Comments on: Existence Theorems are Reductios

By: Walt Pohl

Walt Pohl — Mon, 08 Sep 2003 04:16:07 +0000

I disagree with most of the comments here. The existence of general equilibria is not a priori obvious under any conditions. If general equilibria didn’t exist even for perfect markets, then that would be a significant fact. I didn’t know about the “no money” condition, though. How does the proof depend on that?

By: zizka

zizka — Mon, 08 Sep 2003 03:45:05 +0000

In mathematics I have read people talking about certain solutions or proofs being “possible in principle”, in the sense that they are not impossible like squaring the circle, but would merely require a computer half the size of the known universe to solve. Mathemeticians also speak of “very large numbers” which are still not infinite and thus can be handled just the same as ordinary numbers: for example, if you take the estimated number of electrons in the universe, square it, and keep squaring the result 100 times or so. I agree with the skeptical conclusion that proving that something is formally possible isn’t proving much in a science which is meant to have any real- world (world of historical time, world of historical experience) applications.

By: Michael Kremer

Michael Kremer — Mon, 08 Sep 2003 01:19:11 +0000

But now (after reply to Juan) your terminology (“existence theorems”) is confusing. Amplifying on Juan’s point— the result that IF such and such, then there is a general equilibrum, is an “existence theorem”. The result that the existence of a general equilibrium REQUIRES such and such is NOT an “existence theorem.” It’s really of the form “if there is a general equilibrium, then such and such” and that assumes existence as the antecedent of the if-then, rather than proving it as the consequent. But this means it’s really already an impossibility theorem in disguise. For, notice that: Conditions C are necessary for a K to exist is equivalent to If a K exists, then conditions C hold which is equivalent to If conditions C fail to hold, then no K exists which is an IMPOSSIBILITY theorem. I think that what you’re really noticing is that the search for an existence theorem can lead to both necessary AND sufficient conditions for the existence of something, in other words, a claim of the form: a K exists if and only if conditions C obtain which is really both an existence theorem (if conditions C, then a K exists) and an impossibility theorem (if conditions C fail, then no K exists) at once. This, however, isn’t a defence of existence theorems, but of the search for them.

By: Kieran Healy

Kieran Healy — Mon, 08 Sep 2003 01:06:50 +0000

that a result follows from certain assumptions doesn’t imply that those assumptions are necessary to generate the result. Yes, that’s right, I fudged over that in the post. (I plead “It’s a blog, not the AER.”) The promise of many such theorems is that we’ll find a way to prove a more general case later. Often this never happens.

By: Juan

Juan — Mon, 08 Sep 2003 00:44:47 +0000

Just a small point: that a result follows from certain assumptions doesn’t imply that those assumptions are necessary to generate the result. That may, of course, be true, but it’s being true (if it is) is not part of the proof of the result in question. In general, reductios work when the absurdity is a necessary condition of the result to be reduced, not when it is a sufficient condition. I might just be missing something.