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Benoit Mandelbrot

Benoît B. Mandelbrot is a French and American mathematician, best known as the father of fractal geometry. He is Sterling Professor of Mathematical Sciences, Emeritus at Yale University; IBM Fellow[…]

A conversation with the mathematician and Professor Emeritus at Yale University.

Benoit Mandelbrot: Benoit Mandelbrot, Sterling Professor, Emeritus at Yale University, IBM Fellow Emeritus at IBM Research Center.

Question: What is fractal geometry? 

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Benoit Mandelbrot: Well, regular geometry, the geometry ofrnEuclid, is concerned with shapes which are smooth, except perhaps for cornersrnand lines, special lines which are singularities, but some shapes in nature arernso complicated that they are equally complicated at the big scale and comerncloser and closer and they don’t become any less complicated.  Closer and closer, or you go farther orrnfarther, they remain equally complicated. rnThere is never a plane, never a straight line, never anything smooth andrnordinary.  The idea is very, veryrnvague, is expressed – it’s an expression of reality. 

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Fractal geometry is a new subject and each definition I tryrnto give for it has turned out to be inappropriate.  So I’m now being cagey and saying there are very complexrnshapes which would be the same from close by and far away.

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Question: What does it mean to say that fractal shapes arernself-similar?

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Benoit Mandelbrot: Well, if you look at a shape like arnstraight line, what’s remarkable is that if you look at a straight line fromrnclose by, from far away, it is the same; it is a straight line.  That is, the straight line has arnproperty of self-similarity.  Eachrnpiece of the straight line is the same as the whole line when used to a big orrnsmall extent.  The plane again hasrnthe same property.  For a longrntime, it was widely believed that the only shapes having these extraordinaryrnproperties are the straight line, the whole plane, the whole space.  Now in a certain sense, self-similarityrnis a dull subject because you are used to very familiar shapes.  But that is not the case.  Now many shapes which are self-similarrnagain, the same seen from close by and far away, and which are far from beingrnstraight or plane or solid.  Andrnthose shapes, which I studied and collected and put together and applied inrnmany, many domains, I called fractals.

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Question: How can complex natural shapes be representedrnmathematically?

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Benoit Mandelbrot: Well, historically, a mountain could notrnbe represented, except for a few mountains which are almost like cones. Mountains are very complicated.  Ifrnyou look closer and closer, you find greater and greater details.  If you look away until you find thatrnbigger details become visible, and in a certain sense this same structurernappears at those scales.  If yournlook at coastlines, if you look at that them from far away, from an airplane,rnwell, you don’t see details, you see a certain complication.  When you come closer, the complicationrnbecomes more local, but again continues. rnAnd come closer and closer and closer, the coastline becomes longer andrnlonger and longer because it has more detail entering in.  However, these details amazingly enoughrnenters this certain this certain regular fashion.  Therefore, one can study a coastline **** object because therngeometry for that existed for a long time, and then I put it together and appliedrnit to many domains.

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Question: What was the discovery process behind thernMandelbrot set?

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Benoit Mandelbrot: The Mandelbrot set in a certain sense isrna **** of a dream I had and an uncle of mine had since I was about 20.  I was a student of mathematics, but notrnhappy with mathematics that I was taught in France.  Therefore, looking for other topics, an uncle of mine, whornwas a very well-known pure mathematician, wanted me to study a certain theoryrnwhich was then many years old, 30 years old or something, but had in a wayrnstopped developing.  When he wasrnyoung he had tried to get this theory out of a rut and he didn’t succeed,rnnobody succeeded.  So, there was arncase of two men, Julia, a teacher of mine, and Fatu, who had died, had a veryrngood year in 1910 and then nothing was happening.  My uncle was telling me, if you look at that, if you findrnsomething new, it would be a wonderful thing because I couldn’t – nobodyrncould. 

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I looked at it and found it too difficult.  I just could see nothing I couldrndo.  Then over the years, I putrnthat a bit in the back of my mind until one day I read an obituary.  It is an interesting story that I wasrnmotivated by an obituary, an obituary of a great man named Poincaré, and in that obituary thisrnquestion was raised again.  At thatrntime, I had a computer and I had become quite an expert in the use of therncomputer for mathematics, for physics, and for many sciences.  So, I decided, perhaps the time hasrncome to please my uncle; 35 years later, or something.  To please my uncle and do what my unclernhad been pushing me to do so strongly.  

