The Art of Doing Mathematics

In Berys Gaut & Matthew Kieran (eds.), Creativity and Philosophy. Routledge. pp. 313-330 (2018)

Christian Wenzel
National Taiwan University
Mathematicians often say that their theorems, proofs, and theories can be beautiful. They say mathematics can be like art. They know how to move creatively and freely in their domains. But ordinary people usually cannot do this and do not share this view. They often have unpleasant memories from school and do not have this experience of freedom and creativity in doing mathematics. I myself have been a mathematician, and I wish to highlight some of the creative aspects in doing mathematics. I always had the feeling that there is much freedom in mathematics and that one can do as one pleases as long as one avoids contradictions (and one can even live with contradictions for a while). In mathematics one only needs to define something and there it is! Just imagine it, and it immediately exists. Where else, besides fiction, does one have such power? --- This essay has three parts. In the first, I lay out some historical facts and views that I will need later. Second, I insert an interlude with Kant, pointing out some of his claims and insights in aesthetics, and some aspects in mathematics that I think he overlooked or underestimated. Third, I will bring out some aspects of higher mathematics that I will show are similar to art. These aspects are in the doing and creating of mathematics, not in the finished theories. They are usually not found in textbooks. I want to show that a researcher in mathematics is like a painter or composer, exploring and creating. To see this, one has to adopt a first-person perspective. In this way I will show that aesthetics plays a role and leaves its traces also in mathematics.
Keywords Aesthetics of Mathematics  Aesthetics in Kant  Creativity in Mathematics  Aesthetics  Philosophy of Mathematics
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