The problem he had solved on his afternoon walk by the Speichersee was of particular importance. As regarded his work on the Conjecture it was of course still an intermediate step, not his ultimate goal. Nevertheless, it was a deep, pioneering theorem in its own right, one which opened new vistas in the Theory of Numbers. It shed a new light on the question of Partitions, applying the previous Hardy-Ramanujan theorem in a way that no one had suspected, let alone demonstrated, before. Undoubtedly, its publication would secure him recognition in the mathematical world much greater than that achieved by his method for solving differential equations. In fact, it would probably catapult him to the first ranks of the small but select international community of number theorists, practically on the same level as its great stars, Hadamard, Hardy and Littlewood.
By making his discovery public, he would also be opening the way into the problem to other mathematicians who would build on it by discovering new results and expand the limits of the field in a way a lone researcher, however brilliant, could scarcely hope. The results they would achieve would, in turn, aid him in his pursuit of the proof to the Conjecture. In other words, by publishing the ‘Papachristos Partition Theorem’ (modesty of course obliged him to wait for his colleagues formally to give it this title) he would be acquiring a legion of assistants in his work. Unfortunately there was another side to this coin: one of the new unpaid (also unasked for) assistants might conceivably stumble upon a better way to apply his theorem and manage, God forbid, to prove Goldbach’s Conjecture before him.
He didn’t have to deliberate long. The danger far outweighed the benefit. He wouldn’t publish. The ‘Papachristos Partitions Theorem’ would remain for the time being his private, well-guarded secret.
Reminiscing for my benefit, Uncle Petros marked this decision as a turning point in his life. From then onwards, he said, difficulties began to pile upon difficulties.
By withholding publication of his first truly important contribution to mathematics, he had placed himself under double time-pressure. In addition to the constant, gnawing anxiety of days and weeks and months and years passing without his having achieved the desired final goal, he now also had to worry that someone might arrive at his discovery independently and steal his glory.
The official successes he had achieved until then (a discovery named after him and a university chair) were no mean feats. But time counts differently for mathematicians. He was now at the absolute peak of his powers, in a creative prime that couldn’t last long. This was the time to make his great discovery — if he had it in him to make it at all.
Living as he did a life of near-total isolation, there was no one to ease his pressures.
The loneliness of the researcher doing original mathematics is unlike any other. In a very real sense of the word, he lives in a universe that is totally inaccessible, both to the greater public and to his immediate environment. Even those closest to him cannot partake of his joys and his sorrows in any significant way, since it is all but impossible for them to understand their content.
The only community to which the creative mathematician can truly belong is that of his peers; but from that Petros had wilfully cut himself off. During his first years at Munich he had submitted occasionally to the traditional academic hospitality towards newcomers. When he accepted an invitation, however, it was sheer agony to act with a semblance of normality, behave agreeably and make small talk. He had constantly to curb his tendency to lose himself in number-theoretical thoughts, and fight his frequent impulses to make a mad dash for home and his desk, in the grip of a hunch that required immediate attention. Fortunately, either as a result of his increasingly frequent refusals or his obvious discomfort and awkwardness on those occasions when he did attend social functions, invitations gradually grew fewer and fewer and in the end, to his great relief, ceased altogether.
I don’t need to add that he never married. The rationale he gave me for this, by which getting married to another woman would mean being unfaithful to his great love, ‘dearest Isolde’, was of course no more than an excuse. In truth, he was very much aware that his lifestyle did not allow for the presence of another person. His preoccupation with his research was ceaseless. Goldbach’s Conjecture demanded him whole: his body, his soul and all of his time.
In the summer of 1925, Petros proved a second important result, which in combination with the ‘Partitions Theorem’ opened up a new perspective on many of the classical problems of prime numbers. According to his own, exceedingly fair and well-informed opinion, the work he had done constituted a veritable breakthrough. The temptation to publish was now overwhelming. It tortured him for weeks — once again, though, he managed to resist it. Again, he decided in favour of keeping his secret to himself, lest it open the way to unwelcome intruders. No intermediate result, no matter how important, could sidetrack him from his original aim. He would prove Goldbach’s Conjecture or be damned!
