An unnatural operation

Thanks to the public library I have just had the very great pleasure of reading the story of Uncle Petros & Goldbach’s Conjecture as told by Apostolos Doxiadis.

For some reason – probably too many other things to do - I missed all the glowing reviews when this was first published in English in 2000, & remained completely unaware of its existence until I spotted it, out on one of the displays, not tucked away spine-out on a shelf where it is easy to find if you know what you are looking for.

I wasn’t really expecting the story, about an aged recluse who was once a celebrated mathematician, to be a Good Read, still less a suspenseful page turner, but it surely had curiosity value, worth at least having a look at on the bus home. And it is short – just 208 pages.

A lot is packed in there: number theory, paradox, the nature of the life of a mathematician, family, disappointment.

And it is very funny.

From time to time we are deftly reminded that the story takes place against the background of the upheaval & destruction of C20th European wars, but ‘Petros managed to go through life unhampered by any ideological burden.’

Petros also managed to remain unhampered by women – only two figure in his story at all (three if you count the woman who must have been his mother): Isolde, his only love, who abandoned him for a dashing young army officer & was killed, along with her two daughters, during the bombing of Dresden, & his long-suffering but uncomplaining & resourceful housekeeper. Her response, to the news that he was giving up his pursuit of a proof of Goldbach’s Conjecture & so had no further need of the beans laid out in rectangular arrays across the floor of his study, was to sweep them up, wash them off & turn them into cassoulet; her fate was arrest by the Gestapo & death in Dachau. His lack of action or reaction to this is a starkly harrowing illustration of the loneliness, isolation & social ineptitude (teetering on of the edge of outright insanity) of one doing original research in the inaccessible universe of mathematics. To the researcher numbers populate the real world; the rest is just a dream.

Petros has dreams while he sleeps too - dreams of the Even Numbers as couples of identical twins, a chorus to the Primes, who were ’peculiarly hermaphrodite, semi-human beings’ who executed ‘bizarre dance steps … most likely inspired by a production of Stravinsky’s Rite of Spring that [he] had attended during his early years in Munich, when he still had time for such vanities.’ There can be few other works of C20th classical music which have worked their way into so many other works of art or literature.

One of the joys of the story is that it brings in real mathematical giants of the period. So in 1933 he is working in Cambridge, alongside Hardy & Littlewood, when he is visited in his room by a young Alan Turing seeking his help with the translation of a German academic paper in the, to Petros, unfamiliar field of formal mathematical logic. Petros is so shocked by the news of Godel’s Incompleteness Theorem that he travels post haste to Vienna to interrogate the author.

‘As far as I know, Professor, every unproved statement can in principle be unprovable’ confirmed the young man.
‘At this, Petros saw red. He felt the irrepressible urge to grab the father of the Incompleteness Theorem by the scruff of the neck & bang his head on the shining surface of the table.’

That is exactly my usual reaction to Socrates.

Worse is yet to come. Back in Munich at the end of 1936 Petros receives a telegram from Turing, now at Princeton: I HAVE PROVED THE IMPOSSIBILITY OF A PRIORI DECIDABILITY STOP.

That STOP does not bring an end to the whole story, though it does mark the beginning of Petros’ period of totally reclusive life back in his native Greece, living off his share of the earnings of the family business run by his brothers, until the final denouement.

The narrator of the story is Petros’ young nephew, whose curiosity about this failed member of the family leads eventually to him deciding to pursue a mathematical career of his own. But, once he has come truly to appreciate ‘the dangers of coming too close to Truth in its absolute form’, to understand that ‘the proverbial ‘mad mathematician’ … drawn to an inhuman kind of light, brilliant but scorching & harsh’ is more fact than fancy, he decides to abandon mathematics & opt for graduate studies in Business Economics at Princeton instead, since that is ‘a field that does not traditionally provide material for tragedy.’ Since he then returned to the family business, he presumably knows different now.

I am left with a desire to look into the line that Uncle Petros was pursuing, to see if there have been any developments, or if indeed it had long been known to be a blind alley (we know that Goldbach’s Conjecture remains unproven).

The teacher who had the job of drilling our times tables into us encouraged us to understand that the effort was worthwhile because multiplication provides a quick, shortcut method of adding up: instead of adding up 2+2+2 we would know, quick as a flash, that the answer is 6, and so on.

So I was quite surprised to find, when I dipped my own toe into the Foundations of Arithmetic, that multiplication & addition are considered to be separate operations.

Who knows if Uncle Petros learned the same thing in primary school, but the labours of his mature years certainly led him to the conclusion that ‘Multiplication is unnatural ... It is a contrived, second order concept, no more really than a series of additions of equal elements … To invent a name for this repetition & call it an ‘operation’ is the devil’s work … If multiplication is unnatural, more so is the concept of ‘prime number’ that springs directly from it. The extreme difficulty of the basic problems related to the primes is … a direct outcome of this’.

I am also intrigued by his playing around with rectangular arrays of beans as a way of ‘seeing’ the problem, since I have been playing around with rectangular arrays of coloured cells as a method of ‘showing’ (especially to those who are not very comfortable with numbers) what is going on in the Simpson so-called paradox (which also, at root, explains why there cannot be a single, simple, definitive answer to questions such as ‘nature or nurture?’)