Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-25T05:54:48.750Z Has data issue: false hasContentIssue false
  • English
  • Français

RIGOUR AND PROOF

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

Published online by Cambridge University Press:  21 October 2020

OLIVER TATTON-BROWN
OLIVER TATTON-BROWN*
Affiliation:
DEPARTMENT OF PHILOSOPHY UNIVERSITY OF BRISTOL BRISTOL, UK
*
E-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper puts forward a new account of rigorous mathematical proof and its epistemology. One novel feature is a focus on how the skill of reading and writing valid proofs is learnt, as a way of understanding what validity itself amounts to. The account is used to address two current questions in the literature: that of how mathematicians are so good at resolving disputes about validity, and that of whether rigorous proofs are necessarily formalizable.

Keywords

mathematical rigourmathematical proofvalidityformalizabilitymathematical practicelength of proofs

MSC classification

Primary: 00A30: Philosophy of mathematics
Secondary: 00A35: Methodology of mathematics, didactics 03A05: Philosophical and critical
Type
Research Article
Information
The Review of Symbolic Logic , Volume 16 , Issue 2 , June 2023 , pp. 480 - 508
Check for updates
Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

References

REFERENCES

Aberdein, A., & Dove, I. J., editors (2013). The Argument of Mathematics. Logic, Epistemology, and the Unity of Science. Netherlands: Springer.Google Scholar
Aluffi, P. (2009). Algebra: Chapter 0. Providence, RI: American Mathematical Society.Google Scholar
Andersen, L. E. (2017). On the nature and role of peer review in mathematics. Accountability in Research, 24(3), 177–192.CrossRefGoogle ScholarPubMed
Antonutti Marfori, M. (2010). Informal proofs and mathematical rigour. Studia Logica: An International Journal for Symbolic Logic, 96(2), 261–272.CrossRefGoogle Scholar
Artin, M. (1991). Algebra (first edition). Englewood Cliffs, NJ: Pearson.Google Scholar
Aschbacher, M. (2005). Highly complex proofs and implications of such proofs. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 363(1835), 2401–2406.CrossRefGoogle ScholarPubMed
Avigad, J. (2011). Understanding proofs. In Mancosu, P., editor. The Philosophy of Mathematical Practice. Oxford: Oxford University Press, pp. 317–353.Google Scholar
Awodey, S. (2010). Category Theory (second edition). Oxford: Oxford University Press.Google Scholar
Axler, S. (1997). Linear Algebra Done Right (second edition). New York: Springer.CrossRefGoogle Scholar
Azzouni, J. (2004). The derivation-indicator view of mathematical practice. Philosophia Mathematica, 12(2), 81–106.CrossRefGoogle Scholar
Bak, J., & Newman, D. J. (2010). Complex Analysis (third edition). Undergraduate Texts in Mathematics. New York: Springer-Verlag.CrossRefGoogle Scholar
Banach, S. (1987). Theory of Linear Operations. Netherlands: Elsevier.Google Scholar
Bartle, R. G., & Sherbert, D. R. (2000). Introduction to Real Analysis. Canada: John Wiley & Sons Limited.Google Scholar
Berkeley, G. (1999). The analyst. In Ewald, W. B., editor. From Kant to Hilbert Volume 1: A Source Book in the Foundations of Mathematics. Oxford: Oxford University Press, pp. 60–92.Google Scholar
Brown, B., & Priest, G. (2004). Chunk and permeate, a paraconsistent inference strategy. Part I: The infinitesimal calculus. Journal of Philosophical Logic, 33(4), 379–388.CrossRefGoogle Scholar
Burgess, J. P. (2015). Rigor and Structure. Oxford: Oxford University Press.CrossRefGoogle Scholar
Celluci, C. (2009). Why proof? What is a proof? In Lupacchini, R., and Corsi, G., editors. Deduction, Computation, Experiment. Milano, Italy: Springer-Verlag Gmbh, pp. 1–27.Google Scholar
Conway, J. B. (1978). Functions of One Complex Variable I (second edition). Graduate Texts in Mathematics. New York: Springer-Verlag.CrossRefGoogle Scholar
Cori, R., & Lascar, D. (2000). Mathematical Logic Part 1: Propositional Calculus, Boolean Algebras & Predicate Calculus: A Course with Exercises. Oxford: Oxford University Press.Google Scholar
Coulston, C., editor (1970). Dictionary of Scientific Biography Volume II: Hans Berger—Christoph Buys Ballot. New York: Charles Scribner’s Sons.Google Scholar
