This is what charisma looks like, part 3: Is mathematics democratic?

This post is a series of scattered thoughts in reaction to a  comment by Mike Shulman dated May 26:

I don’t really know whether there are more democratic methods of organizing mathematics that would be as or more effective than the current dependence on charismatic leaders; it seems related to the search for alternative models for recognizing and validating research that could replace the current referee and journal publication system.

The comment was itself in reaction to the implicit assumptions of two articles on HOTT/Univalent Foundations that appeared in close succession in online publications:

What bugs me is when people assume that because of Voevodsky’s charisma, that his interests define the field, or that the rest of us are all following his “research programme”. I used to find such assumptions totally baffling; but after reading chapter 2 of your book, I get the impression that there are fields of mathematics in which it really is the case that the “big name” mathematicians set out “research programmes” that everyone else is expected to follow. It’s not at all what my experience in mathematics has been like (including work in algebraic topology, category theory, and now homotopy type theory), but maybe the problem is that people from fields that do work that way assume that all fields work that way.

I didn’t want the thoughts to be so scattered, but I’m in a hurry to get to the next post before the ink dries, so to speak.

I organized Chapter 2, and most of the rest of the book, around the Langlands program and Grothendieck’s conjectures on motives, as well as the related Guiding Problem of the Birch-Swinnerton-Dyer Conjecture, not only because that’s what I know best, but because they offer an exemplary picture of mathematics as a collective activity, and therefore a social activity, and therefore (at least for the foreseeable future) a human activity.  Probably I should have insisted more strongly on the exceptional character of these research programs, whose expected outcomes have been predicted in unprecedented detail.  There is still plenty of room for surprising proofs, but most mathematical fields seem to leave much more room for surprising results.  I don’t know whether even so structured a field as homotopy theory has long-term research programs or even Guiding Problems.

I have sat on enough hiring committees and editorial committees and other sorts of committees to know, however, that homotopy theory has its share of “big name” mathematicians, and I’d guess no branch of mathematics that attracts the attention of hiring committees and the rest is very different in that regard.  The argument could be made that the very existence of such committees presupposes some hierarchical standards for making decisions.  I wouldn’t want to make such an argument, because it one sense it’s a tautology and in another sense it looks like a defense of hierarchy, which is not at all something I’m inclined to defend.  On the contrary, the main irony of chapter 2 is that, despite my not particular auspicious beginnings, I’ve learned more than I ever wanted to know about how hierarchy works in practice.

Universities in France are run very differently than in the US.   Salaries are public and are determined by the civil service pay scale.  Some university professors reach the top of the scale but the highest salary for a professor (classe exceptionnelle, 2ème échelon) is just slightly over 5200 euros per month, after deductions for national pension and health insurance (but before income tax).   Some state and local universities in the US have a similar system, but in France there are no exceptions and the university administration cannot make special deals to attract professors.  (At least that’s how it was until a few years ago; it is possible that some university presidents are taking advantage of loopholes in the recent laws increasing their “autonomy,” but I haven’t heard of any such cases.)

Within each category, one rises in the ranks on the basis of seniority; then one applies for a promotion to the next category.  The big step is from maître de conférences to professeur — both of these are tenured positions — and this involves a separate hiring procedure.  Most of the promotions within these two large groups are decided by an elected national committee, one for each discipline, and it was on this committee that I served for a few years as an elected bureaucrat.  The number of promotions available to this committee is decided centrally, by the ministry, and there is naturally a fair amount of political maneuvering.  (There are also a few promotions decided locally, in principal for service to the university.)  Political lines are drawn between fields (algebraists vs. analysts) and between the platforms represented on the committee — in mathematics there were trade union lists and one now called Qualité de la science française.  What I want to emphasize here is that, even though the trade union lists are ostensibly committed to promotion strictly based on seniority, while QSF is supposed to be more elitist, in my experience the same charismatic criteria were applied by both groups.  (Maybe it’s different in other fields than mathematics.)

The conflict between democracy and elitism is resolved in France by the paradoxical notion of élitisme républicain.  This is what allows France to run what may well be the world’s most elitist centralized public systems of higher education and to turn out an elite that is sincerely and militantly democratic.  Practically all my French colleagues who were high school or college students in 1968 were either Maoist or Trotskyist (there are still quite a few of the latter), and those of the previous generation were often members of the French Communist Party.  And this is hardly surprising when you consider that Louis Althusser was teaching at the Ecole Normale Supérieure, the innermost sanctum of French elite higher education, where a staggeringly large proportion of my colleagues learned to be mathematicians.

The absence of big salary differences, and of the temptation to move elsewhere in search of a higher salary, makes for a healthy atmosphere, and it’s reassuring to know that most of your colleagues will vote to strike to protest a regressive decision by whatever government is in power.  The regressive decisions go forward nevertheless, and the result is a shrinking budget for higher education and a steady drop in the number of available positions (in spite of an increase in the number of students), and one wonders how much longer the healthy atmosphere will last.

I have the impression that I wanted to say something else in response to Mike Shulman’s comment, but I have forgotten what it was.


9 thoughts on “This is what charisma looks like, part 3: Is mathematics democratic?

    1. Martin Krieger

      Since the 19th century, there have been programs and leaders. Mathematics was now housed in research-oriented colleges/universities.

