Uncle Petros and Goldbach's Conjecture

by Apostolos Doxiadis

  ‘My help on what subject?’

  ‘Oh, in deciphering a difficult German text — a mathematical text.’ The young man apologized again for presuming to take up his time with such a lowly task. This particular article, however, was of such great importance to him that when he heard that a senior mathematician from Germany was at Trinity, he couldn’t resist appealing to him for assistance in its precise translation.

  There was something so childishly eager in his manner that Petros couldn’t refuse him.

  ‘I’d be glad to help you, if I can. What field is the article in?’

  ‘Formal Logic, Professor. The Grundlagen, the Foundations of Mathematics.’

  Petros felt a rush of relief that it wasn’t in Number Theory — he’d feared for a moment the young caller might have wanted to pump him on his work on the Conjecture, using help with the language merely as an excuse. As he was more or less finished with his day’s work, he asked the young visitor to take a seat.

  ‘What did you say your name was?’

  ‘It’s Alan Turing, Professor. I’m an undergraduate.’

  Turing handed him the journal containing the article, opened at the right page.

  ‘Ah, the Monatshefte für Mathematik und Physik,’ said Petros, ‘the Monthly Review for Mathematics and Physics, a highly esteemed publication. The title of the article is, I see, “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme”. In translation this would be … Let’s see … “On the formally undecidable propositions of Principia Mathematica and similar systems”. The author is a Mr Kurt Gödel, from Vienna. Is he well known in this field?’

  Turing looked at him surprised. ‘You don’t mean to say you haven’t heard of this article, Professor?’

  Petros smiled: ‘My dear young man, mathematics too has been infected by the modern plague, overspe-cialization. I’m afraid I have no idea of what’s being accomplished in Formal Logic, or any other field for that matter. Outside of Number Theory I am, alas, a complete innocent.’

  ‘But Professor,’ Turing protested, ‘Gödel’s Theorem is of interest to all mathematicians, and number theorists especially! Its first application is to the very basis of arithmetic, the Peano-Dedekind axiomatic system.’

  To Turing’s amazement, Petros also wasn’t too clear about the Peano-Dedekind axiomatic system. Like most working mathematicians he considered Formal Logic, the field whose main subject is mathematics itself, a preoccupation that was certainly over-fussy and quite possibly altogether unnecessary. Its tireless attempts at rigorous foundation and its endless examination of basic principles he regarded, more or less, as a waste of time. The piece of popular wisdom, ‘If it ain’t broke, don’t fix it,’ could well define this attitude: a mathematician’s job was to try to prove theorems, not perpetually ponder the status of their unspoken and unquestioned basis.

  In spite of this, however, the passion with which his young visitor spoke had aroused Petros’ curiosity. ‘So, what did this young Mr Gödel prove, that is of such interest to number theorists?’

  ‘He solved the Problem of Completeness,’ Turing announced with stars in his eyes.

  Petros smiled. The Problem of Completeness was nothing other than the quest for a formal demonstration of the fact that all true statements are ultimately provable.

  ‘Oh, good,’ Petros said politely. T have to tell you, however — no offence meant to Mr Gödel, of course — that to the active researcher, the completeness of mathematics has always been obvious. Still, it’s nice to know that someone finally sat down and proved it.’

  But Turing was vehemently shaking his head, his face flushed with excitement. ‘That’s exactly the point, Professor Papachristos: Gödel did not prove it!’

  Petros was puzzled. ‘I don’t understand, Mr Turing … You just said this young man solved the Problem of Completeness, didn’t you?’

  ‘Yes, Professor, but contrary to everybody’s expectation — Hilbert’s and Russell’s included — he solved it in the negative! He proved that arithmetic and all mathematical theories are not complete!’

  Petros was not familiar enough with the concepts of Formal Logic immediately to realize the full implications of these words. ‘I beg your pardon?’

  Turing knelt by his armchair, his finger stabbing excitedly at the arcane symbols filling Gödel’s article. ‘Here: this genius proved — conclusively proved! — that no matter what axioms you accept, a theory of numbers will of necessity contain unprovable propositions!’