De Toffoli, S., & Giardino, V. (2016). Envisioning transformations—the practice of topology. In Larvor, B., editor. Mathematical Cultures: The London Meetings 2012–2014 (first edition). New York: Birkhäuser, pp. 25–50.CrossRefGoogle Scholar
Detlefsen, M. (2009). Proof: Its nature and significance. In Gold, B., and Simons, R., editors. Proof and Other Dilemmas: Mathematics and Philosophy (UK edition). Cambridge: Cambridge University Press, pp. 3–32.Google Scholar
Eisenbud, D. (1995). Commutative Algebra: With a View Toward Algebraic Geometry. Berlin, Germany: Springer Science & Business Media.CrossRefGoogle Scholar
Eisenbud, D., & Harris, J. (2000). The Geometry of Schemes. Graduate Texts in Mathematics. New York: Springer-Verlag.Google Scholar
Epp, S. S. (2003). The role of logic in teaching proof. The American Mathematical Monthly, 110(10), 886–899.CrossRefGoogle Scholar
Erdös, P. (1947). Some remarks on the theory of graphs. Bulletin of the American Mathematical Society, 53(4), 292–294.CrossRefGoogle Scholar
Fremlin, D. (2010). Measure Theory, Vol. 1. Torres Fremlin.Google Scholar
Fulton, W., & Harris, J. (1991). Representation Theory: A First Course. Berlin, Germany: Springer-Verlag.Google Scholar
Gentzen, G. (1969). The Collected Papers of Gerhard Gentzen. Amsterdam, Netherlands: North-Holland Publishing Company.Google Scholar
Goethe, N. B., & Friend, M. (2010). Confronting ideals of proof with the ways of proving of the research mathematician. Studia Logica, 96(2), 273–288.CrossRefGoogle Scholar
Gowers, T., Barrow-Green, J., & Leader, I., editors (2008). The Princeton Companion to Mathematics. Princeton, NJ: Princeton University Press.Google Scholar
Grcar, J. (2013). Errors and corrections in mathematics literature. Notices of the American Mathematical Society, 60(4), 418–425.CrossRefGoogle Scholar
Hales, T. C. (2007). Jordan’s proof of the Jordan Curve theorem. Studies in Logic, Grammar and Rhetoric, 10(23), 45–60.Google Scholar
Halmos, P. R. (2011). Naive Set Theory. Mansfield, CT: Martino Fine Books.Google Scholar
Hamami, Y. (2019). Mathematical rigor and proof. The Review of Symbolic Logic, 1–41. doi: 10.1017/S1755020319000443 Google Scholar
Hartshorne, R. (1977). Algebraic Geometry . Graduate Texts in Mathematics. New York: Springer-Verlag.CrossRefGoogle Scholar
Hatcher, A. (2001). Algebraic Topology by Allen Hatcher (first edition). Cambridge: Cambridge University Press.Google Scholar
Hirsch, M. W. (1976). Differential Topology. Graduate Texts in Mathematics. New York: Springer-Verlag.CrossRefGoogle Scholar
Jaffe, A., & Quinn, F. (1993). “Theoretical mathematics”: toward a cultural synthesis of mathematics and theoretical physics. Bulletin of the American Mathematical Society, 29(1), 1–13.CrossRefGoogle Scholar
James, G., & Liebeck, M. (2001). Representations and Characters of Groups (second edition). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Jech, T. (2006). Set Theory: The Third Millennium Edition (revised and expanded third edition). Berlin, Germany: Springer.Google Scholar
Jones, V. (1998). A credo of sorts. In Dales, H. G., and Oliveri, G., editors. Truth in Mathematics. Oxford: Clarendon Press, pp. 203–214.Google Scholar
Jordan, C. (1887). Cours d’analyse de l’ecole polytechnique.Google Scholar
Kline, M. (1990a). Mathematical Thought from Ancient to Modern Times (new edition), Vol. 1. New York: Oxford University Press.Google Scholar
Kline, M. (1990b). Mathematical Thought from Ancient to Modern Times (new edition), Vol. 3. New York: Oxford University Press.Google Scholar
Konyndyk, K. (1986). Introductory Modal Logic (first edition). Notre Dame, IN: University of Notre Dame Press.Google Scholar
Larvor, B. (2012). How to think about informal proofs. Synthese, 187(2), 715–730.CrossRefGoogle Scholar
Larvor, B. (2019). From Euclidean geometry to knots and nets. Synthese, 196(7), 2715–2736.CrossRefGoogle Scholar
Lee, J. (2012). Introduction to Smooth Manifolds (2012 edition). New York: Springer.CrossRefGoogle Scholar
Lee, J. M. (2000). Introduction to Topological Manifolds. Graduate Texts in Mathematics. New York: Springer-Verlag.Google Scholar
Leitgeb, H. (2009). On formal and informal provability. In Bueno, O., and Linnebo, Ø., editors. New Waves in Philosophy of Mathematics (2009 edition). Basingstoke: Palgrave Macmillan, pp. 263–299.CrossRefGoogle Scholar
