Uncle Petros and Goldbach

He wasn't famous, he wasn't even well known and, except at his almost nightly appearances at the university chess club, he hadn't exchanged two words with anyone other than Hardy and Littlewood during his stay.

'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.

Td 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 Conjecrure, 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, overspecialization. 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 urmecessary. Its tireless attempts at rigorous foundation and its endless examination of basic prindples 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. 'I 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 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?