Apostolos Doxiadis - Uncle Petros and Goldbach

'So that's the explanation,' I murmured, as he sat down.

'What explanation?' he asked absently.

I told him of Sammy Epstein and his failure to find any mention of the name Petros Papachristos in the bibliographical index for Number Theory, with the exception of the early joint publications with Hardy and Littlewood on the Riemann Zeta Function. I repeated the 'burnout theory' suggested to my friend by the 'distinguished professor' at our university: that his supposed occupation with Goldbach's Conjecture had been a fabrication to disguise his inactivity.

Uncle Petros laughed bitterly.

'Oh no! It was true enough, most favoured of nephews! You can tell your friend and his "distinguished professor" that I did indeed work on trying to prove Goldbach's Conjecture – and how and for how longl Yes, and I did get intermediate results – wonderful, important results – but I didn't publish them when I should have done and others got in there ahead of me. Unfortunately, in mathematics there's no silver medal. The first to announce and publish gets all the glory. There's nothing left for anyone eise.' He paused.

'As the saying goes, a bird in the hand is worth two in the bush and I, while pursuing the two, lost the one…'

Somehow I didn't think the resigned serenity with which he stated this conclusion was sincere.

'But, Uncle Petros,’ I asked him, 'weren't you horribly upset when you heard from Hardy?'

'Naturally I was – and "horribly" is exactly the word. I was desperate; I was overcome with anger and frustration and grief; I even briefly contemplated suicide. That was back then, however, another time, another seif. Now, assessing my life in retrospect, I don't regret anything I did, or did not do.'

'You don't? You mean you don't regret the opportunity you missed to become famous, to be acknowledged as a great mathematician?'

He lifted a warning finger. 'A very good mathematician perhaps, but not a great one! I had discovered two good theorems, that's all.'

"That's no mean achievement, surely!'

Uncle Petros shook his head. 'Success in life is to be measured by the goals you've set yourself. There are tens of thousands of new theorems published every year the world over, but no more than a handful per century that make history!'

'Still, Uncle, you yourself say your theorems were important.'

'Look at the young man,' he countered, 'the Austrian who published my – as I still think of it – Partitions Theorem before me: was he raised with this result to the pedestal of a Hubert, a Poincare? Of course not! Perhaps he managed to secure a small niche for his portrait, somewhere in a back room of the Edifice of Mathematics… but if he did, so what? Or, for that matter, take Hardy and Littlewood, top-class mathematicians both of them. They possibly made the Hall of Fame – a very large Hall of Farne, mind you – but even they did not get their statues erected at the grand entrance alongside Euclid, Archimedes, Newton, Euler, Gauss… That had been my only ambition and nothing short of the proof of Goldbach's Conjecture, which also meant cracking the deeper mystery of the primes, could possibly have lead me there…’ There was now a gleam in his eyes, a deep, focused intensity as he concluded: ‘I, Petros Papachristos, never having published anything of value, will go down in mathematical history – or rather will not go down in it – as having achieved nothing. This suits me fine, you know. I have no regrets. Mediocrity would never have satisfied me. To an ersatz, footnote kind of immortality, I prefer my flowers, my orchard, my chessboard, the conversation I'm having with you today. Total obscurity!'

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.