MacLane, S. (1998). Categories for the Working Mathematician (second edition). New York: Springer.Google Scholar
Mancosu, P. (1989). The metaphysics of the calculus: A foundational debate in the Paris academy of sciences, 1700–1706. Historia Mathematica, 16(3), 224–248.CrossRefGoogle Scholar
Morgan, J., & Tian, G. (2007). Ricci Flow and the Poincare Conjecture (reprint edition). Providence, RI: American Mathematical Society.Google Scholar
Munkres, J. (2000). Topology (second edition). Upper Saddle River, NJ: Pearson.Google Scholar
Nathanson, M. (2008). Desperately seeking mathematical truth. Notices of the American Mathematical Society, 55(7), 773.Google Scholar
Nelson, E. (1977). Internal set theory: A new approach to nonstandard analysis. Bulletin of the American Mathematical Society, 83(6), 1165–1198.CrossRefGoogle Scholar
Niven, I. M., Niven, A., & Zuckerman, H. S. (1991). An Introduction to the Theory of Numbers (fifth edition). New York: John Wiley & Sons.Google Scholar
Pelc, A. (2009). Why do we believe theorems? Philosophia Mathematica, 17(1), 84–94.CrossRefGoogle Scholar
Perelman, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math/0211159.Google Scholar
Perelman, G. (2003a). Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arXiv:math/0307245.Google Scholar
Perelman, G. (2003b). Ricci flow with surgery on three-manifolds. arXiv:math/0303109.Google Scholar
Prawitz, D. (1965). Natural Deduction: A Proof-Theoretical Study. Mineola, NY: Dover Publications.Google Scholar
Rav, Y. (1999). Why do we prove theorems? Philosophia Mathematica, 7(1), 5–41.CrossRefGoogle Scholar
Rav, Y. (2007). A critique of a formalist-mechanist version of the justification of arguments in mathematicians’ proof practices. Philosophia Mathematica, 15(3), 291–320.CrossRefGoogle Scholar
Robinson, A. (1996). Non-standard Analysis (new edition). Princeton, NJ: Princeton University Press.CrossRefGoogle Scholar
Rudin, W. (1987). Real and Complex Analysis. New York: McGraw-Hill.Google Scholar
Schwartz, J. (1954). The formula for change in variables in a multiple integral. The American Mathematical Monthly, 61(2), 81–85.CrossRefGoogle Scholar
Silverman, J. H. (2012). A Friendly Introduction to Number Theory (fourth edition). Boston, MA: Pearson.Google Scholar
Sweeney, D. J. (2014). Chunk and permeate: the infinitesimals of Isaac Newton. History and Philosophy of Logic, 35(1), 1–23.CrossRefGoogle Scholar
Switzer, R. M. (2002). Algebraic Topology—Homotopy and Homology. Classics in Mathematics. Berlin, Heidelberg/Germany: Springer-Verlag.Google Scholar
Szemerédi, E. (1975). On sets of integers containing no k elements in arithmetic progression. Polska Akademia Nauk. Instytut Matematyczny Acta Arithmetica, 27, 199–245.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 intuition. Erkenntnis (online only early view).Google Scholar
Taylor, J., & Garnier, R. (2014). Understanding Mathematical Proof (first edition). Boca Raton, FL: Routledge.Google Scholar
Thurston, W. P. (1994). On proof and progress in mathematics. Bulletin of the American Mathematical Society, 30(2), 161–177.CrossRefGoogle Scholar
Thurston, W. P. (1997). Three-Dimensional Geometry and Topology, Vol. 1. Princeton, NJ: Princeton University Press.CrossRefGoogle Scholar
Tieszen, R. (1992). What is a proof? In Detlefsen, M., editor. Proof, Logic and Formalization (first edition). London: Routledge, pp. 57–76.Google Scholar
Tragesser, R. (1992). Three insufficiently attended to aspects of most mathematical proofs: Phenomenological studies. In Detlefsen, M., editor. Proof, Logic and Formalization (first edition). London: Routledge, pp. 162–198.Google Scholar
Veblen, O. (1905). Theory on plane curves in non-metrical analysis situs. Transactions of the American Mathematical Society, 6(1), 83–98.CrossRefGoogle Scholar
Velleman, D. J. (2006). How to Prove It: A Structured Approach (second edition). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Walker, R. C. (1974). The Stone-Čech Compactification. New York: Springer-Verlag.CrossRefGoogle Scholar
Weir, A. (2016). Informal proof, formal proof, formalism. The Review of Symbolic Logic, 9(1), 23–43.CrossRefGoogle Scholar