Uncle Petros and Goldbach

by Apostolos Doxiadis

  Yet, I hasten to add, I've never regretted my years as a mathematical hopeful. Learning some real mathematics, even my tiny portion of it, has been for me the most invaluable lesson of life. Obviously, everyday problems can be handled perfectly well without knowledge of the Peano-Dedekind Axiomatic System, and mastery of the Classification of Finite Simple Groups is absolutely no guarantee of success in business. On the other hand, the non-mathematician cannot conceive of the joys that he's beert denied. The amalgam of Truth and Beauty revealed through the understanding of an important theorem cannot be attained through any other human activity, unless it be (I wouldn't know) that of mystical religion. Even if my education was meagre, even if it meant no more than getting my toes wet on the beach of the immense ocean of mathematics, it has marked my life for ever, giving me a small taste of a higher world. Yes, it has made the existence of the Ideal slightly more believable, even tangible.

  For this experience I am forever in Uncle Petros' debt: it's impossible I would have made the choice without him as my dubious role model.

  My decision to abandon plans of a mathematical career came as a joyful surprise to my father (the poor man had fallen into deep despair during my last undergraduate years), a surprise made even happier when he learned I would be going to business school. When, having completed my graduate studies and military service, I joined him in the family business, his happiness was at last complete.

  Despite this volte-face (or maybe because of it?) my relationship with Uncle Petros blossomed anew after I returned to Athens, every vestige of bitterness on my part totally dissipated. As I gradually settled down to the routines of work and family life, visiting him became a frequent habit, if not a necessity. Our contact was an invigorating antidote to the increasing grind of the real world. Seeing him helped me keep alive that part of the self that most people lose, or forget about, with adulthood – call it the Dreamer or the Wonderer or simply the Child Within. On the other hand, I never understood what my friendship offered him, if we exclude the companionship he claimed not to need.

  We wouldn't talk all that much on my visits to Ekali, as we'd found a means of communication better suited to two ex-mathematicians: chess. Uncle Petros was an excellent teacher and soon I came to share his passion (though unfortunately not his talent) for the game.

  In chess, I also had the first direct experience of him as a thinker. As he analysed for my benefit the classic great games, or the more recent contests of the world's best players, I was filled with admiration for the workings of his brilliant mind, its immediate grasp of the most complex problems, its analytical power, the flashes of insight. When he confronted the board his features became fixed in utter concentration, his gaze became sharp and penetrating. Logic and intuition, the instruments with which he'd pursued for two decades the most ambitious intellectual dream, sparkled in his deep-set blue eyes.

  Once, I asked him why he had never entered official competition.

  He shook his head. 'Why should I strive to become a mediocre professional when I can bask in my status as an exceptional amateur?' he said. 'Besides, most favoured of nephews, every life should progress according to its basic axioms and chess wasn't among mine – only mathematics.'

  The first time I ventured to ask him again about his research (after the extensive account of his life he had given me, we'd never again mentioned anything mathematicaL both of us apparently preferring to let our sleeping dogs lie) he immediately dismissed the matter.

  'Let bygones be bygones and tell me what you see on the chessboard. It's a recent game between Petrosian and Spassky, a Sicilian Defence. White takes Knight to f4…'

  More oblique attempts didn't work either. Uncle Petros would not be coaxed into another mathematical discussion – period. Whenever I attempted a direct mention it would always be: 'Let's stick to chess, shall we?'

  His refusals, however, didn't make me give up.

  My wish to draw him once again to the subject of his life's work was not fired by mere curiosity. Although it was a long time since I had any news of my old friend Sammy Epstein (last time I'd heard of him he was an assistant professor in California), I couldn't forget his explanation of Uncle Petros giving up his research. In fact, I'd come to invest it with great existential significance. The development of my own affair with mathematics had taught me an important lesson: one should be brutally honest with oneself about weaknesses, acknowledge them with courage and chart further course accordingly. For myself I had done this, but had Uncle Petros?

