You've touched on something important, and often unrecognized. For many, perhaps most, mathematicians - and I'm one of them - the point is not utility. For us, mathematics is an art form; the point is the discovery - or invention, depending on your philosophy - of beauty. (The highest praise for a proof is for it to be called "elegant".) It's an austere kind of beauty, perhaps, best analogous to the sweeping lines of a great building, but it's there.
On the other hand, there's a famous cautionary tale. G. H. Hardy, of Oxford, was so adamant on this point that he insisted that the only good mathematics was that which had no application whatsoever. In one of his books - or perhaps in a lecture, but I think a book - he pointed to a particular theorem of number theory (which was his own field) as quintessential mathematics; it was utterly inconceivable, he said, that it would ever find application. This was in the teens or twenties of the last century. Sometime in the 1940s, Claude Shannon found an application of the theorem. (It's useful in cryptography and computer science.)
There are too many stories like that for anyone to be too confident that any given piece of mathematics will never find use. But I suspect, for various reasons, that mine will not find application in my lifetime. I just think it's pretty.
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Date: 2007-02-02 05:48 pm (UTC)On the other hand, there's a famous cautionary tale. G. H. Hardy, of Oxford, was so adamant on this point that he insisted that the only good mathematics was that which had no application whatsoever. In one of his books - or perhaps in a lecture, but I think a book - he pointed to a particular theorem of number theory (which was his own field) as quintessential mathematics; it was utterly inconceivable, he said, that it would ever find application. This was in the teens or twenties of the last century. Sometime in the 1940s, Claude Shannon found an application of the theorem. (It's useful in cryptography and computer science.)
There are too many stories like that for anyone to be too confident that any given piece of mathematics will never find use. But I suspect, for various reasons, that mine will not find application in my lifetime. I just think it's pretty.