Philosophers have long pondered explanation in the natural sciences. If they have ignored it in the mathematical sciences, blame lies perhaps with a lingering distinction between 'matters of fact' and 'relations among ideas', the corollary being that mathematics (belonging to the latter class) has nothing to explain. Platonism, no less than empiricism, has also traditionally stressed the differences between natural science and mathematics. The growing acceptance, however, of continuity between the natural and mathematical sciences -urged by Quine, Putnam, and the present author 1 _ has prepared the way for what follows here.
Mathematical explanation exists. Mathematicians routinely distinguish proofs that merely demonstrate from proofs which explain. Solomon Feferman puts it this way: Abstraction and generalization are constantly pursued as the means to reach really satisfactory explanations which account for scattered individual results. In particular, extensive developments in algebra and analysis seem necessary to give us real insight into the behavior of the natural numbers. 2 Chang and Keisler, to cite two more logicians, propose to 'explain' preservation phenomena -i.e. certain theories are such that submodels of their models of their models are again models, or that unions of chains of their models are models, or that homomorphisms of their models are models -to 'explain' these phenomena "just by the syntactical form of the axioms". 3 For example, they prove that a theory is preserved under submodels iff it has a purely universal axiomatization; under unions of chains iffit has a universal-existential axiomatization; and so forth. Let us explore what is common to mathematical explanations such as these; we can always invoke 'family resemblances' later, if we fail.
An obvious suggestion is to identify explanation with generality or abstraction, as Feferman thinks. There is something general and abstract about complex analysis, which at present provides the greatest insight into the