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But I approached this topic in a very different fashion thanrnmy uncle.  My uncle was trying tornthink of something, a new idea, a new problem, a new way of developing therntheory of Fatu and Julia.  I didrnsomething else.  I went to therncomputer and tried to experiment. rnI introduced a very high level of experiment in very purernmathematics.  I was at IBM, I hadrnthe run of computers which then were called very big and powerful, but in factrnwere less powerful than a handheld machine today.  But I had them and I could make the experiments.  The conditions were very, veryrndifficult, but I knew how to look at pictures.  In fact, the reason I did not go into pure mathematicsrnearlier was that I was dominated by visual.  I tried to combine the visual beauty and thernmathematics. 

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So, I looked at the picture for a long time in a veryrnunsystematic fashion just to become acquainted in a kind of physical fashionrnwith those extraordinary difficult and complicated shapes.  Two were extraordinarilyrndifficult.  Computer graphics didrnnot exist back then, but to have a machine which was – made it seemrndoable.  And I started findingrnextraordinary complications, extraordinary structure, extraordinary beauty ofrnboth a theoretical kind, mathematical, and a visual kind.  And collected observations of my triprnin this new territory.  When Irnpresented that work to my colleagues, it was an explosion of interest.  Everybody in mathematics had given uprnfor 100 years or 200 years the idea that you could from pictures, from lookingrnat pictures, find new ideas.  Thatrnwas the case long ago in the Middle Ages, in the Renaissance, in later periods,rnbut then mathematicians had become very abstract.  Pictures were completely eliminated from mathematics; inrnparticular when I was young this happened in a very strong fashion. 

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Some mathematicians didn’t even perceive of the possibilityrnof a picture being helpful.  To therncontrary, I went into an orgy of looking at pictures by the hundreds; thernmachines became a little bit better. rnWe had friends who improved them, who wrote better software to help me,rnwhich was wonderful.  That was the goodrnthing about being at IBM.  And Irnhad this collection of observations, which I gave to my friends in mathematicsrnfor their pleasure and for simulation. rnThe extraordinary fact is that the first idea I had which motivated me,rnthat worked, is conjecture, a mathematical idea which may or may not berntrue.  And that idea is stillrnunproven.  It is the foundation,rnwhat started me and what everybody failed to **** prove has so far defeated therngreatest efforts by experts to be proven. rnIn a certain sense it’s a very, very strange because the object itselfrnis understandable, even for a child. rnIf the object can be drawn by a child with new computers, with newrngraphic devices, and still the basic idea has not been proven. 

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But the development of it has been extraordinary, then itrnwas slowed down a bit, and now again it is going up.  New people are coming in and they prove extraordinaryrnresults which nobody was hoping to prove, and I am astonished and of course,rnvery pleased by this development.

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Question: What is the unproven conjecture that drove you?

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Benoit Mandelbrot: The conjecture itself consists in tworndifferent issues in Mandelbrot set – two alternative definitions which are toorntechnical to describe without a blackboard, but which are both very simple andrnwhich I assumed naively to be equivalent. rnWhy did I assume so? rnBecause on the pictures I could not see any difference.  Obtaining pictures in one way orrnanother way, I couldn’t tell them apart. rnTherefore, I assumed they were identical and I went on studying thisrnpiece.  I found that, again, manyrninteresting observations of which most were very preferred by many other very,rnvery skilled mathematicians.  Butrnthe idea that these two conditions, definitions, are identical is stillrnopen.  So there are two definitionsrnin Mandelbrot set, the usual one and another one, and they may theoretically berndifferent.  People are gettingrncloser, but have not proven it completely.

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Question: Why do people find fractals beautiful?

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Benoit Mandelbrot: Well, first of all, one explanation ofrnthat is that the feeling for fractality is not new.  It is one very surprising and extraordinary discovery I maderngradually, very slowly by looking again at the paintings of the past.  Many painters had a clear idea of whatrnfractals are.  Take a Frenchrnclassic painter named Poussin. rnNow, he painted beautiful landscapes, completely artificial ones,rnimaginary landscapes.  And how didrnhe choose them?  Well, he had thernbalance of trees, of lawns, of houses in the distance.  He had a balance of small objects, bigrnobjects, big trees in front and his balance of objects at every scale is whatrngives to Poussin a special feeling. 