In November of that year he turned thirty, an emblematic age for the research mathematician, practically the first step into middle age.
The sword of Damocles, whose presence Petros had merely sensed all these years hanging in the darkness somewhere high above him (it was labelled: The Waning of his Creative Powers’) now became almost visible. More and more, as he sat hunched over his papers, he could feel its hovering menace. The invisible hourglass measuring out his creative prime became a constant presence at the back of his mind, driving him into bouts of dread and anxiety. During his every waking moment, he was pestered by the worry that he might already be moving away from the apex of his intellectual prowess. Questions buzzed in his mind like mosquitoes: would he be having any more breakthroughs of the same order as the two first important results? Had the inevitable decline, perhaps unbeknown to him, already started? Every little instance of forgetfulness, every tiny slip in a calculation, every short lapse in concentration, brought the ominous refrain: Have I passed my prime?
A brief visit at about this time from his family (already described to me by my father), whom he hadn’t seen in years, was considered by him a gross, violent intrusion. The little time he spent with his parents and younger brothers he felt was stolen from his work, and every moment away from his desk for their benefit he perceived as a small dose of mathematical suicide. By the end of their stay he was inordinately frustrated.
Not wasting time had become a veritable obsession, to the point where he obliterated from his life any activity that was not directly related to Goldbach’s Conjecture — all except the two he couldn’t reduce beyond a certain minimum, teaching and sleep. Yet he now got less sleep than he needed. Constant anxiety had brought insomnia with it, and this was aggravated by his excessive consumption of coffee, the fuel on which mathematicians run. With time, the constant preoccupation with the Conjecture made it impossible for him to relax. Falling or staying asleep became increasingly difficult and often he had to resort to sleeping pills. Occasional use gradually became steady and doses began to increase alarmingly, to the point of dependency, and this without the accompanying beneficial effect.
At about this time, a totally unexpected boost to his spirits came in the unlikely form of a dream. Despite his total disbelief in the supernatural, Petros viewed it as prophetic, a definite omen straight from Mathematical Heaven.
It is not unusual for scientists totally immersed in a difficult problem to carry on their preoccupations into sleep; and although Petros was never honoured by nocturnal visitations from Ramanujan’s Namakiri or any other revelatory deity (a fact that should not surprise us, considering his entrenched agnosticism), after the first year or so of his immersion in the Conjecture he began to have the occasional mathematical dream. In fact, his early visions of amorous bliss in the arms of ‘dearest Isolde’ became less frequent over time, giving their place to dreams of the Even Numbers, which appeared personified as couples of identical twins. They were involved in intricate, unearthly dumbshows, a chorus to the Primes, who were peculiar hermaphrodite, semi-human b
eings. Unlike the speechless Even Numbers, the Primes often chattered among themselves, usually in an unintelligible language, at the same time executing bizarre dance-steps. (By his admission, this dream choreography was most likely inspired by a production of Stravinsky’s Rite of Spring that Petros had attended during his early years in Munich, when he still had time for such vanities.) On rare occasions the singular creatures spoke and then only in classical Greek — perhaps as a tribute to Euclid, who had awarded them infinitude. Even when their utterances made some linguistic sense, however, the content was mathematically either trivial or nonsensical. Petros specifically recalled one such: hapantes protoi perittoi, which means ‘All prime numbers are odd’, an obviously false statement. (By a different reading of the word perittoi, however, it could also mean ‘All prime numbers are useless’, an interpretation which, interestingly, completely escaped my uncle’s attention.)
Yet in a few rare instances there was something of substance in his dreams. He could deduce from the protagonists’ sayings helpful hints that steered his research towards interesting, unexplored paths.*
The dream that lifted his spirits came a few nights after he had proved his second important result. It was not directly mathematical, but laudatory, consisting of no more than a single image, a sparkling tableau vivant, but of such unearthly beauty! Leonard Euler was on the one side and Christian Goldbach (though he’d never seen a portrait, he immediately knew it to be him) on the other. The two men jointly held, from the sides, a golden wreath over the head of the central figure, which was none other than himself, Petros Papa-christos. The triad was bathed in a nimbus of blinding light.
