JohnDoron Zeilberger's 113rd Opinion

Opinion 113: Mathematics is Indeed a Religion, But It has too Many Sects! Let's Unite Under the New God of Experimental Mathematics

By Doron Zeilberger

Written: Jan. 14, 2011

In his acceptance speech of the David P. Robbins prize, during the 2011 Joint Mathematics Meeting, Yuval Peres told the audience that he once overheard his young son Alon asking a friend:

"Do you have a religion? You know, Christian?, Jewish?, Mathematics?"
I don't think that even Yuval realizes how true was Alon's quip. The current official religion of mathematics is centered around the dogma that mathematical knowledge is restricted to statements that have been rigorously proved. For example, in Kannan Soundararajan's wonderful talk he used the phrase "we now know ..." as synonymous to "we now have a complete proof of ...", so Kannan Soundararajan (and 99.99% of mathematicians) do not know whether or not the Riemann Hypothesis, or the Goldbach conjecture, are true, these are just conjectures. But I, for one, know for sure, that they are both true, and while nothing is completely sure in this world, my belief in the truth of RH and Goldbach is much stronger than my belief in FLT or Poincaré , whose long and complicated alleged proofs may contain errors, and whose statements are far less heuristically obvious.

Speaking of certainty, the very same Yuval Peres, in his MAA invited talk, declared, about some scaling limit:

"It's more than certain than most laws of physics that a scaling limit exists, but there is no proof."

To which I would add:"... and who cares?", Oded Schram and collaborators have already found beautiful proofs to some scaling limits, and getting yet-another "rigorous" proof of yet another scaling limit may not be the most optimal use of our time.

Because of this obsession with "rigorous" (or "formal") proofs, Mathematics has gotten so specialized, where no one can see the forest, and even most people can't see the whole tree they sit on. All they can see is their tiny branch. Even in specialized conferences, many people skip the invited talks and only go to their own doubly-specialized session.

I went to almost all the invited talks of the above-mentioned 2011 Joint Mathematics meeting, and while I am glad to report that the quality of the invited talks was greatly improved (compared to last year's, in particular the font size was no longer a problem) the attendance in these invited talks was even worse! Out of more than 4000 participants, the Gibbs lecture (the highlight of the conference, one of the greatest honors, previous Gibbs lecturers include Einstein, Gödel and Wigderson) by George Papanicolau, only had 260 people. The first Colloquium lecture (by Alexander Lubotzky) had a record of 420 people, and the other conflict-free lecture (Sat. 11:10-12:00, by Kannan Soundararajan, that I have already mentioned above) had 360 people. The other invited talks, including the second Colloquium lecture (that had 145 people) averaged 150 people, and one of these invited talks only had 55 people! What a waste of a large lecture hall! The AMS/MAA can do some optimization by moving these "major" talks to smaller rooms, and it is also better for the speakers, since it is depressing to talk to a room with %15 (and sometimes %5) occupancy. [I counted about 1200 available seats].

Since it is a sad fact of life that mathematics has gotten so specialized and the vast majority of the people are unwilling (and often unable) to understand talks outside their own narrow specialty, and most speakers, even when they try hard, are unable to talk to a general audience, it may be a good idea, as long as we are stuck in the current (multi-sect) religion, to stop pretending, and change the format to that of only special sessions. Just change the name "special session on Xi" to "conference on Xi", and change the phrase "invited talk of the JMM" to "key-note talk of the conference on Xi", for i=1 ... 50. Also change the name "Joint Mathematics Meeting" to "Disjoint Union of 50 specialized conferences". This would be much more efficient (and honest!), and we would stop pretending that mathematics is one subject.

And math has gotten to be so splintered and specialized in large part because of the fanatic, entirely obsolete, insistence, on rigorous proofs. One can understand the central notions of a mathematical area without knowing any proofs. "Formal proofs" are just that, a formality. We can pursue mathematics entirely empirically, and proofs should lose their centrality, and become optional.

On another matter, pure mathematicians are so naive! As I have already mentioned, their notion of what is "to know" is very narrow. They also often brag about the "unreasonable effectiveness" of pure mathematics, they may be right, but that same mathematics would be much more useful without wasting time on proofs.

