Uncle Petros and Goldbach

by Apostolos Doxiadis

  This, it must be stressed, was not because he had now decided to be content with less. There was no defeatism whatsoever concerning his ultimate aim of proving Goldbach's Conjecture. In Innsbruck, Petros had calmly reviewed the state of knowledge on his problem. He'd gone over the results arrived at by other mathematicians before him and also he'd analysed the course of his own research. Retracing his steps and coolly assessing his achievement to date, two things became obvious: a) His two theorems on Partitions were important results in their own right, and b) They brought him no closer to the proof of the Conjecture – his initial plan of attack had not yielded results.

  The intellectual peace he had achieved in Innsbruck resulted in a fundamental insight: the fallacy in his approach lay in the adoption of the analytic approach. He realized now that he had been led astray by the success of Hadamard and de la Vallee-Poussin in proving the Prime Number Theorem and also, especially, by Hardy's authority. In other words, he had been misled by the demands of mathematical fashion (oh yes, such a thing does exist!), demands that have no greater right to be considered Mathematical Truth than the annually changing whims of the gurus of haute couture do to be regarded as the Platonic Ideal of Beauty. The theorems arrived at through rigorous proof are indeed absolute and eternal, but the methods used to get to them are definitely not. They represent choices that are by definition circumstantial – which is why they change as often as they do.

  Petros' powerful intuition now told him that the analytic method had all but exhausted itself. The time had come for something new or, to be exact, something old, a return to the ancient, time-honoured approach to the secrets of numbers. The weighty responsibility of redefining the course of Number Theory for the future, he now decided, lay on his shoulders: a proof of Goldbach's Conjecture using the elementary, algebraic techniques would settle the matter once and for all.

  As to his two first results, the Partition Theorem and the other, they could now safely be released to the general mathematical population. Since they had been arrived at through the (no longer seemingly useful to him for proving the Conjecture) analytic method, their publication could not threaten unwelcome infringements on his future research.

  When he returned to Munich, his housekeeper was delighted to see the Herr Professor in such good shape. She hardly recognized him, she said, he 'looked so robust, so flushed with good health'.

  It was mid-summer and, unencumbered by academic obligations, he immediately started to compose the monograph that presented his two important theorems with their proofs. Seeing once again the harvest of his ten-year hard labours with the analytic method in concrete form, with a beginning, a middle and an end, complete and presented and explained in a structured way, Petros now felt deeply satisfied. He realized that, despite the fact that he had not yet managed to prove the Conjecture, he had done excellent mathematics. It was certain that the publication of his two theorems would secure him his first significant scientific laurels. (As already mentioned, he was indifferent to the lesser, applications-oriented interest in the 'Papachristos method for the solution of differential equations'.) He could now even allow himself some gratifying daydreams of what was in store for him. He could almost see the enthusiastic letters from colleagues, the congratulations at the School, the invitations to lecture on his discoveries at all the great universities. He could even envision receiving international honours and prizes. Why not – his theorems certainly deserved them!

  With the beginning of the new academic year (and still working on the monograph) Petros resumed his teaching duties. He was surprised to discover that for the first time he was now enjoying his lectures. The required effort at clarification and explanation for the sake of his students increased his own enjoyment and understanding of the material he was teaching. The Director of the School of Mathematics was obviously satisfied, not only by the improved performance he was hearing about from assistants and students alike, but mainly by the information that Professor Papachristos was preparing a monograph for publication. The two years at Innsbruck had paid off. Even though his forthcoming work apparently did not contain the proof of Goldbach's Conjecture, it was already rumoured in the School that it put forward extremely important results.

  The monograph was finished a little after Christmas and it came to about two hundred pages. It was titied, with the usual slightly hypocritical modesty of many mathematicians when publishing important results, 'Some Observations on the Problem of Partitions'. Petros had it typed at the School and mailed a copy to Hardy and Littlewood, purportedly asking them to go over it lest he had slipped into an undetected pitfall, lest some less-than-obvious deductive error had escaped him. In fact, he knew well that there were no pitfalls and no errors: he just relished the thought of the two paragons of Number Theory's surprise and amazement. In fact, he was already basking in their admiration for his achievement.

