“Whatever is rigorous is insignificant.”

It’s time to add René Thom’s discordant voice to this chorus.  The title of this post is taken from his article La Science Malgré Tout in the 1973 edition of Encyclopaedia Universalis, and more recently from the epigraph of an article by Alain Chenciner entitled Le vrai, le faux, l’insignifiant [The true, the false, the insignificant], which he will be presenting next week at a meeting on “the false vs. the insignificant” at the Sorbonne.  Chenciner, like many of his fellow specialists in dynamical systems, perhaps especially in France, has a deep interest in philosophy, and has also been strongly influenced by Thom’s anti-analytic metaphysics.  (This tradition, by the way, is one reason I find confusing how “analytic” and “synthetic” are used in the discussion of HOTT.)  He sent me his article after being alerted to this post and the surrounding discussion.  The Sorbonne meeting is inspired by another quotation from Thom, this one dating from 1991:

Ce qui limite le vrai, ce n’est pas le faux, c’est l’insignifiant [The true is bounded not by the false but rather by the insignificant]

I’ll try to attend at least one day of the Sorbonne meeting and report on what I learn there, but I can’t promise anything.  The most visible strains in French philosophy of mathematics belongs to the phenomenological and/or structuralist traditions and their attitude to the logicist tendency that is so important in English-language philosophy of mathematics ranges from indifference to contempt.  (There is also an important French school of analytic philosophy, but it’s irrelevant to the present discussion, except as a focus of the hostility of the phenomenologists.)  In my book I only alluded to French philosophy of mathematics once, in a quotation from Gilles Châtelet.  I think it is necessary to have followed the philosophy program in French schools in order to grasp the subtleties of this approach, and it helps to have spent a lot of time around the Ecole Normale Supérieure where, for example, you can participate in the weekly mamuphi seminar (mathematics, music, philosophy).

Thom occupied a peculiar position on the French intellectual scene for about two decades; his catastrophe theory was cited extensively in Jean-François Lyotard’s The Postmodern Condition and for this reason his name must have become familiar to several generations of Comparative Literature majors in the U.S.  For the moment I’m just going to extract two more Thom quotations from Chenciner’s text:

C’est seulement parce qu’on accepte le risque de l’erreur qu’on peut récolter de nouvelles découvertes. [It’s only because one accepts the risk of error that one can harvest new discoveries.]

En refusant le formalsme pur, en exigeant l’intelligible, le futur esprit scientifique va courir, de gaieté de coeur, le risque de l’erreur. Après tout, mieux vaut un univers transparent à l’esprit, translucide, où le contour des choses est un peu flou, qu’un univers aux certitudes précises, écrasantes et incompréhensibles, comme l’est celui de la physique classique. [By refusing pure formalism, by demanding the intelligible, the scientific mind of the future will cheerfully run the risk of error.  After all, a universe that is transparent to the mind, translucid, where contours are a bit blurry, than a universe of precise certainties, crushing and incomprehensible, like that of classical physics.]

In her recent portrait of Voevodsky in a special issue of Nautilis on Error, Siobhan Roberts seems to agree with Voevodsky, and thus to disagree with Thom, that mistakes in mathematics are never a good thing.  She quotes John Conway to the effect that “he has never encountered an error that was in any shape or form advantageous.”  Chenciner is obviously with Thom; he argues that whole branches of mathematics, including his own speciality of modern celestial mechanics, derive from Poincaré’s unusually advantageous errors.

Roberts’s article mainly revisits the material already presented in the Voevodsky article quoted in an earlier post.  The journalistic genre seems to force Roberts to set up an artificial opposition between camps for and against the use of computers in mathematical proofs; without much nuance she refers to the latter as “purists, the old guard.”  As you might expect, the battle lines are not so clear cut.  Voevodsky himself distinguishes between proof assistants and computer-generated proofs; the latter, he says,

…are the proofs which teach us very little. And there is a correct perception that if we go toward computer-generated proofs then we lose all the good that there is in mathematics—mathematics as a spiritual discipline, mathematics as something which helps to form a pure mind.

And Chenciner finds the Univalent Foundations program “encouraging” in that, by “assimilating… the notion (of proof) of equality to a weak homotopy equivalence… geometry returns unexpectedly to the world of abstract symbols.”

Voevodsky has a curious response to the tempting serendipity of error:

Voevodsky dismisses the notion that taking an errant mathematical turn might be like getting lost in New York City and stumbling upon a magnificent hidden garden. “Well, first of all you’d have to be in Manhattan, or you might find something not so desirable.”

