G.H. Hardy on analytic number theory and other matters
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:
 Notice that this implies that in most branches of
mathematics, you can get away with less than absolute rigor. I think
that Hardy is quite correct here. (This is a rather arrogant remark,
since Hardy is much more qualified than I am to be telling you what
counts as worthwhile mathematics and what it is like. But this is my
blog.) In most branches of mathematics, the difficult part is
understanding the objects you are studying. If you understand them
well enough to come up with a plausible conjecture, you are doing
well. And in some mathematical pursuits, the proof may even be
secondary. Consider, for example, linear programming problems. The
point of the theory is to come up with good numerical solutions to the
problems. If you can do that, your understanding of the mathematics
is in some sense unimportant. If you invent a good algorithm that
reliably produces good answers reasonably efficiently, proving that
the algorithm is always efficient is of rather less value. In
fact, there is such an algorithm—the "simplex
algorithm"—and it is known to have exponential time in the worst
case, a fact which is of decidedly limited practical interest.
In analytic number theory, however, two facts weigh in favor of rigor.
First, the objects you are studying are the positive integers. You
already have as much intuitive understanding of them as you are ever
going to have; you are not, through years of study and analysis, going
to come to a clearer intuition of the number 3. And second, analytic
number theory is much more inwardlooking than most mathematics. The
applications to the rest of mathematics are somewhat limited, and to
the wider world even more limited. So a guessed or conjectured
theorem is unlikely to have much value; the value is in understanding
the theorem itself, and if you don't have a rigorous proof, you don't
really understand the theorem.
Hardy's example of the Goldbach conjecture is a good one. In the 18th
Century, Christian Goldbach, who was nobody in particular, conjectured
that every even number is the sum of two primes. Nobody doubts that
this is true. It's certainly true for all small even numbers, and for
large ones, you have lots and lots of primes to choose from. No
proof, however, is in view. (The primes are all about multiplication.
Proving things about their additive properties is swimming upstream.)
And nobody particularly cares whether the conjecture is true or not.
So what if every even number is the sum of two primes? But a
proof would involve startling mathematics, deep understanding of
something not even guessed at now, powerful techniques not currently
devised. The proof itself would have value, but the result
doesn't.
Fermat's theorem (the one about a^{n} +
b^{n} = c^{n}) is another
example of this type. Not that Fermat was in any sense a fool to have
conjectured it. But the result itself is of almost no interest.
Again, all the value is in the proof, and the techniques that were
required to carry it through.
 The Prime Number Theorem that Hardy mentions is the theorem about
the average density of the prime numbers. The Greeks knew that there
were an infinite number of primes. So the next question to ask is
what fraction of integers are prime. Are the primes sparse, like the
squares? Or are they common, like multiples of 7? The answer turns
out to be somewhere in between.
Of the integers 1–10, four (2, 3, 5, 7) are prime, or 40%. Of the
integers 1–100, 25% are prime. Of the integers 1–1000, 16.8% are
prime. What's the relationship?
The relationship turns out to be amazing: Of the integers 1–n,
about 1/log(n) are prime. Here's a graph: the red line is the
fraction of the numbers 1–n that are prime; the green line is
1/log(n):
It's not hard to conjecture this, and I think it's not hard to come up
with offhand arguments why it should be so. But, as Hardy says,
proving it is another matter, and that's where the real value is,
because to prove it requires powerful understanding and sophisticated
technique, and the understanding and technique will be applicable to
other problems.
The theorem of Littlewood that Hardy refers to is a related
matter.
Hardy was an unusual fellow. Toward the end
of his life, he wrote an essay called A Mathematician's
Apology in which he tried to explain why he had devoted his
life for pure mathematics. I found it an extraordinarily compelling
piece of writing. I first read it in my teens, at a time when I
thought I might become a professional mathematician, and it's had a
strong influence on my life. The passage that resonates most for me
is this one:
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.
(1) 'I do what I do because it is the one and only thing I can
do at all well. . . . I agree that it might be better to be a
poet or a mathematician, but unfortunately I have no talents
for such pursuits.'
I am not suggesting that this is a defence which can be made
by most people, since most people can do nothing at all well.
But it is impregnable when it can be made without
absurdity. . . It is a tiny minority who can do anything
really well, and the number of men who can do two
things well is negligible. If a man has any genuine talent,
he should be ready to make almost any sacrifice in order to
cultivate it to the full.
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 secondrate mathematician, with a secure
appointment at some secondrate college, and a series of secondrate
published papers. But as I entered my midtwenties, it became clear
that although I wouldn't ever be a firstrate mathematician, I could
be a firstrate 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 midclass 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
fiftypage(!) 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.” ]
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