  After he sent off the typescript, Petros decided he owed himself a small vacation before he turned once again full-time to his work on the Conjecture. He de-voted the next few days exclusively to chess.

  He joined the best chess club in town, where he discovered to his delight that he could beat all but the very top players and give a hard time to the select few he could not easily overpower. He discovered a small bookshop owned by an enthusiast, where he bought weighty volumes of opening theory and collections of games. He installed the chessboard he'd bought at Innsbruck on a small table in front of his fireplace, next to a comfortable deep armchair upholstered in soft velvet. There he kept his nightly rendezvous with his new white and black friends.

  This lasted for almost two weeks. 'Two very happy weeks,’ he told me, the happiness being made greater by the anticipation of Hardy's and Littlewood's doubtless enthusiastic response to the monograph.

  Yet the response, when it arrived, was anything but enthusiastic and Petros' happiness was cut short. The reaction wasn't at all what he had anticipated. In a rather short note, Hardy informed him that his first important result, the one he'd privately christened the 'Papachristos Partitions Theorem', had been discovered two years before by a young Austrian mathematician. In fact, Hardy expressed his amazement that Petros had not been aware of this, since its publication had caused a sensation in the circles of number theorists and brought great acclaim to its young author. Surely he was following the developments in the field, or wasn't he? As for his second theorem: a rather more general version of it had been proposed without proof by Ramanujan in a letter to Hardy from India, a few days before his death in 1920, one of his last great intuitions. In the years since then, the Hardy-Littlewood partnership had managed to fill in the gaps and their proof had been published in the most recent issue of the Proceedings of the Royal Society, of which he included a copy.

  Hardy concluded his letter on a personal note, expressing his sympathy to Petros for this turn of events. With it there was the suggestion, in the understated fashion of his race and class, that it might in the future be more profitable for him to stay in closer contact with his scientific colleagues. Had Petros been living the normal life of a research mathematician, Hardy pointed out, coming to the international congresses and colloquia, corresponding with his colleagues, finding out from them the progress of their research and letting them know of his, he wouldn't have come in second in both of these otherwise extremely important discoveries. If he continued in his self-imposed isolation, another such 'unfortunate occurrence' was bound to arise.

  At this point in his narrative my uncle stopped. He had been talking for several hours. It was getting dark and the birdsong in the orchard had been gradually tapering off, a solitary cricket now rhythmically piercing the silence. Uncle Petros got up and moved with tired steps to turn on a lamp, a naked bulb that cast a weak light where we were seated. As he walked back towards me, moving slowly in and out of pale yellow light and violet darkness, he looked almost like a ghost.

  'So that's the explanation,' I murmured, as he sat down.

  'What explanation?' h
e asked absently.

  I told him of Sammy Epstein and his failure to find any mention of the name Petros Papachristos in the bibliographical index for Number Theory, with the exception of the early joint publications with Hardy and Littlewood on the Riemann Zeta Function. I repeated the 'burnout theory' suggested to my friend by the 'distinguished professor' at our university: that his supposed occupation with Goldbach's Conjecture had been a fabrication to disguise his inactivity.

  Uncle Petros laughed bitterly.

  'Oh no! It was true enough, most favoured of nephews! You can tell your friend and his "distinguished professor" that I did indeed work on trying to prove Goldbach's Conjecture – and how and for how longl Yes, and I did get intermediate results – wonderful, important results – but I didn't publish them when I should have done and others got in there ahead of me. Unfortunately, in mathematics there's no silver medal. The first to announce and publish gets all the glory. There's nothing left for anyone eise.' He paused.

  'As the saying goes, a bird in the hand is worth two in the bush and I, while pursuing the two, lost the one…'

  Somehow I didn't think the resigned serenity with which he stated this conclusion was sincere.

