BENOIT MANDELBROT ers of chaos theory as it generated man who went deeply into the sub- Mandelbrot Makes Sense: pictures of ever increasing complexi- ject), it was said that “the French did A Book Review Essay ty using a deceptively minuscule quite useful mathematics before recursive rule, one that can be reap- Bourbaki” – as the secretive guild- A discussion of Benoit Mandelbrot’s The plied to itself repeatedly. You can like organization installed a truly (Mis)Behavior of Markets by Nassim look at the set at smaller and smaller top-down view of the subject matter, Nicholas Taleb resolutions without “ever” reaching insuring no corruption by earthly the limit; you will continue to see material. Indeed many physicists the recognizable shapes. have been horrified at the extent I closed this book feeling that it was classroom, may be beautiful and The introduction of fractals was and side effects of such purism, with the first book in economics that pure notions; but they seem more not initially welcomed by the mathe- Murray Gell-Mann calling it the spoke directly to me. Not only that, present in the mind of mathemati- matical establishment. This method “Bourbaki Plague”, and attributing but the astonishing simplicity, real- cians and schoolteachers than in of pictorial presentation did not the divergence between pure mathe- ism, and relevance of the subject nature itself. Mountains are not tri- seem to correspond to what seemed matics and science to the obscure makes it the only general work in angles or pyramids; trees are not cir- “to be mathematics” in the self- language of the Bourbakists5. finance I’ve ever read that seemed to cles; straight lines are almost never defining discipline. It is thanks to its In a way, the separation between make sense. seen anywhere. To figure out how popularity with physicists and other geometry and algebra can be seen as Benoit Mandelbrot makes sense. the world operates, we need a differ- applied scientists, themselves fol- the separation of images and words Just as he used us common readers ent geometry than the classical one lowing the lead of the general public in human expression and thought – outside the ivory tower to force his developed by Euclid of Alexandria (mostly computer “geeks”), that frac- just imagine a world in which fractal ideas into science (where some 2400 years ago. Drawing on a tal geometry vindicated its way into images were barred. The Bourbaki- they became “part of the scientific list of then obscure (but subsequent- the now-broadened field of mathe- inspired purblindness does not just consciousness”1); he may just be the ly made famous) mathematicians, matics. For The Fractal Geometryof limit the tools of analysis. Just like one to help turn economics into BM coined the word fractal geometry Nature made a splash when it came blindness, one of its effects is to something real. to describe these objects that are out a quarter century ago. It spread reduce contact with reality. Platonic This first essay is non-technical jagged yet self-similar in the sense across the artistic circles and led to top-down approaches are interesting and general2 (i.e. can be read by that small parts resemble, to some studies in aesthetics, architectural but they tend to choke under the someone without a mathematical degree, the whole (a more mathe- designs, even large industrial appli- occasional irrelevance of their pur- background) and focuses around matically appropriate designation cations. BM was even offered a posi- suits. It is telling that BM’s hero is the topics covered in this book. The would be the broader “self-affine” tion at a medical school! His talks Antaeus, son of Gaia the mother second one is more technical and it but, somehow, designations are were invaded by all manner of Earth, who needed periodic contact goes deeper into the epistemologi- sticky and, in this discussion, self- artists4, earning him the nickname with earth to replenish his strength. cal problems of “fat tails”, concen- similarity should be held to be “self- “the rock star of mathematics”. The Owing to the vicissitudes of a tration, and extreme events. affine”). Leaves look like branches; computer age thus helped him clandestine life during the Nazi occu- What do fern leaves, commodity branches look like trees; rocks look become one of the most influential pation of France, the young Benoit prices, computer book sales, income like small mountains. If you look at mathematicians in history, in terms was spared some of the conventional distribution, the coast of Britain, the coast of Britain from an air- of the applications of his work, way Gallic education with the uninspir- cauliflowers, and the intricacies of plane, it resembles what you get before his acceptance by the ivory ing algebraic drills, becoming largely the vascular system have to do with using a magnifying glass. This char- tower. We will see that, in addition self-taught with some assistance one another? Mandelbrot’s work acter of self-affinity implies that one to its universality, his