      Perhaps an example from physics will be useful. Geoffrey Chew had the S-matrix program in the 1960s (called “nuclear democracy,” by the way), when quantum field theory did not look too productive for the strong interactions. Lots of students and followers. But eventually the program did not prove productive, AND quantum field theory and what we now call the Standard Model proved very productive (and those who had worked in this arena became leaders of a program). The legacy of S-matrix was important work by others, and nowadays a return to its themes in Zvi Bern and collaborators’ work, but not the same methods.

      My point here is that programs for research have to be productive of interesting work, that charisma must be grounded in actual significant work, and even if a program fails it may well leave a very useful legacy. Strings have quite useful ideas applicable in other than particle physics, not to speak of mathematics. As for “democracy,” what makes that possible in the US is the diversity of institutions, so that some institutions have little concern with strings, others might be more concerned with condensed matter than particle physics, etc.

      In an case, I am not sure what “democracy” has to do with the discussion. There must be a better word.


      1. mathematicswithoutapologies Post author

        There’s this quotation from Chapter 2:

        “Democracy should be used only where it is in place,” wrote Max Weber in the 1920s. “Scientific training … is the affair of an intellectual aristocracy, and we should not hide this from ourselves.”

        And then this, in connection with the Elsevier boycott:

        A few participants in the Math 2.0 blog, launched soon after Gowers announced his Elsevier boycott, wonder why publishing can’t be reorganized along similar lines, eliminating the profit motive and replacing the less democratic features of the charismatic hierarchy by a permanent plebiscite.75 One blogger wrote “…some people like ‘elite communities,’ [others] prefer more democratic communities,” suggesting that the latter are in the majority.

        Of course I agree that “charisma must be grounded in actual significant work,” but if you ask a philosopher or a sociologist how one decides whether or not the work is significant, you are unlikely to be satisfied by the answer you get. And mathematics, unlike physics, cannot rely on the universe for help in drawing the line.


  1. John Sidles

    This particular Mathematics Without Apologies essay can be read as suggesting that progress in mathematics is limited as much by the sparsity of high-quality colleagues as by the sparsity of high-quality appointments.

    By what means (if any) can the skill of collegiality be taught-and-learned? The meditations of the charismatic skateboarder Rodney Mullen, posted as the YouTube video “Rodney Mullen: a Beautiful Mind” — surprisingly including Mullen’s tribute to Cédric Villani (at 00:12:20) — may bring smiles to readers of Mathematics Without Apologies.

    Conclusion  Passion is diminished by shortfalls of collegiality even more than by shortfalls in appointments.


  2. galoisrepresentations

    I think the quote is misleading. I don’t think mathematicians believe that people in HoTT are simply following Voevodsky’s programme, merely that his imprimatur is responsible for the cash/effort being spent in this direction by virtue of his charisma. Number Theory is fortunate in this regard; the charisma associated to the field has built up over centuries (Fermat, Gauss, Euler, etc.). It would also be misleading to imagine that number theorists are simply following a programme laid out by Langlands; while he identified several important questions and guiding principles, much of the progress has come in completely unexpected ways (see Wiles, Andrew). In fact, some of Langlands suggestions on how to answer some of these questions have been mostly ignored, and it’s probably fair to say that he’s slightly pissed off about it.
    I also wanted to stand up for hierarchy, although my current state (iPhone at O’Hare) prevents me from expanding on this point.
    Finally, I wasn’t sure whether readers were supposed to take your defence of the French system ironically or not; at the very least, I was amused.


    1. mathematicswithoutapologies Post author

      I’ll let the author of the post reply to the first part of your comment, and only repeat that chapter 2, apart from serving as an introduction to some of the book’s main themes, and as an extended comment on the irony of having been written by its particular author, was designed to develop the hypothesis (which I don’t necessarily believe) that mathematical ontology is based on a specific social construction, namely the charismatic hierarchy. That’s not what this series of posts is about, however.

      I don’t think I was in any way defending “élitisme républicain”, but my admiration of the democratic aspirations of my French colleagues was perfectly sincere. Living as you do in a city of futures traders whose maps don’t even show the site of the Haymarket massacre, I can understand why you might find this implausible.

      Did you remember to bring a musical instrument?


  3. AV

    This is a bit off-topic relative to your post, but you said about the Langlands program (among others): “…but most mathematical fields seem to leave much more room for surprising results.”

    I think this is part of the unfortunate way that the Langlands program is often defined, namely, the entire goal of the program is to prove a bijection between (some versions of) Galois representations and automorphic representations.

    In practice (say, if you survey people who consider that they work on the Langlands program) it seems to me that the Langlands program is much broader than that – something like a cloud of interesting phenomena surrounding this bijection (e.g. the Gan-Gross-Prasad conjecture, which was certainly not in the original scripture). The bijection itself seems to me just a convenient central reference point.


    1. galoisrepresentations

      Plus one to AV. Also, I decided that a Steinway would be too large to count as a carry on item. The lecture theatre has a serviceable piano, however. (Having read the remainder of the book on the plane, I noticed that you covered in the later chapters some of the points I was trying to make… )


      1. mathematicswithoutapologies Post author

        You were supposed to be sleeping on the plane so you would be awake when I arrive, with two small musical instruments, and ready to help you and your fellow Australians finish off that bottle of wine you mentioned a few weeks ago.

        Speaking of Australians, I have a growing suspicion as to the identity of that AV, whose observation, with which I certainly agree, will probably serve as the basis of a future post. I hope the two (or more) ambitious young historians who have been following this blog will help provide guidance. In the meantime, in spite of appearances, this exchange and those like it should not be read as a genuine conversation involving real individuals but rather as a representative specimen of the kind of persiflage to which number theorists devote their idle hours, especially when some of them are Australian.


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