Someone should tell him that if he’s in Brooklyn he might stumble upon the offices of Cabinet Magazine or n+1 or Triple Canopy — though at the rate things are going, they may soon find themselves priced out of Brooklyn all the way into the Bronx.   But in case you’re wondering how Nautilis can sustain such intellectually rigorous discussions with a Manhattan address, take another look at this post and then scroll down to the bottom of this page.

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8 thoughts on ““Whatever is rigorous is insignificant.”

  1. Martin Krieger

    I observe that rigorous mathematical physics is often revealing of important physical insights, only made clear from that demand for rigor. Dyson and Lenard’s proof of the stability of matter, also showed that if electrons were bosons there would be no stability.

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  2. Pingback: Voevodsky trending | Mathematics without Apologies, by Michael Harris

  3. Phil Koop

    Are you deliberately echoing an old French catchphrase? “All that is excessive is insignificant”? If so, that is rather sly of you.

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    1. mathematicswithoutapologies Post author

      The title of this post is a quotation from Thom. I didn’t know the original expression, and having spent a few minutes looking for a source I’m still unsure. There are two versions:

      Tout ce qui est excessif est dérisoire

      frequently attributed to Beaumarchais (but without any reference, which leads me to suspect there isn’t any), and

      Tout ce qui est excessif est insignifiant

      even more frequently attributed to Talleyrand, again without any reference.

      It’s certain that someone said it, but I’m just copying Chenciner’s quotation from Thom.

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  4. Emilio J. Gallego

    Other interesting mistakes were by Church and Martin-Lof.

    Church’s original lambda calculus was found to be inconsistent by one of his students. Reportedly, this ashamed him so much that he stopped working on it.

    Lof’s type theory was also found inconsistent (I believe that by Girard).

    Whether this mistakes were beneficial or hamrful, I have no clue.

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  5. Holybear

    Thank you Michael for this post. I just would like to add that I do not think that you need to have followed the philosophy program in French schools in order to understand the subtleties of French philosophy of mathematics. The main idea is very basic and wide spread for example in the US (but more in Literature department, where for example they study Deleuze than in Philosophy Dpts when they only stick to analytic philosophy) and it simply states that mathematics is not logic and that most philosophies of mathematics mix up mathematics and logic. Gilles Châtelet’s book, Figuring Space, condemn this confusion and tries to build not only a philosophy of real mathematics paying much attention, prior to formalization to the process of intuition and discovery in mathematics, but even more tries to build a dialogue, not to say a philosophy of nature, on an equal footing between mathematics, physics and philosophy in which the latter is not a servant condemned to paraphrase(1) . Your colleague here in Paris-Jussieu, Pierre Lochak, published very recently a book closely related to those issues : http://www.vrin.fr/book.php?code=9782841747023
    Best,
    Alexis

    1. “Sad destiny for philosophy ; the discipline that formerly sat at the head of the table finds itself reduced to the role of a Cinderella taken up with verification as its thrilling problems of directing the circulation of commonsensical ideas.”
    “A thrusting philosophy cannot be content with continual ratiocinations on the status of scientific objects. It has to position itself as the outposts of the obscure, looking upon the irrational not as diabolical and resistant to articulation, but rather as the means by which new dimensions come into being.”

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  6. Cody

    Surely no one is arguing that exploration and speculation, even when it leads to erroneous conclusion, is blamable. Not only can such erroneous suppositions (or conjectures) be very fruitful, they are inevitable by the very nature of mathematical exploration!

    What’s more doubtful is the utility of erroneous *results* in mathematical *peer reviewed* publications. I feel you would be harder pressed to make a good case for their fundamental utility. The inconsistency of Church’s calculus was fruitful, both because it led to fundamental insights about the nature of computation (looping combinators are what makes the lambda calculus both inconsistent and turing-complete) but also, I think, because it was caught early and didn’t lead a generation of logicians to rely on the (untyped) calculus as a viable foundation for mathematics.

    I feel Frege’s (similar) mistake was less happy, for him at least.

    I still feel like there is a false dichotomy here somewhere, where proponents of machine checked formal proofs are accused of forbidding “fruitful mistakes”, whereas they are adopting the more conservative position of trying to prevent established, published work from containing factual errors (and alleviating the burden for the reviewer!).

    I certainly can’t talk for Roberts though, and maybe she is arguing for the more extreme position.

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