Uncle Petros and Goldbach

by Apostolos Doxiadis

  The drama played out during Petros Papachristos' final days is the last in a triad of episodes from the history of mathematics, unified by a single theme: the Mystery-solution to a Famous Problem by an Important Mathematician. [16]

  By majority consent, the three most famous unsolved mathematical problems are: (a) Fermat's Last Theorem, (b) the Riemann Hypothesis and (c) Goldbach's Conjecture.

  In the case of Fermat's Last Theorem, the mystery-solution existed from its first statement: in 1637, while he was studying Diophantus' Arithmetica, Pierre de Fermat made a note in the margin of his personal copy, right next to proposition II.8 referring to the Pythagorean theorem, in the form x^2 + y^2 = z^2. He wrote: ‘It is impossible to separate a cube into two cubes, or a biquadrate (fourth power) into two biquadrates, or generally any power except a square into two powers with the same exponent. I have discovered a truly marvellous proof of this, which, however, this margin is not large enough to contain.'

  After the death of Fermat his son collected and published his notes. A thorough search of his papers, however, failed to reveal the demonstratio mirabilis, the 'marvellous proof’ that his father claimed to have found. Equally in vain have mathematicians ever since sought to rediscover it. [17] As for the verdict of history on the existence of the mystery-solution: it's ambiguous. Most mathematidans today doubt that Fermat indeed had a proof. The worst-case theory has it that he was consciously lying, that he had not verified his guess and his margin-note was mere bragging. What's likelier, however, is that he was mistaken, the demonstratio mirabilis crippled by an undetected fault.

  In the case of the Riemann Hypothesis, the mystery-solution was in fact a metaphysical practical joke, with G. H. Hardy as its perpetrator. This is how it happened:

  Preparing to board a cross-Channel ferry during a bad storm, the confirmed atheist Hardy sent off to a colleague a postcard with the message: 'I have the proof to the Riemann Hypothesis.' His reasoning was that the Almighty, whose sworn enemy he was, would not permit him to reap such an exalted undeserved reward and would therefore see to his safe arrival, in order to have the falsity of his claim exposed.

  The mystery-solution of Goldbach's Conjecture completes the triad.

  On the morning after our last lesson, I telephoned Uncle Petros. At my insistence, he had recently agreed to have a line installed, on the condition that only I, and no one eise, would know the number.

  He answered sounding tense and distant. 'What do you want?'

  'Oh, I just called to say hello,' I said. 'Also to apologize. I think I was unnecessarily rude last night.'

  There was a pause.

  'Well,’ he said, 'actually I'm busy at the moment. Why don't we talk again… shall we say next week?'

  I wanted to assume that his coldness was due to the fact that he was upset with me (as he had every reason to be, after all) and merely expressing his resentment. Still, I feit a nagging unease.

  'Busy with what, Uncle?' I persisted.

  Another pause.

  'I-I'll tell you about it some other time.'

  He was obviously eager to hang up so, before he could cut me off, I impulsively blurted out the suspicion that had taken shape during the night.

  'You wouldn't by any chance have resumed your researches, would you, Uncle Petros?'

  I heard a sharp intake of breath. 'Who – who told you that?' he said hoarsely.

  I tried to sound casual. 'Oh, come on, give me some credit for having come to know you. As if it needed telling!'

  I heard the click of his hanging up. My God – I was right! The crazy old fool had gone off his rocker. He was trying to prove Goldbach's Conjecture!

  My guilty conscience stung me. What had I done? Humankind indeed cannot stand very much reality – Sammy's theory of Kurt Gödel's insanity also applied, in a different way, to Uncle Petras. I had obviously pushed the poor old man to his uttermost limit and then beyond it. I'd aimed straight at his Achilles heel and hit it. My ridiculous simple-minded scheme to force him into self-confrontation had destroyed his fragile defences. Heedlessly, irresponsibly, I had robbed him of the carefully nurtured justification of his failure: the Incompleteness Theorem. But I had put nothing in its place to sustain his shattered self-image. As his extreme reaction now showed, the exposure of his failure (to himself, more than to me) had been more than he could bear. Stripped of his cherished excuse he had taken, of necessity, the only way left for him to go: madness. For what else was the endeavour to search, in his late seventies, for the proof that he had failed to find when he was at the peak of his powers? If that wasn't total irrationality, what was?

  I walked into my father's office filled with apprehension. Much as I hated to allow him into the charmed circle of my bond with Uncle Petros, I feit obliged to let him know what had happened. He was after all his brother, and any suspicion of serious illness was certainly a family matter. My father dismissed my self-recriminations about causing the crisis as so much poppycock. According to the official Papachristos world-view, a man had only himself to blame for his psychological condition, the only acceptable external reason for emotional discomfort being a serious drop in the price of stocks. As far as he was concerned, his older brother's behaviour had always been bizarre, and one more instance of eccentricity was definitely not to be taken seriously.

