Abstract
The question that is the subject of this article is not intended to be a sociological or statistical question about the practice of today’s mathematicians, but a philosophical question about the nature of mathematics, and specifically the method of mathematics. Since antiquity, saying that mathematics is problem solving has been an expression of the view that the method of mathematics is the analytic method, while saying that mathematics is theorem proving has been an expression of the view that the method of mathematics is the axiomatic method. In this article it is argued that these two views of the mathematical method are really opposed. In order to answer the question whether mathematics is problem solving or theorem proving, the article retraces the Greek origins of the question and Hilbert’s answer. Then it argues that, by Gödel’s incompleteness results and other reasons, only the view that mathematics is problem solving is tenable.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Atiyah, M., & Singer, I. (2004). Interview by M. Raussen and C. Skau. Notices of the American Mathematical Society, 52(2), 223–231.
Auslander, J. (2008). On the roles of proof in mathematics. In B. Gold & R. A. Simons (Eds.), Proof and other dilemmas. Mathematics and philosophy (pp. 62–77). Washington: The Mathematical Association of America.
Bostock, D. (2009). Philosophies of mathematics. An introduction. Madden: Wiley.
Cellucci, C. (2007). La filosofia della matematica del Novecento. Roma: Laterza.
Cellucci, C. (2013). Rethinking logic. Logic in relation to mathematics, evolution, and method. Dordrecht: Springer.
Curry, H. B. (1951). Outlines of a formalist philosophy of mathematics. Amsterdam: North-Holland.
Davis, P. J. (1972). Fidelity in mathematical discourse: Is one and one really two? The American Mathematical Monthly, 79, 252–263.
Davis, M. (2005). What did Gödel believe and when did he believe it? The Bulletin of Symbolic Logic, 11, 194–206.
Detlefsen, M. (1992). On an alleged refutation of Hilbert’s program using Gödel’s first incompleteness theorem. In M. Detlefsen (Ed.), Proof, logic and formalization (pp. 199–235). London: Routledge.
Diels, H. & W. Kranz (Eds.) (2014–2015). Die Fragmente der Vorsokratiker. Hildesheim: Olms-Weidman.
Feferman, S. (1998). In the light of logic. Oxford: Oxford University Press.
Franks, J. (1989). Review of J. Gleick, Chaos: Making a new science. The Mathematical Intelligencer, 11(1), 65–69.
Gödel, K. (1986–2002). Collected works. Oxford: Oxford University Press.
Gowers, T. (2002). Mathematics. A very short introduction. Oxford: Oxford University Press.
Gowers, T. (2006). Does mathematics need a philosophy? In R. Hersh (Ed.), 18 unconventional essays on the nature of mathematics (pp. 182–200). New York: Springer.
Halmos, P. R. (1968). Mathematics as a creative art. American Scientist, 56, 375–389.
Halmos, P. R. (1980). The heart of mathematics. The American Mathematical Monthly, 87, 519–524.
Hamming, R. W. (1980). The unreasonable effectiveness of mathematics. The American Mathematical Monthly, 87, 81–90.
Hamming, R. W. (1998). Mathematics on a distant planet. The American Mathematical Monthly, 105, 640–650.
Hersh, R. (1997). What is mathematics, really?. Oxford: Oxford University Press.
Hersh, R. (2014). Experiencing mathematics. What do we do, when we do mathematics?. Providence: American Mathematical Society.
Hilbert, D. (1902–1903). Über den Satz von der Gleichheit der Basiswinkel im gleichschenkligen Dreieck. Proceedings of the London Mathematical Society 35: 50–68.
Hilbert, D. (1905). Logische Prinzipien des mathematisches Denkens. Göttingen: Bibliothek des Mathematischen Instituts der Universität.
Hilbert, D. (1967). The foundations of mathematics. In J. van Heijenoort (Ed.), From Frege to Gödel. A source book in mathematical logic, 1879–1931 (pp. 464–479). Cambridge: Harvard University Press.
Hilbert, D. (1980). Letter to Frege, December 29, 1899. In G. Frege (Ed.), Philosophical and mathematical correspondence (pp. 41–43). Oxford: Blackwell.
Hilbert, D. (1992). Natur und mathematischen Erkennen. Vorlesungen, gehalten 1919–1920 in Göttingen. Basel: Birkhäuer.
