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 same 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 cafe 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. 'Is 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 Gödeclass="underline" 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 bis 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 cafe's distinguished clientele 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 fe-feel, 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 2^100, 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.