  'In fact,’ he said, 'the condition you describe – absent-mindedness, self-absorption, abrupt changes of mood, irrational demands for beans in the middle of the night, nervous tics, etc. – reminds me of how he was carrying on when we visited him in Munich, back in the late twenties. Then, too, he was behaving like a madman. We'd be at a nice restaurant enjoying our wurst and he'd be squirming around as if there were nails in his chair, his face twitching like mad.'

  'Quod erat demonstrandum,' I said. "That's exactly it. He's back doing mathematics. In fact, he's back working on Goldbach's Conjecture – ridiculous as that may sound at his age.'

  My father shrugged. 'It's ridiculous at any age,’ he said. 'But why worry? Goldbach's Conjecture has already done him all the harm possible. Nothing worse can come of it.'

  But I wasn't so sure about that. In fact, I was quite certain that a lot worse things could be in store for us. Goldbach's resurrection was bound to stir up unfulfilled passions, to aggravate deep-buried, terrible, unhealed wounds. His absurd new application to the old problem boded no good.

  After work that evening, I drove to Ekali. The ancient VW beetle was parked outside the house. I crossed the front yard and rang the bell. There was no response, so I shouted: 'Open up, Uncle Petros; it's me!'

  For a few moments I feared the worst, but then he appeared at a window and stared vaguely in my direction. There was no sign of his usual pleasure at seeing me, no surprise, no greeting – he just stared.

  'Good afternoon,’ I said. 'I just came by to say hello.'

  His normally serene face, the face of a stranger to life's usual worries, was now marked by extreme tension, his skin pale, his eyes red with sleeplessness, his brow furrowed with concern. He was also unshaven, the first time I'd seen him so. His stare continued absent, unfocused. I wasn't even sure he knew who I was.

  'Come on, Uncle dear, please open up for the most favoured,’ I said with a fatuous smile.

  He disappeared and after a while the door creaked open. He stood there, blocking my entry, wearing his pyjama bottoms and a wrinkled vest. It was evident he didn't want me to enter.

  'What's wrong, Uncle?' I asked. ‘I’m worried about you.'

  'Why should you be worried?' he said, now forcing himself to sound normal. 'Everything's fine.'

  'Are you sure?'

  'Of course I'm sure.'

  Then, with a snappy gesture, he beckoned me closer. After quickly, anxiously glancing around, he leaned towards me, his lips almost touching my ear, and whispered: 'I saw them again.'

  I didn't understand. 'Who did you see?'

  'The girls! The twins, the number 2^100!'

  I remembered the strange appa
ritions of his dreams.

  'Well,' I said, trying to sound as casual as possible. ‘If you are once again involved with mathematical research, you are once again having mathematical dreams. Nothing strange about that…'

  I wanted to keep him talking so as to (figuratively, but if need be also literally) put a foot in the door. I had to get some sense of how bad his condition was.

  'So what happened, Uncle,' I asked, feigning great interest in the matter. 'Did the girls speak to you?'

  'Yes,’ he said, 'they gave me a…' His voice quickly trailed off, as if he was afraid he'd said too much.

  'A what?' I asked. 'A clue?'

  He became suspicious again. 'You mustn't tell,’ he said sternly.

  'Mum's the word,' I said.

  He had started to close the door. Convinced now that his situation was extremely serious and that the time had come for emergency action, I grasped the knob and started to push. As he felt my force, he tensed up, gritted his teeth and struggled to prevent me from entering, his face contorted to a grimace of desperation. Fearing the effort might be too much for him (he was nearing eighty, after all) I reduced the pressure a bit for a final attempt at reason.

  Of all the possible stupid things I could have said to him, I chose this: 'Remember Kurt Gödel, Uncle Petros! Remember the Incompleteness Theorem – Goldbach's Conjecture is unprovable!'

  Instantly, his expression changed from despair to wrath. 'Fuck Kurt Gödel,' he barked, 'and fuck his Incompleteness Theorem!' With an unexpected upsurge of strength, he overcame my resistance and slammed the door shut in my face.

  I rang the bell again and again, banged the door with my fist and shouted. I tried threats, reasoning and pleading, but nothing worked. When a torrential October rain began to fall I hoped that, mad or not, Uncle Petros might be moved by mercy and let me in. But he wasn't. I left, soaking wet and very worried.

  From Ekali I drove straight to our family doctor and explained the Situation. Without altogether ruling out serious mental disturbance (possibly triggered by my unwarranted interference in his defence mechanisms) he suggested two or three organic problems as likelier causes of my uncle's transformation. We decided to go to his house first thing the next morning, force our way in if necessary, and submit him to a thorough medical examination.

  That night I couldn't sleep. The rain was getting stronger, it was past two o'clock and I was sitting at home hunched in front of the chessboard, just as Uncle Petros must have been on innumerable sleepless nights, studying a game from the recent world championship. Yet my concern kept interfering and I couldn't concentrate.

  When I heard the ringing I knew it was he, even though he'd never yet initiated a call on his newly installed telephone.

  I jumped up and answered.

  'Is that you, Nephew?' He was obviously all worked up about something.

  'Of course it's me, Uncle. What's wrong?'

  'You must send me someone. Now!'

