A few weeks ago my daughter sent me a Calvin and Hobbes cartoon. The dialogue went like this:
“You know, I don’t think math is a science. I think it’s a religion.”
“Yeah. All these equations are like miracles. You take two numbers and when you add them they magically become one new number. No one can say how it happens. You either believe it or you don’t. The whole book is full of things that have to be accepted on faith. It’s a religion!”
“And in the public schools no less. Call a lawyer.”
“As a math atheist, I should be excused from this.”
At first, it is funny because it lampoons such a badly misguided attitude. Mathematics is the opposite of revealed religion because you are required to take nothing on faith: something only becomes a mathematical truth when a proof is found, and each individual mathematician is responsible for following the proof to the point of becoming convinced of the truth. The only major religion which seems in any way similar is Buddhism, whose founder told questioners (perplexed about the multiplicity of religious teachers at the time) to examine what they said; accept what is good, and reject what is not good.
One can see shadows of the attitude among some students, simply because mathematics to them is a complete mystery. Lacking the attitude that they should only accept it if the proof is convincing (most likely, like Calvin, their teachers never told them this), it all becomes rather mysterious, and there is a human tendency to put the mysterious, the supernatural, and the religious in the same mental box.
But perhaps, like all good jokes, there is some element of truth in it. Perhaps it is something like this. Mathematicians create their own mental universes, after all; maybe there is not so much difference from an algebraist beginning a lecture with “Let G be a group” and the God of Genesis saying “Let there be light”. The second statement, we are told, called light into existence in the real world; the first calls a group into existence in the mental universes of the lecturer and audience.
Mathematicians, for example, are one group of people who are (mostly) quite comfortable with the notion of infinity.
A few years ago the BBC World Service made a programme about infinity, to which I made a small contribution (a half-hour interview was reduced to three short soundbites in the final programme). After my last appearance, the psychiatrist Raj Persaud came on to explain that in his clinic he sees many people who have gone mad thinking about infinity; the clear implication was that the crazy mathematician who has just been talking is likely to be one.
But in fact most mathematicians work with the infinite every day, and as far as I know, we don’t have a higher rate of madness than any other profession. (It is true that Georg Cantor, who devised the theory of infinite numbers we use now, went mad; but this was perhaps partly because implacable opposition to his ideas from conservative mathematicians such as Kronecker essentially destroyed his career.) We devise ways of thinking about, and making mental pictures of, infinite sets in the same way that we do for numbers or groups.
An excellent recent book, Naming Infinity, by Loren Graham and Jean-Michel Kantor, describes the founding of descriptive set theory in the first half of the twentieth century. Part of their story revolves around the fact that, while the French mathematicians Borel, Lebesgue and Baire drew back from the implications of their discoveries, the Russians Egorov and Lusin pressed boldly on (even despite ideological opposition to their views from Stalin’s regime, which caused them enormous hardship). Their boldness sprang in part from the influence on them of Pavel Florensky, who was both a mathematician and an Orthodox monk, a leader in the “Name worshipping” movement in the Russian Orthodox Church. They were not afraid to name God, and likewise to name infinity; and once named, it could become the subject of mathematical analysis.
Does mathematics combine the mystical and the rational, as no other subject can? Maybe.
Or maybe not. I had a student in Oxford who read Mathematics and Philosophy because he had read Robert M. Pirsig’s Zen and the Art of Motorcycle Maintenance and decided that he would find the answers to the ultimate questions in the region between mathematics and philosophy. What he found, of course, was a very substantial dose of logic!