Tor­tured ge­nius not made for this world

The Weekend Australian - Review - - Books - Alan Saun­ders

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.

Il­lus­tra­tion: John Tiede­mann

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