Abstract
Many judgments are made about mathematical proofs, and among these we find the claim that a certain proof is beautiful or not. Beauty in mathematics is rarely denied, but it is also rarely explained. What is meant by the term, what criteria are needed for a proof to be beautiful, could there be objective qualities of a mathematical proof that map more or less onto the subjective experiences we might have when we read and understand it? While the concept of beauty itself is nebulous, the claim in this paper is that it might be related to a slightly more tractable quality of fit. We will discuss how fit arises in mathematics by looking at contrasting examples of proofs commonly held to be beautiful or not. We then take up the question about whether fit is a value, arguing that like justice and beauty, fit describes meaningful relationships and coherence, which make it a candidate for something to be valued not only in our mathematics classrooms, but also in the world at large.
The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way.
G.H. Hardy.
This work was done in part at the Australian National University in Canberra.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
How does one know how a lilac tree should be shaped? Here we take some sort of assumption about its natural form—what the tree would look like if left to its own devices. It might be argued that this stance is subjective, that there could be several forms a lilac could take that are equally beautiful. This question of objectivity is tricky and will not be dealt with thoroughly here.
- 2.
- 3.
- 4.
See (Raman and Öhman 2011) for more discussion of this example.
- 5.
The original proof is found in (Pick 1899). A short historical account is given at http://jsoles.myweb.uga.edu/history.html. Thanks to Bjorn Poonen for this example.
- 6.
Leaving aside the question of whether the theory of natural selection is really a theory.
References
Aigner, M., & Ziegler, G. (2010). Proofs from the Book, 4edn. Berlin: Springer.
Beardsley, M. C. (1958). Aesthetics: problems in the philosophy of criticism. New York: Harcourt, Brace, and World.
Beardsley, M. C. (1982). The aesthetic point of view. Selected essays. Ithaca, NY: Cornell University Press.
Ernest, P. (this volume) (2016). Mathematics and Values.
Ferreirós, J. (this volume) (2016). Purity as a Value in the German-speaking area.
Hardy, G. H. (1940). A Mathematician’s Apology. Cambridge: Cambridge University Press.
Hamming, R. W. (1980). The unreasonable effectiveness of mathematics. The American Mathematical Monthly, 87(2), 81–90.
Hume, D. (1740). A treatise of human nature. Oxford: Oxford University Press.
Pick, G. (1899). Geometrisches zur Zahlenlehre. Sitzungsberichte des deutschen naturwissenschaftlich-medicinischen Vereines fŸr Bšhmen Lotos in Prag. (Neue Folge). 19, 311–319.
Polya, G. (1954). Induction and analogy in mathematics, volume i of mathematics and plausible reasoning. Princeton, NJ: Princeton University Press.
Prinz, J. (2011). Emotion and Aesthetic Value. In E. Schellekens & P. Goldie (Eds.), The aesthetic mind: Philosophy and psychology. Oxford: Oxford University Press.
Raman, M., & Öhman, L.-D. (2011). Two beautiful proofs of Pick’s Theorem. In Proceedings of the Seventh Conference of European Research in Mathematics Education. Rzeszow, Poland. Feb. 9–13.
Raman, M., & Öhman, L.-D. (2013). Beauty as fit: A metaphor in mathematics? Research in Mathematics Education 15(2), 199–200.
Rota, G. C. (1997). Phenomenology of mathematical beauty. Synthese, 111(2), 171–182.
Rolls, E. T. (2011). A neurobiological basis for affective feelings and aesthetics. In E. Schellekens & P. Goldie (Eds.), The aesthetic mind: philosophy and psychology (pp. 116–165). Oxford: Oxford University Press.
Santayana, G. (1896). The sense of beauty. Charles Scribner’s Sons.
Scarry, E. (1999). On beauty and being just. Princeton University Press.
Schoenfeld, A. (2013). Classroom observations in theory and practice ZDM. Mathematics Education, 14, 607–621.
Sinclair, N. (2002). The kissing triangles: The aesthetics of mathematical discovery. International Journal of Computers for Mathematical Learning, 7, 45–63.
Sinclair, N. (2004). The roles of the aesthetic in mathematical inquiry. Mathematical Thinking and Learning, 6(3), 261–284.
Steiner, M. (1978). Mathematical explanation. Philosophical Studies, 34, 135–151.
Tatarkiewicz, W. (1972). The great theory of beauty and its decline. The Journal of Aesthetics and Art Criticism, 31(2), 165–180.
Toronchuk, J. A., & Ellis, G. F. R. (2012). Affective neuronal Darwinism: The nature of the primary emotional systems. Frontiers in Psychology, 3, 589.
Vickers, G. (1978). Rationality and Intuition. In On Aesthetics and Science ed. Weschler: MIT Press.
Weschler, J. (1978). On aesthetics in science. Cambridge, MA: MIT Press.
Zimba, J. (2009). On the possibility of trigonometric proofs of the Pythagorean theorem. Forum Geometricorum, 9, 1–4.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Raman-Sundström, M. (2016). The Notion of Fit as a Mathematical Value. In: Larvor, B. (eds) Mathematical Cultures. Trends in the History of Science. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-28582-5_16
Download citation
DOI: https://doi.org/10.1007/978-3-319-28582-5_16
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-28580-1
Online ISBN: 978-3-319-28582-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)