### Why Do I Bother?

#### Posted by David Corfield

Terence Tao has placed on the ArXiv an article submitted to the Bulletin of the AMS, ‘What is good mathematics?’ After a substantial list of criteria of what constitutes a good piece of mathematics, there follows a lengthy case study of Szemerédi’s theorem to support a conjecture:

It may seem from the above discussion that the problem of evaluating mathematical quality, while important, is a hopelessly complicated one, especially since many good mathematical achievements may score highly on some of the qualities listed above but not on others. However, there is the remarkable phenomenon that good mathematics in one of the above senses tends to beget more good mathematics in many of the other senses as well, leading to the tentative conjecture that perhaps there is, after all, a universal notion of good quality mathematics, and all the specific metrics listed above represent different routes to uncover new mathematics, or difference stages or aspects of the evolution of a mathematical story.

As readers will know, I’ve been pushing for philosophers of mathematics to address the problems of values other than truth. In ‘Towards a Philosophy of Real Mathematics’, I moved beyond a Lakatosian conception of progress to tackle the arguments used for and against the extension of groups to groupoids. And in the past couple of years I’ve been advocating MacIntyre’s idea of a rational tradition of enquiry as governed by overarching dramatic narratives.

I’ve also been working on a conception of mathematical reality close to Michael Polanyi’s:

A new mathematical conception may be said to have reality if its assumption leads to a wide range of new interesting ideas. (Personal Knowledge: 116)

…while in the natural sciences the feeling of making contact with reality is an augury of as yet undreamed of future empirical confirmations of an immanent discovery, in mathematics it betokens an indeterminate range of future germinations within mathematics itself. (Personal Knowledge: 189)

But why need MacIntyre or Polanyi or I bother, if, without reading us, a mathematician can come to very similar conclusions?:

…the very best examples of good mathematics do not merely fulfil one or more of the criteria of mathematical quality listed at the beginning of the article, but are more importantly part of a greater mathematical

story, which then unfurls to generate many further pieces of good mathematics of many different types. Indeed, one can view the history of entire fields of mathematics as being primarily generated by a handful of these great stories, their evolution through time, and their interaction with each other. I would thus conclude that good mathematics is not merely measured by one or more of the “local” qualities listed previously (though these are certainly important, and worth pursuing and debating), but also depends on the more “global” question of how it fits with other pieces of good mathematics, either by building upon earlier achievements or encouraging the development of future breakthroughs.

Well, for one, further to what Tao says, I would like to insist that if the mathematical community does not act to promote the telling of these ‘great stories’, then it is failing to be fully rational. Second, there’s the question of how these stories are to be told. This is perhaps not as straightforward as one might think (see here). Lastly, thinking about MacIntyre’s original motivation, mathematical rationality involving value judgements as it does, this ought to tell us something about the way other rational communities should be organised. Try running through Tao’s quotations, substituting ‘political-ethical’ for ‘mathematical’ and you have something approaching MacIntyre’s moral realism.

## Re: Why Do I Bother?

At first, from reading your title, I feared this would be a lament about the hopelessness of getting analytic philosophers of mathematics to take seriously the questions you’re raising. I’m glad it’s not!

Philosophy departments should take these questions

veryseriously. They should also hire you!