### Michael Harris on Virtues of Priority

#### Posted by David Corfield

Michael Harris has an interesting new article on the arXiv today - Virtues of Priority. He wrote it for an edition of a philosophy journal on virtues in mathematics, but, as he explains in the footnote on the first page, it has ended up being published on the arXiv rather than in that journal. I think it provides interested philosophers of mathematics with excellent material to think through issues concerning the role of the virtues in intellectual lives.

Abstract: The conjecture that every elliptic curve with rational coefficients is a so-called modular curve – since 2000 a theorem due in large part to Andrew Wiles and, in complete generality, to Breuil-Conrad-Diamond-Taylor – has been known by various names: Weil Conjecture, Taniyama-Weil Conjecture, Shimura-Taniyama-Weil Conjecture, or Shimura-Taniyama Conjecture, among others. The question of the authorship of this conjecture, one of whose corollaries is Fermat’s Last Theorem, has been the subject of a priority dispute that has often been quite bitter, but the principles behind one attribution or another have (almost) never been made explicit. The author proposes a reading inspired in part by the “virtue ethics” of Alasdair MacIntyre, analyzing each of the attributions as the expression of a specific value, or virtue, appreciated by the community of mathematicians.

MacIntyre doesn’t appear directly in the article, but he does in Michael’s book Mathematics Without Apologies, a work which, along with other writings, reveals Michael’s broad interests in philosophy. (Some may recall, or even have participated in, discussions about HoTT on his site. Search for ‘HoTT’ there.)

I came to know Michael via a group known as Thales and Friends, led by Apostolos Doxiades. Apostolos held a large conference on Mykonos in 2005, which was followed 2 years later by a smaller meeting at Delphi. The latter I attended with Michael (and John Baez). Papers written for this latter meeting were collected in the volume A. Doxiadis and B. Mazur (eds.), Circles Disturbed, Princeton, 2012. My own contribution is here, largely devoted to exploring what MacIntyre’s ethical theory can say to mathematics as an example of rational enquiry. An intriguing further aspect of the meeting was that the 16 participants interviewed each other in pairs. You can read my interview of Barry Mazur and Colin McLarty’s interview of me here.

For MacIntyre, membership of communities of rational enquiry requires the possession of many virtues, both moral and intellectual. A task for the community is the ever-improved telling of a “true dramatic narrative” of their passage to the present moment, and directions for the future. At the core of Michael’s case study from number theory are details of the problems of achieving a commonly agreed such narrative. Something I’d like to explore is, on the one hand, what we can attribute to the natural differences between participants, their differences in the emphases they place on various attributes valued in mathematical work, and, on the other, what we can attribute to the failure to some degree to exhibit certain virtues, say, lack of justice in according proper due to others; a lack of willingness to engage with rival understandings, and to expose one’s own understanding to challenge; a lack of the enjoyment that is appropriate when one is shown to be wrong; a failure to share understandings; and so on. If discussions of priority are “quite bitter”, I should imagine we will find at least some lack of virtue.

## Re: Michael Harris on Virtues of Priority

For background on the (non-)publication of Michael’s article, see his blog post.

We could see what Michael has written in the article as contributing to his community’s efforts to tell its story correctly, and this not primarily for the fair attribution of praise, but for the good of the current members in finding the best paths forward. This could play some role in modifying the understanding of the community.

More broadly, his reflections on different understandings of the achievements of historical characters could provide useful material for reflection by mathematicians in other fields and by philosophers. In particular, I was interested in the distinction: Lang’s realism, Shimura’s phenomenology, Weil’s falsificationism (on Serre’s reading).

A couple of initial reflections: perhaps from the discussion there, with its talk of logical positivism and of Popper, some support is provided for my use of ideas from the philosophy of science throughout my book – Towards a Philosophy of Real Mathematics. In particular, chapters on inductivism, plausibility via analogy, Bayesianism, research programmes.

Second, the “changing the prevailing psychology” concept applied to Shimura seems to me close to what I was driving at in my reformulation of the research programme idea to allow higher-level beliefs at their core (pp. 181-182). It seems to me also close to what Weil meant by métaphysique in his 1960 paper ‘De la Métaphysique aux Mathématiques’ (p.90 in my book).