Apostolos Doxiadis - Uncle Petros and Goldbach

'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 of mathematics!’

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 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 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 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.