Kieran Healy responds to my limited defence of existence theorems with the following challenge to economists

As John says, existence theorems are the negative form of impossibility theorems. The classic existence theorems in economics ÷ such as those for general equilibrium , also due to Kenneth Arrow, along with Gerard Debreu ÷ illustrate the point neatly. We begin with the result . Roughly speaking, Arrow and Debreu wanted to show that supply and demand could be in balance in all markets at once. We then move backward to the assumptions necessary to make possible such a result. These include (1) All individuals are perfectly rational, (2) All trades take place simultaneously and instantaneously, (3) There is perfect information about all markets for all products in all conditions both now and at any point in the future, (4) Money does not exist. With these (and other) assumptions in place, the existence of a general equilibrium can be proved. The proof is striking because the initial assumptions are so implausible, even absurd, but they must all be satisfied together in order for the desirable result to be possible. And so we give up our quest for what we now recognise is a chimera ÷ the idea that our world could ever contain economies capable of general equilibrium.

This attempt at a reductio ad absurdam doesn’t work properly, though, because Kieran is treating the sufficient conditions found by Arrow and Debreu as if they were necessary conditions. A lot of effort (too much, McCloskey would say) has gone into demonstrating that a general equilibrium can exist under weaker conditions than

I’ll grossly oversimplify and make the claim that an existence result for competitive general equilibrium like that of Arrow and Debreu will hold even if all of the conditions mentioned by Kieran are violated, provided only that technology sets are ultimately convex (that is, provided economies of scale run out in the end). Of course, even this condition is implausible and suggests that some activities where economies of scale are unbounded (for example, the dissemination of knowledge) must be excluded. But (if you accept my claim that convexity is the crucial necessary condition) at least the theorem is telling us where to look for problems.

The other main results proved by Arrow and Debreu are the First and Second Welfare theorems showing that
(1) a competitive general equilibrium is Pareto-optimal [you can’t make anyone better off without making someone else worse off]
(2) any Pareto-optimal outcome is the competitive equilibrium arising from some initial allocation of wealth and other endowments
For these theorems, the conditions found by Arrow and Debreu are, for all practical purposes, necessary and sufficient. The Big Argument in mainstream economics is whether the differences between the Arrow-Debreu conditions and the real world (often called, in a rather question-begging fashion, “imperfections”) are large enough to justify extensive intervention by governments that are, themselves, necessarily imperfect.