I think that is the wrong question. The goal of pure mathematical economics isn’t understanding the economy better, it`s understanding economics better. I think in that respect we have learned much. There is no going back to the naivite of the old days. And I don’t know of any aspect we don’t know more now. All the insights brought by game theory came later.

“To put the problem another way: Who has taught us more, Aumann or Schelling?”

Well Aumann has brought us correlated equilibria, the epistemic foundations of game theory, much of the theory of repeated games, a new way to think about decision theory, the agreement-theorem with many economic applications. Economics would be much worse without the contributions of Aumann.

First, I think many of us would agree that there is a thriving market in economics for clever formalism independent of real-world relevance. This has always struck me as a colosal waste of energy.

Second, I am not advocating any particular approach; e.g., “neoclassical economics” (whatever that means), game theory, NK models, etc. Rather, progress in social science (any science, actually) requires unambiguous claims, rigorously derived. Theoretically, one could use natural language to proceed, but practically speaking, humans aren’t very good at keeping their logic straight when working through complex arguments. (Which is clearly why mathematics was invented in the first place.) My claim is simply that, if one wishes to identify general principles using deductive theory, then formal methods, of whatever stripe, are a necessary condition for scientific progress.

Third, this debate is somewhat pointless. I mean this is the sense that we all get to put our theories out there in the marketplace for ideas (however derived) and, at the end of the day, will see which ones are left standing. This suits me fine.

Finally, Mike's conjecture — progress in economics "would have been even slower had the profession attempted to construct theories relying solely upon natural language as has been the case in other areas — is an empirical claim. How would we go about evaluating such a claim? 

If you need any demonstration of the need for formalism in theory-building within the social sciences, take a look at the train wreck that is Strategy. As we are discovering using formal methods, most of what passes for foundational theory in strategy is wrong.

Conversely, consider how formalization completely transformed Finance - both in academic and practice circles.

Progress in economics is slow because the problem domain is hard (as a physicist once pointed out, our “atoms” think). My guess is that it would have been even slower had the profession attempted to construct theories relying solely upon natural language as has been the case in other areas.