work possesses from his uncle Szolem, a prominent revolves around the simple practical deceivingly short and simple rule of an unusual attribute: it is remark- member of the French mathematical application of a concept called “frac- iteration can be used, either by a ably easy to understand. hierarchy and professor at the tal” in replacement for more com- computer, or more randomly, by A Polish-Lithuanian Jew who College de France. Instead, he devel- plicated mathematical tools that Mother Nature, to build shapes of found refuge in France as a child, BM oped an encyclopedic knowledge of are universally used without empiri- seemingly large complexity. He is also a refugee from the French the history of mathematical cal justification. designed, or rather, according to Sir mathematical establishment protec- thought. He also gave free course to Triangles, squares, circles, and Roger Penrose3, discovered an object, tive of the “purity” of mathematics. his geometric bent. Untrained in the other geometric concepts that known as the “Mandelbrot set”, To borrow from the late probabilist usual equation solving techniques, caused many of us to yawn in the which became popular with follow- and probability thinker E. T. Jaynes (a he passed the entrance exam to the 50 Wilmott magazine BENOIT MANDELBROT elite École Normale using purely geo- thinker who had the luxury to take than the theories of financial econo- strategy of going straight to practi- metric intuitions (this should be a his time to grow his ideas. mists, and, which is worrisome, tioners and the general public and hint for educators: consider how (Charmingly, BM, in his scientific more understandable by the common bypassing the academic establish- much more intuition you can devel- writings, when discussing a contri- man more than by the classically ment, a task that might appear easy op with images instead of words). But bution made by a mature mathe- trained economist – just as the com- with economics given that the he left after two days. Already stub- matician, mentions his age, such as puter graphic designer or a comput- public and professional standing of born, unruly and unmanageable, he “Cauchy, at the age of 64...”). It is erized teenager could get the point economists in general and finance moved to the more engineering-ori- thanks to such maturation that he far more easily than a classically academics in particular is one of ented École Polytechnique. He then joins that category of the classical, trained mathematician. the lowest of any specialty. So the settled in the United States, working pre-academic specialization of It is not a well-known fact that mission of toppling these fake most of his life as an industrial scien- the wisdom-generating natural before his involvement with the and empirically invalid beliefs tist for IBM, with a few transitory and philosophers. roughness in the geometry in seems trivial. varied academic appointments. What does it all have to do with nature, BM started his career focus- Or is it? Finance academia, Indeed, thanks to the computer, he finance? Can we extend the concept ing on problems in social science unlike the physics establishment, could let the potent machine express of fractals and self-similarity to sta- and finance; it is certainly there that seems to work more like a religion his geometric hunches and lead tistical frequencies? It would make most of his ideas were refined. He than a science, with beliefs that have through the subject matter’s natural the concept of astonishing universal- initially wrote papers in the 1960s so far resisted any amount of empiri- course. Indeed, the computer played ity. This would make BM the true presenting his ideas on “infinite cal evidence (actually this statement two roles in the new science he Kepler of the social sciences. The helped conceive. First, these fractal analogy to Kepler is at two levels, objects, as we will see, could be gen- first in the building of insights This would make BM the true erated with a simple rule applied to rather than mere circuitry, second itself which is ideal for the automa- because you can step on his shoul- Kepler of the social sciences ... tion of a computer (or mother ders – the title of Kepler or “Newton nature). Second, in the generation of of the social sciences” is one so many visual intuitions lies a dialectic thinkers with grand ideas have tried first in the building of insights between the mathematician and the to grab (Marx for one aimed at being objects generated. A mathematical the Newton of the sciences of man). I rather than mere circuitry, scientist par excellence, in a subject am not in the business of defining matter that did not (then) exist insti- genius, but it seems to me that the second because you can step tutionally, he was held to be a mathe- mark of a genius is the ability to pick matician for scientists and a scientist up pieces that are fragmented in on his shoulders (particularly a physicist) by the math- people’s mind and binding them ematical establishment.