  ‘You mean, of course, the false propositions?’

  ‘No, I mean true propositions — true yet impossible to prove!’

  Petros jumped to his feet. ‘This is not possible!’

  ‘Oh yes it is, and the proof of it is right here, in these fifteen pages: “Truth is not always provable!”’

  My uncle now felt a sudden dizziness overcome him. ‘But … but this cannot be.’

  He flipped hurriedly through the pages, striving to absorb in a single moment, if possible, the article’s intricate argument, mumbling on, indifferent to the young man’s presence.

  ‘It is obscene … an abnormality … an aberration …’

  Turing was smiling smugly. ‘That’s how all mathematicians react at first … But Russell and Whitehead have examined Gödel’s proof and proclaimed it to be flawless. In fact, the term they used was “exquisite”.’

  Petros grimaced. ‘“Exquisite”? But what it proves — if it really proves it, which I refuse to believe — is the end ofmathematics!’

  For hours he pored over the brief but extremely dense text. He translated as Turing explained to him the underlying concepts of Formal Logic, with which he was unfamiliar. When they’d finished they took it again from the top, going over the proof step by step, Petros desperately seeking a faulty step in the deduction.

  This was the beginning of the end.

  It was past midnight when Turing left. Petros couldn’t sleep. First thing the next morning he went to see Littlewood. To his great surprise, he already knew of Gödel’s Incompleteness Theorem.

  ‘How could you not have mentioned it even once?’ Petros asked him. ‘How could you know of the existence of something like that and be so calm about it?’

  Littlewood didn’t understand: ‘What are you so upset about, old chap? Gödel is researching some very special cases; he’s looking into paradoxes apparently inherent in all axiomatic systems. What does this have to do with us line-of-combat mathematicians?’

  However, Petros was not so easily appeased. ‘But, don’t you see, Littlewood? From now on, we have to ask of every statement still unproved whether it can be a case of application of the Incompleteness Theorem … Every outstanding hypothesis or conjecture can be a priori undemonstrable! Hilbert’s “in mathematics there is no ignorabimus” no longer applies; the very ground that we stood on has been pulled out from under our feet!’

  Littlewood shrugged. ‘I don’t see the point of getting all worked up about the few unprovable truths, when there are billions of provable ones to tackle!’

  ‘Yes, damn it, but how do we know which is which?’

  Although Littlewood’s calm reaction should have been comforting, a welcome note of optimism after the previous evening’s disaster, it didn’t provide Petros with a definite answer to the one and only, dizzying, terrifying question that had jumped into his mind the moment he’d heard of Gödel’s result. The question was so horrible he hardly dared formulate it: what if the Incompleteness Theorem also applied to his problem? What if Goldbach’s Conjecture was unprovable?

  From Littlewood’s rooms he went straight to Alan Turing, at his college, and asked him whether there had been any further progress in the matter of the Incompleteness Theorem, after Gödel’s original paper. Turing didn’t know. Apparently, there was only one person in the world who could answer his question.

  Petros left a note to Hardy and Littlewood saying he had some urgent business in Munich and crossed the Channel that s
ame evening. The next day he was in Vienna. He tracked his man down through an academic acquaintance. They spoke on the telephone and, since Petros didn’t want to be seen at the university, they made an appointment to meet at the café of the Sacher Hotel.

  Kurt Gödel arrived precisely on time, a thin young man of average height, with small myopic eyes behind thick glasses.

  Petros didn’t waste any time: ‘There is something I want to ask you, Herr Gödel, in strict confidentiality.’

  Gödel, by nature uncomfortable at social intercourse, was now even more so. Ts this a personal matter, Herr Professor?’

  ‘It is professional, but as it refers to my personal research I would appreciate it — indeed, I would demand! — that it remain strictly between you and me. Please let me know;. Herr Gödel: is there a procedure for determining whether your theorem applies to a given hypothesis?’

  Gödel gave him the answer he’d feared. ‘No.’

  ‘So you cannot, in fact, a priori determine which statements are provable and which are not?’

  ‘As far as I know, Professor, every unproved statement can in principle be unprovable.’

