Sun, 29 Jan 2006
A while back I was in the Penn math and physics library browsing in the old books, and I ran across Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work by G.H. Hardy. Srinivasa Ramanujan was an unknown amateur mathematician in India; one day he sent Hardy some of the theorems he had been proving. Hardy was boggled; many of Ramanujan's theorems were unlike anything he had ever seen before. Hardy said that the formulas in the letter must be true, because if they were not true, no one would have had the imagination to invent them. Here's a typical example:
Hardy says that it was clear that Ramanujan was either a genius or a confidence trickster, and that confidence tricksters of that caliber were much rarer than geniuses, so he was prepared to give him the benefit of the doubt.
But anyway, the main point of this note is to present the following quotation from Hardy. He is discussing analytic number theory:
The fact remains that hardly any of Ramanujan's work in this field had any permanent value. The analytic theory of numbers is one of those exceptional branches of mathematics in which proof really is everything and nothing short of absolute rigour counts. The achievement of the mathematicians who found the Prime Number Theorem was quite a small thing compared with that of those who found the proof. It is not merely that in this theory (as Littlewood's theorem shows) you can never be quite sure of the facts without the proof, though this is important enough. The whole history of the Prime Number Theorem, and the other big theorems of the subject, shows that you cannot reach any real understanding of the structure and meaning of the theory, or have any sound instincts to guide you in further research, until you have mastered the proofs. It is comparatively easy to make clever guesses; indeed there are theorems like "Goldbach's Theorem", which have never been proved and which any fool could have guessed.(G.H. Hardy, Ramanujan.)
Some notes about this:
A man who sets out to justify his existence and his activities has to distinguish two different questions. The first is whether the work which he does is worth doing; and the second is why he does it, whatever its value may be, The first question is often very difficult, and the answer very discouraging, but most people will find the second easy enough even then. Their answers, if they are honest, will usually take one or another of two forms . . . the first . . . is the only answer which we need consider seriously.And that, ultimately, is why I didn't become a mathematician. I don't have the talent for it. I have no doubt that I could have become a quite competent second-rate mathematician, with a secure appointment at some second-rate college, and a series of second-rate published papers. But as I entered my mid-twenties, it became clear that although I wouldn't ever be a first-rate mathematician, I could be a first-rate computer programmer and teacher of computer programming. I don't think the world is any worse off for the lack of my mediocre mathematical contributions. But by teaching I've been able to give entertainment and skill to a lot of people.
When I teach classes, I sometimes come back from the mid-class break and ask if there are any questions about anything at all. Not infrequently, some wag in the audience asks why the sky is blue, or what the meaning of life is. If you're going to do something as risky as asking for unconstrained questions, you need to be ready with answers. When people ask why the sky is blue, I reply "because it reflects the sea." And the first time I got the question about the meaning of life, I was glad that I had thought about this beforehand and so had an answer ready. "Find out what your work is," I said, "and then do it as well as you can." I am sure that this idea owes a lot to Hardy. I wouldn't want to say that's the meaning of life for everyone, but it seems to me to be a good answer, so if you are looking for a meaning of life, you might try that one and see how you like it.
(Incidentally, I'm not sure it makes sense to buy a copy of this book, since it's really just a long essay. My copy, which is the same as the one I've linked above, ekes it out to book length by setting it in a very large font with very large margins, and by prepending a fifty-page(!) introduction by C.P. Snow.)
[ Addendum 20210727: regarding theorems “which have never been proved and which any fool could have guessed” Vladimir Arnol'd once said something similar: “There is a general principle that a stupid man can ask such questions to which one hundred wise men would not be able to answer.” ]
Addendum 20230810I was quoting from memory above. Hardy's exact words were:
[Formulas] (1.10)–(1.12) defeated me completely; I had never seen anything in the least like them before. A single look at them is enough to show that they could only be written down by a mathematician of the highest class. They must be true because, if they were not true, no one would have had the imagination to invent them. Finally… the writer must be completely honest, because great mathematicians are commoner than thieves or humbugs of such incredible skill.This is from page 9 of Ramanujan: Twelve Lectures on Subjects Suggested by his Life and Work, Cambridge University Press, 1940. The two relevant pages may be downloaded here, and the entire book is available in the Internet Archive.