The reasonable effectiveness of mathematics... for Ptolemy

The class I am teaching this term covers the emergence of Early Modern philosophy and science. The first five weeks are devoted to a whirlwind tour of Ancient Greek natural philosophy (plus a bit of Renaissance thought), and the last 10 weeks cover 17th century philosophy and the scientific revolution.

We spent half of the past week discussing Ptolemy, and I was struck by something that I had noticed before, but never really appreciated. It is very natural for Ptolemy to use fully 'mathematized' explanations for astronomical phenomena, but not for (most) other physical processes. Why? On Ptolemy's view, astronomical objects share more properties with mathematical objects than they do with terrestrial objects. He thought that astronomical objects are eternal and their properties are unchanging -- like the number 5, but unlike terrestial ones. We give a mathematical treatment of astronomical phenomena because they exhibit properties of mathematical objects.

The application of mathematical methods in Ptolemaic astronomy helps bring into focus the so-called problem of the unreasonable effectiveness of mathematics, which some days appears to me to be an unequivocal pseudo-problem. Ptolemy's application of mathematics to physical phenomena, I think, appears extremely well-justified compared to our own: astronomical phenomena can be mathematized because they share peculiar features with mathematical objects, features that the mundane, material objects in our immediate surroundings lack. During and after the scientific revolution, we preserved and expanded Ptolemy's mathematizing proclivities, but we apparently relinquished his justification for treating the natural world mathematically.

Update (10/02/05): Kenny over at Antimeta just put up an interesting post on the (un)reasonable effectiveness of mathematics too, and it is in (small) part a comment on my post.


Einstein and the Units of Selection

No, the title of this post is not a typo. I just finished reading through the first three articles in the most recent issue of Philosophy of Science. They are an argument-response-rebuttal between Elisabeth Lloyd ("Why the Gene Will not Return"; "Pluralism without Genic Causes?") and Ken Waters ("Why Genic and Multilevel Selection Theories Are Here to Stay"), who is one of her targets in the original essay. As the biologically-inclined among you will have inferred, this is the latest installment in the long-standing units of selection debate; very roughly, the question in these debates is: Upon what does natural selection operate? Organisms? Genes? Groups of organisms?

Ken Waters' basic response to this question -- which he first articulated in "Tempered Realism about the Force of Selection" (Philosophy of Science 1991) -- is that there is no determinate fact of the matter about whether selection is really acting at the level of the gene or the organism/ genotype. Mathematical models can be constructed in terms of genes and in terms of genotypes, and both kinds of model suffice to represent the facts of dynamic changes in populations. (See "The Dimensions of Selection," P. Godfrey-Smith and R. Lewontin, Philosophy of Science 2002, for an excellent treatment of the niceties of of the situation.) Since these different models do not represent different facts, Waters concludes that we will choose between them on pragmatic grounds. In the language of his current paper, Waters says that different models "parse" the causal structure differently.

For the purposes of this post, I will assume Waters is correct to maintain that there is no fact of the matter about whether the true cause of any particular evolutionary change lies at the level of the gene or the genotype. What I want to do is to compare this situation with Einstein's reaction in (what I consider) an analogous situation.

At the beginning of Einstein's 1905 paper that introduces special relativity, he asks us to imagine a conductor and a magnet in relative motion with respect to each other. If the take the conductor to be at rest and the magnet moving, then Maxwell's theory says that an electromotive force is generated in the conductor, which gives rise to an observable electric current C. If, on the other hand, we assume the conductor is moving and the magnet is at rest, then Maxwell's theory says that no electromotive force is generated in the conductor, but an electric field is generated around the magnet -- and this field induces exactly the same electric current C as before. Einstein's conclusion is that we are not actually dealing with two physically different situations here; rather, our theoretically distinct models are representing one and the same set of facts. This is exactly Einstein's argumentative maneuver in his famous elevator thought-experiment as well: though the pre-Einsteinian theory would distinguish between the cases in which I am being uniformly accelerated through a gravitation-free region and in which I am at rest in a homogenous gravitational field, Einstein maintains that there is in fact no difference between these two cases. This is (one version of) the Principle of Equivalence.

Note that Einstein does not say is that 'we choose between the competing descriptions of the magnet-and-conductor case on pragmatic grounds,' or that 'we parse the causes differently: either as an electromotive force or as a electric field.' Rather, he re-arranges the permitted causal structures of the theory to eliminate these pseudo-differences, so that the theory no longer "leads to asymmetries which do not appear to be inherent in the phenomena." He replaces the separate categories of 'inertial effects' and 'gravitational effects' with a single category (which we could call gravitational-inertial effects) via his principle of equivalence.

