Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

March 19, 2020

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.

Posted at March 19, 2020 1:41 PM UTC

TrackBack URL for this Entry:   https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/3203

10 Comments & 0 Trackbacks

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).

Posted by: David Corfield on March 20, 2020 2:13 PM | Permalink | Reply to this

Re: Michael Harris on Virtues of Priority

This sentence from your book is a good starting point for examining the meaning of “changing the prevailing psychology”:

“What this amounts to is a shift of perspective from seeing a mathematical theory as a collection of statements making truth claims, to seeing it as the clarification and elaboration of certain central ideas by providing definitions to isolate classes of relevant entities and ways of categorising and organising information about these entities.”

Shimura (and those who elaborated, clarified, and publicized his idea) accomplished something else as well, gradually establishing a new research program that reoriented practice. This seems compatible with your perspective and maybe you develop this idea in the book as well (my copy is across the ocean but Columbia has access to the electronic version).

Most of the discussion among mathematicians has centered on the extent to which it was Shimura’s intervention (as opposed to the publication of Weil’s article, for example) that drove this reorientation. Several colleagues have written to me in private to share their thoughts on this question, but I want to stress that I have nothing to say about the controversy and I am happy to leave it to historians to sort it out. The question of “who said what when” interests me only insofar as the analysis of how whoever said whatever reflects the values underlying the rhetorical strategies employed to convince sectors of the community of the validity of claims to priority. I think this kind of analysis is properly philosophical (rather than sociological or psychological).

Posted by: Michael Harris on March 20, 2020 3:02 PM | Permalink | Reply to this

Re: Michael Harris on Virtues of Priority

…gradually establishing a new research program that reoriented practice. This seems compatible with your perspective and maybe you develop this idea in the book as well…

Along with softening up Lakatos’s hard core via a shift from axioms (as he wanted it) to something more conceptual (as in your quotation), I also raise the problem of individuation of research programmes. Of course, there’s a tremendous amount of interweaving, emergence, amalgamation. I even tried a three-levelled scheme – research traditions, research programs, and research projects – to approach the idea that commitments come with different lifespans.

Posted by: David Corfield on March 20, 2020 3:31 PM | Permalink | Reply to this

Re: Michael Harris on Virtues of Priority

To reenact one of the oppositions discussed in my article, Serre’s argument seems to be that “prevailing psychology,” or disposition to believe, should not be modified by a conjecture that is not “checkable” – and, implicitly, that has already begun to be checked. Many colleagues find this convincing. Most of the ones I know happen to be French. So is this just a matter of national traditions, or a sociological observation, or is there a real underlying philosophical distinction?

And how far are what you call “higher-level beliefs” conditioned by their checkability, as opposed to, say, considerations of parsimony – in this case, the “higher-level belief” that there should not be two completely independent ways to define L-functions, especially in view of the Hasse-Weil conjecture (what my article calls “Hasse’s conjecture”)?

Posted by: Michael Harris on March 20, 2020 10:32 PM | Permalink | Reply to this

Re: Michael Harris on Virtues of Priority

Lakatos’s hard core was designed to contain the ‘essence’ of the programme, in the sense that it identified it. Were one to give up on commitment to a part of it, one would no longer belong to the same programme. He was thinking of the natural sciences. Any experience that went against the programme’s expectation would require modification to auxiliary assumptions. So the Newtonian doesn’t give up on Newton’s 3 Laws or the Law of Gravitation.

There are problems with this model and its ways of identifying programmes, but there’s something right in distinguishing between the definitional and the testable. A classic case is ‘This rod is 1 metre in length’. Said about the standard Paris metre rule in the 19th century, this is definitional, unfalsifiable. Now that we’ve changed our definition of the metre, the length of the old relic becomes an empirical question.

Now, I take it that something of the definitional is there in mathematics at the heart of a research programme. Take one I know you’ve encountered, the HoTT programme. In its hard core is the slogan

Homotopy type theory is the internal language of an elementary (,1)(\infty, 1)-topos.

You can declare this without even having the component definitions in place. You can see things are still up in the air at elementary (infinity,1)-topos. Experience will play its role in prompting theoretical change, but the slogan will remain.

Of course, in number theory you make contact with the ‘empirical’ (counting solutions to elliptic curves, and the like). Do you have examples of adjusting definitions to preserve some larger conceptualisation?

This whole question of definitional adjustment in the face of experience I find intriguing.

