Uncle Petros and
Goldbach’s Conjecture
A Novel of Mathematical Obsession
APOSTOLOS DOXIADIS
BLOOMSBURY
NEW YORK • BERLIN • LONDON
Archimedes will be remembered when Aeschylus
is forgotten, because languages die and mathematical
ideas do not. ‘Immortality’ may be a silly word,
but probably a mathematician has the best
chance of whatever it may mean.
G. H. HARDY, A Mathematician’s Apology
Uncle Petros and
Goldbach’s Conjecture
Contents
One
Two
Three
Acknowledgements
A Note on the Author
One
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 Monatsheftefür 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 outhouse attached to the tool-shed. 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 garden 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 fireplace 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 ’s and ’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, Fm 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 Buler’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 else!
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 his 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 tell me about his eldest brother. I, of course, was all ears.
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