The dream’s message could not be clearer: the proof of Goldbach’s Conjecture would be ultimately his.
Spurred by the glorious spirit of this vision, his mood swung back to optimism and he coaxed himself onwards with added zest. Now, he should concentrate all his powers on his research. He could afford absolutely no distractions.
The painful gastrointestinal symptoms he had been having for some time (most of them by some strange coincidence occurring at times when they interfered with his university duties), a result of the constant, self-imposed pressure, gave him the pretext he needed. Armed with the opinion of a specialist, he went to see the Director of the School of Mathematics and requested a two-year, unpaid leave of absence.
The Director, an insignificant mathematician but a ferocious bureaucrat, was apparently waiting for an occasion to level with Professor Papachristos.
‘I have read your doctor’s recommendation, Herr Professor,’ he said in a sour tone. ‘Apparently you suffer — like many in our School — from gastritis, a condition that is not exactly terminal. Isn’t a two-year leave rather excessive?’
‘Well, Herr Director,’ mumbled Petros, ‘I also happen to be at a critical point in my research. While on my two-year leave I can complete it.’
The Director appeared genuinely surprised. ‘Research? Oh, I had no idea! You see, the fact that you haven’t published anything during all your years with us had led your colleagues to think that you were scientifically inactive.’
Petros knew the next question was inevitable:
‘By the way, what exactly is it you are researching, Herr Professor?’
‘We-ell,’ he replied meekly, ‘I am investigating certain questions in Number Theory.’
The Director, an eminently practical man, considered Number Theory, a field notorious for the inapplicability of its results to the physical sciences, a complete waste of time. His own interest lay in differential equations and, years back, he had hoped that the addition of the inventor of the Papachristos Method to the faculty would perhaps put his own name on some joint publications. This, of course, had never come about.
‘You mean Number Theory in general, Herr Professor?’
Petros suffered the ensuing cat-and-mouse game for a while, trying desperately to prevaricate concerning his real object. When, however, he realized he had not the slightest hope unless he convinced the Director of the importance of his work, he revealed the truth.
‘I’m working on Goldbach’s Conjecture, Herr Director. But please don’t tell anyone!’
The Director appeared startled. ‘Oh? And how are you progressing?’
‘Quite well, actually.’
‘Which means you have arrived at some very interesting intermediate results. Am I right?’
Petros felt as if he were walking on a tightrope. How much could he safely reveal?
‘Well…er…’ He was fidgeting in his seat, sweating profusely. ‘In fact, Herr Director, I believe I’m only one step away from the proof. If you would let me have my two years of unpaid leave, I will try to complete it.’
The Director knew Goldbach’s Conjecture — who didn’t? Despite the fact that it belonged to the cloud-cuckoo-land world of Number Theory it had the advantage of being an exceedingly famous problem. A success by Professor Papachristos (he was reputed to have, after all, a first-class mind) would definitely be to the great benefit of the university, the School of Mathematics and of course himself, its director. After pondering the matter for a while, he gave him a big smile and declared he wasn’t unfavourable to the request.
When Petros went to thank him and say goodbye, the Director was all smiles.
‘Good luck with the Conjecture, Herr Professor. I expect you back with great results!’
Having secured his two-year period of grace, he moved to the outskirts of Innsbruck, in the Austrian Tyrol, where he had rented a small cottage. As a forwarding address he left only the local poste restante. In his new, temporary abode he was a complete stranger. Here, he needn’t fear even the minor distractions of Munich, a chance encounter with an acquaintance in the street or the solicitude of his housekeeper, whom he left behind to look after the empty apartment. His isolation would remain absolutely inviolate.
During his stay in Innsbruck, there was a development in Petros’ life that turned out to have a beneficial effect both on his mood and, as a consequence, on his work: he discovered chess.
