'Unlucky? You mean unlucky to have chosen such a difficult problem?'
'No,’ he said, now looking totally amazed at my inability to grasp an obvious point. 'Unlucky – that, by the way, is a mild word for it – to have chosen a problem that had no solution. Weren't you listening?' He sighed heavily. 'By and by, my suspicions were confirmed: Goldbach's Conjecture is unprovable!'
'But how can you be so sure about it?' I asked.
'Intuition,’ he answered with a shrug. 'It is the only tool left to the mathematidan in the absence of proof. For a truth to be so fundamental, so simple to state, and yet so unimaginably resistant to any form of systematic reasoning, there could have been no other explanation. Unbeknown to me I had undertaken a Sisyphean task.'
I frowned. 'I don't know about that,’ I said. 'But the way I see it -'
Now, however, Uncle Petros interrupted me with a laugh. 'You may be a bright boy,’ he said, but mathematically you are still no more than a foetus – whereas I, in my time, was a veritable, full-blown giant. So, don't go weighing your intuition against mine, most favoured of nephews!' Against that, of course, I couldn't argue.
Three
My first reaction to this extensive autobiographical account was one of admiration. Uncle Petros had given me the facts of his life with remarkable honesty. It wasn't until a few days later, when the oppressive influence of his melancholy narrative diminished, that I realized everything he'd told me had been beside the point.
Remember that our meeting had been initially arranged so that he could try to justify himself. His life's story was only relevant to the extent that it explained his atrocious behaviour, assigning me in all my adolescent mathematical innocence the task of proving Goldbach's Conjecture. Yet, during his long narrative he had not even touched on his cruel prank. He'd ranted on and on about his own failure (or maybe I should do him the favour of calling it 'bad luck'), but about his decision to turn me away from studying mathematics and the method he had chosen to implement it, not a single word. Did he expect me automatically to draw the conclusion that his behaviour to me was determined by his own bitter life-experiences? It didn't follow: although his life story was indeed a valid cautionary tale, it taught a future mathematician what pitfalls to avoid so as to make the most of his career – not how to terminate it.
I let a few days go by before I went back to Ekali and asked him point-blank: could he now explain why he had attempted to dissuade me from following my inclination.
Uncle Petros shrugged. 'Do you want the truth?'
'Of course, Uncle.' I said. 'What eise?'
'All right then. I believed from the first moment – and still do, I'm sorry to say – that you have no special gift for great mathematics.'
I became, once again, furious. 'Oh? And how on earth could you have known that? Did you ask me a single mathematical question? Did you ever set me a problem to solve, other than the unprovable, as you termed it, Conjecture of Christian Goldbach? I certainly hope you don't have the nerve to tell me that you deduced my lack of mathematical ability from that!'
He smiled, sadly. 'You know the popular saying that the three conditions impossible to conceal are a cough, wealth and being in love? Well, to me there is a fourth: mathematical gift.'
I laughed contemptuously. 'Oh, and you can no doubt identify it at a glance, eh? Is it a look in the eye or a certain je ne sais quoi that betrays to your ultra-fine sensibility the presence of mathematical genius? Can you perhaps also determine one's IQ with a hand-shake?'
'Actually there is an element of that "look in the eye",’ he replied, ignoring my sarcasm. 'But in your case physiognomy was only a small part of it. The necessary – but not sufficient, mind you – precondition for supreme achievement is single-minded devotion. If you had the gift that you yourself would like to have had, dear boy, you wouldn't have come asking for my blessing to study mathematics; you would have gone ahead and done it. That was the first tell-tale sign!'
The more he explained himself, the angrier I got. 'If you were so certain I wasn't gifted, Uncle, why did you put me through the horrific experience of that summer? Why did I have to be subjected to the totally unnecessary hurmliation of thinking myself a near-idiot?'
'But, don't you see?' he answered merrily. 'Goldbach's Conjecture was my security! If by some remote chance I'd been wrong about you and, in the most unlikely instance, you were indeed earmarked for greatness, then the experience wouldn't have crushed you. In fact it would not have been at all "horrific", as you significantly termed it, but exciting and inspiring and invigorating. I gave you an ultimate trial of determination, you see: if, after failing to solve the problem I'd set you – as, of course, I knew you would – you came back eager to learn more, to persist in your attempt for better or for worse, then I'd see you might have it in you to become a mathematician. But you… you weren't even curious enough to ask the solution! Indeed, you even gave me a signed declaration of your incompetence!'