  I was alarmed. '"Someone"? A doctor you mean?'

  'What use would a doctor be? A mathematician, of course!'

  I humoured him: 'I'm a mathematician, Uncle; I’ll come right away! Just promise to open the door, so I won't catch pneumonia and -'

  He obviously didn't have time for irrelevancies. 'Oh hell!' he grunted and then: 'All right, all right, you come, but bring another one as well!'

  'Another mathematician?'

  'Yes! I must have two witnesses! Hurry!'

  'But why do the witnesses have to be mathematicians?'

  Naively, I had thought at first he wanted to write his will.

  'To understand my proof!'

  ‘Proof of what?’

  'Goldbach's Conjecture, you idiot – what else!'

  I chose my next words very carefully. 'Look, Uncle Petros,' I said, 'I promise to be with you as soon as my car will get me there. Let's be reasonable, mathematicians aren't kept on call – how on earth can I get one at two o'clock in the morning? You'll tell me all about your proof tonight and tomorrow we will go together -'

  But he cut me off, screaming. 'No, no, no! There's no time for any of that! I need my two witnesses and I need them now!’ Then he broke down and started sobbing. 'O nephew, it's so… it's so…'

  'So what, Uncle? Tell me!'

  'Oh, it's so simple, so simple, my dearest boy! How is it possible that all those years, those endless years, I hadn't realized how blessedly simple it was!'

  I cut him off. ‘I’ll be there as soon as I can.'

  'Wait! Wait! Waaaaa-it!!!' He was now in panic. 'Swear you won't come alone! Get the other witness! Hurry… Hurry up, I implore you! Get the witness! There's no time!'

  I tried to appease him: 'Oh, come on, Uncle; there can't be such a rush. The proof won't go away, you know!'

  These were his last words: 'You don't understand, dear boy – there's no time left!' His voice then dropped to a low, conspiratorial whisper, as if he didn't want to be overheard by someone close by: 'You see, the girls are here. They are waiting to take me.'

  By the time I arrived in Ekali, breaking all speed records, it was too late. Our family doctor (I had picked him up on the way) and I found Uncle Petros' lifeless body slumped on the paving of his little terrace. The torso was leaning against the wall, the legs spread open, the head turned towards us as if in welcome. A flash of distant lightning revealed his features fixed in a wonderful smile of deep, absolute contentment – I imagine it was that which guided the doctor in his instant diagnosis of a stroke. All around him were hundreds of lima beans. The rain had destroyed their neat parallelograms and now they were scattered all over the wet terrace, sparkling like precious jewels.

  The rain had just stopped and the air was infused with the invigorating smell of wet earth and pine trees.

  Our last exchange over the telephone is the only evidence of Petros Papachristos' mystery-solution to Goldbach's Conjecture.

  Unlike Pierre de Fermat's illustrious marginal note, however, it is extremely unlikely that my uncle's demonstratio mirabilis of his famous problem will tempt a host of mathematical hopefuls to attempt to reproduce it. (No rise in the price of beans is expected.) This is as it should be. Fermat's sanity was never in question; no one ever had reason to believe he was in anything less than total possession of his senses when he stated his Last Theorem. Unfortunately, the same cannot be said of my Uncle Petros. When he announced his triumph to me he was probably as mad as a harter. His last words were uttered in a state of terminal confusion, the total relinquishment of logic, the Night of Reason that dimmed the light of his final moments. It would thus be extremely unfair to have him posthumously declared a charlatan by attributing a serious intention to a declaration obviously made in a half-delirious state, his brain most probably already ravaged by the stroke that, a short while later, killed him.

  So: did Petros Papachristos prove Goldbach's Conjecture in his final moments? The wish to protect his memory from any chance of ridicule obliges me to state it as unequivocally as possible: the official answer must be 'No'. (My own opinion need not concern mathematical history – I will therefore keep it to myself.)

  The funeral was strictly family, with only a wreath and a single representative from the Hellenic Mathematical Society.

  The epitaph later carved on Petros Papachristos' tomb, below the dates marking the limits of his earthly existence, was chosen by me, after I had overcome the initial objections of the family elders. They form one further addition to the collection of posthumous messages that make the First Cemetery of Athens one of the world's most poetic:

  EVERY EVEN NUMBER GREATER THAN 2

  IS THE SUM OF TWO PRIMES

  Post Scriptum

  At the time this book was completed, Goldbach's Conjecture was two hundred and fifty years old. To this day it remains unproven.

  Acknowledgements

  I wish to thank Professors Ken Ribet and Keith Conrad, who carefully read the revised manuscript and corrected numerous mistakes, as well as Dr Kevin Buzzard for the
clarification of various points – obviously, any remaining mathematical flaws are my own. Also my sister, Cali Doxiadis, for her invaluable linguistic and editorial advice.

  APOSTOLOS DOXIADIS

  ***

  [1] A method for locating the primes, invented by the Greek mathematician Eratosthenes

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  [2] 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.

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  [3] 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.

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  [4] 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.

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  [5] Principia Mathematica: 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.