by Miriam Sved
A great many of us, perhaps twelve or fourteen, had met at the streetcar. I was nervous about this, afraid that we would be detained for congregating in a group, and the talk on the journey into the hills kept turning to risky subjects: there seemed to be a value, among the men, in having been detained by the police; a few of them claimed to have spent time in the cells, and they boasted about their membership of underground organisations and the interrogations they had faced in a way that filled me with horror. A few of them claimed to be communists.
When we got up into the hills things were better. We set off along a track threading between the beech trees. I walked beside Ildiko and felt freed of the watchfulness and social constrictions of the university. Conversation seemed freer as well, less dominated by the same two or three voices as we all spread out along the path, and it turned from politics to mathematics. Naturally, in our university, most of the group were studying in the sciences. A familiar conversation started about practical applications for our work: I had heard the arguments before and was not paying close attention. A boy ahead of us, a boy named Sámuel – well dressed and with a sharply handsome face – was talking about Hilbert space and shaped charges: something about developments in Germany bringing mathematics and physics closer together, a new hybrid field, no more pottering around in Peano arithmetic and useless number theory.
Ildiko, walking beside me, said, ‘But who are we to know what will and won’t be ultimately useful? I suppose the Greeks thought they had stumbled upon a charming but perfectly abstract pattern when they discovered the ellipse, and only thousands of years later it turns out they were charting the passage of the planets.’
I thought she was very brave, speaking up so confidently in the large group of men. But that is Ildiko.
Sámuel laughed in an indulgent way. ‘I will give you that example,’ he said. ‘But it is an exceptional one, only a special case in geometry that turned out to have unexpected applications.’
‘All right.’ Ildiko returned his laugh, but there was a splinter of hardness in hers. ‘If you want an unexpected application of pure number theory, what about e? I suppose Euler wasn’t trying to predict the rate of growth and decay or Newton’s laws.’
‘Yes, all right, these examples are brilliant,’ Sámuel said, ‘but they are descriptive applications only. The hyperbola describes the rise and descent of heavenly bodies, e describes the rate of growth and decay. I’m talking about a mathematics that works directly on the world. About calculating the hydrodynamics of an implosion lens, the ability to start a uranium chain reaction. This is work that might change the whole world in a moment.’
Ildiko started to say something in reply but Tibor, who was walking behind us, said, ‘What you are describing might produce effects, but that is not the same thing as being a productive branch of mathematics. We should use our energies constructively, for the good of the people of the world, rather than adding to the destructive power of governments.’ At that time I knew him by sight only as the imposing, earnest engineering student with whom Ildiko had corresponded over the years, but when he spoke up in the large group he had the authority of someone used to being listened to.
Sámuel looked back over his shoulder. We were on an ascent and I noticed that he was flushed and the hair around his temples was sticking to his head. ‘The destructive power of governments?’ he said, with a bead of something contemptuous in his voice, and I felt a slight bristling of anxiety. For all the boasting about underground political activity, there were always those who would curry favour with the authorities by turning in dissidents.
Ildiko said, ‘I agree with Tibor. We should not fall in with whatever branch of mathematics makes the most noise in the world, without even knowing the likely repercussions. We have a higher responsibility than that.’
‘Hear, hear; well spoken,’ said a boy walking beside Tibor – a very good-looking boy, sandy-haired and square-shouldered with a face that was all striking planes.
Sámuel started to respond and I sensed that the argument was becoming entrenched, and I could have wished to be walking quietly and only enjoying the beauty of the autumn day and the hills. Then, from a little further behind us, there came a loud laugh, just a single note: Ha! I turned to see who it was. A slim, wiry boy, with rimmed spectacles and a wild head of hair that branched at odd angles. He did the barking laugh again: Ha!
Sámuel twisted around as well. ‘Is something funny?’
The boy said, ‘Only that you think you are discussing mathematics.’
There was silence for a few paces and it became quite hard to follow the path without losing my footing while looking from Sámuel to the boy with the hair.
‘Don’t misunderstand me,’ he continued. ‘I’m sure those problems, particle fusion and so forth, are quite clever and diverting. But they are mostly mechanics and computation. There is a simple test: if your problem arises from the materials of the physical world, if you need machines or particles or anything other than numbers and perhaps a pencil, then it cannot be called true mathematics. You should do it or not do it as you wish, but do not call it that.’ His words were arrogant but he didn’t sound unfriendly.
Sámuel stopped walking and turned, halting us and everyone behind. He looked the small boy up and down quite overtly, and I found myself examining him too. I had seen him in lectures – memorable because of the hair, and because he tended to ask questions of our lecturers: complex, probing questions that I often failed to follow. Sámuel said, ‘Funny that you should speak about true mathematics. If I understand correctly, you are not even a student of mathematics of any kind at all.’
‘Do you mean because I am not enrolled?’ the boy said. ‘Why is that funny?’ It did not seem to be a rhetorical question; he looked interested and a little perplexed.
‘You are not even …’ Sámuel started, and then brandished his hands in the air, as though conjuring some self-evident truth. ‘You do not even have a place at the university, you have no right to be here among us making statements like that.’
