'Oh yes?' he snapped back. 'More important than Goldbach's Conjecture?'
I burst into incredulous laughter, despite myself. 'But you didn't prove Goldbach's Conjecture, Uncle Petros!'
'I didn't prove it, but -'
He broke off in mid-sentence. His expression betrayed he'd said more than he wanted to.
'You didn't prove it but what?’ I pressed him. 'Come on, Uncle, complete what you were going to say! You didn't prove it but were very dose to it? I'm right – am I not?'
Suddenly, he stared at me as if he were Hamlet and I his father's ghost. It was now or never. I leapt up from my seat.
'Oh, for God's sake, Uncle,' I cried. ‘I’m not my father or Uncle Anargyros or grandfather Papachristos! I know some mathematics, remember? Don't give me that crap about Gödel and the Incompleteness Theorem! Do you think I swallowed for a single moment that fairy tale of your "intuition telling you the Conjecture was unprovable"! No – I knew it from the very start for what it was, a pathetic excuse for your failure. Sourgrapes!’
His mouth opened in wonder – from ghost I must have been transformed into a celestial vision.
'I know the whole truth, Uncle Petros,’ I continued fervently. 'You got to within a hair's breadth of the proof! You were almost there… Almost… All but the final step…' – my voice was coming out in a humming, deep chant -'… and then, you lost your nerve! You chickened out, Uncle dearest, didn't you? What happened! Did you run out of willpower or were you just too scared to follow the path to its ultimate conclusion? Whatever the case, you'd always known it deep inside: the fault is not with the Incompleteness of Mathematics!'
My last words had made him recoil and I thought I might as well play the part to the hilt: I grabbed him by the shoulders and shouted straight into his face.
'Face it, Uncle! You owe it to yourself, can't you see that? To your courage, to your brilliance, to all those long, fruitless, lonely years! The blame for not proving Goldbach's Conjecture is all your own – just as the triumph would have been totally yours if you'd succeeded! But you didn't succeed! Goldbach's Conjecture is provable and you knew that all along! It's just that you didn't manage to prove it! You failed -you failed, God damn it, and you've got to admit it, at last!'
I had run out of breath.
As for Uncle Petros, for a slight moment his eyes closed and he wavered. I thought that he was going to pass out, but no – he instantly came to, his inner turmoil now unexpectedly melting into a soft, mellow smile.
I smiled too: naively, I thought that my wild ranting had miraculously achieved its purpose. In fact, at that moment I would have made a bet that his next words would be something like: 'You are absolutely right. I failed. I admit it. Thank you for helping me do it, most favoured of nephews. Now, I can die happy'
Alas, what he actually said was: 'Will you be a good boy and go get me five more kilos of beans?'
I was stunned – all of a sudden he was the ghost and I Hamlet.
'We – we must finish our discussion first,' I faltered, too shocked for anything stronger.
But then he started pleading: 'Please! Please, please, please get me some more beans!'
His tone was so intolerably pathetic that my defences crumbled to dust. For better or for worse, I knew that my experiment in enforced self-confrontation had ended.
Buying uncooked beans in a country where people don't do their grocery shopping in the middle of the night was a worthy challenge to my developing entrepreneurial skills. I drove from taverna to taverna, beguiling the cooks into selling me from their pantry stock a kilo here, half a kilo there, until I accumulated the required quantity. (It was probably the most expensive five kilos of beans ever.)
When I got back to Ekali, it was past midnight. I found Uncle Petros waiting for me at the garden gate.
'You are late!' was his only greeting.
I could see that he was in a state of tremendous agitation.
'Everything all right, Uncle?'
'Are these the beans?'
'They are, but what's the matter? What are you so worked up about?'
Without answering he grabbed the bag. 'Thank you,' he said and began to close the gate.
'Shan't I come in?' I asked, surprised.
'It's too late,’ he said.
I was reluctant to leave him until I found out what was going on.
'We don't have to talk mathematics,’ I said. 'We can have a little game of chess or, even better, drink some herbal tea and gossip about the family.'
'No,’ he said with finality. 'Goodnight.' He walked fast towards his small house.
'When is the next lesson?' I shouted af ter him.
‘I’ll call you,’ he said, went in and banged the door behind him.
I remained standing on the pavement for a while, wondering what to do, whether to attempt once again to enter the house, to talk to him, to see if he was all right. But I knew he could be stubborn as a mule. Anyway, our lesson and my noctumal search for beans had drained me of all energy.
Driving back to Athens I was pestered by my conscience. For the first time, I questioned my course of action. Could my high-handed stance, supposedly intended to lead Uncle Petros into a therapeutic showdown, have been nothing more than my own need to get even, an attempt to avenge the trauma he'd inflicted on my teenage seif? And, even if that weren't so, what right did I have to make the poor old man face
the phantoms of his past, despite himself? Had I seriously considered the consequences of my inexcusable immaturity? The unanswered questions abounded, but still, by the time I got home I had rationalized myself out of the moral tight spot: the distress I'd obviously caused Uncle Petros had most probably been the necessary – the obligatory – step in the process of his redemption. What I'd told him was, after all, too much to digest at one go. Obviously the poor man only needed a chance to think things over in peace. He had to admit his failure to himself, before he could do so to me…
But if that was the case, why the extra five kilos of beans?
A hypothesis had begun to form in my mind, but it was too outrageous to be given serious consideration – until morning anyway.
Nothing in this world is truly new – certainly not the high dramas of the human spirit. Even when one such appears to be an original, on closer examination you realize it's been enacted before, with different protagonists, of course, and quite possibly with many variations in its development. But the main argument, the basic premise, repeats the same old story.
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.
[16] Mystery-solutions to famous problems by charlatans are two-a-penny
[17] Fermat's Last Theorem was, amazingly, proved in 1993. Gerhard Frey first proposed that the problem could possibly be reduced to an unproven hypothesis in the theory of elliptic curves, called the Taniyama-Shimura Conjecture, an insight later conclusively proven by Ken Ribet. The crucial proof of the Taniyama-Shimura Conjecture itself (and thus, as its corollary, Fermat's Last Theorem) was achieved by Andrew Wiles; in the final stage of his work he collaborated with Richard Taylor