Following a tip from [REDACTED] I discovered that my Paris colleague Pierre Schapira has translated and published an introductory article on sets, categories, and higher categories, on an anonymously edited website called Inference and subtitled International Review of Science. The caricature of Schapira that accompanies the article is terrible but the article is well-written — not surprisingly — and a logical next step for anyone (a literary theorist, for instance) whose first exposure to categories came in Jim Holt’s review of MWA. The hypothetical literary theorist, if lazy, will not find it easy reading; Schapira prefaces the article with a dedication that is also a warning:
This essay is written for people like Gilles [Châtelet] open to new languages and new ideas—needless to say, not so many people, and not especially mathematicians.
This blog will undoubtedly get around to writing about Gilles Châtelet, the philosopher, gay activist, and professor of mathematics. For the moment I will just mention that this “singular personality,” as Schapira calls him, was widely admired by people who otherwise would probably not agree on much of anything, especially for his book Les enjeux du mobile (available — barely — in translation as Figuring Space), which is quoted several times in MWA. Schapira’s essay is much more demanding than the few paragraphs MWA devotes to categories but the determined literary theorist or social scientist will certain find it rewarding to read, and even well-informed computer scientists may find some passages enlightening.
The last section of Schapira’s essay bears the title Towards the Revolution, and the “major conceptual revolution” he has in mind is Voevodsky’s univalent foundations, much discussed on this blog.
In this new setting, the ∞-categories would become the usual categories, but in a totally new set-theoretical context. This would be a major conceptual revolution. But this idea of homotopy can also be found at the origin of another, much more tangible, revolution that will eventually allow for the validity of proofs to be checked using computers.
I find that last sentence surprising — in the context of Schapira’s essay, that is, not in reference to Voevodsky — and the next time I see Schapira I’ll have to ask him what he imagines will happen to pre-revolutionary mathematics if and when this comes to pass. Not to mention counter-revolutionary mathematicians. Because while Inference is reasonably eclectic — among its mathematical entries are a highly readable introduction by Gregory Chaitin to his thoughts on set theory, and a reprint of one of Pierre Cartier’s excellent articles on Grothendieck, it attracted some attention early on with a 3-part series by counter-evolutionary Michael Denton, called, appropriately enough, Evolution: A Theory in Crisis Revisited. This series has been welcomed by the Intelligent Design crowd, not so much, apparently, by anyone else. And this, in turn, has fueled speculation about the mind behind Inference. The leading theory (or maybe the only theory, because it’s not clear how many people are paying attention) seems to be that the new online journal is the brainchild of the prolific David Berlinski, who, among his other accomplishments, managed the difficult feat of making Richard Dawkins look sympathetic.
Whoever Inference‘s intelligent designer may be, he or she (or possibly an object of some alternative category) also appears to be an enthusiast for hidden variables theories in quantum mechanics. The first issue reprinted an epistolary exchange on this topic between Sheldon Goldstein and Steven Weinberg, that dates back to 1996, when the Science Wars raged most furiously. Bohmian mechanics may be one of the hidden variables behind the Inference editorial board, as it was behind the Science Wars themselves; Gross and Levitt, the authors of Higher Superstition, referred favorably (in one of their later publications, if I’m not mistaken) to an article by Dürr, Goldstein, and Zanghi (same Goldstein, and I think it was this article), and Jean Bricmont, author (with Alan Sokal) of Fashionable Nonsense, was also apparently sympathetic to hidden variables theories at the time, though not, I take it, sufficiently sympathetic to commit himself to one version or another.
I don’t know enough about quantum mechanics to have any opinion whatsoever about hidden variables, but I have observed that the main motivation of supporters of hidden variables theories seems to be the desire for objectivity — Goldstein writes that a physical theory
…should not involve any subjective notions in its very formulation, nor should it involve axioms concerned with measurement, since the notion of measurement is much too vague…
To my mind, a striking aspect of the “major conceptual revolution” that was category theory is that it radically calls into question the centrality of objectivity in philosophy of mathematics; that, at least, is the point of view of Chapter 7 of MWA. It may be that Schapira sees the univalent axiom as a way to restore objectivity, and that would be a major conceptual revolution; but this is mere speculation. Inference is eclectic enough not to require consistency with regard to such matters from one article to the next.