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MATHEMATICAL RIGOR AND PROOF

Part of: Philosophical aspects of logic and foundations General and miscellaneous specific topics

Published online by Cambridge University Press:  04 October 2019

YACIN HAMAMI
YACIN HAMAMI*
Affiliation:
CENTRE FOR LOGIC AND PHILOSOPHY OF SCIENCE VRIJE UNIVERSITEIT BRUSSEL BRUSSELS B-1050, BELGIUM E-mail: yacin.hamami@gmail.com
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Abstract

Mathematical proof is the primary form of justification for mathematical knowledge, but in order to count as a proper justification for a piece of mathematical knowledge, a mathematical proof must be rigorous. What does it mean then for a mathematical proof to be rigorous? According to what I shall call the standard view, a mathematical proof is rigorous if and only if it can be routinely translated into a formal proof. The standard view is almost an orthodoxy among contemporary mathematicians, and is endorsed by many logicians and philosophers, but it has also been heavily criticized in the philosophy of mathematics literature. Progress on the debate between the proponents and opponents of the standard view is, however, currently blocked by a major obstacle, namely, the absence of a precise formulation of it. To remedy this deficiency, I undertake in this paper to provide a precise formulation and a thorough evaluation of the standard view of mathematical rigor. The upshot of this study is that the standard view is more robust to criticisms than it transpires from the various arguments advanced against it, but that it also requires a certain conception of how mathematical proofs are judged to be rigorous in mathematical practice, a conception that can be challenged on empirical grounds by exhibiting rigor judgments of mathematical proofs in mathematical practice conflicting with it.

Keywords

mathematical rigormathematical proofformal proofmathematical practice

MSC classification

Primary: 00A30: Philosophy of mathematics
Secondary: 00A35: Methodology of mathematics, didactics 03A05: Philosophical and critical

Information

Type
Research Article
Information
The Review of Symbolic Logic , Volume 15 , Issue 2 , June 2022 , pp. 409 - 449
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Copyright
© The Author(s), 2019. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