  'But, Uncle Petros,’ I asked him, 'weren't you horribly upset when you heard from Hardy?'

  'Naturally I was – and "horribly" is exactly the word. I was desperate; I was overcome with anger and frustration and grief; I even briefly contemplated suicide. That was back then, however, another time, another seif. Now, assessing my life in retrospect, I don't regret anything I did, or did not do.'

  'You don't? You mean you don't regret the opportunity you missed to become famous, to be acknowledged as a great mathematician?'

  He lifted a warning finger. 'A very good mathematician perhaps, but not a great one! I had discovered two good theorems, that's all.'

  "That's no mean achievement, surely!'

  Uncle Petros shook his head. 'Success in life is to be measured by the goals you've set yourself. There are tens of thousands of new theorems published every year the world over, but no more than a handful per century that make history!'

  'Still, Uncle, you yourself say your theorems were important.'

  'Look at the young man,' he countered, 'the Austrian who published my – as I still think of it – Partitions Theorem before me: was he raised with this result to the pedestal of a Hubert, a Poincare? Of course not! Perhaps he managed to secure a small niche for his portrait, somewhere in a back room of the Edifice of Mathematics… but if he did, so what? Or, for that matter, take Hardy and Littlewood, top-class mathematicians both of them. They possibly made the Hall of Fame – a very large Hall of Farne, mind you – but even they did not get their statues erected at the grand entrance alongside Euclid, Archimedes, Newton, Euler, Gauss… That had been my only ambition and nothing short of the proof of Goldbach's Conjecture, which also meant cracking the deeper mystery of the primes, could possibly have lead me there…’ There was now a gleam in his eyes, a deep, focused intensity as he concluded: ‘I, Petros Papachristos, never having published anything of value, will go down in mathematical history – or rather will not go down in it – as having achieved nothing. This suits me fine, you know. I have no regrets. Mediocrity would never have satisfied me. To an ersatz, footnote kind of immortality, I prefer my flowers, my orchard, my chessboard, the conversation I'm having with you today. Total obscurity!'

  With these words, my adolescent admiration for him as Ideal Romantic Hero was rekindled. But now it was marked by large doses of realism.

  'So, Uncle, it was really a question of all or nothing, eh?'

  He nodded slowly. 'You could put it that way, yes.'

  'And was this the end of your creative life? Did you ever again work on Goldbach's Conjecture?'

  He gave me a surprised look. 'Of course I did! In fact it was after that I did my most important work.' He smiled. 'We'll come to that by and by, dear boy. Don't worry, in my story there shall be no ignorabimus!’

  Suddenly he laughed loudly at his own joke, too loudly for comfort, I thought. Then he leaned towards me and asked me in a low voice: 'Did you learn Gödel's Incompleteness Theorem?'

  'I did,' I replied, 'but I don't see what it has to do with -'

  He lifted his hand roughly, cutting me short.

  ' "Wir müssen wissen, wir werden wissen! In der Mathematik gibt es kein ignorabimus" ' he declaimed stridently, so loudly that his voice echoed against the pine trees and returned, to menace and haunt me. Sammy's theory of insanity instantly flashed through my mind. Could all this reminiscing have aggravated his condition? Could my uncle have finally become unhinged?

  I was relieved when he continued in a more normal tone: '"We must know, we shall know! In mathematics there is no ignorabimus!” Thus spake the great David

  Hubert in the International Congress, in 1900. A proclamation of mathematics as the heaven of Absolute Truth. The vision of Euclid, the vision of Consistency and Completeness…'

  Uncle Petros resumed his story.

  The vision of Euclid had been the transformation of a random collection of numerical and geometric observations into a well-articulated system, where one can proceed from the a priori accepted elementary truths and advance, applying logical operations, step by step, to rigorous proof of all true statements: mathematics as a tree with strong roots (the Axioms), a solid trunk (Rigorous Proof) and ever growing branches blooming with wondrous flowers (the Theorems). All later mathematicians, geometers, number theorists, algebraists, and more recently analysts, topologists, algebraic geometers, group-theorists, etc., the practitioners of all the new disciplines that keep emerging to this day (new branches of the same ancient tree) never veered from the great pioneer's course: Axioms-Rigorous Proof-Theorems.

