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.