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Take Hokusai, a famous Chinese painter of 1800.  He did not have any mathematicalrntraining; he left no followers because his way of painting or drawing was toornspecial to him.  But it was quiternclear by looking at how Hokusai, the eye, which had been trained from thernfractals, that Hokusai understood fractal structure.  And again, had this balance of big, small, and intermediaterndetails, and you come close to these marvelous drawings, you find that hernunderstood perfectly fractality. rnBut he never expressed it. rnNobody ever expressed it, and then the next stage of Japanese imagernexperts did some other things. 

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So humanity has known for a long time what fractalsrnare.  It is a very strangernsituation in which an idea which each time I look at all documents have deeperrnand deeper roots, never (how to say it), jelled.  Never got together until I started playing with the computerrnand playing with topics which nobody was touching because they were justrndesperate and hopeless.

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Question: How has computer technology impacted your work?

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Benoit Mandelbrot: Well, the computer had been sort of spoken about since thernearly 19th century, even before. rnBut until the electronic computers came, which was in reality duringrnWorld War II, or shortly afterwards. rnThey could not be used for any purpose in science.  They were just too slow, too limited inrntheir capacity.  My chance was thatrnI was myself a very visual person. rnAgain, a mathematician who had started a very unconventional careerrnbecause my interest was both mathematics and in the eye.  And with IBM very primitivernpicture-making machines became available and we had to program everything.  It was heroic.  And my friends at IBM who helped merndeserve a great thank you.  Withrnthese two, I could begin to do things which before had been impossible.  I could begin to implement an idea ofrnhow a mountain looked like.  Tornreduce a mountain, which is something most complicated to a very simple idea –rnhow do you do it?  Well you make arnconjecture, have positives about shapes of mountains, and you don’t think aboutrnthe mathematics of it, you must make a picture of it. 

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If the picture is – everybody to be a mountain, then there’srnsomething true about it.  Or arncloud.  It was astonishing when atrnone point, I got the idea of how to make artifical clouds with a collaborator,rnwe had pictures made which were theoretically completely artificial picturesrnbased upon that one very simple idea. rnAnd this picture everybody views as being clouds.  People don’t believe that they aren’trnphotographs.  So, we have certainlyrnfound something true about nature. rn

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And on the other side, the completely artificial shapes, thernshapes that don’t exist in nature, which, for example, the Mandelbrot set,rnwhich was completely came out of the blue out of a very simple formula which isrnabout one inch long and which gives us an endless, endless stream of questionsrnand results.  There what happened isrnthat to everybody’s surprise there’s a very strong inner resemblance betweenrnthose shapes and the shapes of nature, which I have been studying.  And again, I spent half my life,rnroughly speaking, doing the study of nature in many aspects and half of my lifernstudying completely artificial shapes. rnAnd the two are extraordinarily close; in one way both are fractal.

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Question: What does the word “chaos” mean to mathematicians?

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Benoit Mandelbrot: The theory of chaos and theory ofrnfractals are separate, but have very strong intersections.  That is one part of chaos theory isrngeometrically expressed by fractal shapes.  Another part of chaos theory is not expressed by fractalrnshapes.  And other part of fractalsrndoes not belong to chaos theory so that two theories which overlap veryrnstrongly and do not coincide.  Onernof them, chaos theory, is based on behavior of systems defined byrnequations.  Equations of motion,rnfor example, and classical mathematics, and around 1900, Poincaré and ****, two greatrnmathematicians at the time, have realized that sometimes the solution of veryrnsimple looking equations can be extremely complicated.  But in 1900, it was too early torndevelop that idea.  It was veryrnwell expressed and very much discussed, but did not – could not grow. 

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Much later, of course, with computers this idea came to lifernagain and became the very important part of science.  So both chaos theory and fractal have had contacts in thernpast when they are both impossible to develop and in a certain sense not readyrnto be developed.  And again, theyrnintersect very strongly but they are very distinct.

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Question: Do mathematical descriptions of chaos define somernorder within chaos?

