Every family has its black sheep – in ours it was Uncle Petros.
My father and Uncle Anargyros, his two younger brothers, made sure that my cousins and I should inherit their opinion of him unchallenged.
“That no-good brother of mine, Petros, is one of life's failures”, my father would say at every opportunity. And Uncle Anargyros, during the family get-togethers from which Uncle Petros routinely absented himself, always accompanied mention of his name with snorts and grimaces expressing disapproval, disdain or simple resignation, depending on his mood.
However, I must say this for them: both brothers treated him with scrupulous fairness in financial matters. Despite the fact that he never shared even a slight part of the labour and the responsibilities involved in running the factory that the three inherited jointly from my grandfather, Father and Uncle Anargyros unfailingly paid Uncle Petros his share of the profits.
(This was due to a strong sense of family, another common legacy.) As for Uncle Petros, he repaid them in the same measure. Not having had a family of his own, upon his death he left us, his nephews, the children of his magnanimous brothers, the fortune that had been multiplying in his bank account practically untouched in its entirety.
Specifically to me, his 'most favoured of nephews' (his own words), he additionally bequeathed his huge library which I, in turn, donated to the Hellenic Mathematical Society. For myself I retained only two of its items, volume seventeen of Leonard Euler's Opera Omnia and issue number thirty-eight of the German scientific Journal Monatshefte für Mathematik und Physik. These humble memorabilia were symbolic, as they defined the boundaries of Uncle Petros' essential life-story. Its starting-point is in a letter written in 1742, contained in the former, wherein the minor mathematician Christian Goldbach brings to the attention of the great Euler a certain arithmetical observation. And its termination, so to speak, is to be found in pages 183-98 of the erudite Germanic Journal, in a study entitled 'On Formally Undecidable Propositions in Principia Mathematica and Related Systems', authored in 1931 by the until then totally unknown Viennese mathematician Kurt Gödel.
Until I reached mid-adolescence I would see Uncle Petros only once a year, during the ritual visit on his name day, the feast of Saints Peter and Paul on the twenty-ninth of June. The custom of this annual meeting had been initiated by my grandfather and as a consequence had become an inviolable obligation in our tradition-ridden family. We journeyed to Ekali, a suburb of Athens today but in those days more of an isolated sylvan hamlet, where Uncle Petros lived alone in a small house surrounded by a large garden and orchard.
The contemptuous dismissal of their older brother by Father and Uncle Anargyros had puzzled me from my earliest years and had gradually become for me a veritable mystery. The discrepancy between the picture they painted of him and the impression I formed through our scant personal contact was so glaring that even an immature mind like mine was compelled to wonder.
In vain did I observe Uncle Petros during our annual visit, seeking in his appearance or behaviour signs of dissoluteness, indolence or other characteristics of the reprobate. On the contrary, any comparison weighed unquestionably in his favour: the younger brothers were short-tempered and often outright rude in their dealings with people while Uncle Petros was tactful and considerate, his deep-set blue eyes always kindling with kindness. They were both heavy drinkers and smokers; he drank nothing stronger than water and inhaled only the scented air of his garden. Furthermore, unlike Father, who was portly, and Uncle Anargyros, who was outright obese, Petros had the healthy wiriness resulting from a physically active and abstemious lifestyle.
My curiosity increased with each passing year. To my great disappointment, however, my father refused to disclose any further information about Uncle Petros beyond his dismissive incantation, 'one of life's failures'. From my mother I learned of his daily activities (one could hardly speak of an occupation): he got up every morning at the crack of dawn and spent most daylight hours slaving away in his garden, without help from a gardener or any modern labour-saving contraptions – his brothers erroneously attributed this to stinginess. He seldom left his house, except for a monthly visit to a small philanthropic institution founded by my grandfather, where he volunteered his services as treasurer. In addition, he sometimes went to 'another place', never specified by her. His house was a true hermitage; with the exception of the annual family invasion there were never any visitors. Uncle Petros had no social life of any kind. In the evenings he stayed at home and – here mother had lowered her voice almost to a whisper – 'immersed himself in his studies'.
At this my attention suddenly peaked. 'Studies? What studies?'
“God only knows”, answered Mother, conjuring up in my boyish imagination visions of esoterica, alchemy or worse.
A further unexpected piece of information came to identify the mysterious 'other place' that Uncle Petros visited. It was offered one evening by a dinner-guest of my father's.
“I saw your brother Petros at the club the other day. He demolished me with a Karo-Cann”, said the guest, and I interjected, earning an angry look from my father: 'What do you mean? What's a Karo-Cann?'
Our guest explained that he was referring to a particular way of opening the game of chess, named after its two inventors, Messrs Karo and Cann. Apparently, Uncle Petros was in the habit of paying occasional visits to a chess club in Patissia where he routinely routed his unfortunate opponents.
'What a player!' the guest sighed admiringly. 'If only he'd entered formal competition he'd be a Grand Master today!'
At this point Father changed the subject.
The annual family reunion was held in the garden. The grown-ups sat around a table that had been set up in a paved patio, drinking, snacking and making small-talk, the two younger brothers routinely exerting themselves (as a rule, not altogether successfully) to be gracious to the honouree. My cousins and I played among the trees in the orchard.
On one occasion, having made the decision to seek an answer to the mystery of Uncle Petros, I asked to use the bathroom; I was hoping I would get a chance to examine the inside of the house. To my great disappointment, however, our host indicated a small out-house attached to the toolshed. The next year (by that time I was fourteen) the weather came in aid of my curiosity. A summer storm forced my uncle to open the French windows and lead us into a space that had obviously been intended by the architect to serve as a living room. Equally obviously, however, the owner did not use it to receive guests. Although it did contain a couch, it was totally inappropriately positioned facing a blank wall. Chairs were brought in from the gar-den and placed in a semi-circle, where we sat like the mourners at a provincial wake.
I made a hasty reconnaissance, with quick glances all around. The only pieces of furniture apparently put to daily use were a deep, shabby armchair by the fire-place with a small table at its side; on it was a chessboard with the pieces set out as for a game in progress. Next to the table, on the floor, was a large pile of chess books and periodicals. This, then, was where Uncle Petros sat every night. The studies mentioned by my mother must have been studies of chess. But were they?
I couldn't allow myself to jump to facile conclusions, as there were now new speculative possibilities. The main feature of the room we sat in – what made it so different from the living room in our house – was the overwhelming presence of books, countless books everywhere. Not only were all the visible walls of the room, corridor and entrance hall dressed from floor to ceiling with shelves crammed to overflowing, but books in tall piles covered most of the floor area as well. Most of them looked old and overused.
At first, I chose the most direct route to answering my question about their content: I asked, 'What are all these books, Uncle Petros?'
There was a frozen silence, exactly as if I had spoken of rope in the house of the hanged man.
“They are… old”, he mumbled hesitantly, after casting a quick glance in the direction of my father. He seemed so flustered in his search for an answer, however, and the accompanying smile was so wan that I couldn't bring myself to ask for further explanations.
