I rushed to fetch paper and pencil before he changed his mind.
‘You’ll have to be a little patient,’ he began. ‘A lot of water has gone under the bridge since then. Let’s see now,’ he murmured and began to scribble. ‘Let us assume we have a partial differential equation in the Clairaut form … there! We now take…’
I followed his scribbles and explanations for almost an hour. Although I couldn’t completely follow the argument, I showed exaggerated appreciation at every step.
‘It’s absolutely brilliant, Uncle!’ I cried when he’d finished.
‘Nonsense.’ He brushed my praise aside, but I could see this modesty was not totally sincere. ‘Sheer calculation of the grocery-bill variety, not real mathematics!’
The moment I was waiting for had arrived. ‘Then talk to me about real mathematics, Uncle Petros. Talk to me about your work on Goldbach’s Conjecture!’
He shot me a sideways glance, cunning, inquisitive and at the same time tentative. I held my breath.
‘And what, if I may ask, is the purpose of your interest, Mr Almost-Mathematician?’
I had planned my answer to this beforehand, so as to put him in an emotional impasse.
‘You owe it to me, Uncle! If for nothing else, to compensate me for that summer of anguish in my sixteenth year, when I struggled for three months to prove it myself, floundering in my abysmal ignorance!’
He appeared to be considering this for a while, as if to make a point of not giving in all too easily. When he smiled I knew I had won.
‘What exactly do you want to know about my work on Goldbach’s Conjecture?’
I left Ekali after midnight with a copy of An Introduction to Number Theory by Hardy and Wright. (I had to prepare myself by learning ‘some fundamentals’, he’d said.) I should point out to the non-specialist that mathematical books cannot normally be enjoyed like novels, in bed, in the bathtub, sprawled in an easy chair, or perched on the commode. To ‘read’ here means to understand, and for that you normally need a hard surface, paper, pencil and quality time. Since I had no intention of becoming a number theorist at the advanced age of thirty, I went through the Hardy-Wright book with only moderate attention (‘moderate’ in mathematics is ‘considerable’ by any other measure), without persisting on fully comprehending those details that resisted the initial assault. Even so, and taking into account that the study of the book was not my main occupation, it took me almost a month.
When I returned to Ekali, Uncle Petros, bless his soul, started to examine me as if I were a schoolchild.
‘Have you read the whole book?’
‘I have.’
‘State Landau’s Theorem.’
I did.
‘Write out for me the proof of Euler ‘s Theorem of the φ-function, the extension of Fermat’s Little Theorem.’
I took paper and pencil and proceeded to do so, as best as I could.
‘Now prove to me that the non-trivial zeros of the Riemann Zeta Function have real part equal to ½!’
I burst out laughing and he did too.
‘Oh no, you don’t!’ I said. ‘Not again, Uncle Petros! It’s enough that you set me to prove Goldbach’s Conjecture. Find somebody else to assign the Riemann Hypothesis!’
In the following two and a half months we had our ten ‘Lessons on Goldbach’s Conjecture’, as he called them. What transpired in them is down on paper, with dates and times. Since I was now moving steadily towards the fulfilment of my main aim (his coming face to face with the reason for abandoning his research), I thought I’d also attain a secondary goal while at it: I kept meticulous notes so that, after his death, I could publish a short account of his odyssey, perhaps an insignificant footnote to mathematical history, but still a worthy tribute to Uncle Petros — if not, alas, to his ultimate success, then certainly to his ingenuity and, more importantly, his dedication and single-minded persistence.
During the course of the lessons I witnessed an amazing metamorphosis. The mild, kindly, elderly gentleman I had known since my childhood, one easily mistaken for a retired civil servant, turned before my eyes into a man illuminated by a fierce intelligence and driven by an inner power of unfathomable depth. I’d caught small glimpses of this species of being before, during mathematical discussions with my old room-mate, Sammy Epstein, or even with Uncle Petros himself, when he sat before his chessboard. Listening to him unravel the mysteries of Number Theory, however, I experienced for the first and only time in my life the real thing. You didn’t have to know mathematics to feel it. The sparkle in his eyes and an unspoken power emanating from his whole being were testimony enough. He was the absolute thoroughbred, pure unadulterated genius.
An unexpected fringe benefit was that the last remaining trace of ambivalence (apparently it had been there, dormant, all those years) regarding the wisdom of my decision to abandon mathematics was now dispelled. Watching my uncle do mathematics was enough to confirm it to the full. I was not made of the same mettle as he — this I realized now beyond the shadow of a doubt. Faced with the incarnation of what I definitely was not, I accepted at last the truth of the dictum: Mathematicus nascitur non fit. The true mathematician is born, not made. I had not been born a mathematician and it was just as well that I had given up.