  These were the facts: a) From an early age he had chosen to invest all his energy and time in an incredibly, but most probably not impossibly, difficult problem, a decision which I still continued to regard as basically noble; b) As might reasonably have been expected (by others, if not by himself) he had not achieved his goal; c) He had blamed his failure on the incompleteness of mathematics, deeming Goldbach's Conjecture unprovable.

  Of this much I was now certain: the validity of his excuse had to be judged by the strict standards of the trade and, according to these, I accepted Sammy Epstein's opinion as final – a final verdict of unprovability a la Kurt Gödel is just not an acceptable conclusion of the attempt to prove a mathematical statement. My old friend's explanation was much closer to the point. It wasn't because of his 'bad luck' Uncle Petros hadn't managed to achieve his dream. The appeal to the Incompleteness Theorem was indeed a sophisticated form of 'sour grapes', meant only to shelter him from the truth.

  With the passing of the years, I had learned to recognize the profound sadness that pervaded my uncle's life. His absorption in gardening, his kindly smiles or his brilliance as a chess player couldn't disguise the fact that he was a broken man. And the closer to him I got, the more I realized that the reason for his condition lay in his profound insincerity. Uncle Petros had lied to himself about the most crucial event in his life and this lie had become a cancerous growth that stifled his essence, eating away at the very roots of his psyche. His sin, indeed, had been Pride. And the pride was still there, nowhere more apparent than in his inability to come face to face with himself.

  I've never been a religious man, yet I believe there is great underlying wisdom in the ritual of Absolution: Petros Papachristos, like every human being, deserved to end his life unburdened of unnecessary suffering. In his case, however, this had the necessary prerequisite of his admitting the mea culpa of his failure.

  The context here not being religious, a priest could not do the job.

  The only person fit to absolve Uncle Petros was I myself, for only I had understood the essence of his transgression. (The pride inherent in my own assumption I did not realize until it was too late.) But how could I absolve him if he did not first confess? And how could I lead him to confession unless we started once again to talk mathematics, a thing he persistently refused to do?

  In 1971, I found unexpected assistance in my task.

  The military dictatorship that then ruled the country, in a campaign to appear as a benevolent patron of culture and science, proposed to award a 'Gold Medal of Excellence' to a number of rather obscure Greek scholars who had distinguished themselves abroad. The list was short, since most of the prospective honourees, forewarned of the impending distinction, had hastened to exclude themselves; but topmost in it was 'the great mathematician of international fame, Professor Petros Papachristos'.

  My father and Uncle Anargyros, in a totally uncharacteristic frenzy of democratic passion, strove to convince him to turn down this dubious honour. Talk of 'that old fool becoming the junta's lackey', 'giving the colonels an alibi', etc., filled our business offices and family homes. At moments of greater honesty the two younger brothers (both old men, by now) confessed to a less noble motive: the traditional reluctance of the businessman to be too closely identified with one political faction for fear of what will happen when another comes to power. Yet I, an experienced Papachristos family observer, could also discern a strong need for them to be proved right in their negative evaluati
on of his life, also tinged with an element of envy. Father's and Uncle Anargyros' world-view had always been founded on the simple premise that Uncle Petros was bad and they good, a black-and-white cosmology that distinguished between the grasshoppers and the ants, the dilettantes and 'responsible men'. It didn't sit at all well with them that the country's official government, Junta or no Junta, should honour 'one of life's failures', when the only rewards they ever got for their labours (labours, mind you, that also put food on his table) were financial.

  I, however, took a different position. Beyond my belief that Uncle Petros deserved the honour (he did, after all, rate some recognition of his life's work, even if it came from the colonels) I had an ulterior motive. So I went to Ekali and, exercising to the full my influence as 'most favoured of nephews', convinced him to overcome his brothers' hypocritical appeals to democratic duty as well as his own misgivings and accept his Gold Medal of Excellence.