Once again I resorted to the call of nature. This time Uncle Petros led me to a small toilet next to the kitchen. On my way back to the living room, alone and unobserved, I seized the opportunity I had created. I picked up the top book of the nearest pile in the corridor and flipped hurriedly through the pages. Unfortunately it was in German, a language I was (and still am) totally unfamiliar with. On top of it, most of the pages were adorned with mysterious symbols such as I'd never seen before: "'s and $'s and f’s and Ï's. Among them, I discerned some more intelligible signs, +'s, ='s and ÷'s interspersed with numerals and letters both Latin and Greek. My rational mind overcame cabbalistic fantasies: it was mathematics!
I left Ekali that day totally preoccupied with my discovery, indifferent to the scolding I received from my father on the way back to Athens and to his hypocritical reprimands about my 'rudeness to my uncle' and 'my busybody, prying questions'. As if it was the breach in savoir-vivre that had bothered him!
My curiosity about Uncle Petros' dark, unknown side developed in the next few months into something approaching obsession. I remember compulsively drawing doodles combining mathematical and chess symbols in my notebooks during school classes. Maths and chess: in one of these most probably lay the solution to the mystery surrounding him, yet neither offered a totally satisfactory explanation, neither being reconcilable with his brothers' contemptuously dismissive attitude. Surely, these two fields of interest (or was it more than mere interest?) were not in themselves objectionable. Whichever way you looked at it, being a chess player at Grand Master level or a mathematician who had devoured hundreds of formidable tomes did not immediately classify you as 'one of life's failures'.
I needed to find out, and in order to do so I even contemplated for a while a venture in the style of the exploits of my favourite literary heroes, a project worthy of Enid Blyton's Secret Seven, the Hardy Boys, or their Greek soulmate, the 'heroic Phantom Boy'. I planned, down to the smallest detail, a break-in at my uncle's house during one of his expeditions to the philanthropic institution or the chess club, so I could lay my hands on palpable evidence of transgression.
As things turned out, I did not have to resort to crime to satisfy my curiosity. The answer I was seeking came and hit me, so to speak, over the head. Here's how it happened:
One afternoon, while I was alone at home doing my homework, the phone rang and I answered it.
“Good evening”, said an unfamiliar male voice. I’m calling from the Hellenic Mathematical Society. May I speak to the Professor please?'
Unthinking at first, I corrected the caller: 'You must have dialled the wrong number. There is no professor here.'
“Oh, I'm sorry”, he said. 'I should have inquired first. Isn't that the Papachristos residence?'
I had a sudden flash of inspiration. 'Do you, perhaps, mean Mr Petros Papachristos?' I asked.
'Yes’, said the caller, 'Professor Papachristos.'
'Professor'! The receiver nearly dropped from my hand. However, I suppressed my excitement, lest this windfall opportunity go to waste.
'Oh, I didn't realize you were referring to Professor Papachristos’, I said ingratiatingly. 'You see, this is his brother's home, but as the Professor does not have a telephone' – (fact) – 'we take his calls for him' (blatant lie).
'Could I then have his address?' the caller asked, but by now I had regained my composure and he was no match for me.
'The Professor likes to maintain his privacy’, I said haughtily. 'We also receive his mail.'
I had left the poor man no options. 'Then be so kind as to give me your address. On behalf of the Hellenic Mathematical Society, we would like to send him an invitation.'
The next few days I played sick so as to be at home at the usual time of mail delivery. I didn't have to wait long. On the third day after the phone call I had the precious envelope in my hand. I waited till after midnight for my parents to go to sleep and then tiptoed to the kitchen and steamed it open (another lesson culled from boys' fiction).
I unfolded the letter and read:
Mr Petros Papachristos
f. Professor of Analysis
University of Munich
Honourable Professor,
Our Society is planning a Special Session to commemorate Leonard Euler's two hundred and fiftieth birthday with a lecture on 'Formal Logic and the Foundations of Mathematics'.
We would be greatly honoured, dear Professor, if you would attend and address a short greeting to the Society…
So: the man routinely dismissed by my dear father as 'one of life's failures' was a Professor of Analysis at the University of Munich – the significance of the little 'f.' preceding his unexpectedly prestigious title still escaped me. As to the achievements of this Leonard Euler, still remembered and honoured two hundred and fifty years after his birth, I hadn't the slightest clue.
The next Sunday morning I left home wearing my Boy Scout uniform, but instead of going to the weekly meeting I boarded the bus for Ekali, the letter from the Hellenic Mathematical Society safely in my pocket. I found my uncle in an old hat and rolled-up sleeves, spade in hand, turning the soil in a vegetable plot. He was surprised to see me.
'What brings you here?' he asked.
I gave him the sealed envelope.
'You needn't have gone to the trouble’, he said, barely glancing at it. 'You could have put it in the mail.' Then he smiled kindly. "Thank you anyway, Boy Scout. Does your father know you've come?'
'Uh, no,' I muttered.
'Then I better drive you home; your parents will be worried.'
I protested that it wasn't necessary, but he insisted. He climbed into his ancient, beat-up VW beetle, muddy boots and all, and we set out for Athens. On the way I attempted more than once to start a conversation on the subject of the invitation, but he switched to irrelevant matters like the weather, the correct season for tree-pruning and scouting.
He dropped me off at the corner nearest our house. 'Should I come upstairs and provide excuses?'
'No thanks, Uncle, that won't be necessary.'
However, it turned out that excuses were necessary. As my ill luck would have it, Father had called the club to ask me to pick something up on the way home and been informed of my absence. Naively, I blurted out the whole truth. As it turned out, this was the worst possible choice. If I'd lied and told him that I played truant from the meeting in Order to indulge in forbidden cigarettes in the park, or even visit a house of ill-repute, he wouldn't have been quite so upset.
'Haven't I expressly forbidden you to have anything to do with that man?' he yelled at me, getting so red in the face that my mother started pleading with him to think of his blood pressure.
'No, Father’, I replied truthfully. 'As a matter of fact, you never have. Never!'
'But don't you know about him? Haven't I told you a thousand times about my brother Petros?'
'Oh, you've told me a thousand times that he's "one of life's failures", but so what? He's still your brother – my uncle. Was it so terrible to take the poor fellow his letter? And, come to think of it, I don't see how being "one of life's failures" applies to someone with the rank of Professor of Analysis at a great university!'
"The rank of former Professor of Analysis,’ growled my father, settling the matter of the little 'f.'
Still fuming, he pronounced the sentence for what he termed my 'abominable act of inexcusable disobedience'. I could hardly believe the severity: for a month I would be confined to my room at all hours except those spent at school. Even my meals would have to be taken there and I would be allowed no spoken communication with himself, my mother, or anybody eise!
I went to my room to begin my sentence, feeling a martyr for Truth.
Late that same night, my father knocked softly on my door and entered. I was at my desk reading and, obedient to his decree, didn't speak a word of greeting. He seated himself across from me on the bed and I knew from bis expression something had changed. He now appeared calm, even slightly guilt-ridden. He began by announcing that the punishment he had meted out was 'perhaps a bit too harsh' and thus no longer applied, and subsequently asked my pardon for his manner – a piece of behaviour unprecedented and totally uncharacteristic of the man. He realized that his outburst had been unjust. It was unreasonable, he said – and of course I agreed with him – to expect me to understand something he had never taken the trouble to explain. He had never spoken openly to me about the matter of Uncle Petros and now the time had come for his 'grievous error' to be corrected. He wanted to teil me about his eldest brother. I, of course, was all ears.
