I’ve just started reading Paul J. Nahin’s book Doctor Euler’s Fabulous Formula about one of Leonhard Euler‘s famous equations. The preface establishes the author’s thesis that the great thing about Euler’s formula is that it is beautiful because it is the result of skill (or as the author says “disciplined reason”). He goes on to compare the beauty of a mathematical formula, and specifically mathematical formulae as expressions of concepts in physics, to the work of great artists. Beauty in art is something we can all relate to. What surprised me is that he based this comparison on the dichotomy between “disciplined artists,” such as Michelangelo and “undisciplined artists,” such as Jackson Pollock. To call what “two-year olds routinely do” on a daily basis art, he says, “is delusional or at least deeply confused…” (xix).
Nahin was wrong about Jackson Pollock. He knows he is wrong, i.e. I don’t think he really believes this, but this false dichotomy is highly illustrative. He happens to be wrong in a way that we can learn a lot from. I would like to show that he’s illustrating the particular value of mathematics in evolutionary biology. The value of mathematics in biology can only be seen if the level of application of models matches the proper place of the forces it models. If our application of broad theories is too narrow, then we will not find beautiful theories useful. On the other hand, if we view selection as a very broad force, as Stephen Jay Gould did in Wonderful Life, then we can see the “usefulness” of evolutionary theory.
Jackson Pollock was disciplined and skilled. Consider several facts about his career, training and technique. Firstly, he was a trained artist. He worked hard to develop his techniques. If you look at his early work (e.g. “She-wolf”), it’s certainly not what he’s most remembered for, but it is distinctive. If he could have just splattered paint then why would he bother to develop all that technique? People took him seriously as an artist before he developed his splattering technique. Next consider the drip paintings that he’s remembered for. These paintings were the result of careful consideration and a disciplined technique. Just a few examples: he positioned his canvas on the floor; he carefully mixed the paint to a particular consistency to achieve the kind of “splatter” that he needed; he used carefully chosen brushes and other devices. Finally he chose colors that made sense in order to make a particular artistic statement.
Life, the subject of biology, is a Jackson Pollock painting. I enjoy a mathematical argument because it is logical, internally consistent, one thing builds on another. I can see what it is made out of. I know the goals the entire time the argument is building. Wouldn’t it be great to apply that kind of thinking to understanding living things and their origins? Yeah, that would be great, but it’s incredibly hard for two reasons. Since life is a Jackson Pollock painting, the constraints are incredibly broad; so broad that life has huge leeway to accomplish the same thing in different ways. “Higher fitness” is not very specific. The paint spatters in different ways every time, regardless of its consistency and color palette being the same. The other reason is a confusion about where the predictions of theory are supposed to lie. Because selection is very very broad, we need to make predictions that are similarly broad. We can use the theory we have now to predict that selection would favor earlier age of maturity. That doesn’t mean we can predict whether that age is six months or two years. There are also different developmental ways to accomplish that. The theory we have doesn’t predict whether faster cell growth or slackened mate preferences will produce earlier age of maturity.
Biological mathematical models, if they are to be beautiful in the same way that physical models are, need to be broad, in the same way that Jackson Pollock’s constraints were broad. Take the most beautiful equation in biology, the Price Equation, for example. The Price Equation is beautiful because it is simple, so easy to derive that it seems it must be true, and you can do all kinds of things with it. You can derive almost any model of evolution from the Price Equation (try it!). You can also deal with almost every kind of selection at any level, and genetic drift. Why? Because it is so broad that actually applying it to a population would be ridiculous. The Price Equation is exact, which means that the equation includes more information than we can ever hope to measure in a natural population. The only way to make the application of the equation exact is to dial down the constraints so much that the prediction the equation makes is almost worthless. In other words, to make the equation exact, you have to impose such strong selection that you can predict the result without the equation. Mike Wade might disagree with me about this, but I can bet he won’t disagree about the equation’s beauty. The Price Equation is so broad that Martin Nowak has gone so far as to call it a “mathematical tautology” (also available at arXiv).
Another example, slightly less contentious, but not quite as obvious, are the predictions of life-history theory. Russ Lande and Brian Charlesworth‘s equations (derived from Guess What) make definite, broad predictions about the evolution of age at first reproduction, reproductive output and other life-history traits. The breadth of these predictions matches the strength of selection to enact changes at the genotypic level. What I mean is that as long as evolution is proceeding in the particular direction predicted by Lande and Charlesworth, then the model is justified. The actual genotypic or even morphological changes, or the diversity of adaptations at a particular level, doesn’t matter to the predictions of the theory. It also doesn’t matter whether you’re talking about yeast, fruit flies, elephant birds or dinosaurs. That’s the level of prediction we have with the theory of selection. It’s not precise like Rembrandt or even early Picasso. It’s a lot more like Jackson Pollock. It’s a mess, but it’s a carefully constrained, disciplined mess that is beautiful itself.
This morning I’ve realized yet again that the reason I love mathematics is largely aesthetic. I love math for the same reason that I love listening to a great piece of music. Beautiful models and theories are useful, Nahin and I contend, not because they are inherently “correct,” but because they inspire people to work hard at verifying and refining their logical consistency. Nahin points out that Newton and Einstein’s theories of graviation are “wrong” in the sense of making predictions that still hold true. But they are beautiful. We all say “yuck!” when we see someone try to fit a logistic curve to real data, but we don’t always know why we cringe. We cringe because we all know that the logistic wasn’t so well-studied because it was correct. It was so favored because it’s a beautiful equation. It’s as beautiful as the Mandelbrot set or Euler’s Formula, and it’s easy to teach to undergraduates. It makes sense. However, demographers and ecologists all know that it doesn’t really describe human populations. A.J. Lotka didn’t live to see that, but that doesn’t mean his time was wasted (see Lotka, Alfred J. 1998. Analytical theory of biological populations. New York: Plenum Press).