Apostolos Doxiadis - Uncle Petros and Goldbach

In November of that year he turned thirty, an emblematic age for the research mathematician, practically the first step into middle age.

The sword of Damocles, whose presence Petros had merely sensed all these years hanging in the darkness somewhere high above him (it was labelled: 'The Waning of his Creative Powers') now became almost visible. More and more, as he sat hunched over his papers, he could feel its hovering menace. The invisible hourglass measuring out his creative prime became a constant presence at the back of his mind, driving him into bouts of dread and anxiety. During his every waking moment, he was pestered by the worry that he might already be moving away from the apex of his intellectual prowess. Questions buzzed in his mind like mosquitoes: would he be having any more breakthroughs of the same order as the two first important results? Had the inevitable decline, perhaps unbeknown to him, already started? Every little instance of forgetfulness, every tiny slip in a calculation, every short lapse in concentration, brought the ominous refrain: Have I passed my prime?

A brief visit at about this time from his family (already described to me by my father), whom he hadn't seen in years, was considered by him a gross, violent intrusion. The little time he spent with his parents and younger brothers he felt was stolen from his work, and every moment away from his desk for their benefit he perceived as a small dose of mathematical suicide. By the end of their stay he was inordinately frustrated.

Not wasting time had become a veritable obsession, to the point where he obliterated from his life any activity that was not directly related to Goldbach's Conjecture – all except the two he couldn't reduce beyond a certain minimum, teaching and sleep. Yet he now got less sleep than he needed. Constant anxiety had brought insomnia with it, and this was aggravated by his excessive consumption of coffee, the fuel on which mathematicians run. With time, the constant preoccupation with the Conjecture made it impossible for him to relax. Falling or staying asleep became increasingly difficult and often he had to resort to sleeping pills. Occasional use gradually became steady and doses began to increase alarmingly, to the point of dependency, and this without the accompanying beneficial effect.

At about this time, a totally unexpected boost to his spirits came in the unlikely form of a dream. Despite his total disbelief in the supematural, Petros viewed it as prophetic, a definite omen straight from Mathematical Heaven.

It is not unusual for scientists totally immersed in a difficult problem to carry on their preoccupations into sleep; and although Petros was never honoured by nocturnal visitations from Ramanujan's Namakiri or any other revelatory deity (a fact that should not surprise us, considering his entrenched agnosticism), after the first year or so of his immersion in the Conjecture he began to have the occasional mathematical dream. In fact, his early visions of amorous bliss in the arms of 'dearest Isolde' became less frequent over time, giving their place to dreams of the Even Numbers, which appeared personified as couples of identical twins. They were involved in intricate, unearthly dumbshows, a chorus to the Primes, who were peculiar hermaphrodite, semi-human beings. Unlike the speechless Even Numbers, the Primes often chattered among themselves, usually in an unintelligible language, at the same time executing bizarre dance-steps. (By his admission, this dream choreography was most likely inspired by a production of Stravinsky's Rite of Spring that Petros had attended during his early years in Munich, when he still had time for such vanities.) On rare occasions the singular creatures spoke and then only in classical Greek – perhaps as a tribute to Euclid, who had awarded them infinitude. Even when their utterances made some linguistic sense, however, the content was mathematically either trivial or non-sensical. Petros specifically recalled one such: hapantes protoi perittoi , which means 'All prime numbers are odd', an obviously false Statement. (By a different reading of the word perittoi , however, it could also mean 'All prime numbers are useless', an interpretation which, interestingly, completely escaped my uncle's attention.)

Yet in a few rare instances there was something of substance in his dreams. He could deduce from the protagonists' sayings helpful hints that steered his research towards interesting, unexplored paths [10].

The dream that lifted his spirits came a few nights after he had proved his second important result. It was not directly mathematical, but laudatory, consisting of no more than a single image, a sparkling tableau vivant, but of such unearthly beauty! Leonard Euler was on the one side and Christian Goldbach (though he'd never seen a portrait, he immediately knew it to be him) on the other. The two men jointly held, from the sides, a golden wreath over the head of the central figure, which was none other than himself, Petros Papachristos. The triad was bathed in a nimbus of blinding light.

The dream's message could not be clearer: the proof of Goldbach's Conjecture would be ultimately his.

Spurred by the glorious spirit of this vision, his mood swung back to optimism and he coaxed himself onwards with added zest. Now, he should concentrate all his powers on his research. He could afford absolutely no distractions.

The painful gastrointestinal symptoms he had been having for some time (most of them by some strange coincidence occurring at times when they interfered with his university duties), a result of the constant, self-imposed pressure, gave him the pretext he needed. Armed with the opinion of a specialist, he went to see the Director of the School of Mathematics and requested a two-year, unpaid leave of absence.

The Director, an insignificant mathematician but a ferocious bureaucrat, was apparently waiting for an occasion to level with Professor Papachristos.

'I have read your doctor's recommendation, Herr Professor,’ he said in a sour tone. 'Apparently you suffer – like many in our School – from gastritis, a condition that is not exactly terminal. Isn't a two-year leave rather excessive?'

'Well, Herr Director,’ mumbled Petros, 'I also happen to be at a critical point in my research. While on my two-year leave I can complete it.'

The Director appeared genuinely surprised. 'Research? Oh, I had no idea! You see, the fact that you haven't published anything during all your years with us had led your colleagues to think that you were scientifically inactive.'

Petros knew the next question was inevitable:

'By the way, what exactly is it you are researching, Herr Professor?'

'We-ell,’ he replied meekly, 'I am investigating certain questions in Number Theory.'

The Director, an eminently practical man, considered Number Theory, a field notorious for the inapplicability of its results to the physical sciences, a complete waste of time. His own interest lay in differential equations and, years back, he had hoped that the addition of the inventor of the Papachristos Method to the faculty would perhaps put his own name on some joint publications. This, of course, had never come about.

'You mean Number Theory in general, Herr Professor?'

Petros suffered the ensuing cat-and-mouse game for a while, trying desperately to prevaricate concerning his real object. When, however, he realized he had not the slightest hope unless he convinced the Director of the importance of his work, he revealed the truth.

‘I’m working on Goldbach's Conjecture, Herr Director. But please don't tell anyone!'

The Director appeared startled. 'Oh? And how are you progressing?'

'Quite well, actually.'

'Which means you have arrived at some very interesting intermediate results. Am I right?'