Apostolos Doxiadis - Uncle Petros and Goldbach

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 habitue 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 ultrarationalist Sherlock Holmes he found nothing eise 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.

The relaxing effect of chess also helped Petros to wean himself from sleeping pills. From then on, if some night he were overcome by fruitless anxiety connected with the Conjecture, his tired brain twisting and wandering in endless mathematical mazes, he would get up from bed, seat himself before the chess-board and go over the moves of an interesting game. Immersing himself in it, he would temporarily forget his mathematics, his eyelids would grow heavy and he would sleep like a baby in his armchair till morning.

Before his two years of unpaid leave were up, Petros took a momentous decision: he would publish his two important discoveries, the 'Papachristos Partition Theorem' and the other one.

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.