Written: June 28, 2008.
During the Enlightenment, a new breed of thinkers came along, so-called "free"-thinkers, who trashed religion as superstition, and hailed mathematics and science as the only ways to true knowledge. In order to rebut them, Bishop Berkeley had a brilliant idea. Instead of being on the defensive, and meekly apologizing for apparent contradictions in the scriptures and their apparent incompatibility with scientific knowledge, he found the best way to fight back, attack!
What he did was show how Newton's Calculus, based on infinitesimals ("fluxions"), was just as logically flawed and utter "nonsense" as the bible. His critique still holds today, and the so-called "rigorization" of mathematical analysis, by Cauchy and Weierstrass, if analyzed carefully, has its own unstated metaphysical assumptions, and the Hilbertian gambit of "meaningless axioms" could equally apply to religion, by making the existence of God into an axiom.
Speaking of axioms, when I was a student, about 1972, I went to a talk by Rabbi Shlomo Goren, and he said that "the existence of God is an axiom". Of course he meant it in the sense of Euclid, as "a self-evident truth", not in the sense of Hilbert, as an "arbitrary convention in a deductive game".
The great majority of mathematicians are naive fundamentalist platonists. They believe that mathematics is universal, and independent of them. Can they prove it? Of course not. To them it is just as self-evident as the existence of God is to Rabbi Goren.
But in this piece I want to talk about another fundamental, unproved and unprovable, basic belief of most mathematicians, that they simply take for granted, and that is "every coincidence must have a reason".
Ironically, when non mathematicians (and even some religious mathematicians) make a big deal of "amazing coincidences" (like the Bible Codes), that purport to prove the existence of God, they ridicule them as innumerate, to use the phrase made popular by that great anti-religious crusader, John Allen Paulos. Like all of us, mathematicians have their double-standards, and are unaware that when they pursue their craft, they commit the same blunders.
But first, let me put-in a good word for coincidences, both in real life and in mathematics. Coincidences are the spice of life! Even if they are to be expected, and the most amazing coincidence is having no coincidences and the greatest possible surprise is having no surprises, they are fun to discover. They are the stuff that jokes and puns are made of.
So to all those mathematicians who ridicule people for attaching undue significance to coincidences, one could retort: What you say is what you are!. You are just as naive religious believers, except that your God is the God of mathematics. Most mathematicians, when they do their work, are at least as "innumerate" as those they criticize. The strong faith that coincidences are explainable- "if something is nice, there must be a good reason for it"-is their driving force, and did lead to lots of good and deep mathematics. It also lead to lots of wasted time and garbage, but by and large, it did more good than harm, at least in the human phase of mathematics. This is analogous to religion in humankind, that regardless of its truth-value, did lots of harm, but also lots of good, and the balance is still positive, I hope.
Mathematicians have their own "God", the innate belief in the universality and inevitability of mathematics, and there is no hard evidence for its truth, only their self-serving testimony. But, at least as likely is the scenario that there is no [mathematical] God, and along with genuine, explainable identities, like Analytic Index=Topological Index, and Moonshine, there are at least as many times when the coincidence is indeed a coincidence. I call it contingent beauty. Why do four colors suffice? Just Because!. Why is the Optimal Packing of 3D oranges face-centric cubic? Just Because!. Why is 25 the smallest size of a party in which you are guaranteed that either you can find five people who mutually love each other or four people who mutually hate each other? Just Because! Why are (decimal) digits 3-6 identical with digits 7-10 of e? Just Because!
Take for example the first Rogers Ramanujan identity that states that a certain class of integer partitions has the same number of elements as another class of partitions. None of the existing proofs is convincing to most mathematicians, since, as Hardy put it, they are "essentially verification", and while logically implying its truth, they do not "explain". But I know many prominent mathematicians who keep trying to find such a "deeper" explanation. Maybe they would find it, but maybe not! There is a very good chance that this too is true Just Because!
Another striking example, that motivated this piece, is an amazingly simple conjecture due to Ira Gessel, that a certain class of lattice walks is enumerated by a very simple closed-form formula. Many leading enumerators, for example, Ira Gessel himself, and Mireille Bousquet-Melou, couldn't find any proof, let alone an "explanation". In a recent article, I and my collaborators, Manuel Kauers and Christoph Koutschan, along with their very fast computer, proved it. The proof is so "ugly" that it is beautiful. But the take-home message is that, at least from our perspective, there is no a priori reason why the answer should be nice. "Religious" mathematicians would be sure that "if the answer is nice, there must be a nice reason for it". They may be right, but, just as possibly, they may be wrong. It is very possible that Gessel's conjecture is true Just Because! If you take other classes of lattice paths, their counting function will be extremely complicated, and morally, the Gessel walks should have been just as complicated, but it lucked-out, and turned out to be simple, Just Because!
At the end of that article I am offering a prize of 100 dollars for a "short, human, computer-free proof". Unlike Erdös and other people who offer money for problems, I hope that no one would win it, since I believe (and hope!) that such a proof does not exist. But even if I do have to pay, it would only show that this particular case was not a good example of contingent beauty. But I strongly believe that there are lots of other examples, and that contingent beauty is just as prevalent as necessary beauty.