Wednesday, May 8, 2013
The Platonism of Benoit Mandelbrot
One of the joys of the present era is the cross pollination between realms of thought and expertise that electronic communications have enabled. I get the New York Review of Books on my kindle. I have read NYRB for many years. While I am frequently infuriated by its ridiculous and irresponsible biases, it is still one of the best journals for accessible and penetrating articles on history, literature, and science.
Tonight I read a review of The Fractalist: Memoir of a Scientific Maverick, by Benoit Mandelbrot. Jim Holt’s review gives us an excellent explanation of Mandelbrot’s contribution to mathematics. There is a lot in this review to encourage the Platonist who is need of encouragement, but I would focus on this simple explanation of Mandelbrot’s ground breaking insight:
Benoit Mandelbrot, the brilliant Polish-French-American mathematician who died in 2010, had a poet’s taste for complexity and strangeness. His genius for noticing deep links among far-flung phenomena led him to create a new branch of geometry, one that has deepened our understanding of both natural forms and patterns of human behavior. The key to it is a simple yet elusive idea, that of self-similarity.
To see what self-similarity means, consider a homely example: the cauliflower. Take a head of this vegetable and observe its form—the way it is composed of florets. Pull off one of those florets. What does it look like? It looks like a little head of cauliflower, with its own subflorets. Now pull off one of those subflorets. What does that look like? A still tinier cauliflower. If you continue this process—and you may soon need a magnifying glass—you’ll find that the smaller and smaller pieces all resemble the head you started with. The cauliflower is thus said to be self-similar. Each of its parts echoes the whole.
I am going to have to plug that back into my reading of Aristotle’s biology, but one thing strikes me as very important right now. It provides a new reading of the age old argument between reductionists and non-reductionists.
Reductionists argue that the more basic levels of any phenomena are the most fundamental and real levels. Thus a human being is really just a conglomeration of cells and cells. Organisms are just vehicles for molecules called genes and molecules are really just atoms in motion. Anti-reductionists (Aristotle comes to mind) argue that this blinds one to the most important phenomena. Looking at a pond at the level of molecules may be useful, but it is scarcely possible in this view to distinguish between a fish and the water it swims in. If you want to understand what is going on, you have to resist reduction and pay attention to the trout.
Both of sides of this old debate agree on one thing: that the whole and the parts are fundamentally different. An elk doesn’t look anything like or behave much like the cells of which it is composed. A car doesn’t have a lot in common with a break pad. This makes for an easy either/or choice when it comes to looking for the fundamental reality.
The opposite is true of something that is self-similar. If you have flown over a beach and then walked along it, and you pay attention to what you see, it may have occurred to you that a yard of shore line up close looks pretty much like miles of it from the air.
Other self-similar phenomena, each with its distinctive form, include clouds, coastlines, bolts of lightning, clusters of galaxies, the network of blood vessels in our bodies, and, quite possibly, the pattern of ups and downs in financial markets. The closer you look at a coastline, the more you find it is jagged, not smooth, and each jagged segment contains smaller, similarly jagged segments that can be described by Mandelbrot’s methods.
What I find immediately interesting about this notion of self-similar phenomena is that it doesn’t allow for reduction or anti-reduction. The parts of the whole recapitulate the whole and vice versa.
As Holt indicates repeatedly in his review, this points to Platonism. In certain phenomena, at least, elegant mathematical forms are persistently expressed in both structure and behavior. Our conception of the cauliflower is engendered by the image of that thing. One of the Platonic Socrates’ terms for the form is just another word for image. Socrates recognized that the form went much deeper than that but was, somehow, contained in the initial, common sense grasp of what was on the table. The form that is expressed in the head of cauliflower is expressed again in the floret and yet again in the subfloret.
Socrates would be fascinated by Mandelbrot’s fractals; indeed, he would eat them up. He would also remark that “I always said it was something like this”.