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”.