  The award ceremony – that 'ultimate familial disgrace', according to Uncle Anargyros the late-blooming radical – was held in the main auditorium of the University of Athens. The Rector of the School of Physics and Mathematics, in his ceremonial robes, gave a short lecture on Uncle Petros' contribution to science. Predictably enough he referred almost exclusively to the Papachristos Method for the Solution of Differential Equations, which he lauded with elaborate rhetorical effusions. Still, I was agreeably surprised to hear him also make passing reference to Hardy and Littlewood and their 'appealing to our great fellow-countryman for assistance with their most difficult problems'. While all this was being propounded I stole side-glances at Uncle Petros and saw him blushing red with shame again and again, all the time retreating further into the throne-like, gilded armchair where they had him installed. The Prime Minister (the arch-dictator) then bestowed the Gold Medal of Excellence and afterwards there was a short reception, during which my poor uncle was required to pose for photographs with all the top brass of the Junta. (I have to confess that at this stage of the ceremony I feit a slight dose of guilt about the defining role I had played in his acceptance of the honour.)

  When it was all over, he asked me to go back home for some chess, 'for purposes of recovery'. We started a game. I was a good enough player by that time to offer him decent resistance but not so good as to hold his interest after the ordeal he'd been through.

  'What did you think of that circus?' he asked me, finally looking up from the board.

  'The award ceremony? Oh, it was a bit boring, but I'm still glad you went through with it. Tomorrow it will be in all the newspapers.'

  'Yes,’ he said, 'how the Papachristos Method for the Solution of Differential Equations is almost on a par with Einstein's Theory of Relativity and Heisenberg's Uncertainty Principle, one of the crowning achievements of twentieth-century science… How that fool of a Rector carried on! Did you notice, by the way,’ he added with a sour smile, 'the pregnant silence following the "ooohs" and "aaahs" and "ts-ts-ts's" of admiration at my extreme youth when I made the "great discovery"? You could almost hear everybody wondering: But how did the honouree spend the next fifty-five years of his life?'

  Any sign of self-pity on his part bothered me inordinately.

  'You know, Uncle,’ I provoked him, 'it's not anybody's fault but your own that people don't know of your work on Goldbach's Conjecture. How could they – you've never told! Had you ever written up a report of your research, things would be different.

  The story of your quest itself would make a worth-while publication.'

  'Yes,’ he sneered, 'a full footnote in Great Mathematical Failures of Our Century.'

  'Well,' I mused, 'science advances by failures as well as successes. And anyway, it was a good thing your work in differential equations was acknowledged. I was proud to hear our family name associated with something other than money.'

  Unexpectedly, a bright smile on his face, Uncle Petros asked me: 'Do you know it?'

  'Do I know what?'

  'The Papachristos Method for the Solution of Differential Equations?'

  I'd been taken completely unawares and answered without thinking: 'No, I don't.'

  His smile went away: 'Well, I expect they don't teach it anymore…'

  I feit an upsurge of excitement – this was the chance I was waiting for. Although I had, in fact, ascertained while at university that the Papachristos Method was no longer taught (the advent of electronic calculation had rendered it obsolete), I lied to him, and with great vehemence: 'Of course they teach it, Uncle! It's just that I never took an elective in differential equations.'

  'Get paper and pencil then, and I’ll tell you about it!'

  I held back a triumphant cry. It was precisely what I'd hoped for when I had convinced him to accept the medal: that the honour might reawaken his mathematical vanity and rekindle his interest in his art, enough of it anyway to lure him into a discussion of Goldbach's Conjecture and beyond… to his real reason for abandoning it. Explaining to me the Papachristos Method was an excellent introduction.

  I rushed to fetch paper and pencil before he changed his mind.

  'You'll have to be a little patient,' he began. 'A lot of water has gone under the bridge since then. Let's see now,’ he murmured and began to scribble. 'Let us assume we have a partial differential equation in the Clairaut form… there! We now take…'

  I followed his scribbles and explanations for almost an hour. Although I couldn't completely follow the argument, I showed exaggerated appreciation at every step.

