# Tortured genius not made for this world

THERE are very few novels about mathematical logicians, and probably for good reason. Take, for example, Kurt Godel, perhaps the greatest logician since Aristotle. He starved himself to death at Princeton University in 1978, which suggests a degree of despair offering plentiful possibilities for any fiction maker. But what did he do, what did he stand for?

What he did was devise two incompleteness theorems and it’s fairly easy to sum them up: any axiomatic system rich enough to contain arithmetic must be incomplete or inconsistent, it can’t be both. Put differently, either there will be true sentences of the system that cannot be proved within it, or it will be possible to derive contradictions within the system.

If Godel was right — and he was — it’s easy to see that his results have devastating consequences for logic and mathematics.

Unfortunately, it’s rather less easy to see that his proofs are of extraordinary ingenuity and beauty.

Logic, aspiring to perfect lucidity, is curiously autistic, curiously incapable in its complexity of reaching out to the rest of the world.

A Madman Dreams of Turing Machines, a remarkable first novel by Janna Levin, an American professor of physics and astronomy, tells the story of two autistic men devoted to logic and mathematics. One is Godel, whom we first encounter in a Viennese coffee house in 1931, where the philosophers of the Vienna Circle meet to discuss reality and meaning, to distinguish science from superstition and to swap equations and mathematical symbols, written directly on to the marble tabletop.

The other is Alan Turing, an English mathematician, shambolic in appearance and manner, gay at a time when it was illegal to be gay and, I truly believe, one of the saviours of his country. His work as a cryptographer during World War II gave the Allies access to German naval codes and thus shortened the war. His country’s gratitude was, you might say, moderate. Hardly anybody knew what he had done: his work, and that of those with whom he worked, remained top secret for years.

In 1952 he was, like Oscar Wilde before him, arrested for gross indecency with a member of his own sex. Unlike Wilde, he didn’t go to jail. Instead, he was chemically castrated with doses of estrogen that left him impotent and the possessor of considerably enlarged breasts. He killed himself by eating a poisoned apple.

Apples are among the things that connect

A Madman Dreams of Turing Machines By Janna Levin Orion Publishing, 230pp, $ 29.99

Godel and Turing in Levin’s story. Vegetables, lumpy or random in texture, cause the young Turing great anxiety. He loves apples: ‘‘ Some apples are as round as globes, lovely and geometric, the symmetry pierced by the stem, orienting the fruit to the north.’’ Godel throws an apple away as, in his anguish and neurosis, he begins to starve himself to death.

Then there is the logical paradox of the liar. I tell you that the statement I am now making is false. This statement refers to its own falsity, which means that if it is true, it must be false and vice versa.

Can this possibly matter? Godel thinks it does and by translating an equivalent phrase, ‘‘ this statement is unprovable’’, into a code of pure numbers, derives his famous theorems.

The great Ludwig Wittgenstein is the connection between Godel and Turing, who never met in life. Wittgenstein dominates the thoughts of the Viennese philosophers and when he comes to England, Turing and he discuss the liar. Wittgenstein thinks that contradictions don’t matter, but Turing, inspired by Godel’s work, knows they do.

So the liar lies behind Godel’s theorem, and Godel’s theorem, in what it says and what it leaves unsaid, haunts Turing’s great idea, the Turing machine. The machine was a conceptual machine — something on paper, not of metal and wire — but it helped to bring us the modern computer and drew Turing to ask the question: Can machines think?

Turing and Godel could think with extraordinary intensity, and one of Levin’s triumphs is the grace with which she conveys their ideas. They gave much to the world, but the world was not their home and Levin is most moving in the way she tells us how their failure to grasp it killed them.

This is a very fine book. In its grave beauty and its sensitivity to the messy human realities from which philosophical ideas emerge, it recalls Penelope Fitzgerald’s great novel, The Blue Flower . For me there can be no higher praise than that.

Alan Saunders presents By Design and The Philosopher’s Zone on ABC Radio National.