Posted by: David Corfield on March 21, 2020 9:22 AM | Permalink | Reply to this

Re: Michael Harris on Virtues of Priority

At some point we should think about returning to Alasdair MacIntyre, although I’m not sure I have anything to add to what I wrote in my book, which itself was a continuation of what I understood from your article. Right now I want to explore Lakatos’s hard core in the context of the Modularity Conjecture – without consulting my copy of Lakatos’s text, which is currently inaccessible.

A hard core is hidden in my article on Virtues of Priority, namely the conjecture of Hasse about the analytic continuation and functional equation of the L-function of an elliptic curve. All three protagonists were guided to the conjecture as a consequence of Hasse’s conjecture, plus the belief that L-functions satisfying functional equations must necessarily be attached to modular forms. This consequence was spelled out in what my text calls Taniyama’s “theorem.”

Admitting this for the moment, we can ask: why was it rational, during the period between the Tokyo-Nikko Conference and the publication of Weil’s paper, to believe Hasse’s conjecture? I think most number theorists would mention Deuring’s work on the L-functions of elliptic curves with complex multiplication, and we would have to add Shimura’s theorem mentioned in footnote 23, although it was published after this period. Still, it’s a long way from elliptic curves with complex multiplication to general elliptic curves over Q – that’s the point Weil was making in the comment Lang called “stupid” and “inane.” So the protagonists were motivated by something more than Deuring’s (empirical) confirmation of Hasse’s expectation. We can call this something more the “hard core” of the research program on the arithmetic of elliptic curves as it stood in the 1950s and 1960s.

By the middle of the 1960s there was a much more general conjecture, called the Hasse-Weil Conjecture; Weil had defined zeta functions of general varieties over number fields by the early 1950s, and Serre’s article “Zeta functions and L-functions,” published in the mid 1960s, was a standard reference when I was a student. One can again ask about the rationality of believing the Hasse-Weil Conjecture during this period, when it was hardly “checkable.”

The Langlands program changed the context dramatically at the end of the 1960s. Twenty years later, the work of Frey and Ribet made it irrational not to believe Fermat’s Last Theorem, in the semse that its failure would be an “experience that went against the [Langlands program’s] expectations.” What this means to me is that the Langlands program had by then been integrated into the hard core of the research program on the arithmetic of algebraic varieties, including elliptic curves.

At the very least, this suggests to me that research programs in mathematics evolve by expanding their scope and by incorporating the hard cores of neighboring programs. This would also mean that the hard core is not so hard as Lakatos may have wished it to be. The recent work on elliptic curves over imaginary quadratic fields (the “ten-author paper”) and on abelian surfaces over Q (Boxer-Calegari-Gee-Pilloni) has expanded the scope of this research program once again by incorporating (among other insights) recent developments in p-adic Hodge theory inspired by and developed by Scholze and others, and homotopy-theoretic methods – but not HoTT, at least for now – that were scheduled to be studied in depth at an Oberwolfach meeting (https://www.mfo.de/occasion/2014/www_view) unfortunately now cancelled.

This is long enough, but at some point I would like to describe the evolution of the hard core of the research program around the Gan-Gross-Prasad Conjecture. In the course of a (virtual) conversation about the contents of my Virtues of Priority paper, Dick Gross sent me a copy of a letter by Dipendra Prasad that can plausibly be identified as the official beginning of this particular line of work as a conscious research program.

Posted by: Michael Harris on March 22, 2020 7:44 PM | Permalink | Reply to this

Re: Michael Harris on Virtues of Priority

Something if we return to MacIntyre is the issue of Lakatos’s “rational reconstructions” which leave out the irrationalities of actual humans. MacIntyre insisted on the telling of true histories of intellectual enquiry.

this suggests to me that research programs in mathematics evolve by expanding their scope and by incorporating the hard cores of neighboring programs.

In Chap. 8 of Towards I considered algebraic topology as a research program as a continuation of the Euler conjecture from Proofs and Refutations. His suggestion that axioms made up the hard core, seemed then to be wrong. We wouldn’t say that, say, the Eilenberg-Steenrod axioms define the field. I was led then to consider differences of scale, hence what I wrote above:

there’s a tremendous amount of interweaving, emergence, amalgamation. I even tried a three-levelled scheme – research traditions, research programs, and research projects – to approach the idea that commitments come with different lifespans.