‘Do I need to be enrolled in the university to have a brain that works? Do you only think about things when your lecturers tell you to?’
Again I had the impression that he wasn’t trying to be provocative; he sounded genuinely curious. But there was a mottled redness working its way up Sámuel’s neck, and he stepped towards the smaller boy in a way that looked threatening. Ildiko, beside me, let out an incredulous little laugh; she must have been thinking the same thing as me: were we really going to have a physical confrontation, this group of students on a gentle hike through the Buda hills?
The next thing I knew, Tibor had stepped in front of Sámuel. In a level voice, he said, ‘You are right that my friend Pali is not enrolled in the diploma. But he has as much right to be here as any of us.’
‘Thank you, Tibor,’ said Pali. And then, ‘Though why anyone would not have a right to be here is a mystery to me. We are on public land.’
I saw Tibor roll his eyes very slightly. The tension was still high, Sámuel with his hands hanging wide at his sides. He said, ‘If he is your friend, Tibor, then I will not humiliate him. But I don’t see why I should have to put up with that kind of insolence from someone who is not even a member of the university.’
Tibor said, ‘Have you heard of uncountable compactness?’
‘Of course,’ Sámuel said loftily.
‘That was Pali’s.’
At this everybody – not just Sámuel and Tibor but the little group who had walked behind us and been stopped by the altercation on the path, perhaps six or seven in all – turned to look at Pali. I had also heard of the work – it had created a stir some years earlier when it came to light that someone had proved an uncountable case of Gödel’s compactness theorem. There had been an air of mystery surrounding the provenance of the work, which was a joint publication between one of the high-standing members of th
e university and someone else, someone no-one had heard of.
‘You’re Pali Kalmar?’ Sámuel said, the redness still creeping up his neck.
‘That is my name,’ the boy said. His hands moved compulsively through the air when he spoke, and there was something jerky and strange about his face.
‘Did you really disprove Gödel?’ Sámuel asked.
‘Of course,’ with a wave of his hand. ‘Although the last steps were a little too tedious and obvious to bother with.’ (Typical Pali, I now know: this failure to follow through with the grunt work was what robbed him of full credit and authorship for the proof.)
Sámuel still looked forbidding, but he gave a peremptory nod to the smaller boy. He began to say something but seemed to think better of it, and then he turned and continued along the path and the little confrontation was over.
It was over but it changed everything. We were not, after that, an undifferentiated mass. I felt a sense of the personalities in the group emerging: Sámuel and his clique of loud, smoothly confident men – all quite well off, as I later learned, many of their families with the kind of influence that could circumvent the numerus clausus. And Tibor, whom I already knew of through Ildiko, and now knew as the defender of that thin, strange figure whom everyone knew by reputation, Pali Kalmar.
Ildiko and I found ourselves walking with Tibor and his handsome friend, who introduced himself as Levi. Levi turned his focus completely on to us, asking what we thought of this or that play, of the colourful Hollywood movies and the earnest Hungarian ones, of the latest talk at the Americana and the Balázs. He seemed genuinely fascinated by our opinions of these things, but there was something light and satirical about him: he addressed me and Ildiko with a formality that seemed self-mocking but also gallant, some resonance of the old empire that Nagymama used to talk of from before the war. I remember at one point he handed Ildiko up onto a rocky platform beside the path – from which there was a beautiful view across Buda – and when she was beside him he lifted her hand quickly to his mouth and kissed the palm. Ildiko raised her eyebrows at me, incredulous and amused. They made a striking couple, pictured there against the skyline, but it didn’t occur to me that they would come together. Ildiko didn’t seem to take him seriously enough.
I found that the four of us – Ildiko and I, Tibor and Levi – had slightly detached from the larger group. The other boy, Pali Kalmar, was there too but adhering loosely, a circling electron. He would pick up speed and move a little way ahead, and wait for us without appearing to do so, examining some flower by the path. Then he would walk behind us for a while, offering occasional non sequiturs. (Once he fell in step next to me and said, ‘Your name is Eszter, isn’t it? Epszi, from the Greek epsilon, a small positive quantity.’) The others pulled away ahead of us and their chatter sounded remote, the hills all around us peaceful and bright, the woods opening up to meadows flooded with primroses. Ildiko linked her arm through mine and we walked in silence for a while.
Pali broke the spell. He said, ‘Do you know, I think I have found a new simple proof for Bertrand’s postulate.’
I laughed, not yet used to his abrupt mathematical leaps, but Tibor answered as though he had been thinking of that exact topic. ‘Have you? By what method?’
Without paper or pencil, as though the moss-smelling hills were the most natural workplace in the world, Pali started to draw out the proof for us, moving his hands in illustration of each step. I have never been much good at following workings without notation; there were many steps in Pali’s proof that I could not quite hold on to. But still there was something revelatory about it, even then: listening to him talk about numbers was like hearing someone speak about their close family, loved ones full of quirk and charm. I wish I could capture it, but I have never known a number so well. His voice was nasal but strangely compelling, almost hypnotic, and we drifted along with the proof expanding around us in the open air. When he came to the end (and so there is no counterexample for the postulate) there was silence for a few moments. Then Tibor said, ‘If you have this right, and I think that you do, you must publish it.’
