I do math. Pretty much from sun up to sun down. A lot of people can’t relate to that. Why would a person care so much about a thing like that? Math isn’t going to save the world. The math I do — as is the case for a great many mathematicians — has no practical applications whatsoever. From the “outside view,” it looks like a lot of work that just doesn’t need to be done.
I’ve puzzled over why math might be something worth caring about that much — something worth sacrificing one’s life for. Part of the answer is, art. Math is beautiful. I have been bowled over in aesthetic ecstasy by theorems. I have reacted to proofs and definitions in exactly the same way that I react to a brilliant song or a painting. This isn’t a common experience for those not intimately involved with math; but it is reason enough to justify the funding that math departments get. To keep that experience available to humans is an important social service.
Suppose we were to end war, end poverty, end disease. What then? We would have an awful lot of time to kill, and it would be a sorry situation if we had nothing to beautiful with which to fill that time. Some arts seem to suffer these days from the artists having run out of ideas. But math is not out of ideas.
Despite the outside impression that every mathematical question has been answered, in actual fact we have more open problems than we could possibly begin to approach; and the number only increases as we learn more. I do not doubt that math will continue to keep humans fervently preoccupied until the implosion of the universe. And the indications are that it will only grow more beautiful and more fascinating.
But there are many things of which this could be said. There are many entertaining things other than math; there are many beautiful and fascinating things, other than math. Most of them are more accessible and less stressful to do. What does math uniquely offer to humanity?
I think the unique and central contribution of math is not beauty. It the fact that math is the clearest window which humans have onto the truth. Every academic discipline is a window on the truth; but all agree that math is the brightest and clearest. In math alone, truths can be established beyond all doubt, left to withstand the test of eternity without the possibility of revision or improvement. In all other sciences, truth is tentative and can be overturned by new discoveries and new ways of thinking. In math alone, everything is already perfect, and cannot be any different than it is.
Mathematicians do not disagree for long; every disagreement is quickly resolved, in the direction of the person who is actually right. A totally unknown person can, with no budget, make a mathematical discovery and show it to the top mathematicians in the area. If the result is correct, it will be accepted. There is no room for politics and social games; in math alone, the truth shines so bright that it outshines all of that.
The truths of math usually do not matter, in a worldly sense. But this is, in a sense, beside the point. Humans need truth, independent of all other concerns. We do not need truth only so that we can develop new medicines and navigate the oceans. We need truth simply to know truth and be in relation with reality, and know with assurance that we are in such a relation. Math is the only thing which can provide us with that, with the purity and fullness which it does. And that is why I do it.