Apostolos Doxiadis - Uncle Petros and Goldbach

The drama played out during Petros Papachristos' final days is the last in a triad of episodes from the history of mathematics, unified by a single theme: the Mystery-solution to a Famous Problem by an Important Mathematician. [16]

By majority consent, the three most famous unsolved mathematical problems are: (a) Fermat's Last Theorem, (b) the Riemann Hypothesis and (c) Goldbach's Conjecture.

In the case of Fermat's Last Theorem, the mystery-solution existed from its first statement: in 1637, while he was studying Diophantus' Arithmetica, Pierre de Fermat made a note in the margin of his personal copy, right next to proposition II.8 referring to the Pythagorean theorem, in the form x^2 + y^2 = z^2. He wrote: ‘It is impossible to separate a cube into two cubes, or a biquadrate (fourth power) into two biquadrates, or generally any power except a square into two powers with the same exponent. I have discovered a truly marvellous proof of this, which, however, this margin is not large enough to contain.'

After the death of Fermat his son collected and published his notes. A thorough search of his papers, however, failed to reveal the demonstratio mirabilis, the 'marvellous proof’ that his father claimed to have found. Equally in vain have mathematicians ever since sought to rediscover it. [17]As for the verdict of history on the existence of the mystery-solution: it's ambiguous. Most mathematidans today doubt that Fermat indeed had a proof. The worst-case theory has it that he was consciously lying, that he had not verified his guess and his margin-note was mere bragging. What's likelier, however, is that he was mistaken, the demonstratio mirabilis crippled by an undetected fault.

In the case of the Riemann Hypothesis, the mystery-solution was in fact a metaphysical practical joke, with G. H. Hardy as its perpetrator. This is how it happened:

Preparing to board a cross-Channel ferry during a bad storm, the confirmed atheist Hardy sent off to a colleague a postcard with the message: 'I have the proof to the Riemann Hypothesis.' His reasoning was that the Almighty, whose sworn enemy he was, would not permit him to reap such an exalted undeserved reward and would therefore see to his safe arrival, in order to have the falsity of his claim exposed.

The mystery-solution of Goldbach's Conjecture completes the triad.

On the morning after our last lesson, I telephoned Uncle Petros. At my insistence, he had recently agreed to have a line installed, on the condition that only I, and no one eise, would know the number.

He answered sounding tense and distant. 'What do you want?'

'Oh, I just called to say hello,' I said. 'Also to apologize. I think I was unnecessarily rude last night.'

There was a pause.

'Well,’ he said, 'actually I'm busy at the moment. Why don't we talk again… shall we say next week?'

I wanted to assume that his coldness was due to the fact that he was upset with me (as he had every reason to be, after all) and merely expressing his resentment. Still, I feit a nagging unease.

'Busy with what, Uncle?' I persisted.

Another pause.

'I-I'll tell you about it some other time.'

He was obviously eager to hang up so, before he could cut me off, I impulsively blurted out the suspicion that had taken shape during the night.

'You wouldn't by any chance have resumed your researches, would you, Uncle Petros?'

I heard a sharp intake of breath. 'Who – who told you that?' he said hoarsely.

I tried to sound casual. 'Oh, come on, give me some credit for having come to know you. As if it needed telling!'

I heard the click of his hanging up. My God – I was right! The crazy old fool had gone off his rocker. He was trying to prove Goldbach's Conjecture!

My guilty conscience stung me. What had I done? Humankind indeed cannot stand very much reality – Sammy's theory of Kurt Gödel's insanity also applied, in a different way, to Uncle Petras. I had obviously pushed the poor old man to his uttermost limit and then beyond it. I'd aimed straight at his Achilles heel and hit it. My ridiculous simple-minded scheme to force him into self-confrontation had destroyed his fragile defences. Heedlessly, irresponsibly, I had robbed him of the carefully nurtured justification of his failure: the Incompleteness Theorem. But I had put nothing in its place to sustain his shattered self-image. As his extreme reaction now showed, the exposure of his failure (to himself, more than to me) had been more than he could bear. Stripped of his cherished excuse he had taken, of necessity, the only way left for him to go: madness. For what else was the endeavour to search, in his late seventies, for the proof that he had failed to find when he was at the peak of his powers? If that wasn't total irrationality, what was?

I walked into my father's office filled with apprehension. Much as I hated to allow him into the charmed circle of my bond with Uncle Petros, I feit obliged to let him know what had happened. He was after all his brother, and any suspicion of serious illness was certainly a family matter. My father dismissed my self-recriminations about causing the crisis as so much poppycock. According to the official Papachristos world-view, a man had only himself to blame for his psychological condition, the only acceptable external reason for emotional discomfort being a serious drop in the price of stocks. As far as he was concerned, his older brother's behaviour had always been bizarre, and one more instance of eccentricity was definitely not to be taken seriously.

'In fact,’ he said, 'the condition you describe – absent-mindedness, self-absorption, abrupt changes of mood, irrational demands for beans in the middle of the night, nervous tics, etc. – reminds me of how he was carrying on when we visited him in Munich, back in the late twenties. Then, too, he was behaving like a madman. We'd be at a nice restaurant enjoying our wurst and he'd be squirming around as if there were nails in his chair, his face twitching like mad.'

'Quod erat demonstrandum,' I said. "That's exactly it. He's back doing mathematics. In fact, he's back working on Goldbach's Conjecture – ridiculous as that may sound at his age.'

My father shrugged. 'It's ridiculous at any age,’ he said. 'But why worry? Goldbach's Conjecture has already done him all the harm possible. Nothing worse can come of it.'

But I wasn't so sure about that. In fact, I was quite certain that a lot worse things could be in store for us. Goldbach's resurrection was bound to stir up unfulfilled passions, to aggravate deep-buried, terrible, unhealed wounds. His absurd new application to the old problem boded no good.

After work that evening, I drove to Ekali. The ancient VW beetle was parked outside the house. I crossed the front yard and rang the bell. There was no response, so I shouted: 'Open up, Uncle Petros; it's me!'

For a few moments I feared the worst, but then he appeared at a window and stared vaguely in my direction. There was no sign of his usual pleasure at seeing me, no surprise, no greeting – he just stared.

'Good afternoon,’ I said. 'I just came by to say hello.'

His normally serene face, the face of a stranger to life's usual worries, was now marked by extreme tension, his skin pale, his eyes red with sleeplessness, his brow furrowed with concern. He was also unshaven, the first time I'd seen him so. His stare continued absent, unfocused. I wasn't even sure he knew who I was.

'Come on, Uncle dear, please open up for the most favoured,’ I said with a fatuous smile.

He disappeared and after a while the door creaked open. He stood there, blocking my entry, wearing his pyjama bottoms and a wrinkled vest. It was evident he didn't want me to enter.

'What's wrong, Uncle?' I asked. ‘I’m worried about you.'

'Why should you be worried?' he said, now forcing himself to sound normal. 'Everything's fine.'