From an Uneven Life, a Theory of Roughness

As the celebrated mathematician writes his memoirs, he reflects on the combination of good luck, hard luck, and constant dreaming that made his life a success.
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TRANSCRIPT

Question: As you write your memoirs, which memories are the most fun and the most difficult to look back on? 

Benoit Mandelbrot: Well, my life has been extremely complicated.  Not by choice at the beginning at all, but later on, I had become used to complication and went on accepting things that other people would have found too difficult to accept.  I was born in Poland and moved to France as a child shortly before World War II.  During World War II, I was lucky to live in the French equivalent of Appalachia, a region which is sort of not very high mountains, but very, very poor, and Appalachia we are poorer even, so poorer than Appalachia of the United States.  And for me, I was in high school where things were very easy.  It was a small high school way up in the hills and had mostly a private intellectual life.  I read many books; there were many books, a very good library.  I had many books and I had dreams of all kinds.  Dreams in which were in a certain sense, how to say, easy to make because the near future was always extremely threatening.  It was a very dangerous period.  But since I had nothing to lose, I was dreaming of what I could do. 

Then the war ended.  I had very, very little training in taking an exam to determine a scientist’s life in France.  There were two schools, both very small.  One tiny, and one small, which in a certain sense was the place that I was sure I wanted to go.  I had only a few months of finding out how the exam proceeded, but I took the exam and perhaps because of inherited gifts, I did very well. In fact, I barely missed being number one in France in both schools.  In particular I did very well in mathematical problems.  The physics I could not guess, other things I could not guess.  But then I had a big choice, should I go into mathematics in a small and **** school.  Or should I go to a bigger school in which, in a certain sense would give me time to decide what I wanted to do? 

First I entered the small school where I was, as a matter of fact, number one of the students who entered then.  But immediately, I left because that school, again, was going to teach me something which I did not fully believe, namely mathematics separate from everything else.  It was excellent mathematics, French mathematics was very high level, but in everything else it was not even present.  And I didn’t want to become a pure mathematician, as a matter of fact, my uncle was one, so I knew what the pure mathematician was and I did not want to be a pure – I wanted to do something different.  Not less, not more but different.  Namely, combine pure mathematics at which I was very good, with the real world of which I was very, very curious. 

And so, I did not go to École Polytechnique.  It was a very rough decision, and the year when I took this decision remembers my memory very, very strongly.  Then for several years, I just was lost a bit.  I was looking for a good place.  I spent my time very nicely in many ways, but not fully satisfactory.  Then I became Professor in France, but realized that I was not – for the job that I should spend my life in.  Fortunately, IBM was building a research center, I went there for a summer thing, for a summer only.  I knew this summer, decided to stay.  It was a very big gamble.  I lost my job in France, I received a job in which was extremely uncertain, how long would IBM be interested in research, but the gamble was taken and very shortly afterwards, I had this extraordinary fortune of stopping at Harvard to do a lecture and learning about the price variation in just the right way.  At a time when nobody was looking, was realizing that either one needed, or one could make a theory of price variation other than the theory of 1900 at which Bachelier had proposed, which was very, very far from being representative of the actual thing. 

So, I went to IBM and I was fortunate in being allowed – to be successful as to go from field to field, which in a way was what I had been hoping for.  I didn’t feel comfortable at first with pure mathematics, or as a professor of pure mathematics.  I wanted to do a little bit of everything and explore the world.  And IBM let me do so.  I touched on far more topics than anybody would have found reasonable.  I was often told, “Settle down, stay in one field, don’t go all the time to another field.”  But I was just compelled to move from one thing to another.  

And fractal geometry was not an idea which I had early on, for something was developed progressively.  I didn’t choose to go into the topic because of any compelling reason, but because the problems there seemed to be somehow similar to the ones I knew how to handle.  I had experienced this kind of problem and gradually realized that I was truly putting together a new theory.  A theory of roughness.  What is roughness?  Everybody knows what is roughness.  When was roughness discovered?  Well, prehistory.  Everything is roughness, except for the circles.  How many circles are there in nature?  Very, very few.  The straight lines.  Very shapes are very, very smooth.  But geometry had laid them aside because they were too complicated.  And physics had laid them aside because they were too complicated.  One couldn’t even measure roughness.  So, by luck, and by reward for persistence, I did found the theory of roughness, which certainly I didn’t expect and expecting to found one would have been pure madness. 

So, one of the high points of my life was when I suddenly realized that this dream I had in my late adolescence of combining pure mathematics, very pure mathematics with very hard things which had been long a nuisance to scientists and to engineers, that this combination was possible and I put together this new geometry of nature, the fractal geometry of nature.

Recorded on February 17, 2010
Interviewed by Austin Allen