George Steiner – The Observer
Go forth and multiply
by George Steiner
Pure maths is so technical that few have dared to use it as vehicle for fiction. Until Apostolos Doxiadis arrives with Uncle Petros and Goldbach’s Conjecture.
Does your heart skip a beat when you realise that 220 hooks up with 284 since the sum of the integer divisors of each one is equal to the other? Do you, at a glance, read 256 as 2 to the eighth power?
Can you gauge the enigmatic flash of recognition which allowed the dying mathematical genius Ramanujan to identify the number 1729, culled from a taxi license plate, as the smallest integer that can be expressed as the sum of two cubes in two different ways?
Can you sense why Cambridge went electric at the announcement, not very long ago, of the solution of Fermat’s Last Theorem, a conundrum which had blocked some of humanity’s finest intellects for centuries? If so, you will, even as the lowliest of laymen, have some rudimentary glimpse into the universe of pure mathematics.
Together with music, poetry and metaphysical thought, pure mathematics constitutes the highest reach of the human ape. That elevation is inseparable from its magnificent uselessness (subsequent application of a theorem to such trivia as nuclear physics, economic statistics, engineering, are, as the great algebraist André Weil was so charitable as to inform me on my first day at the Institute for Advanced Studies in Princeton, ‘nothing but bottle-washing’).
And within the arcana of pure mathematics, Number Theory has pride of place, if only because many of its overwhelmingly difficult problems can be formulated on the back of an envelope. Just like one of Mozart’s unforgettable tunes or Descartes’s cogito ergo sum.
Moreover, an utterly mesmerising philosophic problem underlies the entire edifice of mathematical proceedings. Does the mathematician discover truths, results, proofs which pre-exist his search, which exist ‘out there’? Has God ‘created prime numbers’ for us to collide and wrestle with?
Or is pure mathematics, like chess, a human artefact, a formally infinite construct bred by Homo sapiens? In which case, its truths form part of a self-perpetuating sequence, sovereignly intriguing, but with no necessary relations to the empirical world. Both beliefs have had their advocates.
Literature has, on inspired occasions, communicated the genesis of a painting, of a sculpture (Balzac is a master of transfer); far more rarely, it has come close to enacting music in language. Thomas Mann’s Doctor Faustus remains pre-eminent in this respect, but Proust and Joyce come close.
The translation of the languages of mathematics, of the mathematician’s works and days, into common speech is exceedingly rare, if not impossible, the more so as mathematics has progressed into deepening abstruseness and technicality after the seventeenth century.
It has been possible to make plausible anecdotes of Archimedes’s work on conic sections or of Galileo’s applications of Euclid. After that, the homework became too arduous, and the minds capable of taking in the material too few. When a topologist speaks of the ‘beauty’ of a solution, he is using that word in a perfectly concrete, demonstrable sense, but a sense achingly inaccessible to the vast majority of us. Indeed, it is in mathematics, not in our ordinary lives and literary verbiage, that Keats’s ‘truth is beauty’ makes sense.
The Goldbach-Euler conjecture states that every even number greater than two is the sum of two primes. Simplicity itself. Computers have looked at mammoth series of primes and no counter-example has ever been found. Nor has any proof. There lies a challenge which, to certain minds, has made the Holy Grail seem hollow.
The protagonist of this fact-fiction, Uncle Petros Papachristos (the irony here is a touch heavy), devotes his fierce mathematical talent, his personal existence, his sanity, to one single purpose: the proof of Goldbach-Euler’s seemingly elementary, intuitively obvious proposition. The tale, as told by his nephew, ends in predictable ruin.
Uncle Petros has led the life of a fanatical addict, cut off from normal humanity. Refusing to publish his interim results lest rivals beat him to his goal, turning down the collaborative offers of such mathematicians of genius as Hardy and Littlewood, Uncle Petros ends up as an almost forgotten footnote in the long chronicle of algebraic analysis.
At the perhaps unavoidably melodramatic finale, he collapses mentally and physically. Was he, wonders his appalled yet profoundly admiring nephew, on the very verge of a proof? Had Uncle Petros’s near-lunatic solitude and labours been at the threshold of immortality, of a demonstration which would add his name to those of Euclid, of Gauss, of Hilbert?
What ‘intrigue’ there is in this (surely autobiographical) fable turns on the question of Uncle Petros’s failure. Was this defeat due to some defect in his approach, to some cruel lack of supreme mental insight? Or was it due, as Petros believed, in despairing apologia, to Gödel’s fatal theorem?
In 1933, in one of the highest moments in human abstract thought, Kurt Gödel showed the incompletion of all mathematical systems, the fact that within them not every true statement is provable. Three years later, Alan Turing, father of modern computers, confirms this axiom of undecidability in a telegram to Papachristos. Could it follow that there can never be a proof for Goldbach-Euler? That only a near-infinite probability – a wretched compromise – is possible for the human intellect? Or is the resort to Gödel’s interdict only a lofty way in which Uncle Petros gets himself off the hook? His nephew’s almost sadistic challenge prevails. At the close, Petros is hunting still.
There are very few fictions which attempt a theme of this order. Apostolos Doxiadis’s concise novel is deeply generous. It allows the lay-reader lucid access to intrinsically closed worlds. The narrator speaks for almost all of us: ‘Even if my education was meagre, even if it meant no more than getting my toes wet on the beach of the immense ocean of mathematics, it has marked my life for ever, giving me a small taste of a higher world.’ Even if only once in one’s life, Number Theory opens the gates ‘to the real thing’, whose forbidding radiance is, perhaps, kindred to religious ecstacy. Uncle Petros has not, after all, lived in vain.
So take a scrap of paper: 4 is indeed the sum of 1 + 3… 250 years is not all that long. Perhaps tomorrow. And do let Mr. Doxiadis know if you come up with a QED.
March 5, 2000: George Steiner – The Observer