Although mathematicians often use it, mathematical beauty is a philosophically challenging concept. How can abstract objects be evaluated as beautiful? Is this related to their visualisations? Using an example from graph theory, this paper argues that, in making aesthetic judgements, mathematicians may be responding to a combination of perceptual properties of visual representations and mathematical properties of abstract structures; the latter seem to carry greater weight. Mathematical beauty thus primarily involves mathematicians’ sensitivity to aesthetics of the abstract.

Scientists often use aesthetic values in the evaluation and choice of theories. Aesthetic values are not only regarded as leading to practically more useful theories but are often taken to stand in a special epistemic relation to the truth of a theory such that the aesthetic merit of a theory is evidence of its truth. This paper explores what aesthetic considerations influence scientists' reasoning, how such aesthetic values relate to the utility of a scientific theory, and how one can justify (...) the epistemic role for such values. The paper examines ways in which the link between beauty and truth can be defended, the challenges facing such accounts, and explores alternative epistemic roles for aesthetic values in scientific practice. (shrink)

In recent years, philosophical work directly concerned with the practice of mathematics has intensified, giving rise to a movement known as the philosophy of mathematical practice . In this paper we offer a survey of this movement aimed at mathematics educators. We first describe the core questions philosophers of mathematical practice investigate as well as the philosophical methods they use to tackle them. We then provide a selective overview of work in the philosophy of mathematical practice covering topics including the (...) distinction between formal and informal proofs, visualization and artefacts, mathematical explanation and understanding, value judgments, and mathematical design. We conclude with some remarks on the potential connections between the philosophy of mathematical practice and mathematics education. (shrink)

Mathematicians judge proofs to possess, or lack, a variety of different qualities, including, for example, explanatory power, depth, purity, beauty and fit. Philosophers of mathematical practice have begun to investigate the nature of such qualities. However, mathematicians frequently draw attention to another desirable proof quality: being motivated. Intuitively, motivated proofs contain no "puzzling" steps, but they have received little further analysis. In this paper, I begin a philosophical investigation into motivated proofs. I suggest that a proof is motivated if and (...) only if mathematicians can identify (i) the tasks each step is intended to perform; and (ii) where each step could have reasonably come from. I argue that motivated proofs promote understanding, convey new mathematical resources and stimulate new discoveries. They thus have significant epistemic benefits and directly contribute to the efficient dissemination and advancement of mathematical knowledge. Given their benefits, I also discuss the more practical matter of how we can produce motivated proofs. Finally I consider the relationship between motivated proofs and proofs which are explanatory, beautiful and fitting. (shrink)

Aesthetics in philosophy of mathematics is too narrowly construed. Beauty is not the only feature in mathematics that is arguably aesthetic. While not the highest aesthetic value, being interesting is a sine qua non for publishability. Of the many ways to be interesting, being explanatory has recently been discussed. The motivational power of what is interesting is important for both directing research and stimulating education. The scientific satisfaction of curiosity and the artistic desire for beautiful results are complementary but both (...) aesthetic. (shrink)