This question has a philosophical bent, but hopefully it will evoke straightforward, mathematical answers that would be appropriate for this list (like my earlier question about beautiful proofs appropriate for high school level.)
The background is this: we routinely distinguish between proofs that explain and proofs that demonstrate. This distinction has been around at least since Aristotle's time, but it is an open question, for instance in contemporary philosophy of mathematics, what explanation really means. One recent suggestion has been that explanation might be related to beauty. It seems reasonable that explanatory proofs are nicer than non-explanatory ones in SOME way, but are they necessarily more beautiful? And similarly, is beauty necessary for explanation? It seems a good way to attempt to answer these question is to look at a bunch of good examples. And it seems a good way to get examples is to ask mathematicians, which is why I post the question here.
To clarify: the question is not at all to discuss the nature of explanation or beauty (if you want to discuss, I can give you my email address and we can chat offline). The purpose to is collect some good mathematical examples that help understand the relation between beauty and explanation in mathematics.
Examples that would be relevant: beautiful proofs that are not explanatory, explanatory proofs that are not beautiful.
Thanks in advance.
Best Answer
A proof that many people say they find beautiful, but in my view is not at all explanatory, is Zagier's one-sentence proof of the sum of two squares theorem.