BIBLIOGRAPHY

Anacona, M., Arboleda, L. C., & Pérez-Fernández, F. J. (2014). On Bourbaki’s axiomatic system for set theory. Synthese, 191 (17), 4069–4098.CrossRefGoogle Scholar
Antonutti Marfori, M. (2010). Informal proofs and mathematical rigour. Studia Logica, 96(2), 261–272.CrossRefGoogle Scholar
Avigad, J. (2006). Mathematical method and proof. Synthese, 153(1), 105–159.CrossRefGoogle Scholar
Avigad, J. (2008). Understanding proofs. In Mancosu, P., editor. The Philosophy of Mathematical Practice. Oxford: Oxford University Press, pp. 317–353.CrossRefGoogle Scholar
Avigad, J. (2018). The mechanization of mathematics. Notices of the American Mathematical Society, 65(6), 681–690.CrossRefGoogle Scholar
Azzouni, J. (2004). The derivation-indicator view of mathematical practice. Philosophia Mathematica, 12(3), 81–105.CrossRefGoogle Scholar
Azzouni, J. (2006). Tracking Reason: Proof, Consequence, and Truth. Oxford: Oxford University Press.CrossRefGoogle Scholar
Azzouni, J. (2009). Why do informal proofs conform to formal norms? Foundations of Science, 14(1–2), 9–26.CrossRefGoogle Scholar
Azzouni, J. (2013). The relationship of derivations in artificial languages to ordinary rigorous mathematical proof. Philosophia Mathematica, 21(2), 247–254.CrossRefGoogle Scholar
Bourbaki, N. (1970). Éléments de Mathématique, Théorie des Ensembles. Paris: Hermann. Translated to English as Elements of Mathematics, Theory of Sets. Addison-Wesley Publishing Company, Reading, MA, 1968. (Citations are to translation).Google Scholar
Burgess, J. P. (2015). Rigor and Structure. Oxford: Oxford University Press.CrossRefGoogle Scholar
Chartrand, G., Polimeni, A. D., & Zhang, P. (2018). Mathematical Proofs: A Transition to Advanced Mathematics (fourth edition). Boston: Pearson.Google Scholar
Corcoran, J. (2014). Schema. In Zalta, E. N., editor. The Stanford Encyclopedia of Philosophy (Spring 2014 Edition). Available at: http://plato.stanford.edu/archives/spr2014/entries/schema/.Google Scholar
Detlefsen, M. (2009). Proof: Its nature and significance. In Gold, B. and Simons, R. A., editors. Proof and Other Dilemmas: Mathematics and Philosophy. Washington, DC: The Mathematical Association of America, pp. 3–32.Google Scholar
Dutilh Novaes, C. (2016). Reductio ad absurdum from a dialogical perspective. Philosophical Studies, 173(10), 2605–2628.CrossRefGoogle Scholar
Fallis, D. (2003). Intentional gaps in mathematical proofs. Synthese, 134(1–2), 45–69.CrossRefGoogle Scholar
Hales, T. (2012). Dense Sphere Packings: A Blueprint for Formal Proofs. London Mathematical Society Lecture Notes Series, Vol. 400. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Hamami, Y. (2018). Mathematical inference and logical inference. The Review of Symbolic Logic, 11(4), 665–704.CrossRefGoogle Scholar
Hardy, G. H. & Wright, E. M. (1975). An Introduction to the Theory of Numbers (fourth edition). Oxford: Oxford University Press.Google Scholar
Hausdorff, F. (1914). Grundzüge der Mengenlehre. Leipzig: Verlag Von Veit.Google Scholar
Hersh, R. (1997). Prove—Once more and again. Philosophia Mathematica, 5(3), 153–165.CrossRefGoogle Scholar
Hilbert, D. (1920/2013). Lectures on Logic (1920). In Ewald, W. and Sieg, W., editors. David Hilbert’s Lectures on the Foundations of Arithmetic and Logic 1917–1933. Berlin: Springer-Verlag, pp. 275–414.Google Scholar
Hintikka, J. & Sandu, G. (1997). Game-theoretical semantics. In van Benthem, J. and ter Meulen, A. G. B., editors. Handbook of Logic and Language. Amsterdam: Elsevier, pp. 361–410.CrossRefGoogle Scholar
Inglis, M. & Alcock, L. (2012). Expert and novice approaches to reading mathematical proofs. Journal for Research in Mathematics Education, 43(4), 358–390.CrossRefGoogle Scholar
Inglis, M., Mejia-Ramos, J. P., Weber, K., & Alcock, L. (2013). On mathematicians’ different standards when evaluating elementary proofs. Topics in Cognitive Science, 5(2), 270–282.CrossRefGoogle ScholarPubMed
Keiff, L. (2011). Dialogical logic. In Zalta, E. N., editor. The Stanford Encyclopedia of Philosophy (Summer 2011 Edition). Available at: https://plato.stanford.edu/archives/sum2011/entries/logic-dialogical/.Google Scholar
Landau, E. (1930). Grundlagen der Analysis. Leipzig: Akademische Verlagsgesellschaft M.B.H.Google Scholar
Larvor, B. (2012). How to think about informal proofs. Synthese, 187(2), 715–730.CrossRefGoogle Scholar
Larvor, B. (2016). Why the naïve derivation recipe model cannot explain how mathematicians’ proofs secure mathematical knowledge. Philosophia Mathematica, 24(3), 401–404.CrossRefGoogle Scholar
Lehman, H. (1980). An examination of Imre Lakatos’ philosophy of mathematics. The Philosophical Forum, 12(1), 33–48.Google Scholar
Mac Lane, S. (1934). Abgekürzte Beweise im Logikkalkul. Ph.D. Thesis, Georg August-Universität zu Göttingen, Göttingen. Reprinted in Kaplansky, I., editor. Saunders Mac Lane Selected Papers. New York: Springer-Verlag. pp. 1–62. 1979.Google Scholar
Mac Lane, S. (1935). A logical analysis of mathematical structure. The Monist, 45(1), 118–130.CrossRefGoogle Scholar
Mac Lane, S. (1979). A late return to a thesis in logic. In Kaplansky, I., editor. Saunders Mac Lane Selected Papers. New York: Springer-Verlag, pp. 63–66.CrossRefGoogle Scholar
Mac Lane, S. (1986). Mathematics: Form and Function. New York: Springer-Verlag.CrossRefGoogle Scholar
McLarty, C. (2007). The last mathematician from Hilbert’s Göttingen: Saunders Mac Lane as philosopher of mathematics. The British Journal for the Philosophy of Science, 58(1), 77–112.CrossRefGoogle Scholar
Rav, Y. (1999). Why do we prove theorems? Philosophia Mathematica, 7(3), 5–41.CrossRefGoogle Scholar
Robinson, J. A. (1997). Informal rigor and mathematical understanding. In Gottlob, G., Leitsch, A., and Mundici, D., editors. Computational Logic and Proof Theory: Proceedings of the 5th Annual Kurt Gödel Colloquium, August 25–29, 1997. Lecture Notes in Computer Science, Vol. 1289. Heidelberg & New York: Springer, pp. 54–64.CrossRefGoogle Scholar
Rosen, K. H. (2012). Discrete Mathematics and its Applications (seventh edition). New York: McGraw-Hill.Google Scholar
Sieg, W. & Walsh, P. (2018). Natural formalization: Deriving the Cantor-Bernstein theorem in ZF. Unpublished manuscript.Google Scholar
Solow, D. (2014). How to Read and Do Proofs: An Introduction to Mathematical Thought Processes (sixth edition). Hoboken, NJ: John Wiley.Google Scholar
Sørensen, M. H. & Urzyczyn, P. (2006). Lectures on the Curry-Howard Isomorphism. Studies in Logic and the Foundations of Mathematics, Vol. 149. Amsterdam: Elsevier.Google Scholar
Steiner, M. (1975). Mathematical Knowledge. Ithaca, NY: Cornell University Press.Google Scholar
Tanswell, F. (2015). A problem with the dependence of informal proofs on formal proofs. Philosophia Mathematica, 23(3), 295–310.CrossRefGoogle Scholar
Tatton-Brown, O. (2019). Rigour and proof. Unpublished manuscript.Google Scholar
Velleman, D. J. (2006). How to Prove It: A Structured Approach. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Weber, K. (2008). How mathematicians determine if an argument is a valid proof. Journal for Research in Mathematics Education, 39(4), 431–459.CrossRefGoogle Scholar
Weir, A. (2016). Informal proof, formal proof, formalism. The Review of Symbolic Logic, 9(1), 23–43.CrossRefGoogle Scholar
Wiedijk, F. (2008). Formal proof—Getting started. Notices of the American Mathematical Society, 55(11), 1408–1414.Google Scholar