Uncle Petros and Goldbach's Conjecture

by Apostolos Doxiadis

  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 Pet-ros 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 resumé 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 maǐtre 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.*

  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,’ Ilaughed, ‘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 Petrosisafraud!’

  ‘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 Gold-bach’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 Hilbert 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 reductio 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 Mathematical* 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!’

  * * *

  * A method for locating the primes, invented by the Greek mathematician Eratosthenes.

  * According to the American system, a student can go through the first two years of university without being obliged to declare an area of major concentration for his degree or, if he does so, is free to change his mind until the beginning of the Junior (third) year.

  * In fact, Christian Goldbach’s letter of 1742 contains the conjecture that ‘every integer can be expressed as the sum of three primes’. However, as (if this is true) one of the three such primes expressing even numbers will be 2 (the addition of three odd primes would be of necessity odd, and 2 is the only even prime number), it is an obvious corollary that every even number is the sum of two primes. Ironically, it was not Goldbach but Euler who phrased the conjecture that bears the other’s name — a little known fact, even among mathematicians.

  * The main purpose of this narrative is not autobiographical, so I will not burden the reader further with details of my own mathematical progress. (To satisfy the curious I could sum it up as ‘slow but steady’.) Henceforth, my own story will be referred to only to the extent to which it is relevant to that of Uncle Petros.

  * Principia Mathematical the monumental work of logicians Russell and Whitehead, first published in 1910, in which they attempt the titanic task of founding the edifice of mathematical theories on the firm foundations of logic.

  Two

&nbs
p; I went to Ekali on the second day after my arrival in Greece for the summer vacation. Not wanting to catch him unawares, I’d already arranged the meeting with Uncle Petros by correspondence. To continue with the judicial analogy, I’d granted him ample time to prepare his defence.

  I arrived at the arranged time and we sat in the garden.

  ‘So then, most favoured of nephews’ (this was the first time he called me that), ‘what news do you bring me from the New World?’

  If he thought I’d let him pretend this was a mere social occasion, a visit by dutiful nephew to caring uncle, he was mistaken.

  ‘So then, Uncle,’ I said belligerently, ‘in a year’s time I’m getting my degree and I’m already preparing applications for graduate school. Your ploy has failed. Whether it is to your liking or not, I will be a mathematician.’

  He shrugged his shoulders while raising the palms of his hands heavenwards in a gesture of inevitability.

  ‘“He who is fated to drown will never die in his bed”,’ he intoned — a popular Greek proverb. ‘Have you told your father? Is he pleased?’

  ‘Why this sudden interest in my father?’ I snarled. ‘Was it he who put you up to our so-called “deal”? Was it his perverse idea to make me prove myself worthy by tackling Goldbach’s Conjecture? Or do you feel so much in his debt for supporting you all these years that you repaid him by bringing his upstart son to heel?’

  Uncle Petros accepted the blows under the belt without changing expression.