Uncle Petros and Goldbach

by Apostolos Doxiadis

  An unexpected fringe benefit was that the last remaining trace of ambivalence (apparently it had been there, dormant, all those years) regarding the wisdom of my decision to abandon mathematics was now dispelled. Watching my uncle do mathematics was enough to confirm it to the full. I was not made of the same mettle as he – this I realized now beyond the shadow of a doubt. Faced with the incamation of what I definitely was not, I accepted at last the truth of the dictum: Mathematicus nascitur non fit. The true mathematician is born, not made. I had not been born a mathematician and it was just as well that I had given up.

  The exact content of the ten lessons is not within the scope of our story and I won't even attempt to refer to it. What matters here is that by the eighth we had covered the course of the initial period of Uncle Petros' research on Goldbach's Conjecture, culminating in his brilliant Partitions Theorem, now named after the Austrian who rediscovered it; also his other main result, attributed to Ramanujan, Hardy and Littlewood. In the ninth lesson he explained to me as much as I could understand of his rationale for changing the course of his attack from the analytic to the algebraic.

  For the next he had asked me to bring along two kilos of lima beans. In fact, he had initially asked for navy beans, but then corrected himself, smiling sheepishly: 'Actually make it lima, so I can see them better. I'm not getting any younger, most favoured of nephews.'

  As I drove to Ekali for the tenth (which, although I didn't know it yet, would be the last) lesson, I felt apprehensive: I knew from his narrative that he had given up precisely while working with the 'famous bean method'. Very soon, even in that imminent lesson, we would be reaching the cruciai point, his hearing of Gödel's Theorem and the end of his efforts to prove Goldbach's Conjecture. It would be then that I would have to launch my attack on his dearly held defences and expose his rationalization about unprovability for what it was: a mere excuse.

  When I got to Ekaii he led me without a word to his socalled Irving room, which I found transformed. He'd pushed back what furniture there was against the walls, including even the armchair and the small table with the chessboard, and piled even higher piles of books along the perimeter, to create a wide, empty area in the centre. Without so much as a word he took the bag from my hands and started to arrange the beans on the floor, in a number of rectangles. I watched silently.

  When he had finished he said: 'During our previous lessons we went over my early approach to the Conjecture. In this I had done good, perhaps even excellent, mathematics – but mathematics, nevertheless, of a rather traditional variety. The theorems I had proved were difficult and important, but they followed and extended lines of thought started by others, before me. Today, however, I will present to you my most important and original work, a ground-breaking advance. With the discovery of my geometric method I finally entered virgin, unexplored territory.'

  'All the more pity that you abandoned it,’ I said, preparing the climate from the start for a confrontation.

  He disregarded this and continued: 'The basic premise behind the geometric approach is that multiplication is an unnatural operation.'

  'What on earth do you mean by unnatural?' I asked.

  'Leopold Kronecker once said: "Our dear God made the integers, everything else is the work of man." Well, in the same way he made the integers, I think Kronecker forgot to add, the Almighty created addition and subtraction, or give and take.'

  I laughed. 'I thought I came here for lessons in mathematics, not theology!'

  Again he continued, ignoring the interruption. 'Multiplication is unnatural in the same sense as addition is natural. It is a contrived, second-order concept, no more really than a series of additions of equal elements. 3x5, for example, is nothing more than 5+5+5. To invent a name for this repetition and call it an 'operation' is the devil's work more likely…'

  I didn't risk another facetious comment.

  'If multiplication is unnatural,' he continued, 'more so is the concept of "prime number" that springs directly from it. The extreme difficulty of the basic problems related to the primes is in fact a direct outcome of this. The reason there is no visible pattern in their distribution is that the very notion of multiplication – and thus of primes – is unnecessarily complex. This is the basic premise. My geometric method is motivated simply by the desire to construct a natural way of viewing the primes.'

  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!'

  '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 fou
nd 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.