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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

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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

Re: Michael Harris on Virtues of Priority

I have some general reflections here. The first is that it is more or less impossible for a career mathematician to be able to make a ‘free’ philosophical reflection upon mathematics. A successful mathematician has necessarily been subservient to certain norms which are necessary for that success. This is the same for any discipline. It takes a very rare personality, absolutely necessarily on the outside, to think more freely. Nietzsche would be a good example within philosophy.

The possibility of a Nietzsche-like figure arising for mathematics is remote, because mathematics is highly conservative and elitist. This is uncomfortable reading for the typical mathematician, who would prefer to be thought of in very different terms. But it is absolutely necessary to conform to some extent to the mathematics-as-logic view to succeed as a mathematician, not only in the sense of being able to pursue a career as a mathematician, but even in looser senses which are important for the philosophical view David is taking following MacIntyre, for instance of being able to communicate to other mathematicians. Even if a particular mathematician is aware that they are conforming in this way and try to rebel against it in some way (Thurston might be an example here), they still evidently are of a psychology that is fundamentally willing to conform, and such a psychology can never be that of a Nietzschean figure unless that figure is really on the outside (which Thurston for example certainly was not).

In my view, the entire conception of mathematics-as-logic or truth is fundamentally flawed. Two people can for example communicate deeply and achieve some fundamental understanding in exploring some question, even if that understanding might not be able to be communicated more widely, and even if it might be wrong or not logical in all respects by the usual conservative-elitist standards. This is not a ‘genealogical’ position in the sense of MacIntyre either; those two people fundamentally agree at a particular point in time, there is no will-to-power involved. One might say it is a more human view of what mathematics is. It is also a more artistic one.

More so than in mathematics, one might look to didactics for inspiration to understand what I am getting at. Children may express an opinion on something that is wrong in a logical sense or even factual sense, but is brilliant in its own way, and has the essence of logic about it.

I would emphasise finally that, though I ultimately do not take a genealogical view of what mathematical practise is, the genealogical position can never be dismissed lightly. In a highly conservative and elitist discipline, conforming to the value judgements of the elite is likely to lead to success in one form or another (recognition from peers, career advancement, …). Though it may be uncomfortable reading, this is will-to-power. Take Scholze’s work for example. All of the standard narratives can and have been written for this. But fundamentally it is will-to-power. One takes problems that the elite consider hard, and one solves them in ways that impress the elite. Since the elite are so important within mathematics, the masses take their opinion from the elite, hence the masses are impressed too, and thus the narrative which all kinds of mathematicians construct for themselves begins to touch on Scholze’s work. This is not a criticism of Scholze, it is just the way things are.

A slightly different example is that of the work of Yitang Zhang. All kinds of narratives were constructed around this; e.g. how in mathematics truth will out, no matter how unknown the mathematician. This is a myth. Zhang’s work impressed the elite because it conformed to their expectations of what work on a difficult problem should look like. If Zhang had merely sketched a solution, his paper would have been dismissed out of hand. But the mathematics he had done in the years working on the problem, even when he did not have a solution, would have been no lesser mathematics, though, being a solitary endeavour, it would have contributed nothing to mathematics from the point of view of David and MacIntyre either, never mind from the traditional encyclopedic point of view.

Posted by: Richard Williamson on March 29, 2020 2:18 AM | Permalink | Reply to this

Re: Michael Harris on Virtues of Priority

The original title of that MacIntyre article of mine was ‘How mathematicians fail to be fully rational’. The complaint was that there was not enough exposition of conceptual understanding. My motivation in co-founding this blog was to provide a space for the sharing of understanding in the way of your third and fourth paragraphs. There’s a nice opportunity I’ve just made available to revive the old times.

As for the paragraph on the elite, I never saw how Feyerabend thought we could manage without a reasonable degree of assessment by leaders in a field. Of course, this risks narrowness of vision, but there is the other pitfall of ‘anything goes’. One would hope that the leaders are trying their hardest to act for the good of the discipline. And I also hope that this isn’t a pious hope.