  'It's absolutely brilliant, Uncle!' I cried when he'd finished.

  'Nonsense.' He brushed my praise aside, but I could see this modesty was not totally sincere. 'Sheer calculation of the grocery-bill variety, not real mathematics!'

  The moment I was waiting for had arrived. 'Then talk to me about real mathematics, Uncle Petros. Talk to me about your work on Goldbach's Conjecture!'

  He shot me a sideways glance, cunning, inquisitive and at the same time tentative. I held my breath.

  'And what, if I may ask, is the purpose of your interest, Mr Almost-Mathematician?'

  I had planned my answer to this beforehand, so as to put him in an emotional impasse.

  'You owe it to me, Uncle! If for nothing eise, to compensate me for that summer of anguish in my sixteenth year, when I struggled for three months to prove it myself, floundering in my abysmal ignorance!'

  He appeared to be considering this for a while, as if to make a point of not giving in all too easily. When he smiled I knew I had won.

  'What exactly do you want to know about my work on Goldbach's Conjecture?'

  I left Ekali after midnight with a copy of An Introduction to Number Theory by Hardy and Wright. (I had to prepare myself by learning 'some fundamentals’ he'd said.) I should point out to the non-specialist that mathematical books cannot normally be enjoyed like novels, in bed, in the bathtub, sprawled in an easy chair, or perched on the commode. To 'read' here means to understand, and for that you normally need a hard surface, paper, pencil and quality time. Since I had no intention of becoming a number theorist at the advanced age of thirty, I went through the Hardy-Wright book with only moderate attention ('moderate' in mathematics is 'considerable' by any other measure), without persisting on fully comprehending those details that resisted the initial assault. Even so, and taking into account that the study of the book was not my main occupation, it took me almost a month.

  When I returned to Ekali, Uncle Petros, bless his soul, started to examine me as if I were a schoolchild.

  'Have you read the whole book?'

  'I have.'

  'State Landau's Theorem.'

  I did.

  'Write out for me the proof of Euler 's Theorem of the phi-function, the extension of Fermat's Little Theorem.'

  I took paper and pencil and proceeded to do so, as best as I could.

  'Now prove to me that the non-trivial zeros of the Riemann Zeta Function have real part equal to 1/2!'

  I burst out laughing and h
e did too.

  'Oh no, you don't!' I said. 'Not again, Uncle Petros! It's enough that you set me to prove Goldbach's Conjecture. Find somebody else to assign the Riemann Hypothesis!'

  In the following two and a half months we had our ten 'Lessons on Goldbach's Conjecture', as he called them. What transpired in them is down on paper, with dates and times. Since I was now moving steadily towards the fulfilment of my main aim (his coming face to face with the reason for abandoning his research), I thought I'd also attain a secondary goal while at it: I kept meticulous notes so that, after his death, I could publish a short account of his Odyssey, perhaps an insignificant footnote to mathematical history, but still a worthy tribute to Uncle Petros – if not, alas, to his ultimate success, then certainly to his ingenuity and, more importantly, his dedication and single-minded persistence.

  During the course of the lessons I witnessed an amazing metamorphosis. The mild, kindly, elderly gentleman I had known since my childhood, one easily mistaken for a retired civil servant, turned before my eyes into a man illuminated by a fierce intelligence and driven by an inner power of unfathomable depth. I'd caught small glimpses of this species of being before, during mathematical discussions with my old room-mate, Sammy Epstein, or even with Uncle Petros himself, when he sat before his chessboard. Listening to him unravel the mysteries of Number Theory, however, I experienced for the first and only time in my life the real thing. You didn't have to know mathematics to feel it. The sparkle in his eyes and an unspoken power emanating from his whole being were testimony enough. He was the absolute thorough-bred, pure unadulterated genius.