Promenade Through a Life's Work: The Child and its Mother

1. The Magic of Things

When I was a child I loved going to school. The same instructor taught us reading, writing and arithmetic, singing ( he played upon a little violin to accompany us), the archaeology of prehistoric man and the discovery of fire. I don't recall anyone ever being bored at school. There was the magic of numbers and the magic of words, signs and sounds. And the magic of rhyme, in songs or little poems. In rhyming there appeared to be a mystery that went beyond the words. I believed this until the day on which it was explained to me that this was just a 'trick': all one had to do in making a rhyme was to end two consecutive statements with the same syllable. As it by miracle, this turned ordinary speech into verse. What a revelation! In conversations at home I amused myself for weeks and months in spontaneously making verses. For awhile everything I said was in rhyme. Happily that's past. Yet even today, every now and then, I find myself making poems - but without bothering to search for rhymes when they do not arise spontaneously.

On another occasion a buddy who was a bit older than me, who was already going to the primary school , instructed me in negative numbers. This was another amusing game, yet one which lost its interest more quickly. And then there were crossword puzzles. I passed many a day in making them up, making them more and more complicated. This particular game combined the magic of forms with those of signs and words. Yet this new passion also passed away without a trace.

I was a good student in primary school, in Germany for the first year and then in France, although I wasn't what would be considered 'brilliant'. I became thoroughly absorbed in whatever interested me , to the detriment of all else, without concerning myself with winning the appreciation of the teacher. For my first year of schooling in France, 1940., I was interned with my mother in a concentration camp, at Rieucros, near Mende. It was wartime and we were foreigners - "undesirables" as they put it. But the camp administration looked the other way when it came to the children in the camp, undesirable or not. We came and left more or less as we wished. I was the oldest and the only one enrolled in school. It was 4 or 5 kilometers away, and I went in rain, wind and snow, in shoes if I was lucky to find them, that filled up with water.

I can still recall the first "mathematics essay", and that the teacher gave it a bad mark. It was to be a proof of "three cases in which triangles were congruent ". My proof wasn't the official one in the textbook he followed religiously. All the same, I already knew that my proof was neither more nor less convincing than the one in the book, and that it was in accord with the traditional spirit of "gliding this figure over that one". It was self-evident that this man was unable or unwilling to think for himself in judging the worth of a train of reasoning. He needed to lean on some authority, that of a book which he held in his hand. It must have made quite an impression on me that I can now recall it so clearly. Since that time, up to this very day, I've come to see that personalities like his are not the exception but the rule. I have lots to say about that subject in Récoltes et Semailles. Yet even today I continue to be stunned whenever I confront this phenomenon, as if it were for the first time.

During the final years of the war, during which my mother remained interned, I was placed in an orphanage run by the "Secours Suisse", at Chambon sur Lignon. Most of us were Jews, and when we were warned ( by the local police) that the Gestapo was doing a round-up, we all went into the woods to hide for one or two nights, in little groups of two or three without concerning ourselves overmuch if it was good for our health. This region of the Cévennes abounded with Jews in hiding. That so many survived is due to the solidarity of the local population.

What struck me above all at the "Collège Cévenol" ( where I was enrolled) was the extent to which my fellows had no interest in anything they were learning. As for myself I devoured all of my textbooks right from the beginning of each school year, convinced that this year, at last, we were really going to learn really interesting. Then for the rest of the year I had to figure out ways to employ my time as the program unfolded itself with tedious slowness over the course of the semester. However I should say that there were some really great teachers. Monsieur Friedal our instructor for Biology, was a man of high personal and intellectual qualities. However he was totally incapable of administering discipline, so that his class was in an interminable turmoil. So loud was the ruckus that it was impossible hear his voice rising above the din. No doubt that explains why I didn't become a biologist!

Much of my time, even during my lessons, (shh!..) was spent working on math problems. It wasn't long before the ones I found in the textbook were inadequate for me. This may have been because they all tended to resemble each other; but mostly because I had the impression that they were plucked out of the blue, without any idea of the context in which they'd emerged. They were 'book problems', not 'my problems'. However, there were questions that arose naturally. For example, when the lengths a,b,c of the three sides of a triangle are known, then the triangle itself is determined ( up to its position in space), therefore there ought to be some explicit formula for expressing the area of that triangle as a function of a, b and c. The same had to be true for a tetrahedron when the 6 sides are known: what is its volume? That caused me no little difficulty, but in the end I did derive the formula after a lot of hard work. At any rate, once a problem "grabbed me", I stopped paying attention to the amount of time I had to spend on it, nor of all the other things that were being sacrificed for its sake ( This remains true to this day).