Hilbert, D. (1996a). On the concept of number. In W. Ewald (Ed.), From Kant to Hilbert. A source book in the foundations of mathematics, II (pp. 1092–1095). Oxford: Oxford University Press.
Hilbert, D. (1996b). The new grounding of mathematics. First report. In W. Ewald (Ed.), From Kant to Hilbert. A source book in the foundations of mathematics, II (pp. 1117–1134). Oxford: Oxford University Press.
Hilbert, D. (1996c). The logical foundations of mathematics. In W. Ewald (Ed.), From Kant to Hilbert. A source book in the foundations of mathematics, II (pp. 1134–1148). Oxford: Oxford University Press.
Hilbert, D. (1996d). The grounding of elementary number theory. In W. Ewald (Ed.), From Kant to Hilbert. A source book in the foundations of mathematics, II (pp. 1148–1157). Oxford: Oxford University Press.
Hilbert, D. (2000). Mathematical problems. In J. J. Gray (Ed.), The Hilbert challenge (pp. 240–282). Oxford: Oxford University Press.
Hilbert, D. (2013). Beweis des Tertium non datur. In D. Hilbert (Ed.), Lectures on the foundations of arithmetic and logic 1917–1933 (pp. 985–989). Berlin: Springer.
Hintikka, J., & Remes, U. (1974). The method of analysis. Its geometrical origin and its general significance. Dordrecht: Springer.
Kant, I. (1998). Critique of pure reason. Cambridge: Cambridge University Press.
Knorr, W. R. (1993). The ancient tradition of geometric problems. New York: Dover.
Lecat, M. (1935). Erreurs de mathématiciens des origines à nos jours. Bruxelles: Castaigne.
Mäenpää, P. (1993). The art of analysis. Logic and history of problem solving. Helsinki: University of Helsinki.
Mäenpää, P. (1997). From backward reduction to configurational analysis. In M. Otte & M. Panza (Eds.), Analysis and synthesis in mathematics. History and philosophy (pp. 201–226). Dordrecht: Springer.
Matiyasevich, Y. (2000). Hilbert’s tenth problem: What was done and what is to be done. In J. Denef, L. Lipshitz, T. Pheidas, & J. Van Geel (Eds.), Hilbert’s tenth problem: Relations with arithmetic and algebraic geometry (pp. 1–47). Providence: American Mathematical Society.
Newton, I. (1714–1716). An account of the book entituled Commercium Epistolicum Collinii et aliorum, de analysi promota. Philosophical Transactions of the Royal Society of London 29: 173–224.
Newton, I. (1967–1981). The mathematical papers. Cambridge: Cambridge University Press.
Otte, M., & Panza, M. (Eds.). (1997). Analysis and synthesis in mathematics. History and philosophy. Dordrecht: Springer.
Pappus of Alexandria (1876–1878). Collectio. Berlin: Weidmann.
Peckhaus, V. (2002). Regressive analysis. In U. Meixner & A. Newen (Eds.), Philosophiegeschichte und logische analyse. Logical analysis and history of philosophy, V (pp. 97–110). Paderborn: Mentis.
Poincaré, H. (1921). The foundations of science. New York: The Science Press.
Proclus. (1992). In primum Euclidis Elementorum librum commentarii. Hildesheim: Olms.
Rota, G.-C. (1981). Introduction. In P. J. Davis & R. Hersh (Eds.), The mathematical experience, xvii–xix. Boston: Birkhäuser.
Rota, G.-C. (1997). Indiscrete thoughts. Boston: Birkhäuser.
Shoenfield, J. R. (1967). Mathematical logic. Reading: Addison-Wesley.
Singh, S. (2002). Fermat’s last theorem. London: Fourth Estate.
Timmermans, B. (1995). La résolution des problèmes de Descartes à Kant. Paris: Presses Universitaires de France.
Weyl, H. (1998). On the new foundational crisis of mathematics. In P. Mancosu (Ed.), From Brouwer to Hilbert. The debate on the foundations of mathematics in the 1920s (pp. 86–118). Oxford: Oxford University Press.
Acknowledgments
I am grateful to Reuben Hersh, Nathalie Sinclair, Fabio Sterpetti and four anonymous referees for their comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cellucci, C. Is Mathematics Problem Solving or Theorem Proving?. Found Sci 22, 183–199 (2017). https://doi.org/10.1007/s10699-015-9475-2
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10699-015-9475-2