Foundations of Science

, Volume 20, Issue 4, pp 339–355 | Cite as

Mathematical Beauty, Understanding, and Discovery

  • Carlo CellucciEmail author


In a very influential paper Rota stresses the relevance of mathematical beauty to mathematical research, and claims that a piece of mathematics is beautiful when it is enlightening. He stops short, however, of explaining what he means by ‘enlightening’. This paper proposes an alternative approach, according to which a mathematical demonstration or theorem is beautiful when it provides understanding. Mathematical beauty thus considered can have a role in mathematical discovery because it can guide the mathematician in selecting which hypothesis to consider and which to disregard. Thus aesthetic factors can have an epistemic role qua aesthetic factors in mathematical research.


Mathematics Beauty Enlightenment Understanding 



I wish to thank Arthur Bierman, Angela Breitenbach, Arlette Dupuis, Donald Gillies, Reuben Hersh, Hansmichael Hohenegger, Nathalie Sinclair, Semir Zeki and four anonymous reviewers for their comments and suggestions.


  1. Atiyah, M. (1988). Collected works. Oxford: Oxford University Press.Google Scholar
  2. Auslander, J. (2008). On the roles of proof in mathematics. In B. Gold & R. A. Simon (Eds.), Proof and other dilemmas. Mathematics and philosophy (pp. 62–77). Washington: The Mathematical Association of America.Google Scholar
  3. Avigad, J. (2008). Understanding proofs. In P. Mancosu (Ed.), The philosophy of mathematical practice (pp. 317–353). Oxford: Oxford University Press.CrossRefGoogle Scholar
  4. Breitenbach, A. (2013a). Aesthetics in science. A Kantian proposal. Proceedings of the Aristotelian Society, 113(1), 83–100.CrossRefGoogle Scholar
  5. Breitenbach, A. (2013b). Beauty in proofs. Kant on aesthetics in mathematics. European Journal of Philosophy, 1–23. doi: 10.1111/ejop.12021.
  6. Cellucci, C. (2013). Rethinking logic. Logic in relation to mathematics, evolution, and method. Berlin: Springer.CrossRefGoogle Scholar
  7. Cellucci, C. (2014a). Knowledge, truth, and plausibility. Axiomathes. doi: 10.1007/s10516-014-9238-7.
  8. Cellucci, C. (2014b). Explanatory and non-explanatory demonstrations. In P. E. Bour, G. Heinzmann, W. Hodges, & P. Schroeder-Heister (Eds.), Logic, methodology and philosophy of science. Proceedings of the fourteenth international congress London: College Publications (in press).Google Scholar
  9. Clagett, M. (1999). Ancient Egyptian science. A source book, vol. 3: Ancient Egyptian mathematics. Philadelphia: American Mathematical Society.Google Scholar
  10. Conway, J. H., & Guy, R. K. (1996). The book of numbers. New York: Copernicus.CrossRefGoogle Scholar
  11. Davis, P. J., & Hersh, R. (1981). The mathematical experience. Boston: Birkhäuser.Google Scholar
  12. Dirac, P. A. M. (1951). A new classical theory of electrons. Proceedings of the Royal Society of London, Series A, 209, 291–296.CrossRefGoogle Scholar
  13. Dirac, P. A. M. (1963). The evolution of the physicist’s picture of nature. Scientific American, 208(5), 45–53.CrossRefGoogle Scholar
  14. Dirac, P. A. M. (1978). Directions in physics. New York: Wiley.Google Scholar
  15. Frege, G. (1964). The basic laws of arithmetic. Exposition of the system. Berkeley: University of California Press.Google Scholar
  16. Halmos, P. (1992). Mathematics. In C. G. Morris (Ed.), Academic Press dictionary of science and technology (p. 1325). San Diego: Academic Press.Google Scholar
  17. Hardy, G. H. (1992). A mathematician’s apology. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  18. Heisenberg, W. (2001). Science and the beautiful. In K. Wilber (Ed.), Quantum questions (pp. 56–69). Boston: Shambhala.Google Scholar
  19. Hersh, R., & John-Steiner, V. (2011). Loving + hating mathematics. Challenging the myths of mathematical life. Princeton: Princeton University Press.Google Scholar
