Opinion 126: [Contemporary Pure] Math is far LESS than the Sum of its [Too Numerous] Parts, and Beware of Pure Mathematicians Preaching How Important Math is for Science and Technology. Of Course it is, but NOT their, Soon-To-Be Obsolete and PEDANTIC Style of Doing Math, but Rather the Way Physical Scientists Practice It:
Experimentally, Heuristically and Non-Rigorously.

By Doron Zeilberger

Written: July 23, 2012

In a recent newspaper article, entitled "Math is more than the sum of its parts", the great pure mathematician Edward Frenkel, along with mathematics educator Ronald Ross, preach the importance of math, a propos the announcement of the discovery of the Higgs boson.

What Edward and Ronald did not tell us is that the "math" that lead to the discovery of the Higgs boson is not their kind of (pure-and-rigorous) math, but the much more effective, and efficient, non-rigorous mathematics practiced by theoretical physicists called quantum field theory. This highly successful (and precise!) mathematical theory would not be considered mathematics by Edward Frenkel and most members of the American Mathematical Society, since it is completely non-rigorous.

[Another example of a beautiful mathematical theory, with great explanatory power and great predictions, alas "non-rigorous" (hence completely tref for pure mathematicians) is renormalization groups, for which Ken Wilson won a Nobel prize]

The detection of the Higgs boson probably also involved many hours of heavy-duty computer calculations, very far afield from the Langlands program and other esoterica dear to Edward Frenkel and his friends. Ironically, (pure) mathematicians are much more indebted to theoretical physicists than vice versa (e.g. Seiberg-Witten and quantum groups), by giving them fresh ideas to pursue their very possibly beautiful, but completely useless, game.

In the same article, Frenkel and Ross allude to the RSA algorithm. Let me remind you that the "safety" of RSA is only conjectural (from the pedantic standpoint of pure mathematicians). It is possible (but very unlikely!) that tomorrow an assistant professor of computer science (not math!) together with two undergrads, will find a fast algorithm for integer factorization. The rest of the math behind the clever RSA algorithm goes back to Euler. If p and q are prime, and a is divisible by neither p nor q, then

a(p-1)(q-1) ≡ 1 ( mod pq)   ,

and by today's standards is utterly trivial
[ [(a+1)p ≡ ap +1 (mod p) (binomial theorem), hence (since 0p ≡ 0 (mod p)), by induction on a, ap ≡ a (mod p). This much goes back to Fermat. Hence if gcd(a,pq)=1, a(p-1) ≡ 1 (mod p) and a(q-1) ≡ 1 (mod q), hence a(p-1)(q-1) ≡ 1 (mod p), a(q-1)(p-1) ≡ 1 (mod q), and we get a(p-1)(q-1) ≡ 1 (mod pq) by Chinese Remainders],
no need for the Langlands program! While Euler's result has the above two-line "rigorous" proof, the RSA algorithm would have been just as useful had it only an "empirical" proof.

The reason so many mathematically talented students are so turned off from math is that, once they go to university, even the science and engineering students are taught by professional mathematicians, whose rigid, pedantic, "rigorous-or-nothing" philosophy is imposed on them, at least in part.

Even at its "highest" level, conference talks, communicating math is highly dysfunctional. Highly specialized specialists, who attempt to communicate their subject to a "general mathematical audience", just read their highly technical, usually very dry, pre-prepared laptop presentations, and (almost) no one has any clue. Indeed because pure math has gotten so splintered, very few people see the mathematical forest, they can (barely) understand their own tree.

One example, out of many, was my great disappointment at the very same Edward Frenkel, who delivered the prestigious (three) Colloquium lectures at the last (Jan. 2012) Joint Mathematics Meeting. I know from RateMyProfessor (and his calculus classes viewable on-line) that he is a very gifted teacher. So he had the potential to give three talks accessible to a general mathematical audience. Instead he chose to give highly technical talks, with completely unrealistic expectations about the audience's background, and all I got from them was the "subtext":
Look how smart I am, I am collaborating with a Fields medalist!

Mathematics is so useful because physical scientists and engineers have the good sense to largely ignore the "religious" fanaticism of professional mathematicians, and their insistence on so-called rigor, that in many cases is misplaced and hypocritical, since it is based on "axioms" that are completely fictional, i.e. those that involve the so-called infinity.

The purpose of mathematical research should be the increase of mathematical knowledge, broadly defined. We should not be tied-up with the antiquated notions of alleged "rigor". This new philosophy and attitude to mathematics, loosely called experimental math (and looked down on by most of my colleagues, I often hear the phrase "this is only experimental math") should trickle down to all levels of education, from professional math meetings, via grad school, all the way to kindergarten. Should that happen, Wigner's "unreasonable effectiveness of math in science" would be all the more effective!

Let's start right now! A modest beginning would be to have every math major undergrad take a course in experimental mathematics!

Please don't misunderstand me. Personally, I love (quite a few!) rigorous proofs, and it is OK for anyone who loves them to look for them in their spare time, but for the research and teaching that we get paid for, we should adopt the much more open-minded attitude to mathematical truth, in a par with the standards of the "hard" physical sciences, and abandon our fanatic insistence on "rigorous" proofs.

Acknowledgement: I would like to thank my colleague Volodia Retakh for alerting me to the Frenkel-Ross polemic piece.
Added Nov. 19. 2013: A sanitized version of this opinion appreared in the Dec. 2013 issue of the Notices of the Amer. Math. Soc.
Added Nov. 26, 2013: For examples on how one should teach math, watch the following wonderful talks by Steven Wittens:
Opinions of Doron Zeilberger