J. Corcoran, Indianapolis: Hackett Publishing Co., 1983.
Search in Google Scholar Back to article
29. Wang, H. From Mathematics to Philosophy, London: Routledge and Kegan Paul, 1974.
Search in Google Scholar Back to article
30. Wang, H. Reflections on Kurt Gödel, Cambridge, Mass: The MIT Press, 1987.
Search in Google Scholar Back to article
31. Woleński, J. Semantics and Truth, Logic, Epistemology and the Unity of Science 45, Berlin: Springer Verlag, 2019.10.1007/978-3-030-24536-8
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/sh-2020-0025 | Journal eISSN: 2299-0518

Journal RSS Feed

Language: English

Page range: 10 - 18

Published on: Nov 11, 2020

Published by: University of Information Technology and Management in Rzeszow

In partnership with: Paradigm Publishing Services

Publication frequency: 4 issues per year

Keywords:

truth,

logic,

Related subjects:

Business and economics,

Political economics,

Political economics, other,

Mathematics,

Logic and set theory,

Philosophy,

Philosophy, other

© 2020 Roman Murawski, published by University of Information Technology and Management in Rzeszow
This work is licensed under the Creative Commons Attribution 4.0 License.

Volume 9 (2020): Issue 3-4 (October 2020)