Apostolos Doxiadis - Uncle Petros and Goldbach

'No, I don't. Enlighten me.'

'He said: "Why should I kill the goose that lays the golden eggs?" What he meant, you see, was that when great mathematicians attempt to solve great problems a lot of great mathematics – so-called "intermediate results" – is born, and this even though the initial problems may remain unsolved. Just to give you an example you'll understand, the field of Finite Group Theory came into being as a result of Evariste Galois' efforts to solve the equation of the fifth degree in its general form…'

The gist of Sammy's argument was this: there was no way that a top-class professional mathematician, as we had every indication that Uncle Petros was in his youth, could have spent his life wrestling with a great problem such as Goldbach's Conjecture without discovering along the way a single intermediate result of some value. However, since he had never published anything, we necessarily had to conclude (here Sammy was applying a form of the redudio ad absurdum) that he was lying: he never had attempted to prove Goldbach's Conjecture.

'But to what purpose would he tell such a lie?' I asked my friend, perplexed.

'Oh, it's more likely than not that he concocted the Goldbach Conjecture story to explain his mathematical inactivity – this is why I used the harsh word "fraud". You see, this is a problem so notoriously difficult that nobody could hold it against him if he didn't manage to solve it.'

'But this is absurd,’ I protested. 'Mathematics was Uncle Petros' life, his only interest and passion! Why would he want to abandon it and need to make up excuses for his inactivity? It doesn't make sense!'

Sam shook his head. "The explanation, I'm afraid, is rather depressing. A distinguished professor in our department, with whom I discussed the case, suggested it to me.' He must have seen the signs of dismay in my face, for he hastened to add:'… without mentioning your uncle's name, of course!'

Sammy then outlined the 'distinguished professor's' theory: 'It's quite likely that at some point early in his career your uncle lost either the intellectual capacity or the willpower (or possibly both) to do mathematics. Unfortunately, this is quite common with early developers. Burnout and breakdown are the fate of quite a few precocious geniuses…'

The distressing possibility that this sorry fate could possibly also one day await himself had obviously entered his mind: the conclusion was spoken solemnly, sadly even.

'You see, it's not that your poor Uncle Petros didn't want after a certain point to do any more mathematics – it's that he couldn 't.'

After my talk with Sammy on New Year's Eve, my attitude towards Uncle Petros changed once again. The rage I had felt when I first realized he had tricked me into attempting to prove Goldbach's Conjecture had already given way to more charitable feelings. Now, an element of sympathy was added: how terrible it must have been for him, if after such a brilliant beginning he suddenly began to feel his great gift, his only strength in life, his only joy, deserting him. Poor Uncle Petros!

The more I thought about it, the more I became upset at the unnamed 'distinguished professor' who could pronounce such damning indictments of someone he didn't even know, in the total absence of data. At Sammy, too. How could he so lightheartedly accuse him of being a 'fraud'?

I ended up deciding that Uncle Petros should be given the chance to defend himself, and to counter both the facile levelling generalizations of his brothers ('one of life's failures', etc.) as well as the condescending analyses of the 'distinguished professor' and the cocky boy-genius Sammy. The time had come for the accused to speak. Needless to say, I decided the person best qualified to hear his defence was none other than I, his close kin and victim. After all, he owed me.

I needed to prepare myself.

Although I had torn his telegram of apology into little pieces, I hadn't forgotten its content. My uncle had enjoined me to learn Kurt Gödel's Incompleteness Theorem; in some unfathomable way the explanation of his despicable behaviour to me lay in this. (Without knowing the first thing about the Incompleteness Theorem I didn't like the sound of it: the negative particle 'in-' carried a lot of baggage; the vacuum it hinted

at seemed to have metaphorical implications.)

At the first opportunity, which came while selecting my mathematics courses for the next semester, I asked Sammy, careful not to have him suspect that my question had anything to do with Uncle Petros: 'Have you ever heard of Kurt Gödel's Incompleteness Theorem?' Sammy threw his arms in the air, in comic exaggeration. 'Oy vey!’ he exclaimed. 'He asks me if I’ve heard of Kurt Gödel's Incompleteness Theorem!' 'To what branch does it belong? Topology?' Sammy stared at me aghast. 'The Incompleteness Theorem? – to Mathematical Logic, you total ignoramus!' 'Well, stop clowning and tell me about it. Tell me what it says.'

Sammy proceeded to explain along general lines the content of Gödel's great discovery. He began from Euclid and his vision of the solid construction of mathematical theories, starting from axioms as foundations and proceeding by the tools of rigorous logical induction to theorems. Then, he spanned twenty-two centuries to talk of 'Hilbert's Second Problem' and skimmed over the basics of Russell's and Whitehead's Principia Mathematica [5]terminating with the Incompleteness Theorem itself, which he explained in as simple language as he could.

'But is that possible?' I asked when he was finished, staring at him wide-eyed.

'More than possible,’ answered Sammy, 'it's a proven fact!’

I went to Ekali on the second day after my arrival in Greece for the summer vacation. Not wanting to catch him unawares, I'd already arranged the meeting with Uncle Petros by correspondence. To continue with the judicial analogy, I'd granted him ample time to prepare his defence.

I arrived at the arranged time and we sat in the garden.

'So then, most favoured of nephews' (this was the first time he called me that), 'what news do you bring me from the New World?'

If he thought I'd let him pretend this was a mere social occasion, a visit by dutiful nephew to caring uncle, he was mistaken.

'So then, Uncle,' I said belligerently, 'in a year's time I'm getting my degree and I'm already preparing applications for graduate school. Your ploy has failed. Whether it is to your liking or not, I will be a mathematician.'

He shrugged his shoulders while raising the palms of his hands heavenwards in a gesture of inevitability.

"'He who is fated to drown will never die in his bed",' he intoned – a populär Greek proverb. 'Have you told your father? Is he pleased?'

'Why this sudden interest in my father?' I snarled. 'Was it he who put you up to our so-called "deal"? Was it his perverse idea to make me prove myself worthy by tackling Goldbach's Conjecture? Or do you feel so much in his debt for supporting you all these years that you repaid him by bringing his upstart son to heel?'

Uncle Petros accepted the blows under the belt without changing expression.

'I don't blame you for being angry,’ he said. 'Yet you have to try to understand. Although my method was indeed questionable, the motives were as pure as driven snow.'

I laughed scornfully. "There is nothing pure in having your failure determine my life!'

He sighed. 'You have time at your disposal?'

'As much as you want.'

'And you are seated comfortably?'

‘Perfectly.'

'Then listen to my story. Listen and judge for your-self.'

The Story of Petros Papachristos

I cannot pretend to remember as I write now the exact phrasing and expressions my uncle used on that summer afternoon, so many years ago. I have preferred to recreate his narrative in the third person, opting for completeness and coherence. Where memory failed me I consulted his extant correspondence with family and mathematical colleagues as well as the thick, leather-bound volumes of the personal diaries in which he traced the progress of his research.