  20. Hofstadter, D. R. (1999). Gödel, Escher, Bach. An eternal golden braid. New York: Basic Books.Google Scholar
  21. Kant, I. (2000). Critique of the power of judgment. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  22. Kosso, P. (2002). The omniscienter. Beauty and scientific understanding. International Studies in the Philosophy of Science, 16(1), 39–48.CrossRefGoogle Scholar
  23. Kragh, H. S. (1990). Dirac. A scientific biography. Cambridge: Cambridge University Press.Google Scholar
  24. McAllister, J. W. (2005). Mathematical beauty and the evolution of the standards of mathematical proof. In M. Emmer (Ed.), The visual mind II (pp. 15–34). Cambridge: The MIT Press.Google Scholar
  25. Mordell, L. J. (1959). Reflections of a mathematician. Montreal: Canadian Mathematical Congress. McGill University.Google Scholar
  26. Nahin, P. J. (2006). Dr. Euler’s fabulous formula: Cures many mathematical ills. Princeton: Princeton University Press.Google Scholar
  27. Nielsen, M. (2011). Reinventing discovery. The new era of networked science. Princeton: Princeton University Press.Google Scholar
  28. Ó Cairbre, F. (2009). The importance of being beautiful in mathematics. IMTA Newsletter, 109, 29–45.Google Scholar
  29. Poincaré, H. (1914). Science and method. London: Nelson.Google Scholar
  30. Poincaré, H. (1958). The value of science. Mineola: Dover.Google Scholar
  31. Rav, Y. (2005). Reflections on the proliferous growth of mathematical concepts and tools. Some case histories from mathematicians’ workshops. In C. Cellucci & D. Gillies (Eds.), Mathematical reasoning and heuristics (pp. 49–69). London: College Publications.Google Scholar
  32. Reichenbach, H. (1947). Elements of symbolic logic. New York: Macmillan.Google Scholar
  33. Rota, G.-C. (1997). The phenomenology of mathematical beauty. Synthese, 111(2), 171–182.CrossRefGoogle Scholar
  34. Schopenhauer, A. (2010). The world as will and representation (Vol. 1). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  35. Sinclair, N. (2004). The roles of the aesthetic in mathematical inquiry. Mathematical Thinking and Learning, 6(3), 261–284.CrossRefGoogle Scholar
  36. Sinclair, N. (2006). Mathematics and beauty. Aesthetic approaches to teaching children. New York: Teachers College Press.Google Scholar
  37. Sinclair, N. (2011). Aesthetic considerations in mathematics. Journal of Humanistic Mathematics, 1(1), 2–32.CrossRefGoogle Scholar
  38. Snow, C. P. (1973). The classical mind. In J. Mehra (Ed.), The physicist’s conception of nature (pp. 809–813). Dordrecht: Reidel.Google Scholar
  39. Todd, C. S. (2008). Unmasking the truth beneath the beauty. Why the supposed aesthetic judgments made in science may not be aesthetic at all. International Studies in the Philosophy of Science, 22(1), 61–79.CrossRefGoogle Scholar
  40. van Gerwen, R. (2011). Mathematical beauty and perceptual presence. Philosophical Investigations, 34(3), 249–267.CrossRefGoogle Scholar
  41. von Neumann, J. (1961). The mathematician. In J. R. Newman (Ed.), The world of mathematics (Vol. IV, pp. 2053–2063). New York: Simon & Schuster.Google Scholar
  42. Wells, D. (1990). Are these the most beautiful? The Mathematical Intelligencer, 12(3), 37–41.CrossRefGoogle Scholar
  43. Zangwill, N. (1998). Aesthetic/sensory dependence. British Journal of Aesthetics, 38(1), 66–81.CrossRefGoogle Scholar
  44. Zeki, S., Romaya, J. P., Benincasa, D. M. T., & Atiyah, M. F. (2014). The experience of mathematical beauty and its neural correlates. Frontiers in Human Neuroscience. doi: 10.3389/fnhum.2014.00068.

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© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of PhilosophySapienza University of RomeRomeItaly

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