Draft: Scientific Models and Representation

For those interested, I have uploaded the penultimate draft of my entry on scientific models and representation for The Continuum Companion to the Philosophy of Science (edited by fellow blogger Steven French and Juha Saatsi). (You can get a copy of it here)

The piece is meant to be a user-friendly introduction to this very interesting but somewhat baffling topic. As usual, comments (either here or by e-mail) are greatly appreciated.

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....

Bas van Fraassen's Scientific Representation: Paradoxes of Perspectives

As many of you will already know, Bas van Fraassen's new book Scientific Representation: Paradoxes of Perspectives (OUP 2008) has been out for a few months and I highly reccommend it to all of you who haven't read it already.

If you are interested and you would like to have some idea of what the book is about, it looks like a few it'sonlyatheorists have been busy reviewing it lately. So, here is a review of it by Ron Giere, here one by Steven French, and here is one that I've written for NDPR. (If I missed anyone, please let me know!)

Also, Bas has kindly invited me to be one of the contributors to a book symposium on the book that will appear some time next year in Analysis and I'm considering developing some of the points I raise in my review in that piece. So, if you have any comments about my review, please let me know (either commenting on this post or by e-mail).

The Semantic View of Theories Again

I was thinking more about Gabriele's post on the semantic view of theories. Consider this quick argument in favour of a semantic view of theories:

1. Scientists believe (or at least accept) scientific theories;
2. Attitudes like belief and acceptance are held to propositions (certainly belief and acceptance ascriptions involve embedded 'that' clauses which seem to denote propositions);
3. So, scientific theories are propositions.

This is pretty clearly a semantic view of theories. But Gabriele, and I presume others, seem not to believe this is the semantic view of theories, as discussed in the literature. Could someone explain the difference, if there is one? For I do not at this stage see it, for reasons I'll now explain.

The conclusion of this argument is what I have always understood to be the semantic view of theories. If propositions are structured (i.e., not just sets of possible worlds), then the propositions which express scientific theories can easily be models in the model-theoretic sense; if theories have merely qualitative content, no one model will capture the proposition expressed (as qualitatively indistinguishable models will equally satisfy the theory), so the proposition expressed should be a set of models. (Things seem to be a little, but not much, trickier if propositions are unstructured.) In any case, there seems to be a clear correspondence between the propositions expressed by the sentences of some presentation of the theory and the models which satisfy those sentences, a close enough correspondence that reducing the one to the other doesn't seem unreasonable.

Models and Fiction

In a forthcoming paper "Models and Fiction", Roman Frigg gives an argument for the view that scientific models are best understood as fictional entities whose metaphysical commitments are “none” (17). I think this argument is a new and important one, but I don’t agree with it. Frigg first considers the view that models are abstract structures. He points out that an abstract mathematical structure, by itself, is not a model because there is nothing about it that ties it to any purported target system. But "in order for it to be true that a target system possesses a particular structure, a more concrete description must be true of the system as well" (5). The problem is that this more concrete description is not a true description of the abstract structure and it is not a true description of the target system either in the case if idealization. So, for these descriptions to do their job of linking the abstract structure to their targets, they must be descriptions of "hypothetical systems", and it is these systems that Frigg argues are the models after all.

My objection to this argument is that there are things besides Frigg’s descriptions that can do the job of linking abstract structures to target systems. A weaker link is a relation of denotation between some parts of the abstract structure and features of the target systems. This, of course, requires some explanation, but a denotation or reference relation, emphasized, e.g. by Hughes, need not involve a concrete description of any hypothetical system.

(Cross-posted with Honest Toil.)

Two Cases of Underdetermination?

Bryan at Soul Physics has an interesting post about underdetermination here.

What Was Wrong With the Syntactic View of Theories Exactly?

These days most philosophers of science (PoSs) seem to subscribe to the semantic view of scientific theories, according to which scientific theories are collections of models (the question obviously become what kind of thing a scientific models is. For my take on this question see here). In the heydays of logical empiricism however, the prevailing view was the so-called syntactic view of theories, by which I mean here the view that scientific theories were collections of sentences. Logical empiricists, unfortunately, saddled this view with a host of other less plausible views about language and truth, which most philosophers today seem unwilling to accept. However arguments against such views are not arguments against what I call the syntactic view. So, was the rejection of the syntactic view a case of guilt by association or are there any serious arguments against the view itself (rather than the views that were usually held in conjunction with it)? If not, what are the arguments in favour of the semantic view (other than its supposedly being more empirically adequate)?
Thinking about it the only serious argument that I can think of that seems to target what I call the syntactic view (as I am intending it here) is the one according to which the same scientific theory can be formulated by using different sets of sentences (e.g. in English and French or in Lagrangian and Newtonian terms) and, therefore, the theory cannot be identified with any set of sentences. But what if we substitute sets of sentences with sets of propositions? (Would this work in the case of Newtonian and Lagrangean mechanics or would one have to say that the two are distinct theories?) The only obstacle I can see to this way of recasting the syntactic view this way was the logical empiricists' prejudice against propositions. But I don't see any reason to think of propositions as being more metaphysically mysterious than sentences (utterances are physical events but sentences like propositions seem to be abstract entities).
Am I missing something major?