Mancosu and mathematical explanations
Last Friday, Paolo Mancosu was in Pittsburgh to give a talk on explanation in mathematics. His visit gives me the opportunity to correct an oversight in my last post -- Paolo helped me improve my dissertation substantially: he read an early partial draft very carefully, and brought his learned insight to bear on it. He is the only person in the universe who has written on the specific topic of my dissertation, and his comments were extremely helpful.
The basic claim of Paolo's talk was that Philip Kitcher's account of mathematical explanation falls afoul of certain apparently widely-shared intuitions about which proofs are explanatory and which are not. But I was intrigued by something else mentioned in the talk, which came to the fore more in the Q&A and dinner afterwards. When working on explanation in the natural sciences, there seems to be much more widespread agreement about what counts as a good explanation than in the mathematical/ formal sciences. That is, whereas most practitioners of a natural science can mostly agree on which purported explanations are good and which not, two mathematicians are much less likely to agree on whether a given proof counts as explanatory.
So I am wondering what accounts for this difference between the natural and formal sciences. Might this be due (in part) to mathematics lacking the 'onion structure' of the empirical sciences? For example, the claims of fundamental physics are not explained via results in chemistry, and observation reports (or whatever you want to call claims at the phenomenological level [in the physicist's sense]) are not used to explain any theoretical claim, and so on. My intuitions about mathematics are not as well-tutored, but I have the sense that the different branches of mathematics do not have such a clear direction of explanation. (Of course, there is no globally-defined univocal direction of explanation in the natural sciences [the cognoscenti can think of van Fraassen's flagpole-and-the-shadow story here], but there is nonetheless an appreciable difference between math and empirical sciences on this score.) At least in some cases, this clearer direction of explanation probably results from empirical sciences' explaining wholes in terms of their parts -- whereas mathematics lacks that clear part-whole structure. Often two bits of mathematics can each be embedded in one another, but we tend not to find this in the empirical sciences. (The concepts of thermodynamics [temperature, entropy] can be defined using the concepts of statistical mechanics [kinetic energy], while the converse is clearly out of the question.)
Pointing to the onion structure/ clearer direction of explanation in science might be just a re-statement of the original question; I'm not sure. Or maybe it's not relevant. In any case, I have to bury myself beneath a mountain's worth of student essays on the scientific revolution...