The World Wide Web is also the Random Reading Reticulation – the biggest library that ever existed. Obscure texts and ancient manuscripts are now a mouse-click away. A Mathematician’s Apology is neither obscure nor ancient, but it wasn’t easy to get hold of before it became available online. I’ve wanted to read it for a long time. And now I have.
Alas, I was disappointed. G.H. Hardy (1877-1947) was a very good mathematician, but he’s not a very good writer about mathematics. And in fact, he didn’t want to be a writer at all, good, bad or indifferent:
It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done. Statesmen despise publicists, painters despise art-critics, and physiologists, physicists, or mathematicians have usually similar feelings: there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds. (Op. cit., Section 1)
But is philosophy of mathematics work for second-rate minds? At the highest level, I don’t think it is. The relation of mathematics to reality, and vice versa, is a profoundly interesting and important topic, but Hardy doesn’t have anything new or illuminating to say about it:
It may be that modern physics fits best into some framework of idealistic philosophy — I do not believe it, but there are eminent physicists who say so. Pure mathematics, on the other hand, seems to me a rock on which all idealism founders: 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. (Section 24)
He experienced and explored that mathematical reality, but he can’t communicate the excitement or importance of doing so very well. I wasn’t surprised by his confession that, as a boy: “I thought of mathematics in terms of examinations and scholarships: I wanted to beat other boys, and this seemed to be the way in which I could do so most decisively” (sec. 29). He says it wasn’t until he had begun his degree at Cambridge that he “learnt for the first time … what mathematics really meant” (ibid.).
In this, he was very different from someone he helped make much more famous than he now is: an unknown and struggling Indian mathematician called Srinivasa Ramanujan, who sparked Hardy’s interest by sending him theorems of startling originality and depth before the First World War. Hardy brought Ramanujan to England, but barely mentions his protégé here. All the same, his respect and even perhaps his affection are still apparent:
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 [J.E.] Littlewood and Ramanujan on something like equal terms.’ (sec. 29)
Very few people could have done that: a mere handful of the many millions who lived at the time. So it would be wrong to expect that Hardy could both ascend to the highest peaks of mathematics and write well about what he experienced there. He couldn’t and A Mathematician’s Apology supplies the proof. That’s a shame, but the text is short and still worth reading. Hardy had no false modesty, but he had no delusions of grandeur either:
I have never done anything ‘useful’. No 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 is undeniable: the question is about its value.
It was valuable: that is why Hardy is still remembered and celebrated, sixty-seven years after his death. He is also still famous as an atheist, but you could say that he spent his life in the service of Our Lady – Mathematica Magistra Mundi, Mathematics Mistress of the World.