*Numericon*, 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.