As for your point about Zhang, there are two issues in your counterfactual scenario – had he merely sketched a solution and had he kept the work to himself. Since, it’s the latter aspect that concerns your final claim, having never committed myself to problem-solution as the main goal, let’s consider that.

With the emphasis on community endeavour, a healthy community requiring the exercise of the intellectual and moral virtues, it would appear that the work of the isolated, solitary researcher cannot be valued by the tradition-constituted approach. One might extend valuation by use of a counterfactual, “XX’s work is good because, had it been received in timely fashion, it would have influenced the course of the field”. Evaluating such a counterfactual in the case where you’ve already used a counterfactual – i.e., Zhang’s work hadn’t been known – so that we now can say something sensible about the counterfactual, makes it seem that this extension could work. But note that it is only against the backdrop of mathematics being the kind of intellectual enquiry where there is reasonably stable assessment of what counts as a good contribution, that we can even entertain the idea of counterfactual influence. And I would also note that Zhang could only come to work on his problem because there was a tradition of people who developed the field to a certain state.

The work of the individual can only make sense against the background of a community.

Posted by: David Corfield on March 29, 2020 2:13 PM | Permalink | Reply to this

Re: Michael Harris on Virtues of Priority

Richard wrote:

A successful mathematician has necessarily been subservient to certain norms which are necessary for that success.

mathematics is highly conservative and elitist. This is uncomfortable reading for the typical mathematician

As a “typical” career mathematician, I agree with both those sentiments. Or to be more pedantic about it, I agree with them with the interpretation I choose to give them, e.g. interpreting “mathematics” to mean “mathematical society”.

For what it’s worth.

Posted by: Tom Leinster on March 30, 2020 6:01 PM | Permalink | Reply to this

Re: Michael Harris on Virtues of Priority

This is a rather surprising direction for a discussion of priority, but it’s certainly interesting. I wouldn’t say that it’s at all uncomfortable for the “typical mathematician,” whoever that might be, to read mathematics described as “highly conservative and elitist.” On the contrary, in certain narratives, notably the Inclusion/Exclusion blog on the AMS website, and rather widely in the area of mathematics education, the conservatism and elitism of mathematics is discussed on a daily basis, with vocabulary including terms like “patriarchy,” “racism,” “colonialism,” and the like. It was also very much in evidence at plenary and special sessions of January’s Joint Mathematics Meeting in Denver. Roughly as many people signed the letter objecting to Abigail Thompson’s article on diversity statements as signed the letter objecting to the objections. Which of the two groups is “typical”?

While I respect the position of critique of mathematics on grounds of its (sociocultural) elitism, I have preferred to develop a rather different critique, where the vocabulary involves (at least potentially) terms such as “neoliberalism,” “benchmarking,” “social responsibility,” “technocracy,” and “student debt.” I have been thinking quite a lot about these questions as I prepare the materials for online teaching. But that’s not really germane to the question at hand! Elite decision making within the profession was the subject of Chapter 2 of my book. At the end of that chapter I found myself led to the surprising conclusion – surprising for me – that a certain form of elitism is constitutive of the practice (tradition-based, in MacIntyre’s sense) of pure mathematics. And by “constitutive” I don’t mean merely sociologically. This particular fly-bottle has been uncomfortable enough to lead me to hope for a philosophical means of escape, and I wrote the article on Virtues of Priority in part with that in mind.

Posted by: Michael Harris on March 30, 2020 8:30 PM | Permalink | Reply to this

Re: Michael Harris on Virtues of Priority

a certain form of elitism is constitutive of the practice (tradition-based, in MacIntyre’s sense) of pure mathematics.

‘Elitism’ is generally used in a loaded way, with the connotation that members of the elite have not earned their position by proving their worth, although it simply derives from the Latin word ‘eligere’ - to chose, elect.

‘Aristocracy’ has suffered a similar fate. Literally ‘rule of the best’, ‘aristos’ is cognate with ‘arete’ – excellence, which, significantly for this thread, is the word Aristotle uses for ‘virtue’.

