Apostolos Doxiadis - Uncle Petros and Goldbach

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 But I was puzzled: the Petros Papachristos he was referring to was a different man from the self-effacing, withdrawn senior citizen I'd known since childhood.

Of course, it had taken him some time to recover from reading Hardy's letter and its disheartening news. Yet recover he eventually did. He pulled himself together and, his deposits of hope refilled through the belief in 'the existence of the proof somewhere out there', he resumed his quest, a slightly changed man. His misadventure, by exposing an element of vanity in his manic search, had created in him an inner core of peace, a sense of life continuing irrespective of Goldbach's Conjecture. His working schedule now became slightly more relaxed, his mind also aided by interludes of chess, more tranquil despite the constant effort.

In addition, the switch to the algebraic method, already decided in Innsbruck, made him feel once again the excitement of a fresh start, the exhilaration of entering virgin territory.

For a hundred years, from Riemann's paper in the mid-nineteenth Century, the dominant trend in Number Theory had been analytic. By now resorting to the ancient, elementary approach, my uncle was in the vanguard of an important regression, if I may be allowed the oxymoron. The historians of mathematics will do well to remember him for this, if for no other part of his work.

It must be stressed here that, in the context of Number Theory, the word 'elementary' can on no account be considered synonymous with 'simple' and even less so with 'easy'. Its techniques are those of Diophantus', Euclid's, Fermat's, Gauss's and Euler's great results and are elementary only in the sense of deriving from the elements of mathematics, the basic arithmetical operations and the methods of classical algebra on the real numbers. Despite the effectiveness of the analytic techniques, the elementary method stays closer to the fundamental properties of the integers and the results arrived at with it are, in an intuitive way clear to the mathematician, more profound.

Gossip had by now seeped out from Cambridge, that Petros Papachristos of Munich University had had a bit of bad luck, deferring publication of very important work. Fellow number theorists began to seek his opinions. He was invited to their meetings, which from that point on he would invariably attend, enlivening his monotonous lifestyle with occasional travel. The news had also leaked out (thanks here to

the Director of the School of Mathematics) that he was working on the notoriously difficult Conjecture of Goldbach, and that made his colleagues look on him with a mixture of awe and sympathy.

At an international meeting, about a year after his return to Munich, he ran across Littlewood. 'How's the work going on Goldbach, old chap?' he asked Perros.

'Always at it.'

'Is it true what I hear, that you're using algebraic methods?'

'It's true.'

Littlewood expressed his doubts and Petros surprised himself by talking freely about the content of his research. 'After all, Littlewood,’ he concluded, 'I know the problem better than anyone eise. My intuition tells me the truth expressed by the Conjecture is so fundamental that only an elementary approach can reveal it.'

Littlewood shrugged. 'I respect your intuition, Papachristos; it's just that you are totally isolated. Without a constant exchange of ideas, you may find yourself grappling with phantoms before you know it.'

'So what do you recommend,' Petros joked, 'issuing weekly reports of the progress of my research?'

'Listen,' said Littlewood seriously, 'you should find a few people whose judgement and integrity you trust. Start sharing; exchange, old chap!'

The more he thought about this suggestion, the more it made sense. Much to his surprise he realized that, far from frightening him, the prospect of discussing the progress of his work now filled him with pleasurable anticipation. Of course his audience would have to be small, very small indeed. If it was to consist of people 'whose judgement and integrity he trusted', that would of necessity mean an audience of no more than two: Hardy and Littlewood.

He started anew the correspondence with them that he'd interrupted a couple of years after he left Cambridge. Without stating it in so many words, he dropped hints about his intention to bring about a meeting during which he would present his work. Around Christmas of 1931, he received an official invitation to spend the next year at Trinity College. He knew that since, for all practical purposes, he had been absent from the mathematical world for a long long time, Hardy must have used all his influence to secure the offer. Gratitude, combined with the exciting prospect of a creative exchange with the two great number theorists, made him immediately accept.

Petros described his first few months in England, in the academic year 1932-33, as probably the happiest of his life. Memories of his first stay there, fifteen years earlier, infused his days at Cambridge with the enthusiasm of early youth, as yet untainted by the possibility of failure.

Soon after he arrived, he presented to Hardy and Littlewood the outline of his work to date with the algebraic method, and this gave him the first taste, after more than a decade, of the joy of peer recognition. It took him several mornings, standing at the blackboard in Hardy's office, to trace his progress in the three years since his volte-face from the analytic techniques. His two renowned colleagues, who were at first extremely sceptical, now began to see some advantages to his approach, Littlewood more so than Hardy.