What I found most unsatisfactory in my mathematics textbooks was the absence of any serious attempt to tackle the meaning of the idea of the arc-length of a curve, or the area of a surface or the volume of a solid. I resolved therefore to make up for this defect once I found time to do so. In fact I devoted most of my energy to this when I became a student at the University of Montpellier, between 1945 and 1948. The courses offered by the faculty didn't please me in the least. Although I was never told as much, I'd the impression that the professors had gotten into the habit of dictating from their texts, just like they used to do in the lycée at Mende. Consequently I stopped showing up at the mathematics department, and only did so to keep in touch with the official 'program'. For this purpose the textbooks were sufficient, but they had little to do with the questions I was posing, To speak truthfully, what they lacked was insight , even as the textbooks in the lycée were lacking in insight . Once delivered of their formulae for calculating lengths, areas, volumes in terms of simple, double or triple integrals ( higher dimensions carefully avoided), they didn't care to probe further into the intrinsic meaning of these things. And this was as true of my professors as it was of the books from which they taught.

On the basis of my very limited experience I'd the impression that I was the only person in the entire world who was curious to know the answers to such mathematical questions. That was, at least, my private and unspoken opinion during all those years passed in almost total intellectual isolation, which, I should say, did not oppress me overmuch.(*) I don't think I ever gave any deep thought to trying to find out whether or not I was the only person on earth who considered such things important. My energies were sufficiently absorbed in keeping the promise I'd made with myself: to develop a theory that could satisfy me.

(*)Between 1945 and 1948 my mother and I lived in a small hamlet about a dozen kilometers from Montpellier, named Mairargues ( near Vendargues), surrounded by vineyards. ( My father disappeared in Auschwitz in 1942). We lived marginally on the tiny government stipend guaranteed to college students in France. Each year I participated in the grape harvests ( "vendanges". Translators Note: I worked in these briefly, in the summer of 1970, in the region around Dijon. ). After the harvests there was the gathering up of the loose remains of the grapes in the fields ( grapillage), from which we made a more or less acceptable wine ( apparently illegally) . There was in addition, our garden, which, without having to do much work in it, furnished us with figs, spinach and even (in the late Fall ) tomatoes, which had been planted by a well-disposed neighbor right in the middle of a splendid field of poppies. It was 'the good life' . although a little on the short side when it came to getting a new pair of glasses, or having to wear out one's shoes down to the soles. Luckily my mother, chronically invalided from her long term in the internment camps, had the right to free medical care. There was no way we could have paid for doctors.
I never once doubted that I would eventually succeed in getting to the bottom of things, provided only that I took the effort to thoroughly review the things that came to me about them, and which I took pains to write down in black and white. We have, for example, an undeniable intuition of volume . It had to be the reflection of some deeper reality , which for the moment remained elusive, but was ultimately apprehensible. It was this reality, plain and simple, that had to be grasped - a bit, perhaps, the way that the "magic of rhyme" had been grasped one day in a moment of understanding.

In applying myself to this problem at the age of 17 and fresh out of the lycée, I believed that I could succeed in my objective in a matter of weeks. As it was, it preoccupied me fully three years. It even led me to flunk an examination, during my second year in college - in spherical trigonometry! ( for an optional course on 'advanced astronomy') because of a stupid mistake in arithmetic. (I should confess here that I've always been weak in arithmetic, ever since leaving the lycée .)

Because of this I was forced to remain for a third year at Montpellier to obtain my license(*)

(*)Translator's Note: the basic undergraduate degree in the French university system, not quite the same as our B.A. )
, rather than heading immediately up to Paris - the only place, I was told, where one found people who really knew what was important in modern mathematics. The person who said this to me, Monsieur Soula, also assured me that all outstanding issues in mathematics had been stated and resolved, twenty or thirty years before, by a certain " Lebesgue" ! (Translator's Italics ). In fact he'd developed a theory of integration and measure ( decidedly a coincidence!), beyond which nothing more needed to be said.

Soula, it should be said, was my teacher for differential calculus, a good-hearted man and well disposed towards me. But he did not succeed at all in persuading me to his point of view. I must already have possessed the conviction that Mathematics has no limit in grandeur or depth. Does the sea have a "final end" ? The fact remains that at no point did it occur to me to dig out the book by Lebesgue that M. Soula had recommended to me, which furthermore he himself had never looked at! To my point of view, I could see little connection between what one might find in a book and the work I was doing to convince my own curiosity on issues that perplexed and intrigued me.

Promenade continued

2. The Importance of Solitude

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