The population biologist, James Crow, in an article in the latest issue of the Journal of Biology, reminds us of Ernst Mayr's challenge (in 1959) to explain the relevance of mathematical models to evolutionary studies. In Mayr's words the challenge is "what, precisely, has been the contribution of this mathematical school to the evolutionary theory, if I may ask such a provocative question?". Crow's 2009 response is pretty typical of the responses made by mathematical evolutionists (including Haldane) back in the early 1960s: a laundry list of evolutionary problems solved by mathematics. While, this sort of response has value, it isn't general enough for philosophers of science. Further, some of the problems solved were mathematical to begin with. Mayr is clearly asking what role mathematics plays in the biological sciences, not in the mathematical sciences.
How should philosophers respond? Chris Pincock, for one, is working on a book that explores this issue. I hope this post prompts Chris to reply.
For what it is worth, I have a couple of half-backed ideas to initiate a list of how mathematical models contribute to evolutionary biology in particular and perhaps science in general. What I am really hoping is for some of you to point me to some of the relevant philosophical literature. Or, even better, I hope some of you add to the list.
Invoking A. Garfinkel (1982), I think one of the crucial contributions mathematical models make to sciences is that they allow us to consider what could have been otherwise (which, on some theories of causation, help us understand causal relations between events). Garfinkel's example is from population ecology, in particular the Lotka-Volterra equation which tracks the dynamics of population levels between preditors and prey. I can't represent the equation in this webpost but you all know the basic idea: the influences on the population numbers of preditors and prey are modeled in terms of the frequency in which the preditors encounter and eat their prey. From higher frequency of encounters we can predict that the population of prey organisms will go down. From observation alone we might confirm that a particular rabbit was eaten by a particular fox at a particular time. But, what observation doesn't tell us and what the L-V equation does is what would have happened had the particular rabbit not been eaten by the particular fox. If the population of foxes is high enough (and the rabbit popuation is low enough) then it is relatively likely the rabbit would have been eaten anyhow (by a different fox). If the population of foxes is low enough (and the rabbit population is high enough) then the chance of the rabbit getting eaten is relatively lower.
This one is a bit more vague. This time I'm channeling Poisson, Quetelet, LaPlace, and Gauss (and Strevens has an excellent book on the topic--Bigger than Chaos). One of the remarkable discoveries in the empirical sciences is the existence of large scale regularities that emerge from a "chaos" of individual variation. Extinction and adaptive speciation are two examples from evolutionary biology, predator/prey relations is an example from ecology. From physics we have gas laws. From demography there are sex ratio skews (towards boys found in England in the 18th and 19th century), and crime data in Paris that showed consistent crime rates in the 1820s despite the variety of ways crimes are committed (among a host of other found demographic phenomena). In economics Adam Smith hypothesized that well-ordered economies emerge from the variety of ways that individuals strive (unfettered) for their own reproductive success. Statistical data helped us see these patterns but probability theory (law of large numbers, central limit theorem) allowed us to see how these patterns could possibly emerge without the interference of external forces. Placed in historical context this application of mathematics (in the form of probability theory) was crucial in distinguishing the naturalistic sciences from theology. Before Gauss, Poisson, and LaPlace, people thought that, for example, the sex ratio skew towards boys was part of God's plan to make sure there are enough men for women to marry (after all many bachelors die in war). So, mathematics plays a role in explaining how large scale regularities emerge without reference to special external forces.
Incidentally, I think Darwin's version of natural selection relies a bit on both a rather primitive form of probability theory (in the form of what every gambler knows--that even slightly weighted dice gives the player an advantage or disadvantage) and an external force, or a force external to the mere lives, deaths, and reproductive activities of individuals. Crucial for Darwin's theory of natural selection is that a struggle for existence "inevitably follows" the Malthusian crush of population growth against resource restrictions. The struggle due to population growth is the natural selector and a condition that is external to individual life histories. But, modern versions of natural selection have downplayed the role of population growth. Evolution by natural selection has no need for an external force. Adaptive speciation (as shown by more sophisticated probabilistic models) can emerge from individuals who vary in their reproductive qualities.
Well, this has gone on long enough....