The pent-up anger of many years now exploded. 'Do you know something, you old bastard? You may once have been a good mathematician, but as a human being you rate zero! Absolutely, totally zilch!'
To my surprise, this opinion was rewarded with a huge, hearty smile. 'On that, most favoured of nephews, I couldn't agree with you more!'
A month later I returned to the United States to prepare for my Senior year. I now had a new room-mate, unrelated to mathemarics. Sammy had meanwhile graduated and was at Princeton, already deeply involved in the problem that would in due course become his doctoral dissertation – with the exotic title: "The orders of the torsion subgroups of Omega_{n} and the Adams spectral sequence'.
On my first free weekend I took the train and went to visit him. I found him quite changed, much more nervous and irritable than I had known him in the year of our association. He'd also acquired some kind of facial tic. Obviously, the torsion subgroups of Omega_{n (whatever they were) had taken their toll on his nerves. We had dinner at a small pizza place across from the university and there I gave him a shortened version of Uncle Petros' story, as I'd heard it from him. He listened without once interrupting for question or comment.
After I was finished, he summed up his reaction in two words: 'Sour grapes.'
'What?'
'You should know – Aesop was a Greek.'
'What's Aesop got to do with it?'
'Everything. The fable of the fox who couldn't reach a yummy bunch of grapes and therefore decided they were unripe anyway. What a wonderful excuse your uncle found for his failure: he put the blame on Kurt Gödel! Wow!' Sammy burst out laughing. 'Audacious! Unheard of! But I have to grant it to him, it is original; in fact it's unique, it should go into some book of records! Never before has there been a mathematician seriously attributing his failure to find a proof to the Incompleteness Theorem!'
Although Sammy's words echoed my own first doubts, I lacked the mathematical knowledge to understand this immediate verdict.
'So, you think it's impossible that Goldbach's Conjecture is unprovable?'
'Man, what does "impossible" mean in this context?' Sammy sneered. 'As your uncle correctly told you, there is, thanks to Turing, no way of telling with certainty that a statement is a priori unprovable. But if mathematicians involved in advanced research started invoking Gödel, no one would ever go near the interesting problems – you see, in mathematics the interesting is always difficult. The Riemann Hypothesis has not yielded to proof after more than a Century? A case of application of the Incompleteness Theorem! The Four Colour Problem? Likewise! Fermat's Last Theorem still unproved? Blame it on evil Kurt Gödel! No one would ever have touched Hilbert's Twenty-three Problems; [14] indeed it's conceivable that all mathematical research, except the most trivial, would come to an end. Abandoning the study of a particular problem because it might be unprovable is like… like…' His face lit up when he found the appropriate analogy. 'Why, it's like not going out in the street for fear that a brick might fall on your head and kill you!'
'Let's face it,' he concluded, 'your Uncle Petros simply and plainly failed to prove Goldbach's Conjecture, like many greater men before him. But because, unlike them, he had spent his whole creative life on the problem, admitting his failure was unbearable. So, he concocted for himself this far-fetched, extravagant justification.'
Sammy raised his soda-glass in a mock toast. 'Here's to far-fetched excuses,' he said. Then he added in a more serious tone: 'Obviously, for Hardy and Littlewood to have accepted him as a collaborator, your uncle must have been a gifted mathematician. He could have made a great success of his life. Instead, he wilfully chose to throw it away by setting himself an unattainable goal and going after a notoriously difficult problem. His sin was Pride: he presumed that he would succeed where Euler and Gauss had failed.'
I was laughing now.
'What's so funny?' asked Sammy.
'After all these years of grappling with the mystery of Uncle Petros,' I said, ‘I’m back to square one. You just repeated my father's words, which I high-handedly rejected as philistine and coarse in my adolescence: "The secret of life, my son, is to set yourself attainable goals." It's exactly what you are saying now. That he didn't do so is, indeed, the essence of Petros' tragedy!'
