"...Mirowski thinks that only computable mathematics should be used in economic modeling. He is particularly scathing about the fact that the Walrasian correspondence is non-computable, and rejects out of hand the response that Scarf has provided a computable algorithm for finding approximate Walrasian equilibria. If I understand him correctly, the reason is that an approximate Walrasian equilibrium may not be an approximation to an exact Walrasian equilibrium. But this is to misunderstand how mathematical modeling works - not only in economics, but in physics as well.
The equilibria of physics are no more computable than those of economics. It is not just that Scarf cannot think of an algorithm that computes them, but that there is nothing whatever the universe as a whole can do to instantiate them. So how come the non-computable models of physics work? The reason is that they provide mathematically tractable approximations to the way the universe really is. Similarly, in economics, it is wrong-headed to reject Scarf's algorithm because it does not adequately approximate the Arrow-Debreu model. On the contrary, we need to elaborate the Arrow-Debreu model so it approximates better whatever the real-world analogue of Scarf's algorithm happens to be in any particular context. Whether the elaboration turns out to be computable or not matters not in the least. If you think it does, consistency also demands that you advocate abandoning the use of calculus and most other mathematical tools in physics."
- Kenneth Binmore (2004). "A Review of Philip Mirowski's Machine Dreams", Journal of Economic Methodology, V. 11, N. 4 (Dec.): 477-483
- John B. Davis (2004). "Economists' Dreams: Machine Dreams", Journal of Economic Methodology, V. 11, N. 4 (Dec.): 483-488
- Matthias Klaes (2004). "Algorithmic Economics: A Plea for Natural Economic History", Journal of Economic Methodology, V. 11, N. 4 (Dec.): 489-498
- Philip Mirowski (2004). "Philosophizing with a Hammer: Reply to Binmore, Davis and Klaes", Journal of Economic Methodology, V. 11, N. 4 (Dec.): 499-513