This is what he told me:
Uncle Petros had, from early childhood, shown signs of exceptional ability in mathematics. In grade school he had impressed his teachers with his ease in arithmetic and in high school he had mastered abstractions in algebra, geometry and trigonometry with unbelievable facility. Words like 'prodigy' and even 'genius' were applied. Though a man of little formal education, their father, my grandfather, proved himself enlightened. Rather than divert Petros to more practical studies that would prepare him to work at his side in the family business, he had encouraged him to follow his heart. He had enrolled at a precocious age at the University of Berlin, from which he had graduated with honours at nineteen. He had earned his doctorate the next year and joined the faculty at the University of Munich as full professor at the amazing age of twenty-four – the youngest man ever to achieve this rank.
I listened, goggle-eyed. 'Hardly the progress of "one of life's failures",’ I commented.
'I haven't finished yet,’ warned my father.
At this point he digressed from his narrative. Without any prompting from me he spoke of himself and Uncle Anargyros and their feelings towards Petros. The two younger brothers had followed his successes with pride. Never for a moment did they feel the least bit envious – after all they too were doing extremely well at school, though in nowhere near as spectacular a manner as their genius of a brother. Still, they had never feit very close to him. Since early childhood, Petros had been a loner. Even when he'd still lived at home, Father and Uncle Anargyros hardly ever spent time with him; while they played with their friends he was in his room solving geometry problems. When he went abroad to university, Grandfather had them write polite letters to Petros ('Dear brother, We are well… etc.'), to which he would reply, infrequently, with a laconic acknowledgement on a postcard. In 1925, when the whole family travelled to Germany to visit him, he turned up at their few encounters behaving like a total stranger, absent-minded, anxious, obviously impatient to get back to whatever it was he was doing. After that they never saw him again until 1940 when Greece went to war with Germany and he had to return.
'Why?' I asked Father. 'To enlist?'
'Of course not! Your uncle never had patriotic – or any other, for that matter – feelings. It's just that once war was declared he was considered an enemy alien and had to leave Germany.'
'So why didn't he go elsewhere, to England or America, to some other great university? If he was such a great mathematician -'
My father interrupted me with an appreciative grunt, accompanied by a loud slap on his thigh.
"That's the point,' he snapped. "That's the whole point: he was no longer a great mathematician!'
'What do you mean?' I asked. 'How can that be?'
There was a long, pregnant pause, a sign that the critical point in the narrative, the exact locus where the action changes direction from uphill to down, had been reached. My father leaned towards me, frowning ominously, and his next words came in a deep mur-mur, almost a groan:
'Your uncle, my son, committed the greatest of sins.'
'But what did he do, Father, teil me! Did he steal or rob or kill?'
'No, no, all these are simple misdemeanours compared to his crime! Mind you, it isn't I who deem it so but the Gospel, our Lord Himself: "Thou shalt not blaspheme against the Spirit!" Your Uncle Petros cast pearls before swine; he took something holy and sacred and great, and shamelessly defiled it!'
The unexpected theological twist put me for a moment on guard: 'And what exactly was that?'
'His gift, of course!' shouted my father. 'The great, unique gift that God had blessed him with, his phenomenal, unprecedented mathematical talent! The miserable fool wasted it; he squandered it and threw it out with the garbage. Can you imagine it? The ungrateful bastard never did one day's useful work in mathemarics. Never! Nothing! Zero!'
'But why?’ I asked.
'Oh, because his Illustrious Excellence was engaged with "Goldbach's Conjecture".'
‘With what?’
Father made a distasteful grimace. 'Oh, a riddle of some sort, something of no interest to anyone except a handful of idlers playing intellectual games.'
' A riddle? You mean like a crossword puzzle?'
'No, a mathematical problem – but not just any problem: this "Goldbach's Conjecture" thing is considered to be one of the most difficult in the whole of mathematics. Can you imagine? The greatest minds on this planet had failed to solve it, but your smart aleck uncle decided at the age of twenty-one that he would be the one… Then, he proceeded to waste his life on it!'
I was rather confused by the course of his reasoning. 'Wait a minute, Father,' I said. 'Is that his crime? Pursuing the solution of the most difficult problem in the history of mathematics? Are you serious? Why, this is magnificent; it is absolutely fantastic!'
Father glared at me. 'Had he managed to solve it, it might be "magnificent" or "absolutely fantastic" or what have you – although it would still be totally useless, of course. But he didn't!’
He now got impatient with me, once again his usual seif. 'Son, do you know the Secret of Life?' he asked with a scowl.
'No,I don't.'
Before divulging it to me he blew his nose with a trumpeting sound into his monogrammed silk handkerchief.
'The Secret of Life is always to set yourself attainable goals. They may be easy or difficult, depending on the circumstances and your character and abilities, but they should always be at-tai-na-ble! In fact, I think I'll hang your Uncle Petros' portrait in your room, with a caption: example to be avoided!'
It's impossible as I write now, in middle age, to describe the turbulence caused in my adolescent heart by this first, however prejudiced and incomplete, account of Uncle Petros' story. My father had obviously intended it to serve as a cautionary tale and yet for me his words had exactly the opposite effect: instead of steering me away from his aberrant older brother, they drew me towards him as to a brilliantly shining star.
I was awestruck by what I'd learned. Exactly what this famous 'Goldbach's Conjecture' was I didn't know, nor at that time did I care very much to learn. What fascinated me was that the kindly, withdrawn and seemingly unassuming uncle of mine was in fact a man who, by his own deliberate choice, had struggled for years on end at the outermost boundaries of human ambition. This man whom I'd known all my life, who was in fact my close blood relative, had spent his whole life striving to solve One of the Most Difficult Problems in the History of Mathematics! While his brothers were studying and getting married, raising children and running the family business, wearing out their lives along with the rest of nameless humanity in the daily routines of subsistence, procreation and killing time, he, Prometheus-like, had striven to cast light into the darkest and most inaccessible corner of knowledge.
The fact that he had finally failed in his endeavour not only did not lower him in my eyes but, on the contrary, raised him to the highest peak of excellence. Was this not, after all, the very definition of the plight of the Ideal Romantic Hero, to Fight the Great Battle Although You Know It To Be Desperate? In fact, was my uncle any different from Leonidas and his Spartan troops guarding Thermopylae? The last verses of Cavafy's poem I had learned at school seemed ideally applicable to him:
… But greatest honour befits them that foresee,
As many do indeed foresee,
That Ephialtes the Traitor will finally appear
And thus the Persians will at last
Go through the narrow straits.
Even before I'd heard Uncle Petros' story, his brothers' derogatory remarks, beyond exciting curiosity, had inspired my sympathy. (This, by the way, had been in contrast to my two cousins' reactions, who bought their fathers' contempt wholesale.) Now that I knew the truth – even this highly prejudiced Version of it -I immediately elevated him to role model.
The first consequence of this was a change in my attitude towards mathematical subjects at school, which I had found till then rather boring, with a resultant dramatic improvement in my performance. When Father saw on the next report card that my grades in Algebra, Geometry and Trigonometry had shot up to honours level, he raised a perplexed eyebrow and gave me a queer look. It's possible that he even became slightly suspicious, but of course he couldn't make an issue of it. He could hardly criticize me for excelling!
