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...


Steve Petersen said...

Interesting post - I'm interested in unification models of explanation, but never even really thought about mathematical explanation or its differences with the scientific. From outside that literature, a natural instinct on your question, I would think, is that "explanation" is really what's behind induction broadly construed (as Harman first claimed in his IBE paper) - in other words, explanation is our method of inference when we don't have available the deductive methods of math. It may be easy to forget that your puzzle disappears if inference in math isn't "explanation" at all. I can't access Mancosu's recent Topoi article on the topic from my small college, nor the apparently seminal Steiner paper - can you give any easily accessible internet sources for why we should think of math as explanatory?


Greg Frost-Arnold said...


Thanks for stopping by and leaving a comment! I unfortunately don't know of anything accessible everywhere -- in fact, this topic is pretty new to me, and furthermore I hav the sense that the literature on mathematical explanation is orders of magnitude smaller than that on explanation in the empirical sciences.

I certainly sympathize with your intuition that perhaps proofs don't explain at all. But on the other hand, there's a couple things to note: (1) practicing mathematicians do in fact consider some proofs more explanatory than others -- i.e., some proofs merely establish the fact that p, whereas others genuinely explain why p. And someone working under the banner of naturalism usually takes such facts about working mathematicians seriously. (2) Under the old-school, i.e. deductive-nomological (D-N), model of [scientific] explanation, proofs will come out as explanations (or at least they'll be close; presumably, we don't want to demand that mathematical explanations include at least one law of nature, as the Hempel D-N account does).

But, all that said, I still do wonder whether there is a reasonably well-defined target explicandum here -- especially since (unlike in the empirical sciences) there is no consensus among the mathematical community as to what counts as a better or worse explanation... that especially seems to make problems for (1) in the previous paragraph.

Steve Petersen said...

(Thanks - yes I can see how different proofs could be said to be more or less explanatory. But in that case I'm not sure there's asymmetry with standard "scientific" cases of explanation, especially under the unification banner I prefer. I'll try to dig up the Steiner and ponder on it.)

(Oh, and congrats on the successful defense!)