The computational naïveté of even the greatest mathematicians was well illustrated by this year's Colloquium lecturer, Alex Lubotzky, an eminent number theorist. They do not even know how to google correctly! Lubotzky started his first talk [BTW, these talks were excellent, one of the best that I have ever attended, lucid and accessible, and entertaining for the mathematical masses, too bad so few people took advantage of them] by bragging how important his subject, "expanders", is. So he googled expanders, and initially was excited to get more than 4 million hits. Even he soon realized that most of these hits were not about graph expanders, but had to do more with dentistry [laugh], so he tried again and googled graph expanders, and was delighted that there are still about 355000 pages, and hence graph expanders are really popular. But even this is a very gross overestimate, and while the first few pages are indeed about graph expanders, most of them (e.g. this one) are just texts that have both the words "graph" and the word "expander". What he should have googled was: "expander graphs" that yields the still impressive, ca. 24000 hits, but an order-of-magnitude less than claimed by Lubotzky's "second try".

Lubotzky also boasted that pure mathematics is so useful to computer science, since using "deep number theory" (first Margulis, then Lubotzky-Philips-Sarnak) it was possible to construct "explicit" "infinite" families of expander graphs. As far as I know, applied computer scientists do not need these admittedly beautiful constructions, since it is very easy to generate a random graph, that is provably an expander with probability 1- ε, and since all the graphs that they are likely to need do have finitely many vertices, they couldn't care less about the "explicit" and "uniform" constructions. So the only application of (at least this piece of) pure math, is to pure math, and pure mathematicians (who love the truth so much) should not overstate their case.

Pure mathematicians, once again epitomized by Lubotzky, also don't quite understand the practical meaning of "explicit". In his second colloquium talk, he wrote down a formula for the number of prime numbers less than x, due to Lagrange (essentially applying inclusion-exclusion to Eratosthenes) whose "computational complexity" was doubly-exponential! He called it "explicit but useless". If "explicit" means "computable in finite time", then he is right, but it is a bit of a stretch to use the word "explicit" [and pardon my explicit language].

Going back to poor attendance, I didn't see Lubotzky in any of the other invited talks, and I didn't see most of the other invited speakers in his (at least second) talk. People even stopped pretending to be interested in things outside their specialty, so it is about time that the AMS/MAA will get a reality check, and also stop pretending that mathematics is one subject, and adopt my suggestions above.

But there is hope for a grand-unification! In twenty years (perhaps sooner!), mathematics can once again become a unified religion. All we need is to worship the new God of Experimental Mathematics! Even the most abstract mathematics can be made concrete, and we recently saw lots of examples, e.g. Khovanov's "categorification", that can all be computerized, and Denis Auroux, in the first AMS invited talk (on Thurs.) mentioned the complete combinatorization by Lipschitz-Ozsvath-(Dylan)Thurston of something in very abstract topology, and that construction can be understood by undergraduates, and even by "high-school students". I strongly believe that all of mathematics, at least that part that is worth doing, can be similarly reduced to combinatorics and hence to computations. Also the second invited talk, by Chuu-Lian Terng, about solitons, mentioned some very concrete constructions, that at the time were breakthroughs, but that could have been found (today, or even twenty years ago) by computer algebra.

Computer algebra, and experimental mathematics, has the potential to become the new unifying "religion". There is still room for some proofs, especially nice ones, (those from the book, and these too can be obtained experimentally, the computer is a great tool not just for discovering conjectures, but also for discovering proofs) but "formal" proofs should lose their centrality. They are an obsolete relic from a bygone age, just like print-journals, and using a typist to convert your hand-written manuscript to a .tex file. There is so much mathematical knowledge out there that can be discovered empirically (like in the natural sciences, of course it should still be theory-laden, or else it won't go very far). Once we convert to this new religion, we would understand the big picture so much better, and have much more global insight (those that tell me that the purpose of proofs is insight make me laugh, true, the top one percent of proofs give you (local) insight, but the bottom 99 percent are just formal verifications, many of which can already be done by computer, and the rest soon will be [if you are stupid enough to want them]).

Proofs are Dead, Long Live Algorithms (and Meta-Algorithms!).

Added Feb. 14, 2011: Watch the lecture: part 1   part 2 (produced by Edinah Gnang)
Added March 5, 2011: Read the interesting comments by Aram Harrow. I stand corrected about the usefulness of explicit expanders.
Opinions of Doron Zeilberger