On the date when the Hellenic Mathematical Society was due to commemorate Leonard Euler's two hundred and fiftieth birthday, I arrived ahead of time at the auditorium, full of expectation. Although high-school maths was of no help in fathoming its precise meaning, the announced lecture's title, ‘Formal Logic and the Foundations of Mathematics', had intrigued me since first reading the invitation. I knew of 'formal receptions' and 'simple logic' but how did the two concepts combine? I'd learned that buildings have foundations – but mathematics?
I waited in vain, however, as the audience and the Speakers took their places, to see among them the lean, ascetic figure of my uncle. As I should have guessed, he didn't come. I already knew he never accepted invitations; now I'd learned he didn't make exceptions even for mathematics.
The first speaker, the president of the Society, mentioned his name, and with particular respect:
'Professor Petros Papachristos, the world-renowned Greek mathematician, will unfortunately be unable to deliver his short address, because of a slight indisposition.'
I smiled smugly, proud that only I among the audience knew that my uncle's 'slight indisposition' was a diplomatic one, an excuse to protect his peace.
Despite Uncle Petros' absence, I stayed until the end of the event. I listened fascinated to a brief resume of the honouree's life (Leonard Euler, apparently, had made epoch-making discoveries in practically every branch of mathematics). Then, as the main speaker took the podium and started elaborating on the 'Foundations of Mathematical Theories by Formal Logic', I feil into a charmed state. Despite the fact that I didn't completely understand more than the first few words of what he said, my spirit wallowed in the unfamiliar bliss of unknown definitions and concepts, all symbols of a world which, although mysterious, impressed me from the start as almost sacred in its unfathomable wisdom. Magical, previously unheard-of names rolled on and on, enthralling me with their sublime music: the Continuum Problem, Aleph, Tarski, Gottlob Frege, Inductive Reasoning, Hilbert's Programme, Proof Theory, Riemannian Geometry, Verifiability and Non-Verifiability, Consistency Proofs, Completeness Proofs, Sets of Sets, Universal Turing Machines, Von Neumann Automata, Russell's Paradox, Boolean Algebras… At some point, in the midst of these intoxicating verbal waves washing over me, I thought for a moment I discerned the momentous words 'Goldbach's Conjecture'; but before I could focus my attention the subject had evolved along new magical pathways: Peano's Axioms for Arithmetic, the Prime Number Theorem, Closed and Open Systems, Axioms, Euclid, Euler, Cantor, Zeno, Gödel…
Paradoxically, the lecture on the 'Foundations of Mathematical Theories by Formal Logic' worked its insidious magic on my adolescent soul precisely because it disclosed none of the secrets that it introduced – I don't know whether it would have had the same effect had its mysteries been explained in detail. At last I understood the meaning of the sign at the entrance of Plato's Academy: oudeis ageometretos eiseto – 'Let no one ignorant of geometry enter'. The moral of my evening emerged with crystal clarity: mathematics was something infinitely more interesting than solving second-degree equations or calculating the volumes of solids, the menial tasks at which we laboured at school. Its practitioners dwelt in a veritable conceptual heaven, a majestic poetic realm totally inaccessible to the un-mathematical hoi polloi.
The evening at the Hellenic Mathematical Society was the turning point. It was then and there that I first resolved to become a mathematician.
At the end of that school year I was awarded the school prize for highest achievement in Mathematics. My father boasted about it to Uncle Anargyros – as if he could have done otherwise!
By now, I had completed my second-to-last year of high school and it had already been decided that I would be attending university in the United States. As the American System doesn't require students to declare their major field of interest upon registration, I could defer revealing to my father the horrible (as he would no doubt consider it) truth for a few more years. (Luckily, my two cousins had already stated a preference that assured the family business of a new generation of managers.) In fact, I misled him for a while with vague talk of plans to study economics, while I was hatching my plan: once I was safely enrolled in university, with the whole Atlantic Ocean between me and his authority, I could steer my course toward my destiny.
That year, on the feast day of Saints Peter and Paul, I couldn't hold back any longer. At some point I drew Uncle Petros aside and, impulsively, I blurted my intention.
'Uncle, I'm thinking of becoming a mathematician.'
My enthusiasm, however, found no immediate response. My uncle remained silent and impassive, his gaze suddenly focused on my face with intense seriousness – with a shiver I realized that this was what he must have looked like as he was struggling to penetrate the mysteries of Goldbach's Conjecture.
'What do you know of mathematics, young man?' he asked after a short pause.
I didn't like his tone but I went on as planned: 'I was first in my class, Uncle Petros; I received the school prize!'
He seemed to consider this information awhile and then shrugged. 'It's an important decision,’ he said, 'not to be taken without serious deliberation. Why don't you come here one afternoon and we'll talk about it.' Then he added, unnecessarily. 'It's better if you don't tell your father.'
I went a few days later, as soon as I could arrange a good cover story.
Uncle Petros led me to the kitchen and offered me a cold drink made from the sour cherries from his tree. Then he took a seat across from me, looking solemn and professorial.
'So tell me,' he asked, 'what is mathematics in your opinion?' The emphasis on the last word seemed to carry the implication that whatever answer I gave was bound to be wrong.
I spurted out commonplaces about 'the most supreme of sciences' and the wonderful applications in electronics, medicine and space exploration.
Uncle Petros frowned. 'If you're interested in applications why don't you become an engineer? Or a physicist. They too are involved with some sort of mathematics.'
Another emphasis with meaning: obviously he didn't hold this 'sort' in very high esteem. Before I embarrassed myself further, I decided that I was not equipped to spar with him as an equal, and confessed it.
'Uncle, I can't put the "why" into words. All I know is that I want to be a mathematician – I thought you'd understand.'
He considered this for a while and then asked: 'Do you know chess?'
'Sort of, but please don't ask me to play; I can tell you right now I'm going to lose!'
He smiled. 'I wasn't suggesting a game; I just want to give you an example that you'll understand. Look, real mathematics has nothing to do with applications, nor with the calculating procedures that you learn at school. It studies abstract intellectual constructs which, at least while the mathematician is occupied with them, do not in any way touch on the physical, sensible world.'
"That's all right with me,’ I said.
'Mathematicians,’ he continued, 'find the same enjoyment in their studies that chess players find in chess. In fact, the psychological make-up of the true mathematician is closer to that of the poet or the musical composer, in other words of someone concerned with the creation of Beauty and the search for Harmony and Perfection. He is the polar opposite of the practical man, the engineer, the politician or the he paused for a moment seeking something even more abhorred in his scale of values -'… indeed, the businessman.'
If he was telling me all this in order to discourage me, he had chosen the wrong route.
'That's what I'm after too, Uncle Petros,’ I responded excitedly. 'I don't want to be an engineer; I don't want to work in the family business. I want to immerse myself in real mathematics, just like you… just like Goldbach's Conjecture!'
I'd blown it! Before I'd left for Ekali I had decided that any reference to the Conjecture should be avoided like the devil during our conversation. But in my carelessness and excitement I'd let it slip out.
Although Uncle Petros remained expressionless, I noticed a slight tremor run down his hand.' Who's spoken to you about Goldbach's Conjecture?' he asked quietly.