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Benoit Mandelbrot: Well a very strong distinction was madernbetween chaos and fractals.  Forrnexample, the rules which generate most of natural fractals, models ofrnmountains, of clouds, and many other phenomena involve change.  And therefore they are not at allrnchaotic in the ordinary sense of the word, in an ordinary, current, modernrnsense of the word.  Not chaotic inrnthe old sense of the word, which doesn’t have any specific meaning. But I don’trnlike to discuss the question of terms. rnThe term chaos came, but you know something which was very confused, itrnhelped it jell, but the use of a biblical name in a certain sense forces us tornfind the implications which were not important in mathematics.  That’s why when the time came to give arnname to my work, I chose the word fractal which was new.  Before that, there was no need of arnword at all because again there were only a few undeveloped ideas in the veryrnmany great minds.  But when a wordrnbecame necessary, I preferred not to use an old word, but to create a new one.

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Question: How did you come up with the word “fractal”?

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Benoit Mandelbrot: rnWell, it was a very, very interesting story.  At one point, a friend of mine, an older person, told mernthat he saw a paper of mine on a new topic.  And he said, “Look Benoit, I tell you, you must stop writingrnall of these papers in that field, that field, that field, that field.  Nobody knows where you are, what yournare doing.  You just sit down andrnwrite a book.  A short book, arnclear book, a book of things which you have done.”  So, I sat down and wrote the book.  Now, the book had to title, why?  Because the topics I had been studying had not been thernobject of any theory whatsoever. rnAnd there are many words which mean nothing, but many fields which havernno name because they don’t exist. rnSo, the publisher didn’t like this very ponderous title, said, “Look.”  And a friend of mine, another friend,rntold me, “Look, you create a new field, you are entitled to give it arnname.”  So, I had Latin in highrnschool and it turned out that one of my son’s was taking Latin in the UnitedrnStates, and so there was a Latin dictionary in our house, which was anrnexception. 

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I went in there and tried to look for a word which fittedrnwhat I had been working on.  Andrnwhen I was playing with the word fraction, and looked in the dictionary for arnword where fraction came from.  Itrncame from a Latin word which meant, how to say disconnect – rough andrndisconnected, it was a very general – the idea of roughness originally in Latin.  So, I started playing with fractus, which I named it that and coinedrnthe word fractal.  First of all, Irnput it in this book, Objets Fractals,rnin French as it turned out, and then the English translation of the book, andrnthen the word took off.  First ofrnall, people applied it in ways in which I didn’t find sensible, but there wasrnnothing I could say about it.  So,rnthen the dictionary started defining it, each a little bit differently.  And in a certain sense the word becamernalive and independent of me.  Irncould scream and say, I don’t like it, but it made no difference. 

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I had once a curiosity of looking on the Web in differentrncountries having different languages, what is a fractal, and found that in onerncountry, I will not mention, it’s a word that has become applied to somernnightclubs.  A fractal nightclub isrna kind of nightclub.  I don’t knowrnwhich, because I haven’t been there, but, and I don’t know the language, but Irnguess, from what I could guess, what it was.  It’s a word which has its own life.  I gave it a definition, but thatrndefinition became too narrow because some objects I want to go fractal did notrnfit the old definition. 

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So some people asked me would I still believe the definitionrnof whatever – 40 years ago.  Irndon’t.  But I have no control.  It’s something which works byrnitself.  The fact that very manyrnadults I know never heard of it, but the children have, is what gives mernparticular pleasure because high school students, even the bright ones, arernvery resistant to, how to say, imposed terms.  And the combination of pictures and of deep theory, you canrnlook at the picture and find something, some idea about this picture isrnsensible, and then be told that very great scientists either can’t prove it, orrnhas taken 40 years to prove it, or had to be several of them together to provernit because it was so difficult. rnAnd it can be seen by a child, understood by a child.  That aspect is one which very manyrnpeople find particularly attractive in the field. 

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In mathematics and science definition are simple, butrnbare-bones. Until you get to a problem which you understand it takes hundredsrnand hundreds of pages and years and years of learning.  In this case, you have this formula,rnyou track in a computer and from a simple formula, in a very short timernamazingly beautiful things come out, which sometimes people can prove instantlyrnand sometimes great scientists take forever to prove.  Or don’t even succeed in proving it.

