Apostolos Doxiadis - Uncle Petros and Goldbach

He seemed to consider this information awhile and then shrugged. 'It's an important decision,’ he said, 'not to be taken without serious deliberation. Why don't you come here one afternoon and we'll talk about it.' Then he added, unnecessarily. 'It's better if you don't tell your father.'

I went a few days later, as soon as I could arrange a good cover story.

Uncle Petros led me to the kitchen and offered me a cold drink made from the sour cherries from his tree. Then he took a seat across from me, looking solemn and professorial.

'So tell me,' he asked, 'what is mathematics in your opinion?' The emphasis on the last word seemed to carry the implication that whatever answer I gave was bound to be wrong.

I spurted out commonplaces about 'the most supreme of sciences' and the wonderful applications in electronics, medicine and space exploration.

Uncle Petros frowned. 'If you're interested in applications why don't you become an engineer? Or a physicist. They too are involved with some sort of mathematics.'

Another emphasis with meaning: obviously he didn't hold this 'sort' in very high esteem. Before I embarrassed myself further, I decided that I was not equipped to spar with him as an equal, and confessed it.

'Uncle, I can't put the "why" into words. All I know is that I want to be a mathematician – I thought you'd understand.'

He considered this for a while and then asked: 'Do you know chess?'

'Sort of, but please don't ask me to play; I can tell you right now I'm going to lose!'

He smiled. 'I wasn't suggesting a game; I just want to give you an example that you'll understand. Look, real mathematics has nothing to do with applications, nor with the calculating procedures that you learn at school. It studies abstract intellectual constructs which, at least while the mathematician is occupied with them, do not in any way touch on the physical, sensible world.'

"That's all right with me,’ I said.

'Mathematicians,’ he continued, 'find the same enjoyment in their studies that chess players find in chess. In fact, the psychological make-up of the true mathematician is closer to that of the poet or the musical composer, in other words of someone concerned with the creation of Beauty and the search for Harmony and Perfection. He is the polar opposite of the practical man, the engineer, the politician or the he paused for a moment seeking something even more abhorred in his scale of values -'… indeed, the businessman.'

If he was telling me all this in order to discourage me, he had chosen the wrong route.

'That's what I'm after too, Uncle Petros,’ I responded excitedly. 'I don't want to be an engineer; I don't want to work in the family business. I want to immerse myself in real mathematics, just like you… just like Goldbach's Conjecture!'

I'd blown it! Before I'd left for Ekali I had decided that any reference to the Conjecture should be avoided like the devil during our conversation. But in my carelessness and excitement I'd let it slip out.

Although Uncle Petros remained expressionless, I noticed a slight tremor run down his hand.' Who's spoken to you about Goldbach's Conjecture?' he asked quietly.

'My father,' I murmured.

'And what did he say, precisely?'

'That you tried to prove it.'

'Just that?'

'And… and that you didn't succeed.'

His hand was steady again. 'Nothing else?'

'Nothing else.'

'Hm,’ he said. 'Suppose we make a deal?'

'What sort of a deal?'

'Listen to me: the way I see things, in mathematics as in the arts – or in sports, for that matter – if you're not the best, you're nothing. A civil engineer, or a lawyer, or a dentist who is merely capable may yet lead a creative and fulfilling professional life. However, a mathematician who is just average – I'm talking about a researcher, of course, not a high-school teacher – is a living, walking tragedy…'

'But Uncle,’ I interrupted, 'I haven't the slightest intention of being "just average". I want to be Number One!'

He smiled. 'In that at least you definitely resemble me. I too was overambitious. But you see, dear boy, good intentions are, alas, not enough. This is not like many other fields where diligence always pays. To get to the top in mathematics you also need something more, the absolutely necessary condition for success.'

'Which one is that?'

He gave me a puzzled look, for ignoring the obvious.

'Why, the talent! The natural predisposition in its more extreme manifestation. Never forget it: Mathematicus nascitur, non fit – A mathematician is born, not made. If you don't carry the special aptitude in your genes, you will labour in vain all your life and one day you will end up a mediocrity. A golden mediocrity, perhaps, but a mediocrity nevertheless!'

I looked him straight in the eye.

'What's your deal, Uncle?'

He hesitated for a moment, as if thinking it over. Then he said: 'I don't want to see you following a course that will lead to failure and unhappiness. Therefore I'm proposing that you will make a binding promise to me to become a mathematician if and only if you're supremely gifted. Do you accept?'

I was disconcerted. 'But how on earth can I determine that, Uncle?'

'You can't and you don't need to,' he said with a sly little smile. 'I will.'

'You?'

'Yes. I will set you a problem, which you will take home with you and attempt to solve. By your success, or failure, I will measure your potential for mathematical greatness with great accuracy.'

I had mixed feelings for the proposed deal: I hated tests but adored challenges.

'How much time will I have?' I asked. Uncle Petros half-closed his eyes, considering this. 'Mmm… Let's say till the beginning of school, the first of October. That gives you almost three months.'

Ignorant as I was, I believed that in three months I could solve not one but any number of mathematical problems. 'That much!'

'Well, the problem will be difficult,’ he pointed out. 'It's not one just anybody can solve, but if you've got what it takes to become a great mathematician, you will manage. Of course, you will swear that you will seek help from no one and you will not consult any books.' 'I swear,’ I said. He fixed his stare on me. 'Does that mean you accept the deal?'

I heaved a deep sigh. 'I do!'

Without a word, Uncle Petros disappeared briefly and returned with paper and pencil. He now became businesslike, mathematician to mathematician.

'Here's the problem… I assume you already know what a prime number is?'

'Of course I know, Uncle! A prime is an integer greater than 1 that has no divisors other than itself and unity. For example 2,3,5,7,11,13, and so on.'

He appeared pleased with the precision of my definition. 'Wonderful! Now tell me, please, how many prime numbers are there?'

I suddenly feit out of my depth. 'How many?'

'Yes, how many. Haven't they taught you that at school?'

'No.'

My uncle sighed a deep sigh of disappointment at the low quality of modern Greek mathematical education.

'All right, I will tell you this because you will need it: the primes are infinite, a fact first proven by Euclid in the third century BC. His proof is a gem of beauty and simplicity. By using reductio ad absurdum, he first assumes the contrary of what he wants to prove, namely that the primes are finite. So…'

With fast vigorous jabs at the paper and a few explanatory words Uncle Petros laid out for my benefit our wise ancestor's proof, also giving me my first example of real mathematics.

'… which, however,’ he concluded, 'is contrary to our initial assumption. Assuming finiteness leads to a contradiction; ergo the primes are infinite. Quod erat demonstrandum.'

'That's fantastic, Uncle,’ I said, exhilarated by the ingeniousness of the proof. 'It's so simple!'