MacIntyre’s tradition-constituted version of enquiry certainly requires leaders or masters, as he puts it. From my Narrative article:

So we have the movement of a community of enquirers toward a telos, where the best understanding of this movement is through a narrative account of the path to the present position. Becoming a member of the community, you identify with this story and seek to find your place in its unfolding. The understanding of this story is passed on by teachers, who instruct new members in becoming experts in the community.

The authority of a master is both more and other than a matter of exemplifying the best standards so far. It is also and most importantly a matter of knowing how to go further and especially how to direct others towards going further, using what can be learned from the tradition afforded by the past to move towards the telos of fully perfected work. It is thus in knowing how to link past and future that those with authority are able to draw upon tradition, to interpret and reinterpret it, so that its directedness towards the telos of that particular craft becomes apparent in new and characteristically unexpected ways. And it is by the ability to teach others how to learn this type of knowing how that the power of the master within the community of a craft is legitimated as rational authority. (MacIntyre, Three versions of moral enquiry, pp. 65–66)

For the encyclopedist there is no need for such lengthy instruction, for the genealogist what is at stake is indoctrination to maintain power. (Narrative, pp. 247-248)

Of course, an intellectual practice may be led astray – the leaders may lead badly, and this perhaps for political reasons, such as notoriously with Soviet biology under Lysenko.

There will be more subtle defects arising when less good practitioners are promoted over others. Any number of vices may be involved in leading a practice astray, but that there needs to be judgement by leaders as to the quality of work, and that these judgement will have consequences for people in the field, seems evident.

Posted by: David Corfield on March 31, 2020 9:38 AM | Permalink | Reply to this

Re: Michael Harris on Virtues of Priority

My article alludes briefly to the role of leaders in connection with the “prevailing psychology”:

“Lang indirectly hinted at a model of the mathematical subject by assigning to Shimura direct responsibility ‘for changing the prevailing psychology’ by privately conveying his belief in the truth of what we are calling the Modularity Conjecture to two or three individuals. This subject is structured hierarchically, so that impressions acquired by those (like Weil, Serre, or Tate) at the pinnacle of the hierarchy have the power of detectably altering dispositions throughout the subject.”

No one would deny that the five mathematicians mentioned in this paragraph were recognized leaders, but it’s still a problem when they identify the “prevailing psychology” with their own psychology. What “prevails” depends both on positions of authority within the hierarchy and the rhetorical (narrative) skills and willingness to expend energy on the part of those who occupy those positions. This is what is usually called “charisma,” not the same as the notion explored in chapter 2 of my book, which in principle is acquired when one enters the hierarchy, even at its lowest levels.

We are used to developing our tastes under the guidance of leaders. It seems inevitable that they would also determine what counts as virtues. “Checkability” would not be a recognized virtue if someone like Serre (but there’s no one quite like Serre) hadn’t proposed it as a criterion. At this point a historian is needed to tell us whether the virtues of mathematical practice come and go according to the preferences of acknowledged leaders, or whether they show some constancy over different periods.

Posted by: Michael Harris on March 31, 2020 10:18 PM | Permalink | Reply to this

Re: Michael Harris on Virtues of Priority

That there have been profound disagreements among leaders over the centuries as to the right form of mathematics is surely common knowledge – synthetic geometers rejecting the intrusion of algebra into geometry, arithmeticians rejecting the intrusion of geometry into arithmetic, the path leading from Dedekind to Noether to Mac Lane meeting with great resistance along the way, the Hilbert-Brouwer dispute, and so on.

I suppose it might be that members of a fairly fixed family of values are espoused by a varying cast of leaders as time passes. So we might have successive occupation of the roles of the one who insists on ‘purity’, the one who seeks a rootedness in calculation, the one who seeks unification, and so on.

In making such observations, nothing would speak to the question of the rationality of the espousal of particular values at any time. But surely there are things to say here, perhaps even about the desirability of maintaining a pluralism. Something we can do is look back at disputes in the past and see whether we can discern that one party had a better sense of how things should go, bearing in mind that we may be judging from a perspective partly determined by such previous decisions.

Posted by: David Corfield on April 1, 2020 8:45 AM | Permalink | Reply to this

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