Uncle Petros and Goldbach

With these words, my adolescent admiration for him as Ideal Romantic Hero was rekindled. But now it was marked by large doses of realism.

'So, Uncle, it was really a question of all or nothing, eh?'

He nodded slowly. 'You could put it that way, yes.'

'And was this the end of your creative life? Did you ever again work on Goldbach's Conjecture?'

He gave me a surprised look. 'Of course I did! In fact it was after that I did my most important work.' He smiled. 'We'll come to that by and by, dear boy. Don't worry, in my story there shall be no ignorabimus!’

Suddenly he laughed loudly at his own joke, too loudly for comfort, I thought. Then he leaned towards me and asked me in a low voice: 'Did you learn Gödel's Incompleteness Theorem?'

'I did,' I replied, 'but I don't see what it has to do with -'

He lifted his hand roughly, cutting me short.

' "Wir müssen wissen, wir werden wissen! In der Mathematik gibt es kein ignorabimus" ' he declaimed stridently, so loudly that his voice echoed against the pine trees and returned, to menace and haunt me. Sammy's theory of insanity instantly flashed through my mind. Could all this reminiscing have aggravated his condition? Could my uncle have finally become unhinged?

I was relieved when he continued in a more normal tone: '"We must know, we shall know! In mathematics there is no ignorabimus!” Thus spake the great David

Hubert in the International Congress, in 1900. A proclamation of mathematics as the heaven of Absolute Truth. The vision of Euclid, the vision of Consistency and Completeness…'

Uncle Petros resumed his story.

The vision of Euclid had been the transformation of a random collection of numerical and geometric observations into a well-articulated system, where one can proceed from the a priori accepted elementary truths and advance, applying logical operations, step by step, to rigorous proof of all true statements: mathematics as a tree with strong roots (the Axioms), a solid trunk (Rigorous Proof) and ever growing branches blooming with wondrous flowers (the Theorems). All later mathematicians, geometers, number theorists, algebraists, and more recently analysts, topologists, algebraic geometers, group-theorists, etc., the practitioners of all the new disciplines that keep emerging to this day (new branches of the same ancient tree) never veered from the great pioneer's course: Axioms-Rigorous Proof-Theorems.

With a bitter smile, Petros remembered the constant exhortation of Hardy to anyone (especially poor Ramanujan, whose mind produced them like grass on fertile soil) bothering him with hypotheses: 'Prove it! Prove it!' Indeed, Hardy liked saying, if a heraldic motto were needed for a noble family of mathematicians, there could be no better than Quod Erat Demonstrandum.

In 1900, during the Second International Congress of Mathematicians, held in Paris, Hubert announced that the time had come to extend the ancient dream to its ultimate consequences. Mathematicians now had at their disposal, as Euclid had not, the language of Formal Logic, which allowed them to examine, in a rigorous way, mathematics itself. The holy trinity of Axioms-Rigorous Proof-Theorems should hence be applied not only to the numbers, shapes or algebraic identities of the various mathematical theories but to the very theories themselves. Mathematicians could at last rigorously demonstrate what for two millennia had been their central, unquestioned credo, the core of the vision: that in mathematics every true statement is provable.

A few years later, Russell and Whitehead published their monumental Principia Mathematica, proposing for the first time a totally precise way of speaking about deduction, Proof Theory. Yet although this new tool brought with it great promise of a final answer to Hilbert's demand, the two English logicians fell short of actually demonstrating the critical property. The 'completeness of mathematical theories' (i.e. the fact that within them every true statement is provable) had not yet been proven, but there was now not the smallest doubt in anybody's mind or heart that one day, very soon, it would be. Mathematicians continued to believe, as Euclid had believed, that they dwelt in the Realm of Absolute Truth. The victorious cry emerging from the Paris Congress, 'We must know, we shall know, in Mathematics there is no ignorabimus,' still constituted the one unshakable article of faith of every working mathematician.

I interrupted this rather exalted historical excursion: 'I know all this, Uncle. Once you enjoined me to learn Gödel's theorem I obviously also had to find out about its background.'

'It's not the background,' he corrected me; 'it's the psychology. You have to understand the emotional climate in which mathematicians worked in those happy days, before Kurt Gödel. You asked me how I mustered up the courage to continue after my great disap-pointment. Well, here's how…'

Despite the fact that he hadn't yet managed to attain his goal and prove Goldbach's Conjecture, Uncle Petros firmly believed that his goal was attainable. Being himself Euclid's spiritual great-grandson, his trust in this was complete. Since the Conjecture was almost certainly valid (nobody with the exception of Ramanujan and his vague 'hunch' had ever seriously doubted this), the proof of it existed somewhere, in some form.

He continued with an example:

'Suppose a friend states that he has mislaid a key somewhere in his house and asks you to help him find it. If you believe his memory to be faultless and you have absolute trust in his integrity, what does it mean?'

'It means that he has indeed mislaid the key somewhere in his house.'

'And if he further ascertains that no one else entered the house since?'

'Then we can assume that it was not taken out of the house.'

'Ergo?'

'Ergo, the key is still there, and if we search long enough – the house being finite – sooner or later we will find it.'

My uncle applauded. 'Excellent! It is precisely this certainty that fuelled my optimism anew. After I had recovered from my first disappointment I got up one fine morning and said to myself: "What the hell – that proof is still out there, somewhere!"'

'And so?'

'And so, my boy, since the proof existed, one had but to find it!'

I wasn't following his reasoning.

'I don't see how this provided comfort, Uncle Petros: the fact that proof existed didn't in any way imply that you would be the one to discover it!'

He glared at me for not immediately seeing the obvious. 'Was there anyone in the whole wide world better equipped to do so than I, Petros Papachristos?'

The question was obviously rhetorical and so I didn't bother to answer it. But I was puzzled: the Petros Papachristos he was referring to was a different man from the self-effacing, withdrawn senior citizen I'd known since childhood.

Of course, it had taken him some time to recover from reading Hardy's letter and its disheartening news. Yet recover he eventually did. He pulled himself together and, his deposits of hope refilled through the belief in 'the existence of the proof somewhere out there', he resumed his quest, a slightly changed man. His misadventure, by exposing an element of vanity in his manic search, had created in him an inner core of peace, a sense of life continuing irrespective of Goldbach's Conjecture. His working schedule now became slightly more relaxed, his mind also aided by interludes of chess, more tranquil despite the constant effort.

In addition, the switch to the algebraic method, already decided in Innsbruck, made him feel once again the excitement of a fresh start, the exhilaration of entering virgin territory.

For a hundred years, from Riemann's paper in the mid-nineteenth Century, the dominant trend in Number Theory had been analytic. By now resorting to the ancient, elementary approach, my uncle was in the vanguard of an important regression, if I may be allowed the oxymoron. The historians of mathematics will do well to remember him for this, if for no other part of his work.