The exact content of the ten lessons is not within the scope of our story and I won’t even attempt to refer to it. What matters here is that by the eighth we had covered the course of the initial period of Uncle Petros’ research on Goldbach’s Conjecture, culminating in his brilliant Partitions Theorem, now named after the Austrian who rediscovered it; also his other main result, attributed to Ramanujan, Hardy and Littlewood. In the ninth lesson he explained to me as much as I could understand of his rationale for changing the course of his attack from the analytic to the algebraic. For the next he had asked me to bring along two kilos of lima beans. In fact, he had initially asked for navy beans, but then corrected himself, smiling sheepishly: ‘Actually make it lima, so I can see them better. I’m not getting any younger, most favoured of nephews.’
As I drove to Ekali for the tenth (which, although I didn’t know it yet, would be the last) lesson, I felt apprehensive: I knew from his narrative that he had given up precisely while working with the ‘famous bean method’. Very soon, even in that imminent lesson, we would be reaching the crucial point, his hearing of Gödel’s Theorem and the end of his efforts to prove Goldbach’s Conjecture. It would be then that I would have to launch my attack on his dearly held defences and expose his rationalization about unprovability for what it was: a mere excuse.
When I got to Ekali he led me without a word to his so-called living room, which I found transformed. He’d pushed back what furniture there was against the walls, including even the armchair and the small table with the chessboard, and piled even higher piles of books along the perimeter, to create a wide, empty area in the centre. Without so much as a word he took the bag from my hands and started to arrange the beans on the floor, in a number of rectangles. I watched silently.
When he had finished he said: ‘During our previous lessons we went over my early approach to the Conjecture. In this I had done good, perhaps even excellent, mathematics — but mathematics, nevertheless, of a rather traditional variety. The theorems I had proved were difficult and important, but they followed and extended lines of thought started by others, before me. Today, however, I will present to you my most important and original work, a groundbreaking advance. With the discovery of my geometric method I finally entered virgin, unexplored territory.’
‘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 s
ame 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. 3×5, 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 internal 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 else 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 Little wood,’ I added, delivering my first serious blow.
This brought the expected reaction — only to a much greater degree than I’d foreseen.
‘“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) self 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, over-ambitious, 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.
To push it to its limit I used a casual tone: ‘G. H. Hardy’s sexual inclinations do not concern me,’ I said. ‘All that is relevant, vis-à-vis his opinion of your “bean method”, is that he was a great mathematician!’
Uncle Petros’ face went crimson. ‘Bollocks,’ he growled. ‘Prove it!’
‘I don’t have to,’ I said dismissively. ‘His theorems speak for themselves.’
‘Oh? Which one?’
I stated two or three of the results I remembered from his textbook.
‘Ha!’ Uncle Petros snarled. ‘Mere calculations of the grocery-bill variety! But show me one great idea, one inspired insight … You can’t?
That’s because there isn’t one!’ He was fuming now. ‘Oh, and while you’re at it, tell me of a theorem the old pansy proved on his own, without good old Littlewood or poor dear Ramanujan holding his hand — or whatever other part of his anatomy it was they were holding!’
The mounting nastiness signalled that we were approaching a breakthrough. A tiny extra bit of annoyance was probably all that was necessary to bring it about.
‘Really, Uncle,’ I said, trying to sound as haughty as possible. This is beneath you. After all, whatever theorems Hardy proved, they were certainly more important than yours!’
‘Oh yes?’ he snapped back. ‘More important than Goldhach’s Conjecture!’
I burst into incredulous laughter, despite myself. ‘But you didn’t prove Goldbach’s Conjecture, Uncle Petros!’
‘I didn’t prove it, but —’
He broke off in mid-sentence. His expression betrayed he’d said more than he wanted to.
‘You didn’t prove it but what?’ I pressed him. ‘Come on, Uncle, complete what you were going to say! You didn’t prove it but were very close to it! I’m right — am I not?’
Suddenly, he stared at me as if he were Hamlet and I his father’s ghost. It was now or never. I leapt up from my seat.
‘Oh, for God’s sake, Uncle,’ I cried. I’m not my father or Uncle Anargyros or grandfather Papachris-tos! I know some mathematics, remember? Don’t give me that crap about Gödel and the Incompleteness Theorem! Do you think I swallowed for a single moment that fairy tale of your “intuition telling you the Conjecture was unprovable”! No — I knew it from the very start for what it was, a pathetic excuse for your failure. Sour grapes!’
His mouth opened in wonder — from ghost I must have been transformed into a celestial vision.
Uncle Petros and Goldbach's Conjecture Page 13
