The (in)famous rallying cry of Jean Dieudonné is the title of a new blog by “Nick Bornak” (hmmm, I wonder where the name came from…). The author, a student of mathematics and philosophy, started the blog in order to have a place to write about mathematics (for that, WordPress is pretty much the only way to go at the moment).
One of his major interests is nonstandard analysis, the very concept of which I used to despise until I read this brilliant post by Terence Tao. If a Fields Medalist says it’s okay, then it must be okay! Actually, what Tao’s post helped me to realize was that so-called “nonstandard analysis” can be regarded as just another item in the “soft” vocabulary of (“standard”) analysis, like asymptotic notation, or the definition of continuity in terms of inverse images of open sets. Since I absolutely adore “soft” mathematics (to the point where one of my missions in life is to “mollify” as much “hard” mathematics as I possibly can), you can see why I would find this point of view so appealing!
Before this, I used to see nonstandard analysis as, at best, an obfuscatory way of formulating ordinary analysis; or at worst, a sort of contrary position in the ontology/epistemology of mathematics, like constructivism (except perhaps on the “other extreme”). The way nonstandard analysis was described — as a way of making Leibniz’s infinitesimal reasoning rigorous — irritated me to no end. Didn’t these people realize that the work of the likes of Cauchy, Weierstrass, and Cantor had already made that reasoning rigorous? Not to understand this — simply because of Leibniz’s “different” vocabulary — is to exhibit the sort of over-concrete literal-mindedness that is all too typical of certain historians — particularly historians of mathematics — and, for that matter, music theorists. It thus seemed as if (not for the first time, alas) a whole field had been founded on a simple lack of intellectual agility!
Luckily, thanks largely to Terry Tao, I’ve since moved beyond this. Perhaps Nick Bornak will over time be able to still further illuminate the virtues of the “nonstandard” way of thinking. (That word “nonstandard” really is a turn-off, I have to say — to me it carries a suggestion of crackpottery.)
(Incidentally, though I’m a great admirer of Dieudonné, I don’t have anything against triangles. In fact, I think I could discuss them in a way that he would approve of. But that’s for another occasion…)