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Question: How can we understand financial marketrnfluctuations in fractal terms?

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Benoit Mandelbrot: Well, what I discovered quite a while agornin fact, that was my first major piece of work is that a model of pricernvariation which everybody was adopting was very far from being applicable.  It’s a very curious story. 

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In 1900, a Frenchman named Bachelier, who was a student ofrnmathematics, wrote a thesis on the theory of speculation.  It was not at all an acceptable topicrnin pure mathematics and he had a very miserable life.  But his thesis was extraordinary.  Extraordinary in a very strange way.  It applied very well to Brownianrnmotion, which is in physics.  So,rnBachelier was a pioneer of a very marvelous essential theory in physics.  But to economics, it didn’t apply atrnall, it was very ingenious, but Bachelier had no data, in fact no data wasrnavailable at that time in 1900, so he imagined an artificial market in whichrncertain rules may apply. rnUnfortunately, the theory which was developed by economists whenrncomputers came up was Bachelier’s theory. rnIt does not account for any of the major effects in economics.  For example, it assumes prices arerncontinuous when everybody knows the prices are not continuous.  Some people say, well, all right, therernare discontinuities but they are a different kind of economics that we arerndoing, not because certain discontinuities become too complicated and only willrnthe **** look more or less continuous. rnBut it turns out that discontinuities are as important, or morernimportant than the rest. 

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Bachelier assumed that each price changes in compared of thernpreceding price change.  It’s arnvery beautiful assumption, but it’s completely incorrect because we know veryrnwell, especially today that for a long time prices may vary moderately and thenrnsuddenly they begin to vary a great deal. rnSo, even we’re saying that the theory changes or you say the theoryrnwhich exists is not appropriate. rnWhat I found that Bachelier’s theory was defective on both grounds.  That was in 1961, 1962, I forgot thernexact dates and when the development of Bachelier became very, very rapid.  Since nobody wanted to listen to me, Irndid other things.  Many otherrnthings, but I was waiting because it was quite clear that my time would have torncome.  And unfortunately, it hasrncome, that is, the fluctuation of the economy, the stock market, and commodityrnmarkets today are about as they were in historical times.  There was no change which made thernstock market different today than it was long ago.  And the lessons which are drawn from **** peers do representrntoday’s events very accurately. rnBut the situation is much more complicated than Bachelier hadrnassumed.  Bachelier, again, was arngenius, Bachelier had an excellent idea which happened to be very useful inrnphysics, but economics, he just lacked data.  He did not have awareness of discontinuity which isrnessential in this context.  Notrnhaving an awareness of dependence, which is also essential in this context.  So, his theory is very, very differentrnfrom what you observe in reality.

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

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Benoit Mandelbrot: Well, my life has been extremelyrncomplicated.  Not by choice at thernbeginning at all, but later on, I had become used to complication and went onrnaccepting things that other people would have found too difficult tornaccept.  I was born in Poland andrnmoved to France as a child shortly before World War II.  During World War II, I was lucky tornlive in the French equivalent of Appalachia, a region which is sort of not veryrnhigh mountains, but very, very poor, and Appalachia we are poorer even, sornpoorer 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 thernhills and had mostly a private intellectual life.  I read many books; there were many books, a very goodrnlibrary.  I had many books and Irnhad dreams of all kinds.  Dreams inrnwhich were in a certain sense, how to say, easy to make because the near futurernwas always extremely threatening. rnIt was a very dangerous period. rnBut since I had nothing to lose, I was dreaming of what I could do. 

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Then the war ended. rnI had very, very little training in taking an exam to determine arnscientist’s life in France.  Therernwere two schools, both very small. rnOne tiny, and one small, which in a certain sense was the place that Irnwas sure I wanted to go.  I hadrnonly a few months of finding out how the exam proceeded, but I took the examrnand perhaps because of inherited gifts, I did very well. In fact, I barelyrnmissed being number one in France in both schools.  In particular I did very well in mathematical problems.  The physics I could not guess, otherrnthings I could not guess.  But thenrnI had a big choice, should I go into mathematics in a small and ****rnschool.  Or should I go to a biggerrnschool in which, in a certain sense would give me time to decide what I wantedrnto do? 