'My father,' I murmured.
'And what did he say, precisely?'
'That you tried to prove it.'
'Just that?'
'And… and that you didn't succeed.'
His hand was steady again. 'Nothing else?'
'Nothing else.'
'Hm,’ he said. 'Suppose we make a deal?'
'What sort of a deal?'
'Listen to me: the way I see things, in mathematics as in the arts – or in sports, for that matter – if you're not the best, you're nothing. A civil engineer, or a lawyer, or a dentist who is merely capable may yet lead a creative and fulfilling professional life. However, a mathematician who is just average – I'm talking about a researcher, of course, not a high-school teacher – is a living, walking tragedy…'
'But Uncle,’ I interrupted, 'I haven't the slightest intention of being "just average". I want to be Number One!'
He smiled. 'In that at least you definitely resemble me. I too was overambitious. But you see, dear boy, good intentions are, alas, not enough. This is not like many other fields where diligence always pays. To get to the top in mathematics you also need something more, the absolutely necessary condition for success.'
'Which one is that?'
He gave me a puzzled look, for ignoring the obvious.
'Why, the talent! The natural predisposition in its more extreme manifestation. Never forget it: Mathematicus nascitur, non fit – A mathematician is born, not made. If you don't carry the special aptitude in your genes, you will labour in vain all your life and one day you will end up a mediocrity. A golden mediocrity, perhaps, but a mediocrity nevertheless!'
I looked him straight in the eye.
'What's your deal, Uncle?'
He hesitated for a moment, as if thinking it over. Then he said: 'I don't want to see you following a course that will lead to failure and unhappiness. Therefore I'm proposing that you will make a binding promise to me to become a mathematician if and only if you're supremely gifted. Do you accept?'
I was disconcerted. 'But how on earth can I determine that, Uncle?'
'You can't and you don't need to,' he said with a sly little smile. 'I will.'
'You?'
'Yes. I will set you a problem, which you will take home with you and attempt to solve. By your success, or failure, I will measure your potential for mathematical greatness with great accuracy.'
I had mixed feelings for the proposed deal: I hated tests but adored challenges.
'How much time will I have?' I asked. Uncle Petros half-closed his eyes, considering this. 'Mmm… Let's say till the beginning of school, the first of October. That gives you almost three months.'
Ignorant as I was, I believed that in three months I could solve not one but any number of mathematical problems. 'That much!'
'Well, the problem will be difficult,’ he pointed out. 'It's not one just anybody can solve, but if you've got what it takes to become a great mathematician, you will manage. Of course, you will swear that you will seek help from no one and you will not consult any books.' 'I swear,’ I said. He fixed his stare on me. 'Does that mean you accept the deal?'
I heaved a deep sigh. 'I do!'
Without a word, Uncle Petros disappeared briefly and returned with paper and pencil. He now became businesslike, mathematician to mathematician.
'Here's the problem… I assume you already know what a prime number is?'
'Of course I know, Uncle! A prime is an integer greater than 1 that has no divisors other than itself and unity. For example 2,3,5,7,11,13, and so on.'
He appeared pleased with the precision of my definition. 'Wonderful! Now tell me, please, how many prime numbers are there?'
I suddenly feit out of my depth. 'How many?'
'Yes, how many. Haven't they taught you that at school?'
'No.'
My uncle sighed a deep sigh of disappointment at the low quality of modern Greek mathematical education.
'All right, I will tell you this because you will need it: the primes are infinite, a fact first proven by Euclid in the third century BC. His proof is a gem of beauty and simplicity. By using reductio ad absurdum, he first assumes the contrary of what he wants to prove, namely that the primes are finite. So…'
With fast vigorous jabs at the paper and a few explanatory words Uncle Petros laid out for my benefit our wise ancestor's proof, also giving me my first example of real mathematics.
'… which, however,’ he concluded, 'is contrary to our initial assumption. Assuming finiteness leads to a contradiction; ergo the primes are infinite. Quod erat demonstrandum.'
'That's fantastic, Uncle,’ I said, exhilarated by the ingeniousness of the proof. 'It's so simple!'
'Yes,’ he sighed, 'so simple, yet no one had thought of it before Euclid. Consider the lesson behind this: sometimes things appear simple only in retrospect.'
I was in no mood for philosophizing. 'Go on now, Uncle. State the problem I have to solve!'
First he wrote it out on a piece of paper and then he read it to me.
'I want you to try to demonstrate,’ he said, 'that every even number greater than 2 is the sum of two primes.'
I considered it for a moment, fervently praying for a flash of inspiration that would blow him away with an instant solution. As it wasn't forthcoming, however, I just said:'That's all?'
Uncle Petros wagged his finger in warning. 'Ah, it's not that simple! For every particular case you can consider, 4 = 2 +1,6 = 3 + 3,8 = 3 + 5,10 = 3 + 7,12 = 7 + 5, 14 = 7 + 7, etc., it's obvious, although the bigger the numbers get the more extensive the calculating. However, since there is an infinity of evens, a case-by-case approach is not possible. You have to find a general demonstration and this, I suspect, you may find more difficult than you think.'
I got up. 'Difficult or not,’ I said, 'I will do it! I'm going to start work right away.'
As I was on my way to the gate he called from the kitchen window. 'Hey! Aren't you going to take the paper with the problem?'
A cold wind was blowing and I breathed in the exhalation of the moist soil. I don't think that ever in my life, whether before or after that brief moment, have I felt so happy, so full of promise and anticipation and glorious hope.
'I don't need to, Uncle,’ I called back. 'I remember it perfectly: Every even number greater than 2 is the sum of two primes. See you on October the first with the solution!'
His stern reminder found me in the street: 'Don't forget our deal,' he shouted. 'Only if you solve the problem can you become a mathematician!'
A rough summer lay in store for me.
Luckily, my parents always packed me off to my maternal uncle's house in Pylos for the hot months, July and August. That meant that, removed from my father's range, at least I didn't have the additional problem (as if the one Uncle Petros had set me were not enough) of having to conduct my work in secret. As soon as I arrived in Pylos I spread out my papers on the dining-room table (we always ate outdoors in the summer) and declared to my cousins that until further notice I would not be available for swimming, games and visits to the open-air movie theatre. I began to work at the problem from morning to night, with minimal interruption.
My aunt fussed in her good-natured manner: 'You're workirvg too much, dear boy. Take it easy. It's summer vacation. Leave the books aside for a while. You came here to rest.'
I, however, was determined not to rest until final victory. I slaved at my table incessantly, scribbling away on sheet after sheet of paper, approaching the problem from this side and that. Often, when I felt too exhausted for abstract deductive reasoning, I would test specific cases, lest Uncle Petros had set me a trap by asking me to demonstrate something obviously false. After countless divisions I had created a table of the first few hundred primes (a primitive, self-made Eratosthenes' Sieve [1]) which I then proceeded to add, in all possible pairs, to confirm that the principle indeed applied. In vain did I search for an even number within this boundary that didn't fit the required condition – all of them turned out to be expressible as the sum of two primes.
