Apostolos Doxiadis - Uncle Petros and Goldbach

'All the more pity that you abandoned it,’ I said, preparing the climate from the start for a confrontation.

He disregarded this and continued: 'The basic premise behind the geometric approach is that multiplication is an unnatural operation.'

'What on earth do you mean by unnatural?' I asked.

'Leopold Kronecker once said: "Our dear God made the integers, everything else is the work of man." Well, in the same way he made the integers, I think Kronecker forgot to add, the Almighty created addition and subtraction, or give and take.'

I laughed. 'I thought I came here for lessons in mathematics, not theology!'

Again he continued, ignoring the interruption. 'Multiplication is unnatural in the same sense as addition is natural. It is a contrived, second-order concept, no more really than a series of additions of equal elements. 3x5, for example, is nothing more than 5+5+5. To invent a name for this repetition and call it an 'operation' is the devil's work more likely…'

I didn't risk another facetious comment.

'If multiplication is unnatural,' he continued, 'more so is the concept of "prime number" that springs directly from it. The extreme difficulty of the basic problems related to the primes is in fact a direct outcome of this. The reason there is no visible pattern in their distribution is that the very notion of multiplication – and thus of primes – is unnecessarily complex. This is the basic premise. My geometric method is motivated simply by the desire to construct a natural way of viewing the primes.'

Uncle Petros then pointed at what he'd made while he was talking. 'What is that?' he asked me.

'A rectangle made of beans,' I replied. 'Of 7 rows and 5 columns, their product giving us 35, the total number of beans in the rectangle. All right?'

He proceeded to explain how he was struck by an observation which, although totally elementary, seemed to him to have great intuitive depth. Namely, that if you constructed, in theory, all possible rectangles of dots (or beans) this would give you all the integers – except the primes. (Since a prime is never a product, it cannot be represented as a rectangle but only as a single row.) He went on to describe a calculus for operations among the rectangles and gave me some examples. Then he stated and proved some elementary theorems.

After a while I began to notice a change in his style. In our previous lessons he'd been the perfect teacher, varying the tempo of his exposition in inverse proportion to its difficulty, always making sure I had grasped one point before proceeding to the next. As he advanced deeper into the geometric approach, however, his answers became hurried, fragmented and incomplete to the point of total obscurity. In fact, after a certain point my questions were ignored and what might have appeared at first as explanations I recognized now as overheard fragments of his ongoing infernal monologue.

At first, I thought this anomalous form of presentation was a result of his not remembering the details of the geometric approach as clearly as the more conventional mathematics of the analytic, and making desperate efforts to reconstruct it.

I sat back and watched him: he was walking about the living room, rearranging his rectangles, mumbling to himself, going to the mantelpiece where he'd left paper and pencil, scribbling, looking something up in a tattered notebook, mumbling some more, returning to his beans, looking here and there, pausing, thinking, doing some more rearranging, then scribbling some more… Increasingly, references to a 'promising line of thought', 'an extremely elegant lemma' or a 'deep little theorem' (all his own inventions, obviously) made his face light up with a self-satisfied smile and his eyes sparkle with boyish mischievousness. I suddenly realized that the apparent chaos was nothing eise than the outer form of inner, bustling mental activity. Not only did he remember the 'famous bean method' perfectly well – its memory made him positively gloat with pride!

A previously unthought-of possibility quickly entered my mind, only to become a near conviction moments later.

When first discussing Uncle Petros' abandoning Goldbach's Conjecture with Sammy, it had seemed obvious to both of us that the reason was a form of burnout, an extreme case of scientific battle fatigue after years and years of fruitless attacks. The poor man had striven and striven and striven and, after failing each time, was finally too exhausted and too disappointed to continue, Kurt Gödel providing him with a convenient if far-fetched excuse. But now, watching his obvious exhilaration as he played around with his beans, a new and much more exciting scenario presented itself: was it possible that, in direct contrast to what I'd thought until then, his surrender had come at the very peak of his achievement? In fact, precisely at the point when he felt he was ready to solve the problem?

In a flash of memory, the words he had used when describing the period just before Turing's visit came back – words whose real significance I had barely realized when I'd first heard them. Certainly he'd said that the despair and self-doubts he had felt in Cambridge, in that spring of 1933, had been stronger than ever. But had he not interpreted these as the 'inevitable anguish before the final triumph', even as the 'onset of the labour pains leading to the delivery of the great discovery'? And what about what he'd said a little earlier, just a little while ago, about this being his 'most important work', 'important and original work, a groundbreaking advance'? Oh my good God! Fatigue and disillusionment didn't have to be the causes: his surrender could have been the loss of nerve before the great leap into the unknown and his final triumph!

The excitement caused by this realization was such that I could no longer wait for the tactically correct moment. I launched my attack right away.

'I notice,’ I said, my tone accusing rather than observing, 'that you seem to think very highly of the "famous Papachristos bean method".'

I had interrupted his train of thought and it took a few moments for my comment to register.

' You have an amazing command of the obvious,’ he said rudely. 'Of course I think highly of it.'

'… in contrast to Hardy and Littlewood,’ I added, delivering my first seriousblow.

This brought the expected reaction – only to a much greater degree than I'd f oreseen.

'"Can't prove Goldbach with beans, old chap!"' he said in a gruff, boorish tone, obviously parodying Littlewood. Then, he took on the other member of the immortal mathematical pair in a cruel mimicry of effeminacy. "Too elementary for your own good, my dear fellow, infantile even!'"

He banged his fist on the mantelpiece, furious. "That ass Hardy,’ he shouted, 'calling my geometric method "infantile" – as if he understood the first thing about it!'

'Now, now, Uncle,’ I said scoldingly, 'you can't go calling G. H. Hardy an ass!'

He banged his fist again, with greater force.

'An ass he was, and a sodomite too! The "great G. H. Hardy" – the Queen of Number Theory!'

This was so untypical of him I gasped. 'My, my, we are getting nasty, Uncle Petros!'

'Not at all! I'll call a spade a spade and a bugger a bugger!'

If I was startled I was also exhilarated: a totally new man had magically appeared before my eyes. Could it be that, together with the 'famous bean method', his old (I mean his young) seif had at last resurfaced? Could I now be hearing, for the first time, Petros Papachristos' real voice? Eccentricity – even Obsession – was certainly more characteristic of the single-minded, overambitious, brilliant mathematician of his youth than the gentle, civilized manners I'd come to associate with my elderly Uncle Petros. Conceit and malice towards his peers could well be the necessary other side of his genius. After all, both were perfectly suited to his capital sin, as diagnosed by Sammy: Pride.