Theoretical virtues such as simplicity and explanatory power are valued by scientists because they enhance the credibility of scientific theories. Philosophers have long been interested in how these virtues attain their positive epistemic value (Douglas, 2013). Mathematicians also discuss virtues such as simplicity and explanatory power, but they often emphasize mathematical beauty as the primary theoretical virtue that drives their research (Engler, 1990). Although beauty is also valued by scientists (Ivanova, 2017), the ways in which mathematicians rely on beauty appear to be different. As Hardy (1940/2012, §10) put it, there is no permanent place in the world for ugly mathematics.

To explain its alleged epistemic value, mathematicians and philosophers have proposed various definitions of mathematical beauty. However, these definitions often conflict with each other and fail to accurately describe how mathematicians use terms such as “beautiful” or “elegant” (Inglis & Aberdein, 2014). This discrepancy creates a mismatch between the importance mathematicians place on beauty and the apparent failure of philosophical accounts to capture its positive epistemic value.

In this talk, I propose a solution to this mismatch. I argue that mathematicians, like scientists, rely on multiple theoretical virtues when judging proofs, theorems, and conjectures. Not all of these virtues have positive epistemic value: some are truth-conducive, while others may serve merely pragmatic purposes. Theoretical virtues in mathematics are therefore a diverse set, and beauty is only one of them.

To support this argument, I take a two-step approach. First, I’ll review prominent definitions of beauty found in the literature. I’ll argue that they fail to explain its epistemic value and fail to generalize. Second, I’ll argue that this failure can be explained by taking the components of the definitions at face value: they are all to be found in the diverse set of virtues that mathematicians rely on when judging proofs.

Sources

Douglas, H. (2013). The Value of Cognitive Values. Philosophy of Science, 80(5), 796–806. https://doi.org/10.1086/673716

Engler, G. (1990). Aesthetics in Science and in Art. The British Journal of Aesthetics, 30(1), 24–34. https://doi.org/10.1093/bjaesthetics/30.1.24

Hardy, G. H. (2012). A Mathematician’s Apology. Cambridge University Press. https://doi.org/10.1017/cbo9781107295599 (Original work published 1940)

Inglis, M., & Aberdein, A. (2014). Beauty Is Not Simplicity: An Analysisof Mathematicians’ Proof Appraisals. Philosophia Mathematica, 23(1), 87–109. https://doi.org/10.1093/philmat/nku014

Ivanova, M. (2017). Aesthetic values in science. Philosophy Compass, 12(10), e12433. https://doi.org/10.1111/phc3.12433