Algebraic topology became something all-pervading, its ideas used all over the place, so that it becomes hard to identify something to play the role of its hard core. One might try something like a commitment to the idea that important topological distinctions can be made by assigning algebraic invariants. In such a vague form, it becomes all but unfalsifiable.

It seems then that one might take commitments on a scale from the all-but-unfalsifiable to the falsifiable (by explicit counterexample). It sounds as though your suggestion “the conjecture of Hasse about the analytic continuation and functional equation of the L-function of an elliptic curve” as a hard core element is at the falsifiable end. Is it that it was quite clear what it would have meant to provide a counterexample? In your description of the Hasse-Weil Conjecture at a time ‘when it was hardly “checkable” ’, is this just due to the limited calculating reach of the time?

The all-but-falsifiable end is closer to what Lakatos meant. Consider in physics, something like F=ma F = m a for the Newtonian. If we have a situation which appears not to satisfy it, then we consider that there are unknown forces or we’ve measured the mass incorrectly. It’s playing more of a stipulative role. (This heads in the direction of the philosopher Michael Friedman, some slides of a talk are here.)

So I think this is an important dimension: how does one react when expectations are not met. It’s one thing to have a falsifiable conjecture, and then give it up or modify it when a counterexample comes in. But there’s also the level of the more nebulous conception which may be clarified through investigation, but won’t be readily given up. In my Friedman talk, the latter might apply to a claim like ‘cohomology is an important device’. Its thinkable that in 200 years there won’t be any concept that is recognisably what we mean by ‘cohomology’ today, even if it seems unlikely today.

Is there anything in the field you’re describing that has this all-but-unfalsifiable quality?

Posted by: David Corfield on March 23, 2020 9:05 AM | Permalink | Reply to this

Re: Michael Harris on Virtues of Priority

I think the Hasse-Weil conjecture was – and still is – not “checkable” because there is no systematic way to prove the analytic properties of the L-function attached to an algebraic variety. The special case of the Modularity Conjecture proposed an approach in the form of a modular form. The Langlands program generalizes this by assigning adelic representations to the cohomology of the variety and then conjectures that these representations are automorphic. Laurent Lafforgue’s theorem uses the results of the Grothendieck school, among many others, to show that this conjecture is actually true over function fields. But over number fields most of the adelic representations don’t have the shape that lends them to “checking” in the sense intended by Serre.

To answer your last question, I wish I could say that the Langlands conjectures are “all-but-falsifiable,” but I don’t think that will be possible until there is a reason for Langlands duality that is as transparent and natural as the Fourier transform. In that sense the theory remains to find its “hard core.”

But let me pivot to virtue ethics and ask whether you see any behavior in this story that is inconsistent with the professed virtues of the mathematical community. Or perhaps is the point that the community has failed to come to a consensus on the what counts as primary among the possible virtues of priority?

Posted by: Michael Harris on March 25, 2020 2:04 PM | Permalink | Reply to this

Re: Michael Harris on Virtues of Priority

MacIntyre, being a moral philosopher in the virtue ethics tradition, is more interested in naming the virtues of practitioners than the virtues of their work. There were attempts in the philosophy of science to give characterisations of progress in theories. Lakatos hoped for “honest scorekeeping” where members of rival programs could gauge how each other were faring. The other program might be doing well, you should recognise this, but you are within your rights to remain with your program since it may contain untapped heuristic resources. Kuhn emphasised incommensurability between paradigms, including their standards of appraisal – what counts as success for one group might not for the other. Still there are for him some general criteria – accuracy, consistency, scope, simplicity, and fruitfulness.

So, in mathematics what is there to say? I wondered about this when carrying over research programs to mathematics (Chap. 8). Then I looked at the reasons given for using groupoids instead of groups (Chap. 9). While one can roughly define kinds of advance - generalisation, application, naturalness, and so on, I wonder in the end if there’s anything more precise to say.

