I want to pass along this note I received from my friend Paul Davis tonight. When I have a problem I can’t solve I have coffee with Paul. He is the best pure mathematician I know, a real rocket scientist. No really, he worked for NASA after his PhD in Applied Math. Of course he also likes opera and runs 200 mile ultra-marathon races too, but nobody is perfect. By the way, when I got Paul’s email I pulled my marked-up copy of Mandelbrot’s The Misbehavior of Markets from my shelf. Mandelbrot–the guy who invented fractal geometry–uses fractals tools to show why markets are more risky than the modern portfolio theory mavens tell you (although at least some of them have proven so themselves–Long Term Capital Management had their offices in the building next to ours in Greenwich). Don’t believe the dust jacket though. Mandelbrot doesn’t tell you how to understand markets; he tells you why you don’t understand markets and never will, which is even more useful to know. Final footnote, Mandelbrot discusses a concept he refers to as long dependence. This is the same idea Einstein referred to as action at a distance, which he, and other prominent physicists argued was impossible. Yet action at a distance is exactly what arises in the far-from-equilibrium, i.e., irreversible and entropy-producing, systems that comprise the meat of Prigogine’s work that I discussed last weekend. Now will you read Prigogine’s book?
I found your comments on randomness right on the mark. When I was a student at UVA, I had a math professor (in the Engineering School) who argued that Newton had done physics and the world in general a huge disservice by inventing calculus. Calculus requires a continuous universe rather than one made up of discreet components such as quarks, quantum gravity or quantum time. In real life, he argued, nothing is smooth and continuous. In the early 1970’s Benoit Mandelbrot wrote a paper titled “How to Measure an Avocado”. The question was simple, i.e. what is the circumference of an avocado? The answer is: there is no simple answer. It depends on whether you are willing to ignore the fact that the avocado’s skin is not continuous but rather made up at even the macroscopic level of dimples and ridges, at the electron microscope level of molecules which aren’t really solid and then at the atomic level which isn’t solid and so on. So we are left measuring the tangents between peaks, but which peaks? And how does one measure the tangents between subatomic particles which move whenever we “look” at them?
The relevance of this to economics and the stock markets is that those systems are also quantized. Transactions in the economy are ultimately discreet and a continuum of transactions does not exist between transactions. Finally, true randomness can not be continuous as being continuous implies a natural sequence or order. As Ritholtz pointed out, being human we tend to look for patterns and random systems will usually exhibit some patterns. Just look at the “clumpiness” of the universe. We have stars clumping together to form galaxies and galaxies clumping together to form super galaxies and then huge voids in between. But trying to predict when and where the next galaxy will appear based on observations of existing galaxies is a waste of time. Randomness does not imply equal spacing or order.