  With a bitter smile, Petros remembered the constant exhortation of Hardy to anyone (especially poor Ramanujan, whose mind produced them like grass on fertile soil) bothering him with hypotheses: 'Prove it! Prove it!' Indeed, Hardy liked saying, if a heraldic motto were needed for a noble family of mathematicians, there could be no better than Quod Erat Demonstrandum.

  In 1900, during the Second International Congress of Mathematicians, held in Paris, Hubert announced that the time had come to extend the ancient dream to its ultimate consequences. Mathematicians now had at their disposal, as Euclid had not, the language of Formal Logic, which allowed them to examine, in a rigorous way, mathematics itself. The holy trinity of Axioms-Rigorous Proof-Theorems should hence be applied not only to the numbers, shapes or algebraic identities of the various mathematical theories but to the very theories themselves. Mathematicians could at last rigorously demonstrate what for two millennia had been their central, unquestioned credo, the core of the vision: that in mathematics every true statement is provable.

  A few years later, Russell and Whitehead published their monumental Principia Mathematica, proposing for the first time a totally precise way of speaking about deduction, Proof Theory. Yet although this new tool brought with it great promise of a final answer to Hilbert's demand, the two English logicians fell short of actually demonstrating the critical property. The 'completeness of mathematical theories' (i.e. the fact that within them every true statement is provable) had not yet been proven, but there was now not the smallest doubt in anybody's mind or heart that one day, very soon, it would be. Mathematicians continued to believe, as Euclid had believed, that they dwelt in the Realm of Absolute Truth. The victorious cry emerging from the Paris Congress, 'We must know, we shall know, in Mathematics there is no ignorabimus,' still constituted the one unshakable article of faith of every working mathematician.

  I interrupted this rather exalted historical excursion: 'I know all this, Uncle. Once you enjoined me to learn Gödel's theorem I obviously also had to find out about its background.'

  'It's not the background,' he corrected me; 'it's the psychology. You have to understand the emotional climate in which mathematicians worked in those happy days, before Kurt Gödel. You
asked me how I mustered up the courage to continue after my great disap-pointment. Well, here's how…'

  Despite the fact that he hadn't yet managed to attain his goal and prove Goldbach's Conjecture, Uncle Petros firmly believed that his goal was attainable. Being himself Euclid's spiritual great-grandson, his trust in this was complete. Since the Conjecture was almost certainly valid (nobody with the exception of Ramanujan and his vague 'hunch' had ever seriously doubted this), the proof of it existed somewhere, in some form.

  He continued with an example:

  'Suppose a friend states that he has mislaid a key somewhere in his house and asks you to help him find it. If you believe his memory to be faultless and you have absolute trust in his integrity, what does it mean?'

  'It means that he has indeed mislaid the key somewhere in his house.'

  'And if he further ascertains that no one else entered the house since?'

  'Then we can assume that it was not taken out of the house.'

  'Ergo?'

  'Ergo, the key is still there, and if we search long enough – the house being finite – sooner or later we will find it.'

  My uncle applauded. 'Excellent! It is precisely this certainty that fuelled my optimism anew. After I had recovered from my first disappointment I got up one fine morning and said to myself: "What the hell – that proof is still out there, somewhere!"'

  'And so?'

  'And so, my boy, since the proof existed, one had but to find it!'

  I wasn't following his reasoning.

  'I don't see how this provided comfort, Uncle Petros: the fact that proof existed didn't in any way imply that you would be the one to discover it!'

  He glared at me for not immediately seeing the obvious. 'Was there anyone in the whole wide world better equipped to do so than I, Petros Papachristos?'

  The question was obviously rhetorical and so I didn't bother to answer it. But I was puzzled: the Petros Papachristos he was referring to was a different man from the self-effacing, withdrawn senior citizen I'd known since childhood.