Experimentally, Heuristically and Non-Rigorously.

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} ≡ a^{p} +1 (mod p) (binomial theorem),
hence (since 0^{p} ≡ 0 (mod p)), by induction on a,
a^{p} ≡ 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

- procedural generation
- complex numbers
- limits / calculus
- kinematics / filters / splines / quaternions

Opinions of Doron Zeilberger