Sammy nodded. 'Appearances are after all deceptive,’ he said with mock solemnity. 'It turns out the wise elder in the Papachristos family is not your Uncle Petros!'
I slept on the floor of Sammy's room that night, to the familiar sound of his pen scratching on paper accompanied by the occasional sigh or groan, as he struggled to untangle himself from the knots of a difficult topological problem. He left early in the morning to attend a seminar and in the afternoon we met at the Mathematics Library at Fine Hall, as arranged.
'We are going sightseeing,’ he said. 'I have a surprise for you.'
We walked a distance on a long suburban road lined with trees and strewn with yellow leaves.
'What courses are you taking this year?' Sammy asked as we walked towards our mysterious destination.
I started to list them: Introduction to Algebraic Geometry, Advanced Complex Analysis, Group Representation Theory…
'What about Number Theory?' he interrupted.
'No. Why do you ask?'
'Oh, I've been thinking about this business with your uncle. I wouldn't want you getting any crazy ideas into your head about following family tradition and tackling -'
I laughed.''Goldbach 's Conjecture? Not bloody likely!'
Sammy nodded. 'That's good. Because I have a suspicion that you Greeks are attracted to impossible problems.'
'Why? Do you know any others?'
'A famous topologist here, Professor Papakyriakopoulos. He's been struggling for years on end to prove the "Poincare Conjecture" – it's the most famous problem in low-dimensional topology, unproved for more than sixty years… ultra-hyper-difficult!'
I shook my head. 'I wouldn't touch anybody's famous unproved ultra-hyper-difficult problem with a ten-foot pole,’ I assured him.
‘I’m relieved to hear it,' he said.
We had reached a large nondescript building with extensive grounds. Once we had entered, Sammy lowered his voice.
'I got a special permit to come, in your honour,’ he said.
'What is this place?'
'You'll see.'
We walked down a corridor and entered a large, darkish room, with the atmosphere of a slightly shabby but genteel English gentlemen's club. About fifteen men, ranging from middle-aged to elderly, were seated in leather armchairs and couches, some by the windows, reading newspapers in the scanty daylight, others talking in little groups.
We settled ourselves at a little table in a corner.
'See that guy over there?' Sammy said in a low voice, pointing to an old Asian gentleman, quietly stirring his coffee.
'Yes?'
'He is a Nobel Prize in Physics. And that other one at the far end' – he indicated a plump, red-haired man gesturing heatedly as he spoke to his neighbour with a strong accent – 'is a Nobel Prize in Chemistry.' Then he directed my attention to two middle-aged men seated at a table near us. 'The one on the left is Andre Weil -'
'The Andre Weil?'
'Indeed, one of the greatest living mathematicians. And the other one with the pipe is Robert Oppenheimer – yes, the Robert Oppenheimer, the father of the atom bomb. He's the Director.'
'Director of what?'
'Of this place here. You are now in the Institute for Advanced Study, think-tank for the world's greatest scientific minds!'
I was about to ask more when Sammy cut me short. 'Shh! Look! Over there!'
A most odd-looking man had just come in through the door. He was about sixty, of average height and emaciated to an extreme degree, wearing a heavy overcoat and a knitted cap pulled down over his ears. He stood for a moment and peered at the room vaguely through extremely thick glasses. No one paid him any attention: he was obviously a regular. He made his way slowly to the tea and coffee table without greeting anybody, filled a cup with piain boiling water from the kettle and made his way to a seat by a window. He slowly removed his heavy overcoat. Underneath it he was wearing a thick jacket over at least four or five layers of sweaters, visible through his collar.
'Who is that man?' I whispered.
'Take a guess!'
'I haven't the slightest idea – he looks like a street person. Is he mad, or what?'
Sammy giggled. 'That, my friend, is your uncle's nemesis, the man who gave him the pretext for abandoning his mathematical career, none other than the father of the Incompleteness Theorem, the great Kurt Gödel!'
I gasped in amazement. 'My God! That's Kurt Gödel? But, why is he dressed like that?'