At some point in mid-August, after a succession of late nights and countless Greek coffees, I thought for a few happy hours that I'd got it, that I'd found the solution. I filled several pages with my reasoning and mailed them, by special delivery, to Uncle Petros. I had barely enjoyed my triumph for a few days when the postman brought me the telegram:
THE ONLY THING YOU HAVE DEMONSTRATED IS THAT EVERY EVEN NUMBER CAN BE EXPRESSED AS THE SUM OF ONE PRIME AND ONE ODD WHICH HOWEVER IS OBVIOUS STOP
It took me a week to recover from the failure of my first attempt and the blow to my pride. But recover I did and half-heartedly I resumed work, this time employing the redudio ad absurdum:
'Let us assume there is an even number n which cannot be expressed as the sum of two primes. Then…'
The longer I laboured on the problem the more apparent it became that it expressed a fundamental truth regarding the integers, the materia prima of the mathematical universe. Soon I was driven to wondering about the precise way in which the primes are distributed among the other integers or the procedure which, given a certain prime, leads us to the next. I knew that this Information, were I to possess it, would be extremely useful in my plight and once or twice I was tempted to search for it in a book. However, loyal to my commitment not to seek outside help, I never did.
By stating Euclid's demonstration of the infinity of the primes, Uncle Petros said he'd given me the only tool I needed to find the proof. Yet I was making no progress.
At the end of September, a few days before the beginning of my last year in school, I found myself once again in Ekali, morose and crestfallen. Since Uncle Petros didn't have a telephone, I had to go through with this in person.
'Well?' he asked me as soon as we sat down, after I'd stiffly refused his offer of a sour-cherry drink. 'Did you solve the problem?'
'No,' I said. 'As a matter of fact, I didn't.'
The last thing I wanted at that point was to have to trace the course of my failure or have him analyse it for my sake. What's more, I had absolutely no curiosity to learn the solution, the proof of the principle. All I wished was to forget everything even remotely related to numbers, whether odd or even – not to mention prime.
But Uncle Petros wasn't willing to let me off easily. 'That's that then,’ he said. 'You remember our deal, don't you?'
I found his need officially to ratify his victory (as, for some reason, I was certain he viewed my defeat) intensely annoying. Yet I wasn't planning to make it sweeter for him by displaying any hint of hurt feelings.
'Of course I do, Uncle, as I'm sure you do too. Our deal was that I wouldn't become a mathematician unless I solved the problem -'
'No!' he cut me off, with sudden vehemence. 'The deal was that unless you solved the problem you'd make a binding promise not to become a mathematician!'
I scowled at him. 'Precisely,’ I agreed. 'And as I haven't solved the problem -'
'You will now make a binding promise,' he interrupted, a second time completing the sentence, stressing the words as if his life (or mine, rather) depended on it.
'Sure,’ I said, forcing myself to sound nonchalant, 'if it pleases you, I'll make a binding promise.'
His voice became harsh, cruel even. 'It's not a question of pleasing me, young man, but of honouring our agreement! You will pledge to stay away from mathematics!'
My annoyance instantly developed into full-fledged hatred.
'All right, Uncle,' I said coldly. 'I pledge to stay away from mathematics. Happy now?'
But as I got up to go he lifted his hand, menacingly. 'Not so fast!'
With a quick move he got a sheet of paper out of his pocket, unfolded it and stuck it in front of my nose. This was it:
/, the undersigned, being in full possession of my faculties, hereby solemnly pledge that, having failed in my examination for a higher mathematical capability and in accordance with the agreement made with my uncle, Petros Papachristos, I will never work towards a mathematics degree at an Institution of higher learning, nor in any other way attempt to pursue a professional career in mathematics.
I stared at him in disbelief.
'Sign!' he commanded.
'What's the use of this?' I growled, now making no effort to conceal my feelings.
'Sign,’ he repeated unmoved. 'A deal is a deal!'
I left his extended hand holding the fountain pen suspended in mid-air, got out my ballpoint and jabbed in my signature. Before he had time to say anything more I threw the paper at him and made a wild rush to the gate.
'Wait!' he shouted, but I was already outside.
I ran and ran and ran until I was safely out of his hearing and then I stopped and, still breathless, broke down and cried like a baby, tears of anger and frustration and humiliation streaming down my face.
I neither saw nor spoke to Uncle Petros during my last year of school, and in the following June I made up an excuse to my father and stayed home during the traditional family visit to Ekali.
My experience of the previous summer had had the exact result that Uncle Petros had, doubtless, intended and foreseen. Irrespective of any obligation to keep my part of our 'deal', I had lost all desire to become a mathematician. Luckily, the side effects of my failure were not extreme, my rejection was not total and my superior performance at school continued. As a consequence, I was admitted to one of the best universities in the United States. Upon registration I declared a major in Economics, a choice I abided by till my Junior year [2]. Apart from the basic requirements, Elementary Calculus and Linear Algebra (incidentally, I got As in both), I took no other mathematics courses in my first two years.
Uncle Petros' successful (at first, anyway) ploy had been based on the application of the absolute determinism of mathematics to my life. He had taken a risk, of course, but it was a well-calculated one: the possibility of my discovering the identity of the problem he had assigned me in the course of elementary university mathematics was minimal. The field to which it belongs is Number Theory, only taught in electives aimed at mathematics majors. Therefore it was reasonable for him to assume that, as long as I kept my pledge, I would complete my university studies (and conceivably my life) without learning the truth.
Reality, however, is not as dependable as mathematics, and things turned out differently.
On the first day of my Junior year I was informed that Fate (for who else can arrange coincidences such as this?) had assigned that I share my dormitory room with Sammy Epstein, a slightly built boy from Brooklyn renowned among undergraduates as a phenomenal maths prodigy. Sammy would be getting his degree that same year at the age of seventeen and, although he was nominally still an undergraduate, all his classes were already at advanced graduate level. In fact, he had already started work on his doctoral dissertation in Algebraic Topology.
Convinced as I was until that point that the wounds of my short traumatic history as a mathematics hopeful had more or less healed, I was delighted, even amused, when I learned the identity of my new room-mate. As we were dining side by side in the university dining hall on our first evening, to get better acquainted, I said to him casually:
'Since you're a mathematical genius, Sammy, I'm sure you can easily prove that every even number greater than 2 is the sum of two primes.'
He burst out laughing. 'If I could prove that, man, I wouldn't be here eating with you; I'd be a professor already. Maybe I'd even have my Fields Medal, the Nobel Prize of Mathematics!'
Even as he was speaking, in a flash of revelation, I guessed the awful truth. Sammy confirmed it with his next words:
'The statement you just made is Goldbach's Conjecture, one of the most notoriously difficult unsolved problems in the whole of mathematics!'
My reactions went through the phases referred to (if I accurately remember what I learned in my elementary college Psychology course) as the Four Stages of Mourning: Denial, Anger, Depression and Acceptance.
Of these, the first was the most short-lived. 'It… it can't be!' I stammered as soon as Sammy had uttered the horrible words, hoping I'd misheard.
'What do you mean "it can't be"?' he asked. 'It can and it is! Goldbach's Conjecture – that's the name of the hypothesis, for it is only a hypothesis, since it's never been proved – is that all evens are the sum of two primes. It was first stated by a mathematician named Goldbach in a letter to Euler [3]. Although it's been tested and found to be true up to enormous even numbers, no one has managed to find a general proof.'