Ah, I thought I’d had some further ideas about this. I discussed a paper by Tao, ‘What is good mathematics?’ back here at Why do I bother?, thinking it rather good. However, a few years later here, I write:

“You ask whether philosophers have attempted lists of criteria for evaluation. The short answer is ‘no’, apart perhaps from my own attempt in my chapter on groupoids in my book, which was closely related to that one of Ronnie’s you mentioned on MO. From my current perspective, I think the attempt to give anything very prescriptive is misguided. Here’s something from a piece I’m working on:

I have mentioned Atiyah’s reliance on the noun ‘story’ in his survey, but it is clear from his article that it is inseparable from a form of judgement. He is telling us there what he considers to have been the most important developments through the past hundred years, and what he considers to be promising for the future. Recently Terence Tao has written a piece ‘What is good mathematics?’, which after making a list of what ‘good mathematics’ can mean, continues by telling us the story of what he assesses to be some good mathematics. The list, consisting of twenty items, received this harsh assessment from Alain Connes:

It is hard to comment on Tao’s paper, the second part on the specific case of Szemeredi’s theorem is nice and entertaining, but the first part has this painful flavor of an artist trying to define beauty by giving a list of criteria. This type of judgement is so subjective that I really had the impression of learning nothing except the pretty obvious fact about arrogance and hubris…

The story, the ‘second part’, escapes censure.

I think Connes is right to point to what is troublesome about lists of criteria. Criteria seem clumsy. Also in mathematics at least it seems rather easy to do well. In chapter 8 of my book, the Methodology of Mathematical Research Programs, it was too easy to give an account of a program scoring well. What are we left with in Tao’s piece then is a good story. Would this be such a bad thing to conclude: Good mathematics is that which can be described by good mathematical stories? But what is a good mathematical story? Can we capture this except by listing qualitities we would expect to find in it? Fortunately assessment is made easier by a phenomenon Tao notes that when good things happen in a piece of research other good things follow in its wake.

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.

…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.

The primary problem with the list of criteria is that we should expect what it is to be a good story to change. We need a meta-level story of how we have moved on from old stories. These days we expect surprise, such as when Vaughan Jones working on von Neumann algebras very unexpectedly realises he has his hands on a new knot invariant. It would be a very worthwhile task to examine the way overviews have changed in style over the years. For example, how does Klein’s Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert compare with modern surveys?”

Rather a jumble of thoughts. I’m not even sure where this writing of mine ended up. I’ll see if I can straighten them out a little tomorrow.

Posted by: David Corfield on March 25, 2020 8:48 PM | Permalink | Reply to this

Re: Michael Harris on Virtues of Priority

A lot of issues are swirling about together here. There’s the MacIntyrean thought that it is not because one is a great X-ist that one appears in the history of X, but the other way around. A X-ist is great for doing work that appears in the history of X. But then what properties should this history of X possess?

Can there be a history of the ideas of X which abstracts somewhat from the messy details of their human development. In a history of group theory, do we include, say, details of Cauchy’s unfair treatment of Galois and Abel? Lakatos argues for rational reconstructions. Ideas have their heuristic potential and humans may stand in the way of their progress. Kuhn thinks these reconstructions are fairy tales. Many current historians of science would take Kuhn himself as too ‘intellectualist’ and ‘internalist’. Others now think we’ve strayed too far from the history of ideas in focusing on the political and material conditions of science.

Much of the history of mathematics has been done by mathematicians rather than historians. This raises the charge that the latter are engaged in Whiggishness, only understanding the past inasmuch as it leads to the present. Grattan-Guinness wanted to call these activities by different names - ‘heritage’ for what the mathematicians do. In the paper that I can’t lay my hands on (hopefully safely somewhere on my office computer) I was arguing against this claim. In histories of intellectual enquiry there should be an element of considering ‘what something will have become’.

So what to do with two rival histories of an episode of mathematics? Is at least one necessarily wrong? If appraisals of the rational value of a piece of research are tied up with ways of writing its history, competing histories would be expected. Values are not fixed but are contested and are modified over time. Weierstrass (if I recall) thought poorly of Dedekind and Weber’s analogising between rings of polynomials and numbers rings for mixing ideas from different fields (an idea relating to the ‘purity’ we were told about at a talk by Andrew Arana in Paris). Such high-level meta-commitments vary in time so that today cross-field transfer is simply commonplace. One may hope that there is a rational story to tell as to the changes in such commitments.

In the case of your rival histories, what is at stake? Is it that to prefer one will typically make a difference to how one will act now? Is it then that we might see as time unfolds that one history was a better indicator of the more productive way to view things? Had Weierstrass written a history of mathematics in, say, 1895, he might have omitted Dedekind and Weber’s 1882 paper. We would surely judge such a history now as inadequate, even if this was not clear at the time.

Posted by: David Corfield on March 26, 2020 9:38 AM | Permalink | Reply to this

Post a New Comment