  At this, Petros saw red. He felt the irresistible urge to grab the father of the Incompleteness Theorem by the scruff of the neck and bang his head on the shining surface of the table. However, he restrained himself, leaned forward and clasped his arm tightly.

  ‘I’ve spent my whole life trying to prove Goldbach’s Conjecture,’ he told him in a low, intense voice, ‘and now you’re telling me it may be unprovable?’

  Gödel’s already pale face was now totally drained of colour.

  ‘In theory, yes —’

  ‘Damn theory, man!’ Petros’ shout made the heads of the Sacher café’ s distinguished clientèle turn in their direction. ‘I need to be certain, don’t you understand? I have a right to know whether I’m wasting my life!’

  He was squeezing his arm so hard that Gödel grimaced in pain. Suddenly, Petros felt shame at the way he was carrying on. After all, the poor man wasn’t personally responsible for the incompleteness of mathematics — all he had done was discover it! He released his arm, mumbling apologies.

  Gödel was shaking. ‘I un-understand how you fefeel, Professor,’ he stammered, ‘but I-I’m afraid that for the time being there is no way to answer yo-your question.’

  From then on, the vague threat hinted at by Gödel’s Incompleteness Theorem developed into a relentless anxiety that gradually came to shadow his every living moment and finally quench his fighting spirit.

  This didn’t happen overnight, of course. Petros persisted in his research for a few more years, but he was now a changed man. From that point on, when he worked he worked half-heartedly, but when he despaired his despair was total, so insufferable in fact that it took on the form of indifference, a much more bearable feeling.

  ‘You see,’ Petros explained to me, ‘from the first moment I heard of it, the Incompleteness Theorem destroyed the certainty that had fuelled my efforts. It told me there was a definite probability I had been wandering inside a labyrinth whose exit I’d never find, even if I had a hundred lifetimes to give to the search. And this for a very simple reason: because it was possible that the exit didn’t exist, that the labyrinth was an infinity of cul-de-sacs! O, most favoured of nephews, I began to believe that I had wasted my life chasing a chimera!’

  He illustrated his new situation by resorting once again to the example he’d given me earlier. The hypothetical friend who had enlisted his help in seeking a key mislaid in his house might (or again might not, but there was no way to know which) be suffering from amnesia. It was possible that the ‘lost key’ had never existed in the first place!

  The comforting reassurance, on which his efforts of two decades had rested, had, from one moment to the next, ceased to apply, and frequent visitations of the Even Numbers increased his anxiety. Practically every night now they would return, injecting his dreams with evil portent. New images haunted his nightmares, constant variations on themes of failure and defeat. High walls were being erected between him and the Even Numbers, which were retreating in droves, farther and farther away, heads lowered, a sad, vanquished army receding into the darkness of desolate, wide, empty spaces … Yet, the worst of these visions, the one that never failed to wake him trembling and drenched in sweat, was of 2100, the two freckled, dark-eyed, beautiful girls. They gazed at him mutely, their eyes brimming with tears, then slowly turned their heads away, again and again, their features being gradually consumed by darkness.

  The dream’s meaning was clear; its bleak symbolism did not need a soothsayer or a psychoanalyst to decipher it: alas, the Incompleteness Theorem applied to his problem. Goldbach’s Conjecture was a priori unprovable.

  Upon his return to Munich after the year in Cambridge, Petros resumed the external routine he had established before his departure: teaching, chess, and also a minimum of social life; since he now had nothing better to do, he began to accept the occasional invitation. It was the first time since his earliest childhood that preoccupation with mathematical truths didn’t occupy the central role in his life. And although he did continue his research awhile, the old fervour was gone. From then on he spent no more than a few hours a day at it, working half-absently at his geometric method. He’d still wake up before dawn, go to his study and pace slowly up and down, picking his way among the parallelograms of beans laid out on the floor (he had pushed all the furniture against the walls to make room). He picked up a few here, added a few there, muttering absently to himself. This went on for a while and then, sooner or later, he drifted towards the armchair, sat, sighed and turned his attention to the chessboard.