I didn't hear Sammy's next words, for I had already passed into the stage of Anger:
'The old bastard!' I yelled in Greek. "The son of a bitch! God damn him! May he rot in hell!'
My new room-mate, totally bewildered that a hypothesis in Number Theory could provoke such an outburst of violent Mediterranean passion, pleaded with me to tell him what was going on. I, however, was in no state for explanations.
I was nineteen and until then had led a protected life. Except for the single Scotch drunk with Father to celebrate, 'among grown-up men', my graduation from high school and the required sip of wine to toast a relative's wedding, I had never tasted alcohol. Consequently, the great quantities I put down that night at a bar near the university (I started out with beer, moved on to bourbon and ended up with rum) must be multiplied by a rather large n to fully realize their effect.
While on my third or fourth glass of beer and still in moderate possession of my senses, I wrote to Uncle Petros. Later, once into the phase of fatalistic certainty as to my imminent death, and before I passed out, I handed over the letter to the barman with his address and what remained of my monthly allowance, asking him to fulfil my last wish and mail it. The partial amnesia that cloaks the events of that night has obscured for ever the detailed content of the letter. (I did not have the emotional stamina to seek it out from among my uncle's papers, when many years later I inherited his archive.) From the little that I remember, however, there can be no swear-word, vulgarity, insult, condemnation and curse that it didn't contain. The gist of it was that he had destroyed my life and as a consequence upon my return to Greece I would murder him, but this only after torturing him in the most perverse ways human imagination could contrive.
I don't know how long I remained unconscious, struggling with outlandish nightmares. It must have been late afternoon of the following day before I began to be aware of my surroundings. I was in my bed in the dormitory and Sammy was there, at his desk, bent over his books. I groaned. He came over and explained:
I had been brought back by some fellow students who'd found me dead to the world on the lawn in front of the library. They'd hauled me to the infirmary, where the doctor on duty had had no difficulty diagnosing my condition. As a matter of fact, he didn't even have to examine me, as my clothes were covered in vomit and I reeked of alcohol.
My new room-mate, obviously concerned about the future of our cohabitation, asked me whether this sort of thing occurred frequently with me. Humiliated, I mumbled that it was the first time.
'It's all because of Goldbach's Conjecture,' I whispered, and sank back into sleep.
It took me two days to recover from an excruciating headache. After that (it seems the torrent of alcohol had carried me right through Rage) I entered the next stage of my mourning: Depression. For two days and nights I stayed slumped in an armchair in the common room on our floor, listlessly observing the black-and-white images dancing on the TV screen.
It was Sammy who helped me out of this self-inflicted lethargy, displaying a sense of camaraderie totally inconsistent with the caricature of the self-centred, absent-minded mathematician. On the evening of the third day after my bender I saw him standing there, looking down at me.
'Do you know tomorrow is the deadline for registration?' he asked severely.
'Mmmm…' I groaned.
'So, have you registered?'
I shook my head wearily.
'Have you at least selected the courses you'll be taking?'
I shook my head once again and he frowned.
'Not that it's any of my business, but don't you think you better turn your attention to these rather urgent matters, instead of sitting there all day staring at the idiot-box?'
As he later confessed, it wasn't merely the urge to assist a fellow human being in crisis that made him assume responsibility – the curiosity to discover the connection between his new room-mate and the notorious mathematical problem was overwhelming. One thing is certain: regardless of his motives, the long discussion I had that evening with Sammy made all the difference to me. Without his understanding and support, I couldn't have crossed the crucial line. And, what's perhaps more important: it's quite unlikely I would ever have forgiven Uncle Petros.
We started our talk in the dining hall, over dinner, and continued through the night in our room, drinking coffee. I told him everything: about my family, my early fascination with the remote figure of Uncle Petros and my gradual discoveries of his accomplishments, his brilliant chess-playing, his books, the invitation of the Hellenic Mathematical Society and the professorship in Munich. About Father's brief resume of his life, his early successes and the mysterious (to me, at least) role of Goldbach's Conjecture in his later dismal failure. I mentioned my initial decision to study mathematics and the discussion with Uncle Petros that summer afternoon, three years back, in his kitchen in Ekali. Finally, I described our 'deal'.
Sammy listened without interrupting once, his small, deep eyes narrowed intently in focus. Only when I reached the end of my narrative and stated the problem that my uncle had required me to solve to demonstrate my potential for mathematical greatness did he burst out, seized by sudden fury.
'What an ass-hole!' he cried.
'My feelings exactly,' I said.
'The man is a sadist,' Sammy went on. 'Why, he's criminally insane! Only a perverted mind could conceive the plot of making a school-kid spend a summer trying to solve Goldbach's Conjecture, and this under the illusion that he had merely been set a challenging exercise. What a total beast!'
The guilt about the extreme vocabulary I had used in my delirious letter to Uncle Petros led me for a moment to attempt to defend him and find a logical excuse for his behaviour.
'Maybe his intentions were not all bad,’ I muttered. 'Maybe he thought he was protecting me from greater disappointment.'
'With what right?' Sammy said loudly, banging his hand on my desk. (Unlike me, he'd grown up in a society where children were not expected as a rule to conform to the expectations of their parents and elders.) 'Every person has the right to expose himself to whatever disappointment he chooses,’ he said fervently. 'Besides, what's all this crap about "being the best" and "golden mediocrities" and whatnot. You could have become a great -'
Sammy stopped in mid-sentence, his mouth gaping in amazement.' Wait a minute, why am I using the past tense?' he said, beaming. 'You can still become a great mathematician!'
I glanced up, startled. 'What are you talking about, Sammy? It's too late, you know that!'
'Not at all! The deadline for declaring a major is tomorrow.'
'That's not what I mean. I've already lost so much time doing other things and -'
'Nonsense,' he said firmly. 'If you work hard you can make up for lost time. What's important is that you recover your enthusiasm, the passion you had for mathematics before your uncle shamelessly destroyed it for you. Believe me, it can be done – and I'll help you do it!'
Day was breaking outside and the moment had come for the fourth and last stage that would complete the mourning process: Acceptance. The cycle had closed. I would pick up my life from where I'd left off when Uncle Petros, through the appalling trick he'd played on me, steered me away from what I then still considered my true course.
Sammy and I consumed a hearty breakfast in the dining hall and then sat down with the list of courses offered by the Department of Mathematics. He explained the contents of each one the way an experienced maitre d' might present choice items on the menu. I took notes, and in the early afternoon I went to the Registrar's office and filed my selection of courses for the semester just beginning: Introduction to Analysis, Introduction to Complex Analysis, Introduction to Modern Algebra and General Topology.
Naturally, I also declared my new field of major concentration: Mathematics.
A few days after the beginning of classes, during the most difficult phase of my efforts to penetrate into the new discipline, a telegram from Uncle Petros arrived. When I found the notice, I had no doubts as to the identity of the sender and initially considered not claiming it at all. However, curiosity finally prevailed.
