Uncle Petros then pointed at what he'd made while he was talking. 'What is that?' he asked me.
'A rectangle made of beans,' I replied. 'Of 7 rows and 5 columns, their product giving us 35, the total number of beans in the rectangle. All right?'
He proceeded to explain how he was struck by an observation which, although totally elementary, seemed to him to have great intuitive depth. Namely, that if you constructed, in theory, all possible rectangles of dots (or beans) this would give you all the integers – except the primes. (Since a prime is never a product, it cannot be represented as a rectangle but only as a single row.) He went on to describe a calculus for operations among the rectangles and gave me some examples. Then he stated and proved some elementary theorems.
After a while I began to notice a change in his style. In our previous lessons he'd been the perfect teacher, varying the tempo of his exposition in inverse proportion to its difficulty, always making sure I had grasped one point before proceeding to the next. As he advanced deeper into the geometric approach, however, his answers became hurried, fragmented and incomplete to the point of total obscurity. In fact, after a certain point my questions were ignored and what might have appeared at first as explanations I recognized now as overheard fragments of his ongoing infernal monologue.
At first, I thought this anomalous form of presentation was a result of his not remembering the details of the geometric approach as clearly as the more conventional mathematics of the analytic, and making desperate efforts to reconstruct it.
I sat back and watched him: he was walking about the living room, rearranging his rectangles, mumbling to himself, going to the mantelpiece where he'd left paper and pencil, scribbling, looking something up in a tattered notebook, mumbling some more, returning to his beans, looking here and there, pausing, thinking, doing some more rearranging, then scribbling some more… Increasingly, references to a 'promising line of thought', 'an extremely elegant lemma' or a 'deep little theorem' (all his own inventions, obviously) made his face light up with a self-satisfied smile and his eyes sparkle with boyish mischievousness. I suddenly realized that the apparent chaos was nothing eise than the outer form of inner, bustling mental activity. Not only did he remember the 'famous bean method' perfectly well – its memory made him positively gloat with pride!
A previously unthought-of possibility quickly entered my mind, only to become a near conviction moments later.
When first discussing Uncle Petros' abandoning Goldbach's Conjecture with Sammy, it had seemed obvious to both of us that the reason was a form of burnout, an extreme case of scientific battle fatigue after years and years of fruitless attacks. The poor man had striven and striven and striven and, after failing each time, was finally too exhausted and too disappointed to continue, Kurt Gödel providing him with a convenient if far-fetched excuse. But now, watching his obvious exhilaration as he played around with his beans, a new and much more exciting scenario presented itself: was it possible that, in direct contrast to what I'd thought until then, his surrender had come at the very peak of his achievement? In fact, precisely at the point when he felt he was ready to solve the problem?
In a flash of memory, the words he had used when describing the period just before Turing's visit came back – words whose real significance I had barely realized when I'd first heard them. Certainly he'd said that the despair and self-doubts he had felt in Cambridge, in that spring of 1933, had been stronger than ever. But had he not interpreted these as the 'inevitable anguish before the final triumph', even as the 'onset of the labour pains leading to the delivery of the great discovery'? And what about what he'd said a little earlier, just a little while ago, about this being his 'most important work', 'important and original work, a groundbreaking advance'? Oh my good God! Fatigue and disillusionment didn't have to be the causes: his surrender could have been the loss of nerve before the great leap into the unknown and his final triumph!
The excitement caused by this realization was such that I could no longer wait for the tactically correct moment. I launched my attack right away.
'I notice,’ I said, my tone accusing rather than observing, 'that you seem to think very highly of the "famous Papachristos bean method".'
I had interrupted his train of thought and it took a few moments for my comment to register.
' You have an amazing command of the obvious,’ he said rudely. 'Of course I think highly of it.'
'… in contrast to Hardy and Littlewood,’ I added, delivering my first seriousblow.
This brought the expected reaction – only to a much greater degree than I'd f oreseen.
'"Can't prove Goldbach with beans, old chap!"' he said in a gruff, boorish tone, obviously parodying Littlewood. Then, he took on the other member of the immortal mathematical pair in a cruel mimicry of effeminacy. "Too elementary for your own good, my dear fellow, infantile even!'"
He banged his fist on the mantelpiece, furious. "That ass Hardy,’ he shouted, 'calling my geometric method "infantile" – as if he understood the first thing about it!'
'Now, now, Uncle,’ I said scoldingly, 'you can't go calling G. H. Hardy an ass!'
He banged his fist again, with greater force.
'An ass he was, and a sodomite too! The "great G. H. Hardy" – the Queen of Number Theory!'
This was so untypical of him I gasped. 'My, my, we are getting nasty, Uncle Petros!'
'Not at all! I'll call a spade a spade and a bugger a bugger!'
If I was startled I was also exhilarated: a totally new man had magically appeared before my eyes. Could it be that, together with the 'famous bean method', his old (I mean his young) seif had at last resurfaced? Could I now be hearing, for the first time, Petros Papachristos' real voice? Eccentricity – even Obsession – was certainly more characteristic of the single-minded, overambitious, brilliant mathematician of his youth than the gentle, civilized manners I'd come to associate with my elderly Uncle Petros. Conceit and malice towards his peers could well be the necessary other side of his genius. After all, both were perfectly suited to his capital sin, as diagnosed by Sammy: Pride.
To push it to its limit I used a casual tone: 'G. H. Hardy's sexual inclinations do not concern me,' I said. 'All that is relevant, vis-ä-vis his opinion of your "bean method", is that he was a great mathematician!'
Uncle Petros' face went crimson. 'Bollocks,' he growled. 'Prove it!'
'I don't have to,' I said dismissively. 'His theorems speak for themselves.'
'Oh?Which one?'
I stated two or three of the results I remembered from his textbook.
'Ha!' Uncle Petros snarled. 'Mere calculations of the grocery-bill variety! But show me one great idea, one inspired insight… You can't? That's because there isn't one!' He was fuming now. 'Oh, and while you're at it, tell me of a theorem the old pansy proved on his own, without good old Littlewood or poor dear Ramanujan holding his hand – or whatever other part of his anatomy it was they were holding!'
The mounting nastiness signalled that we were approaching a breakthrough. A tiny extra bit of annoyance was probably all that was necessary to bring it about.
'Really, Uncle,’ I said, trying to sound as haughty as possible. This is beneath you. After all, whatever theorems Hardy proved, they were certainly more important than yours!'