One evening, while out for his habitual walk, he stopped for a hot drink at a coffee-house, which happened to be the meeting-place of the local club. He had been taught the rules of chess and played a few games as a child, yet he remained to that day totally unaware of its profundity. Now, as he sipped his cocoa, his attention was caught by the game in progress at the next table and he followed it through with increasing interest. The next evening his footsteps led him to the same place, and the day after that as well. At first through mere observation, he gradually began to grasp the fascinating logic of the game.
After a few visits, he accepted a challenge to play. He lost, which was an irritant to his antagonistic nature, particularly so when he learned that his opponent was a cattle-herder by occupation. He stayed up that night, recreating the moves in his mind, trying to pinpoint his mistakes. The next evenings he lost a few more games, but then he won one and felt immense joy, a feeling that spurred him on towards more victories.
Gradually, he became a habitué of the coffee-house and joined the chess club. One of the members told him about the huge volume of accumulated wisdom on the subject of the game’s first moves, also known as ‘opening theory’. Petros borrowed a basic book and bought the chess set that he was still using in his old age, at his house in Ekali. He’d always kept late nights, but in Innsbruck it wasn’t due to Goldbach. With the pieces set out in front of him and the book in hand, he spent the hours before sleep teaching himself the basic openings, the ‘Ruy Lopez’, the ‘King’s’ and ‘Queen’s Gambits’, the ‘Sicilian Defence’.
Armed with some theoretical knowledge he proceeded to win more and more often, to his huge satisfaction. Indeed, displaying the fanaticism of the recent convert, he went overboard for a while, spending time on the game which belonged to his mathematical research, going to the coffee-house earlier and earlier, even turning to his chessboard during the daylight hours to analyse the previous day’s
games. However, he soon disciplined himself and restricted his chess activity to his nightly outing and an hour or so of study (an opening, or a famous game) before bedtime. Despite this, by the time he left Innsbruck he was the undisputed local champion.
The change brought about in Petros’ life by chess was considerable. From the moment he had first dedicated himself to proving Goldbach’s Conjecture, almost a decade earlier, he had hardly ever relaxed from his work. However, for a mathematician to spend time away from the problem at hand is essential. Mentally to digest the work accomplished and process its results at an unconscious level, the mind needs leisure as well as exertion. Invigorating as the investigation of mathematical concepts can be to a calm intellect, it can become intolerable when the brain is overcome by weariness, exhausted by incessant effort.
Of the mathematicians of his acquaintance, each had his own way of relaxing. For Caratheodory it was his administrative duties at Berlin University. With his colleagues at the School of Mathematics it varied: for family men it was usually the family; for some it was sports; for some, collecting or the theatrical performances, concerts and other cultural events that were on constant offer at Munich. None of these, however, suited Petros — none engaged him sufficiently to provide distraction from his research. At some point he tried reading detective stories, but after he’d exhausted the exploits of the ultra-rationalist Sherlock Holmes he found nothing else to hold his attention. As for his long afternoon walks, they definitely did not count as relaxation. While his body moved, whether in the countryside or the city, by a serene lakeside or on a busy pavement, his mind was totally preoccupied with the Conjecture, the walking itself being no more than a way to focus on his research.
So, chess seemed to have been sent to him from heaven. Being by its nature a cerebral game, it has concentration as a necessary requirement. Unless matched with a much inferior opponent, and sometimes even then, the player’s attention can only wander at a cost. Petros now immersed himself in the recorded encounters between the great players (Steinitz, Alekhine, Capablanca) with a concentration known to him only from his mathematical studies. While trying to defeat Innsbruck’s better players he discovered that it was possible to take total leave of Goldbach, even if only for a few hours. Faced with a strong opponent he realized, to his utter amazement, that for a few hours he could think of nothing but chess. The effect was invigorating. The morning after a challenging game he would tackle the Conjecture with a clear and refreshed mind, new perspectives and connections emerging, just as he’d begun to fear that he was drying up.
Uncle Petros and Goldbach's Conjecture Page 7