To apologize is to express remorse or regret for ones actions. I may apologize for not
uploading videos for weeks at a time, or for never checking my DMs, but an apology can
also be a formal justification or a defence of something. A Mathematician's Apology is
an essay written by English mathematician G H Hardy, in which Hardy provides a justification
that mathematics has value, even if it doesn't have practical applications. If you've seen
my last video about the story of Ramanujan you may remember that Hardy was the Cambridge
professor who invited Ramanujan over from India to come and work with him. And if you've
at all strayed into the philosophy of mathematics, you have likely come across Hardy's apology.
He says "I have never done anything useful. Mo discovery of mine has made, or is likely
to make, directly or indirectly, for good or ill, the least difference to the amenity
of the world. I have helped to train other mathematicians, but mathematicians of the
same kind as myself, and their work has been, so far at any rate as I have helped them to
it, as useless as my own. Judged by all practical standards, the value of my mathematical life
is nil; and outside mathematics it is trivial anyhow. I have just one chance of escaping
a verdict of complete triviality, that I may be judged to have created something worth
creating. And that I have created something in undeniable: the question is about its value."
Hardy finds this value in the beauty of mathematics. He says "The mathematicians patterns, like
the painters or the poets, must be beautiful.The ideas, like the colors or the words, must
fit together in a harmonious way. Beauty is the first test, there is no permanent place
in the world for ugly mathematics." Hardy's specialty was in number theory, and
he remarked it was a field that had particularly few practical applications. Although that
was before it became a prominent feature of cryptography and was used to crack codes.
And perhaps that lack of obvious applications was one of the appeals for Hardy to number
theory. That the mathematical truths were untainted by our reality. He says "317 is
a prime, not because we think so or because our minds are shaped in one way rather than
another, but because it is. Because mathematical reality is built that way."
In the book he goes on to give a couple of examples of what he would call first rate
mathematics. These are proofs which are particularly beautiful and important. Although he is constrained
by giving a proof that a reader who might not have a lot of background in mathematics
would also be able to appreciate. One such proof is from the Greeks, attributed to the
Pythagoras group, and it is of the irrationality of the square root of 2. It is a proof that
the square root of 2 cannot be written as a fraction of integers which have no common
factor. I'll link to this page of proofs in the description but the essence is that we
use reductio ad absurdum, we set up a hypothesis, that you can write it as a fraction of integers
with no common factor, and we go through and see that if we did that, then both a and b
could be shown to be even. But if they are both even then they have a common factor of
2 and our hypothesis is false. Meaning that the square root of 2 indeed cannot be written
in this form. Hardy then goes on to compare the kind of mathematical problem solving you
might see in chess, to these kind of serious proofs that we just looked at. And Hardy says
that the proof of Pythagoras is much more serious, in that it has influenced thought
profoundly, even outside mathematics. The irrationality proof shows that when you have
constructed arithmetic itself, it will not prove sufficient for our needs. Since there
will be things we will be unable to measure, for example the diagonal of a square. This
seriousness of the proof is what makes it beautiful but it doesn't make it useful. Hardy
says that this proof in particular, is not particularly useful to anyone since what would
be useful to an engineer would be an approximation of the value. And he says all approximations
are rational. Another one of the proofs that Hardy describes as being very beautiful is
one to do with there being an infinite number of prime numbers, although he acknowledges
that this proof is also not useful. An engineer would have plenty of prime numbers to work
with that are already known and probably no one is wondering if they're about to run out
of primes. But these proofs have brought great fame and respect to the mathematicians who
wrote them. They might inspire you to want to pursue mathematics and want to be immersed
in solving these very beautiful puzzles. From the outside this book may seem like a happy
celebration of mathematics, but that is not what it is. It is actually a somewhat depressing
autobiography of Hardy, who at age 62, acknowledged that he was well past his prime. And he even
considered his mathematical life to be over. He says "Mathematics is not a contemplative
but a creative subject; no one can draw much consolation from it when he has lost the power
or the desire to create; and that is apt to happen to a mathematician rather soon. It
is a pity, but in that case he does not matter a great deal anyhow, and it would be silly
to bother about him." In this Hardy is claiming that the best work
of a mathematician if often done when they are young and he feels that he is passed that
time and can't go back. He has lost his desire to be creative and he considered mathematics
to be a creative field. Hardy also acknowledges that a lot of his best work was done through
collaborations, notably with Littlewood and Ramanujan. He says "I still say to myself
when I am depressed, and find myself forced to listen to pompous and tiresome people,
'Well, I have done one thing you could never have done, and that is to have collaborated
with both Littlewood and Ramanujan on something like equal terms.'"
Even writing this book at all would have upset Hardy because he acknowledges in one of the
first few pages that he thinks talking about mathematics is a topic only for second rate
minds. I'd hate to know what he thinks of me talking about him talking about mathematics,
but i'd like to end on a quote from the forward written by another mathematician who knew
Hardy quite well. C P Snow. Snow spent time with Hardy in his final days,
he saw Hardy in a depression when he lost his creative abilities and he saw Hardy come
to terms with this. He mentions that the one thing that Hardy did still have a passion
and interest in was cricket. When he no longer cared about maths, he would ask Snow on every
visit to just discuss the cricket scores with him. On the last day that Snow visited Hardy
he said "there was an Indian test team playing in Australia and we talked about them."
"It was in the same week that he told his sister: 'If I knew that I was going to die
today, I think I should still want to hear the cricket scores.' And each evening that
week before she left him, she read a chapter from a history of Cambridge University cricket.
One such chapter contained the last words he heard, for he died suddenly in the early
morning." I hope you have enjoyed that look at A Mathematician's
Apology, a big thank you to my Patreon supporters who make my videos possible. If you can afford
it then please consider supporting me there as well. It is at patreon.com/tibees and the
link will be onscreen and in the description. A special shout out to todays Patron Cat of
the Day Rio, and thank you for watching, I hope to see you next time.