I made a bet with myself as to whether he would be trying to defend himself, or simply scolding me for the tone of my letter. I opted for the latter and lost. He wrote:
I FULLY UNDERSTAND YOUR REACTION STOP IN ORDER TO UNDERSTAND MY BEHAVIOUR YOU SHOULD ACQUAINT YOURSELF WITH KURT GÖDEL's INCOMPLETENESS THEOREM
At that time I had no idea what Kurt Gödel's Incompleteness Theorem was. Also, I had no desire to find out – mastering the theorems of Lagrange, Cauchy, Fatou, Bolzano, Weierstrass, Heine, Borel, Lebesgue, Tychonoff, et al. for my various courses was hard enough. Anyway, by now I had more or less come to accept Sammy's assessment that Uncle Petros' behaviour towards me showed definite signs of derangement. The latest message confirmed this: he was trying to defend his despicable treatment of me by way of a mathematical theorem! The wretched old man's obsessions were of no further interest to me.
I did not mention the telegram to my room-mate, nor did I give it further thought.
I spent that Christmas vacation studying with Sammy at the Mathematics Library [4].
On New Year's Eve he invited me to celebrate with him and his family at their Brooklyn home. We'd been drinking and were feeling quite merry when he took me aside to a quiet corner.
'Could you bear to talk about your uncle a bit?' he asked. Since that first, all-night session, the subject had never again come up, as if by unspoken agreement.
'Sure I can bear it,' I laughed, 'but what more is there to say?'
Sammy took out of his pocket a sheet of paper and unfolded it. 'It's been a while now since I've been doing some discreet research on the subject,’ he said.
I was surprised. 'What kind of "discreet research"?'
'Oh, don't go imagining anything nefarious; mostly bibliographical stuff.'
'And?'
'And I came to the conclusion that your dear Uncle Petros is a fraud!'
'A fraud?' It was the last thing I would have expected to hear about him and, since blood is thicker than water, I immediately jumped to his defence.
'How can you say that, Sammy? It's a proven fact that he was Professor of Analysis at the University of Munich. He is no fraud!'
He explained: 'I went through the bibliographical indexes of all articles published in mathematical Journals in this Century. I only found three items under his name, but nothing – not one single word – on the subject of Goldbach's Conjecture or anything remotely related to it!'
I couldn't understand how this led to accusations of fraud. 'What's so surprising in that? My uncle is the first to admit that he didn't manage to prove the Conjecture: there was nothing to publish. I find it perfectly understandable!'
Sammy smiled condescendingly.
‘That's because you don't know the first thing about research,’ he said. 'Do you know what the great David Hubert answered when questioned by his colleagues as to why he never attempted to prove the so-called "Fermat's Last Theorem", another famous unsolved problem?'
'No, I don't. Enlighten me.'
'He said: "Why should I kill the goose that lays the golden eggs?" What he meant, you see, was that when great mathematicians attempt to solve great problems a lot of great mathematics – so-called "intermediate results" – is born, and this even though the initial problems may remain unsolved. Just to give you an example you'll understand, the field of Finite Group Theory came into being as a result of Evariste Galois' efforts to solve the equation of the fifth degree in its general form…'
The gist of Sammy's argument was this: there was no way that a top-class professional mathematician, as we had every indication that Uncle Petros was in his youth, could have spent his life wrestling with a great problem such as Goldbach's Conjecture without discovering along the way a single intermediate result of some value. However, since he had never published anything, we necessarily had to conclude (here Sammy was applying a form of the redudio ad absurdum) that he was lying: he never had attempted to prove Goldbach's Conjecture.
'But to what purpose would he tell such a lie?' I asked my friend, perplexed.
'Oh, it's more likely than not that he concocted the Goldbach Conjecture story to explain his mathematical inactivity – this is why I used the harsh word "fraud". You see, this is a problem so notoriously difficult that nobody could hold it against him if he didn't manage to solve it.'
'But this is absurd,’ I protested. 'Mathematics was Uncle Petros' life, his only interest and passion! Why would he want to abandon it and need to make up excuses for his inactivity? It doesn't make sense!'
Sam shook his head. "The explanation, I'm afraid, is rather depressing. A distinguished professor in our department, with whom I discussed the case, suggested it to me.' He must have seen the signs of dismay in my face, for he hastened to add:'… without mentioning your uncle's name, of course!'
Sammy then outlined the 'distinguished professor's' theory: 'It's quite likely that at some point early in his career your uncle lost either the intellectual capacity or the willpower (or possibly both) to do mathematics. Unfortunately, this is quite common with early developers. Burnout and breakdown are the fate of quite a few precocious geniuses…'
The distressing possibility that this sorry fate could possibly also one day await himself had obviously entered his mind: the conclusion was spoken solemnly, sadly even.
'You see, it's not that your poor Uncle Petros didn't want after a certain point to do any more mathematics – it's that he couldn 't.'
After my talk with Sammy on New Year's Eve, my attitude towards Uncle Petros changed once again. The rage I had felt when I first realized he had tricked me into attempting to prove Goldbach's Conjecture had already given way to more charitable feelings. Now, an element of sympathy was added: how terrible it must have been for him, if after such a brilliant beginning he suddenly began to feel his great gift, his only strength in life, his only joy, deserting him. Poor Uncle Petros!
The more I thought about it, the more I became upset at the unnamed 'distinguished professor' who could pronounce such damning indictments of someone he didn't even know, in the total absence of data. At Sammy, too. How could he so lightheartedly accuse him of being a 'fraud'?
I ended up deciding that Uncle Petros should be given the chance to defend himself, and to counter both the facile levelling generalizations of his brothers ('one of life's failures', etc.) as well as the condescending analyses of the 'distinguished professor' and the cocky boy-genius Sammy. The time had come for the accused to speak. Needless to say, I decided the person best qualified to hear his defence was none other than I, his close kin and victim. After all, he owed me.
I needed to prepare myself.
Although I had torn his telegram of apology into little pieces, I hadn't forgotten its content. My uncle had enjoined me to learn Kurt Gödel's Incompleteness Theorem; in some unfathomable way the explanation of his despicable behaviour to me lay in this. (Without knowing the first thing about the Incompleteness Theorem I didn't like the sound of it: the negative particle 'in-' carried a lot of baggage; the vacuum it hinted
at seemed to have metaphorical implications.)
At the first opportunity, which came while selecting my mathematics courses for the next semester, I asked Sammy, careful not to have him suspect that my question had anything to do with Uncle Petros: 'Have you ever heard of Kurt Gödel's Incompleteness Theorem?' Sammy threw his arms in the air, in comic exaggeration. 'Oy vey!’ he exclaimed. 'He asks me if I’ve heard of Kurt Gödel's Incompleteness Theorem!' 'To what branch does it belong? Topology?' Sammy stared at me aghast. 'The Incompleteness Theorem? – to Mathematical Logic, you total ignoramus!' 'Well, stop clowning and tell me about it. Tell me what it says.'
Sammy proceeded to explain along general lines the content of Gödel's great discovery. He began from Euclid and his vision of the solid construction of mathematical theories, starting from axioms as foundations and proceeding by the tools of rigorous logical induction to theorems. Then, he spanned twenty-two centuries to talk of 'Hilbert's Second Problem' and skimmed over the basics of Russell's and Whitehead's Principia Mathematica [5] terminating with the Incompleteness Theorem itself, which he explained in as simple language as he could.
'But is that possible?' I asked when he was finished, staring at him wide-eyed.
'More than possible,’ answered Sammy, 'it's a proven fact!’