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First I entered the small school where I was, as a matter ofrnfact, number one of the students who entered then.  But immediately, I left because that school, again, wasrngoing to teach me something which I did not fully believe, namely mathematicsrnseparate from everything else.  Itrnwas excellent mathematics, French mathematics was very high level, but inrneverything else it was not even present. rnAnd I didn’t want to become a pure mathematician, as a matter of fact,rnmy uncle was one, so I knew what the pure mathematician was and I did not wantrnto 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,rnwith the real world of which I was very, very curious. 

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And so, I did not go to École Polytechnique.  It was a very rough decision, and thernyear when I took this decision remembers my memory very, very strongly.  Then for several years, I just was lostrna bit.  I was looking for a goodrnplace.  I spent my time very nicelyrnin many ways, but not fully satisfactory. rnThen I became Professor in France, but realized that I was not – for thernjob that I should spend my life in. rnFortunately, IBM was building a research center, I went there for arnsummer thing, for a summer only.  Irnknew this summer, decided to stay. rnIt was a very big gamble.  Irnlost my job in France, I received a job in which was extremely uncertain, howrnlong would IBM be interested in research, but the gamble was taken and veryrnshortly afterwards, I had this extraordinary fortune of stopping at Harvard torndo a lecture and learning about the price variation in just the right way.  At a time when nobody was looking, was realizingrnthat either one needed, or one could make a theory of price variation otherrnthan the theory of 1900 at which Bachelier had proposed, which was very, veryrnfar from being representative of the actual thing. 

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

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And fractal geometry was not an idea which I had early on,rnfor something was developed progressively.  I didn’t choose to go into the topic because of anyrncompelling reason, but because the problems there seemed to be somehow similarrnto the ones I knew how to handle. rnI had experienced this kind of problem and gradually realized that I wasrntruly putting together a new theory. rnA theory of roughness.  Whatrnis roughness?  Everybody knows whatrnis roughness.  When was roughnessrndiscovered?  Well, prehistory.  Everything is roughness, except for therncircles.  How many circles arernthere in nature?  Very, veryrnfew.  The straight lines.  Very shapes are very, very smooth.  But geometry had laid them asidernbecause they were too complicated. rnAnd physics had laid them aside because they were too complicated.  One couldn’t even measure roughness.  So, by luck, and by reward forrnpersistence, I did found the theory of roughness, which certainly I didn’t expectrnand expecting to found one would have been pure madness. 

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

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Question: Which honor means more to you: your Légionrnd'Honneur medal or the “Mandelbrot Set” rock song?

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Benoit Mandelbrot: Well, I happen to know this song, it wasrnsent to me and I was very impressed by it and by its popularity.  In a certain sense, it is not whichrnone, but the combination.  Whenrnpeople ask me what’s my field?  I say,rnon one hand, a fractalist.  Perhapsrnthe only one, the only full-time one. rnOn the other hand, I’ve been a professor of mathematics at Harvard andrnat Yale.  At Yale for a longrntime.  But I’m not a mathematicianrnonly.  I’m a professor of physics,rnof economics, a long list.  Eachrnelement of this list is normal. rnThe combination of these elements is very rare at best.  And so in a certain sense, it is notrnthe fact that I was a professor of mathematics at these great universities, orrnprofessor of physics at other great universities, or that I received, amongrnother doctorates, one in medicine, believe it or not.  And one in civil engineering.  It is the coexistence of these various aspects that in onernlifetime it is possible, if one takes the kinds of risks which I took, whichrnare colossal, but taking risks, I was rewarded by being able to contribute in arnvery substantial fashion to a variety of fields.  I was able to reawaken and solve some very old problems.  The problems are just so old that in arncertain sense, they were no longer being pursued.  And nobody – I didn’t know anybody who was trying to define roughnessrnof ****. It was a hopeless subject. rnBut I did it and there’s a whole field by which has been created byrnthat. 

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In a certain sense the beauty of what I happened byrnextraordinary chance to put together is that nobody would have believed thatrnthis is possible, and certainly I didn’t expect that it was possible.  I just moved from step to step to step.  Lately I realized that all these thingsrnheld together, and very lately I see that in each field very old problems couldrnbe if not solved, at least advanced or reawakened, and therefore gradually veryrnmuch improved in your understanding.

Recorded on February 17, 2010
Interviewed by Austin Allen

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