Apostolos Doxiadis - Uncle Petros and Goldbach

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.