Written: Feb. 16, 2022
My fellow experimental mathematician Craig Larson emailed me yesterday telling me about a defense of current "conceptual" pure math, by logician-mathematician-philosopher Jeremy Avigad (who btw uses computers a lot!). In that essay Avigad tries to explain why "conceptual math" is superior to computer-assisted, computational, mathematics.
Craig Larson kindly allowed me to post his brilliant response.
Let me just add a few words of my own.
Avigad joins a distinguished list of pure-math-elitists whom I love to hate:
In section 2 of Jeremy Avigad's tirade he puts down the astounding progress in "discrete geometry", including Tom Hales' amazing breakthrough of proving the Kepler conjecture, and Maryna Viazovska's seminal proof of the optimal sphere-packing in 8 dimensions (and its follow-up in 24 dimensions by Henry Cohn, Abhinav Kumar, my Rutgers colleague Stephen D. Miller, and Danylo Radchenko). He also cites two other results that have to do with tilings. In fact he finds some `redeeming features' in Viazovska et.al.'s work, because it is not "only" computational, but does use some parts of "conceptual mathematics".
This is followed, in section 3, by an ode to what Avigad calls "conceptual mathematics", that seems to me rehashing the Bourbaki-style "structural" mathematics. While Bourbaki was supposed to be "out of style", its spirit still lingers. Avigad apparently endorses this `conceptual-structural' superioty, and his "proof" why, for example, the Langlands program, is "important" is because the "elite" considers it important. Seems tautological to me!
I also noticed that Avigad uses the two-letter word "we", and the three letter word "our" much more frequently than in the average essay. By this first-person-plural he means, of course, we [human] mathematicians. And that's the whole point, Avigad is a human supremacist. Luckily computers do not yet have feelings, so they won't be offended, and if they had feelings they would just ignore these rantings by a lowly human.
Another problem with what Avigad calls "conceptual mathematics" is that a lot of it is philosophically flawed, using actual (and hence fictional) infinities. Number theorists use the euphemism "non-effective", but I call it "meaningless".
And this brings me to the main point. The goal of mathematics is not `understanding' (whatever it means), the goal of mathematics is mathematical knowledge, and very soon, we lowly humans will only `understand' (or rather think we understand, understanding is in the eyes of the beholder), the most trivial parts of mathematical knowledge. We (humans) would be lucky if we would be able to meta-understand, i.e. to see the `big-picture' of the algorithms behind the computers who would generate new, really deep math. FLT would become a trivial exercise (for our machine masters).
Current mainstream pure mathematics is an artifact of the fact that computers were only invented recently. Since human pure mathematicians ran out of interesting problems, they created their own sub-optimal artificial in-bred, very specialized body of knowledge, creating contrived problems. They seem `deep', but mostly because these high-brow purists are bad expositors, and perhaps they intentionally make is look foreboding, just to maintain their elite status, same way as lawyers use "legaleze" to intimidate the common people (and force us to hire them).
But what really got my blood-pressure up was (section 4, p. 109) the following quote:
"But to the mathematical elite [sic!] the phrase "experimental mathematics" is itself a contradiction in terms. Many mathematicians look down on such work as being recreational at best, and at worst, detrimental to the subject."
I don't know whether Avigad belongs to the above set of "many mathematicians", but I suspect that he is. I am sure that these sentiments will, very soon, seem so quaint. Mathematics is slowly-but-surely becoming a science, and there is no science without experiments!
Also note the word "elite" that Avigad uses in quite a few places, and always in a positive sense. But that's the whole problem! Most "elites", for example in dictatorial regimes, are "bad guys". If the current mathematical "elite" considers experimental mathematics as "detrimental to the subject", than we really need a new "elite". Better still, why have any "elites"? What's nice about experimental mathematics is that it is egalitarian and does not need any "elite".
In fairness, in the last section (section 5), Avigad offers some reconciliation suggestions, and concedes that mathematics, until now, is a human endeavor and hence there are many sociological and political factors at play, as well as fashions coming and going. Also he quotes the interesting historical work of Guicciardini about Newton's and others' notions of `understanding', that seem quaint to us.
And indeed, the whole of the `conceptual math supremacy' drivel of section 3 would seem to "us" (both humans and computers) so "quaint" and laughable, in fifty years (or sooner!). Perhaps, some of "us" humans will still pursue the `Langlands program', "just for fun" (each to his/her own!), by purely human means, sans ordinateurs, but they will not be called mathematicians, but rather some kind of eccentric hobbyists.
Added Feb. 18, 2022: Read Jeremy Avigad's response.