  This routine went on for another two or three years, the time spent daily at this erratic form of ‘research’ continuously decreasing to almost nil. Then, near the end of 1936, Petros received a telegram from Alan Turing, who was now at Princeton University:

  I HAVE PROVED THE IMPOSSIBILITY OF A PRIORI

  DECIDABILITY STOP.

  Exactly: STOP. This meant, in effect, that it was impossible to know in advance whether a particular mathematical statement is provable: if it is eventually proven, then it obviously is — what Turing had managed to show was that as long as it remains unproven, there is absolutely no way of ascertaining whether its proof is impossible or simply very difficult.

  The immediate corollary of this, which concerned Petros, was that if he chose to pursue the proof of Goldbach’s Conjecture, he would be doing so at his own risk. If he continued with his research, it would have to be out of sheer optimism and positive fighting spirit. Of these two qualities, however — time, exhaustion, ill luck, Kurt Gödel and now Alan Turing assisting — he had run out.

  STOP.

  A few days after Turing’s telegram (the date he gives in his diary is 7 December 1936) Petros informed his housekeeper that the beans would no longer be required. She swept them all up, gave them a good wash and turned them into a hearty cassoulet for the Herr Professor’s dinner.

  *

  Uncle Petros remained silent for a while, looking dejectedly at his hands. Beyond the small circle of pale yellow light around us, cast by the single light-bulb, there was now total darkness.

  ‘So that’s when you gave up?’ I asked softly.

  He nodded.’ Yes.’

  ‘And you never again worked on Goldbach’s Conjecture?’

  ‘Never.’

  ‘What about Isolde?’

  My question seemed to startle him. ‘Isolde? What about her?’

  ‘I thought that it was to win her love you decided to prove the Conjecture — no?’

  Uncle Petros smiled sadly.

  ‘Isolde gave me “the beautiful journey”, as our poet says. Without her I might “never have set out”.* Yet, she was no more than the original stimulus. A few years after I had begun my work on the Conjecture her memory faded, she became no more than a phantasm, a bittersweet recollection … My ambitions became of a higher, more
exalted variety.’

  He sighed. ‘Poor Isolde! She was killed during the Allied bombardment of Dresden, along with her two daughters. Her husband, the “dashing young lieutenant” for whom she’d abandoned me, had died earlier on the Eastern Front.’

  The last part of my uncle’s story had no particular mathematical interest:

  In the years that followed history, not mathematics, became the determining force in his life. World events broke down the protective barrier which till then had kept him safe within the ivory tower of his research. In 1938, the Gestapo arrested his housekeeper and sent her to what was still in those days referred to as a ‘work camp’. He didn’t hire anybody to take her place, naively believing that she’d return soon, her arrest due to some ‘misunderstanding’. (After the war’s end he learned from a surviving relative that she’d died in 1943 in Dachau, just a short distance from Munich.) He started to eat out, returning home only to sleep. When he was not at the university he would hang out at the chess club, playing, watching or analysing games.

  In 1939, the Director of the School of Mathematics, by then a prominent member of the Nazi party, indicated that Petros should immediately apply for German citizenship and formally become a subject of the Third Reich. He refused, not for any reasons of principle (Petros managed to go through life unhampered by any ideological burden) but because the last thing he wanted was to be involved once again with differential equations. Apparently, it was the Ministry of Defence that had suggested he apply for citizenship, with precisely this aim in mind. After his refusal he became in essence a persona non grata. In September 1940, a little before Italy’s declaration of war on Greece would have made him an enemy alien subject to internment, he was fired from his post. After a friendly warning, he left Germany.

  Having, by the strict criterion of published work, been mathematically inactive for more than twenty years, Petros was now academically unemployable and so he had to return to his homeland. During the first years of the country’s occupation by the Axis powers he lived in the family house in central Athens, on Queen Sophia Avenue, with his recently widowed father and his newly-wed brother Anargyros (my parents had moved to their own house), devoting practically all his time to chess. Very soon, however, my newborn cousins with their cries and toddler activities became a much greater annoyance to him than the occupying Fascists and Nazis and he moved to the small, rarely used family cottage in Ekali.