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Numericon by Marianne Freiberger and Rachel ThomasNumericon, Marianne Freiberger and Rachel Thomas (Quercus Editions 2014)

An easy-to-read introduction to some profound and difficult ideas. You can start climbing the foothills of Mount Maths here and train for the icy cliffs and avalanche-scoured slopes above. And perhaps the mountain rises for ever. There may be important mathematical questions that are impossible to answer. But some answers we already have, like the irrationality of √2 and the infinitude of the primes, are astonishing to contemplate. How are finite creatures able to discover facts about infinity? Logic is a lever that can move more than the universe. Euclid described 2,300 years ago how it could be used to prove that primes never ended: for every integer, there is a prime.

Numericon explains his proof, then goes on to describe how the German Georg Cantor (1845-1918) used logic to prove that there is more than one kind of infinity. In fact, there’s an infinite hierarchy of infinities, climbing endlessly into the metaphysical empyrean. That seems like an insane idea and Cantor did end his life in a sanatorium. He’s an extreme example of something Freiberger and Thomas note in their introduction: that mathematicians are often eccentric. It’s a strange subject that doesn’t come naturally to human beings. We can’t escape it, because our brains and bodies are governed by mathematics, but we don’t need to apply it explicitly and consciously to get through life. Those who lift the surface of reality and gaze upon its brilliant mathematical core can find that the light troubles and even subverts their brains.

And some avert their gaze. The German Carl Friedrich Gauss (1777-1855) may have been the greatest mathematician who ever lived, but he declined to publish his revolutionary ideas about Euclid’s fifth axiom: that parallel lines never meet. New geometries arise if the axiom is eliminated, but Gauss didn’t claim the credit for discovering them. As this book describes, it was the younger mathematicians János Bolyai (1802-60) and the Russian Nikolai Lobachevsky (1792-1865). And Bolyai’s mathematician father had tried to warn his son off.

Mathematicians need a sense of adventure. They need recognition and support too, but sometimes even the great ones find both hard to win. Sometimes their greatness is part of the problem, because the work they produce is too new and powerful to be properly appreciated. The Norwegian Niels Henrik Abel (1802-29) died at 26 “from tuberculosis and in abject poverty” (pg. 183). The Frenchman Évariste Galois (1811-32) died even younger, at 21, fatally wounded in a duel. Perhaps it was an affair de cœur, perhaps a plot by his enemies: Galois was revolutionary in both his mathematics and his politics. The Indian Srinivasa Ramanujan (1887-1920) is another great who died young after making discoveries of permanent value.

This book discusses all three and many more, but mathematicians aren’t truly important. Mathematics is necessary; mathematicians are contingent. If advanced civilizations exist on other planets or in parallel universes, their art and literature might be entirely different from our own or might not exist at all. But their mathematics would be recognizable, whatever the nature of the brains that had discovered it. Advanced civilization is impossible to imagine without mathematics and although we can ignore mathematics if we choose, we’re missing something central to existence if we do. As Darwin said: it seems to give one an extra sense. It’s a mind’s eye that can see into infinity – and into infinitesimality.

As though to reflect the importance of mind in maths, this book has relatively few and simple illustrations. Maths can yield gorgeous complexity for real eyes, but the mind’s eye is more important. So this book is eye-candy for the mind. Eye-brandy too: maths can both dazzle you and make you drunk. It’s appropriate that Numericon echoes Necronomicon, because new worlds wait within the pages of both.

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Front cover of A Mathematician's Apology by G.H. HardyA Mathematician’s Apology, G.H. Hardy (1940)

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.

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