Tuesday, March 31, 2009

The role of mathematics in the study of evolution

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....


  1. I think Okasha's book "Evolution and the Levels of Selection" is a good example of how philosophers can inform and expand how mathematical models are used in evolution. His careful dissection of the Price equation sheds light both philosophically and biologically.

  2. André,
    Thanks for raising this issue, which unsurprisingly I think is very important. More generally, I think philosophers of science should think a bit more about mathematics than they often do.
    It is hard to get a grip on the contributions of mathematics to science, even for a special science like biology. There are at least five approaches that I am familiar with, and there is no reason why they couldn't be combined across cases:
    1. Mathematics makes a pragmatic contribution by allowing us to condense things or otherwise work out the implications of our non-mathematical beliefs. See, for example, the Tractatus or Hartry Field.
    2. Mathematics lets us represent features of the world that we could otherwise not represent. This may be your view based on the claim that modal features come from the mathematics. James Franklin has a version of this and it is also a strand of Colyvan's discussion of indispensability.
    3. Mathematics leads to new scientific discoveries. Here we have the Wigner problem, revived in a new form by Mark Steiner.
    4. Mathematics has explanatory power. Discussed by Steiner also, but more recently by Alan Baker and Mark Colyvan.
    5. Finally, I want to emphasize the epistemic contributions from mathematics. The basic idea is that knowing mathematics allows us to calibrate our representations so that they fit with the data we can collect and remain neutral on things we don't understand. I make a start on this here.

  3. I agree with (1) - (3) in regard to the use of mathematics in the biological sciences. However, I am also interested in whether mathematics can distort scientific work as well. For example, in both population genetics and theoretical ecology there is sometimes an emphasis on very simple models which underperform empirically.

  4. On AA's point that mathematics allows to reason counterfactually: That seems to me a feature of any system that has inference rules. The inference rules of mathematics are simply precise enough to decide what would have happened if things had been different, without (much?) disagreement. Of course, the same holds for first order or propositional logic.

    This gain in precision might be what connects the point about counterfactual reasoning to CP's second point about representability: The concepts of mathematics are much more precise than those of ordinary language, with several clear explications of, for example, `not making jumps' (continuity, Lipschitz-continuity, and what-have-you). Accordingly, one can describe a situation much more precisely. In that respect the success of mathematics is an argument for ideal language philosophy (at least against ordinary language philosophy, not obviously against non-linguistic philosophy).

  5. Jay's point is important. The models used in evolution and (especially) ecology with just a little complexity tend not to be robust (in the Wimsatt sense). Tons of models a produced every year in these disciplines, they proliferate, but rarely reference each other in meaningful ways, and if they explore the same system they tend to use wildly different assumptions and parameterizations so comparing them is difficult. They are often hardly touched by field or bench biologists. Someone ought to count the number of models produced in the bio-theory journals and see how many never see the light of day again—I mean actually get used, not just referenced (e.g., “so and so did a model of x which I feel obligated to mention because I'm doing something similar”). On the other hand simple models stay referenced and used for decades (like L-V, Price equation, Levin's metapopulation model) but are not known to exemplify any known real biological system (Charlie Hall did a nice paper on this back in '88). Of course, I'm setting this up as a dichotomy and there are lots of counterexamples -- complex models that get used and simple models that are tossed, but this does seem to be the way things tend.

  6. Thanks for all your responses so far. This is a helpful discussion to me.

    Thanks, Chris, for showing your cards. This lays a better groundwork for further thought then my original post offered.

    Jay's question is very interesting because it makes us think about the balance between simplicity and complexity in a model. Sober and Forester write a bit about this with respect to line drawing problems. I'm referring to their work on Aikaike.

    Jay: you say you agree with 1-3 in Chris's post. What about 4 and 5? What can't mathematical models explain? The Garfinkel case seems to me to be sufficient to show how. What do you think? And, what about 5?

    Sebastian: your point about inference rules is interesting because you offer an explanation for why mathematical models are counter-factually supporting. How about taking this one step further? Which inference rules matter? Can't be all of them, right? They would have to be the ones particular to counter-factual reasoning. On the other hand, I recall that Pearl makes an argument against using some mathematical representations, i.e. equations, for representing interventions (assuming these are counter-factually supporting) because equations don't represent the asymmetry of causation. Have any thoughts about this?

    Steve P: are you arguing for something like an underdetermination problem with mathematical models? For any set of data about a dynamic system there are a huge number of distinct (and maybe even incommensurable?) models that accurately represent them? Maybe I got you wrong. Can you say more?

  7. André: I took Chris' (1) - (5) to be about the contribution of mathematics qua mathematics to the sciences. As such, (1)- (3) concern pragmatic or heuristic contributions that mathematics makes. However, I would need to see evidence that mathematics "shaved off" from the empirical models or theories makes an unique explanatory contribution in the biological sciences. This might be because I generally think of scientific explanation as causal and mathematical explanations as non-causal. But, I am not a philosopher of mathematics and I know what I am saying is contentious.

    Also, to flesh out Steve's point, most models in ecology historically started out as "templates" - here I am thinking of LV, Nicholson-Bailey host-parasitoid, or Levin's metapopulation equations. They are too simplistic to fit to real world data. But, we tweak them through time by adding complexities relevant to specific system types and we have something we can test. From the outside, it seems like numerology - theoreticians are developing models which are never tested. The moral: if you focus on the simple textbook models, then they appear "untestable" but you have to look at the more complex, specific models to see confirmation and disconfirmation at work and it takes theoreticians time to get from the textbook to data. Steve may disagree with this...

  8. André: I don't see why some but not other inference rules should matter for counterfactual inferences. The difference rather is with the status of the descriptions, say, equations. If a description holds subjunctively, it has to support counterfactual reasoning. Accordingly, anything that can be inferred from it has to hold in counterfactual cases. If, on the other hand, the description only holds indicatively, it might be false in a different situation, and inferences from the description can accordingly be false, too.

    I don't know Pearl's argument (that's Judea Pearl, right?). But I'd be surprised if it could be used to distinguish between subjunctively and indicatively true descriptions just by looking at the descriptions, because in that case she would have succeeded in identifying law-like (subjunctive) generalizations by syntactic means. I think the identification of causal and subjunctive descriptions is illicit: Even if a description holds subjunctively, it does not have to describe causes or interventions, and accordingly does not have to represent any asymmetry.

  9. This is a naive question (prompted by reflection on Jay's comment on the implied contrast between mathematical vs causal explanations -- a distinction that goes back to differences between Galileo & Descartes, by the way): how good/important are mathematical models in representing mechanisms (in say, Craver's sense) in biological sciences?

  10. Since the discussion has already expanded into biology in general, here's an interesting case in point. I honestly can't see how this paper could possibly be de-mathematized and communicate its message:

    "Statistical physics of RNA folding"