Pali was a couple of paces ahead of us. He turned with eyebrows raised, face open and guileless as a child. ‘Why?’
‘Why?’ Levi said, and laughed. ‘He has a proof that would make his name – you could get your doctorate on very little more, you know – and he asks us why?’
‘Doctorate!’ said Pali incredulously, as though he had never considered anything so absurd.
Tibor said, ‘And because others should have the benefit of understanding this and of building on your work.’
‘Do you know,’ Pali said, ‘that Socrates never published a word in his lifetime? There is not a jot of writing that exists from his hand.’
Ildiko said, ‘But Socrates had Plato to do it for him.’
‘Then perhaps I must find myself a Plato.’
One of them, I’m sure, would have argued the point further, but just then the track gave way to a clearing and we came upon the others, who had stopped and were settling to eat lunch. There were outcrops of rock here and there, and some of our classmates had chosen one or other of them to sit on. Others lounged on the soft bracken-y ground. A boy was walking through the group taking photographs with a little Model A. He turned to the five of us where we stood looking around the clearing. ‘Smile, you lot.’
As he raised the camera Pali dived to hide behind us in a practised manoeuvre but Tibor, equally quick, grabbed him by the arm and said, ‘No you don’t,’ and to the rest of us: ‘He is notorious for this.’
‘It will steal my soul,’ Pali said theatrically. But he relented quickly and then set about stage-managing the rest of us, Ildiko and I, Tibor and Levi arranged against the smooth side of one of the large crags, himself on the ground at our feet. The sun was bright overhead and we were all squinting slightly when the camera shutter lowered, but still – unusual in a group photograph – we are all smiling, all looking relaxed and leaning close together. I remember the feeling that the camera had not just captured the moment but made it substantial somehow, its little click binding us into a unit, a kind of understanding moving between us all. I felt that my life was knitting itself around me; it was one of those moments that one would like to crawl inside of to know it again, experience it more fully than the present ever allows.
Was I already in the grip of my Pali problem then? I can’t remember; there are only fleeting impressions: of his hair, the surprising way he talked. I remember feeling a little uneasy when he turned his full attention on to me, as though I had stepped into a different element without quite realising it. But there were so many impressions from that brightly lit day, impressions to be dwelled on later in the evening and kept hidden from Mama, whose questions, as always, centred on whether there were any eligible men in the group. I didn’t tell Mama that I had had my first surprised inkling that the tall, impressive engineering student – Tibor – had noticed me as something other than Ildiko’s friend. It was on the way down the hill track, as I stepped clumsily off a rock and almost lost my footing, and found Tibor’s hand supporting me, and he smiled at me. I was surprised when he kept his hand where it was for a few paces, and then withdrew it awkwardly. Ildiko, walking on my other side, saw the moment pass and raised her eyebrows at me.
I do remember that it was Pali, on our way down the track, who made me happy by suggesting – demanding really – that the five of us meet again later in the week. It was also Pali’s suggestion that we make the statue of Anonymous at the front of the castle in the City Park our meeting place. ‘It is usually quiet there,’ he said. ‘Have you noticed? I think people don’t like the statue because it is hooded. They do not understand why anyone would cover his face and not seek fame and recognition; it makes them suspicious of his motives.’
Still holding the photograph, the light beginning to fade outside our carriage, I let my eyes rest on Pali’
s real-life face for a moment. He had his eyes half closed and his head tipped back. He looked unusually tired, and his clothes – in fact all his atoms – seemed to have defaulted to their natural state of chaos: hair reaching for its wild peaks and angles, vest crumpled and some stain on his lapel that didn’t seem to correlate with anything we had eaten that day. He turned to meet my eyes. The way Pali manages eye contact is not quite like other people, although it is hard to name the difference. There is a directness to his gaze, an added intensity, that I think often makes people uncomfortable; he does not look away after the socially appropriate portion of time. We looked at each other and I felt that all the shocks and upheavals of the day were there between us: those poor people scrubbing the cobblestoned street, the young boys jeering happily, Judenschwein. Broken glass and splashes of red paint.
I felt a shiver of something as we kept looking at each other, a tremor not unlike the shaky chill that had overcome me earlier; there was fear in it but something else as well. I looked at the downward curve of his lips and thought: what would he do? Has anybody ever kissed him? The space between us was tremulous, his eyes steady on me but his face unmoving. I would have to be the one to cross the space if anything were to happen, and that distance suddenly seemed teeming with the infinite, a living embodiment of Zeno’s paradox that I could halve and halve and halve and never get to him. I leaned in; he stayed inert. I was sure there was a thickening in the intensity of his gaze. I needed something to throw out across the void, some intimacy more possible than the physical. Without any awareness of my own intentions I said, ‘Has Ildiko showed you the new problem we are working on? An upper limit problem – it combines geometry and combinatorics.’