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 [10].
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 Papachristos. 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 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.
[10] In his seminal work The Nature of Mathematical Discovery, Henri Poincare demolishes the myth of the mathematician as a totally rational being. With examples drawn from history, as well as from his own research experience, he places special emphasis on the role of the unconscious in research. Often, he says, great discoveries happen unexpectedly, in a flash of revelation that comes in a moment of repose – of course, these can occur only to minds that are otherwise prepared through endless months or years of conscious work. It is in this aspect of the workings of a mathematician's mind that revelatory dreams can play an important role, sometimes providing the route through which the unconscious announces its conclusions to the conscious mind.