'Apparently he is convinced – despite his doctors' total disagreement – that he has a very bad heart and that unless he insulates it from the cold with all those clothes it will go into arrest.'
'But it's warm in here!'
'The modern high priest of Logic, the new Aristotle, disagrees with your conclusion. Which of the two should I believe, you or him?'
On our walk back to the university Sammy expounded his theory: ‘I think Gödel's insanity – for unquestionably he is in a certain sense insane – is the price he paid for coming too close to Truth in its absolute form. In some poem it says that "people cannot bear very much reality", or something like that. Think of the biblical Tree of Knowledge or the Prometheus of your mythology. People like him have surpassed the common measure; they've come to know more than is necessary to man, and for this hubris they have to pay.'
There was a wind blowing, lifting dead leaves in whirls around us. I sighed.
I’ll cut a long story (my own) short:
I never did become a mathematician, and this not because of any further scheming by Uncle Petros. Although his 'intuitive' depreciation of my abilities had definitely played a part in the decision by nurturing a constant, nagging sense of self-doubt, the true reason was fear.
The examples of the mathematical enfants terribles mentioned in my uncle's narrative – Srinivasa Ramanujan, Alan Turing, Kurt Gödel and, last but not least, himself – had made me think twice about whether I was indeed equipped for mathematical greatness. These were men who at twenty-five years of age, or even less, had tackled and solved problems of inconceivable difficulty and momentous importance. In this I'd definitely taken after my uncle: I didn't want to become a mediocrity and end up 'a walking tragedy', to use his own words. Mathematics, Petros had taught me, is a field that acknowledges only its greatest; this particular kind of natural selection offers failure as the only alternative to glory. Yet, hopeful as I still was in my ignorance about my abilities, it wasn't professional failure that I feared.
It all started with the sorry sight of the father of the Incompleteness Theorem padded with layers of warm clothing, of the great Kurt Gödel as a pathetic, deranged old man sipping his hot water in total isolation in the lounge of the Institute for Advanced Study.
When I returned
to my university from the visit to Sammy, I looked up the biographies of the great mathematicians who had played a part in my uncle's story. Of the six mentioned in his narrative only two, a mere third, had lived a personal life that could be considered more or less happy and these two, significantly, were comparatively speaking the lesser men of the six, Caratheodory and Littlewood. Hardy and Ramanujan had attempted suicide (Hardy twice), and Turing had succeeded in taking his own life. Gödel's sorry state I've already mentioned. [15] Adding Uncle Petros to the list made the statistics even grimmer. Even if I still admired the romantic courage and persistence of his youth, I couldn't say the same of the way he'd decided to waste the second part of his life. For the first time I saw him for what he had clearly been all along, a sad recluse, with no social life, no friends, no aspirations, killing his time with chess problems. His was definitely not a prototype of the fulfilled life.
Sammy's theory of hubris had haunted me ever since I'd heard it, and after my brief review of mathematical history I embraced it wholeheartedly. His words about the dangers of coming too close to Truth in its absolute form kept echoing in my mind. The proverbial 'mad mathematician' was more fact than fancy. I came increasingly to view the great practitioners of the Queen of Sciences as moths drawn towards an inhuman kind of light, brilliant but scorching and harsh. Some couldn't stand it for long, like Pascal and Newton, who abandoned mathematics for theology.
Others had chosen haphazard, improvised ways out – Evariste Galois' mindless daring that led to his untimely death comes immediately to mind. Finally, some extraordinary minds had given way and broken down. Georg Cantor, the father of the Theory of Sets, led the latter part of his life in a lunatic asylum. Ramanujan, Hardy, Turing, Gödel and so many more were too enamoured of the brilliant light; they got too close, scorched their wings, fell and died.
In a short while I realized that even if I did have their gift (which, after listening to Uncle Petros' story, I began seriously to doubt) I definitely did not want to suffer their personal misery. Thus, with the Scylla of mediocrity on the one side and the Charybdis of insanity on the other, I decided to abandon ship. Although I did, come June, eventually get my BA in Mathematics, Ihad already applied for graduate studies in Business Economics, a field that does not traditionally provide material for tragedy.
Uncle Petros and Goldbach Page 11