What I am curious about is whether Einstein's maneuver can be carried over into the biological case. I am hoping someone better-informed than I am can tell me why this has no prayer of working, or why Einstein's cases are not analgous to the situation in evolutionary biology. Of course, I wouldn't mind hearing suggestions for how this might work, either.

Editorial note. Posting here will probably be sporadic for the next few months: I am going on the job market this year, and that process has been (and, I imagine, will continue to be) time-consuming.


Specialization and collaboration, again

Yesterday Paul Hoyningen-Huene presented a talk entitled "What is Science?" at the Center for Philosophy of Science here. He intends the question in his title to be taken in a very general way, so his target is one of those Big Questions that, in my last post, I bemoaned as a dying breed in our climate of increasing specialization.

Prof. Hoyningen-Huene pointed out a discouraging fact for anyone who wants to attempt an answer to the Big Questions in the philosophy of science and simultaneously remain reasonably close to actual scientific practice: according to Thomson ISI (the citation management company), there are 170 categories of natural science, 54 in the social sciences, and 15 in the formal sciences -- not including subdisciplines, which can vary widely. So if someone makes a general claim about science or scientific practice, and wants to check that claim thoroughly, then 239 different categories of scientific activity -- most of them complex and varigated -- must be checked.

I feel pulled in two directions by the existence of these 239 categories. On the one hand, it seems that collaboration is the only means to make headway on the Big Questions. On the (not-quite-mutually-exclusive) other, it seems likely that the Big Questions just won't admit of anything approximating a (reasonably) general answer. (Hoyningen-Huene's strategy is to describe several examples drawn from across several scientific disciplines that support his thesis, and assert that these examples are paradigmatic.)

Finally, Kieran Setiya has also recently posted about specialization in philosophy over on his blog, Ideas of Imperfection. Since he's much smarter than I am, I recommend you read his post.


Specialization and collaboration

Over the past few decades, philosophy -- and philosophy of science in particular -- has become increasingly specialized: we have philosophy of quantum field theory, philosophy of developmental biology, etc. It seems that even the so-called "generalists" in philosophy of science are becoming a more and more self-contained group. (For example, I went to a session entitled "Confirmation" at the last Philosophy of Science Association meeting, and I had a very difficult time understanding what was being discussed, at least in part because there was a lot of specialized jargon and assumptions shared by the experts used without explanation -- though my limited brainpower certainly played its part in my incomprehension.)

In general, I think this trend of specialization is a Good Thing, primarily because it has led to specific results that we might not have found otherwise. (Thus I disagree with Karl Popper's claim: "For the scientist, specialization is a great temptation, but for the philosopher, it is a mortal sin.") But I think specialization also has its costs -- in particular, we tend to bypass answers to bigger questions. The question "What is a scientific explanation?" is replaced by "What is explanation in quantum information theory?" or "What is an evolutionary explanation?" and so on. (I think both of those questions are very interesting and philosophically important ones!) The philosopher of biology is uncomfortable talking about explanation in the physical sciences, and the philosopher of physics feels likewise about explanations of biological phenomena -- and the generalist is busy worrying about 'grue'some predicates, the barometer and the thunderstorm, or the irrelevant conjunction problem to deal with explanations in particular sciences. (I think this may in part explain why philosophy of science survey classes often begin with writings of logical empiricists: they tried to give genuinely general accounts of notions central to science.)

In keeping with the generally naturalist spirit of philosophy of science and this blog, we can ask ourselves: What Would Scientists Do? Scientists these days are hyperspecialized, and publish their hyperspecialized research in increasingly specialized journals. However, they also answer bigger, broader questions as well, via collaboration with scientists outside their specialty. So I wonder whether the time is ripe now for philosophers of science, armed with the insights about their particular sub-disciplines amassed over the last few decades, to begin collaborating to answer some of the bigger questions again. And the collaborations need not end there -- philosophers of science could also collaborate more with folks working within epistemology and metaphysics proper, or other fields.

I imagine many will say that we have overthrown the logical empiricist myth that there is a single thing, explanation, or confirmation, or even science. I am open to the idea that these might be myths. But I think we should check whether this is the case -- and if they are mythical, we can at least gain clarity and specificity about what the differences are between e.g. the explanatory patterns of physics and biology.