This essay is the Nobel prize speech of Debreu, who is one of the first-ballot Economics Hall of Famers of the 20th century. (I’ll take Marshall, Fisher, Keynes, Ramsey, Hotelling, Hicks, von Neumann, Savage, Nash, Samuelson, Arrow, Becker, Aumann and Myerson. You can see my obvious microeconomic bias: am I missing anyone here? I’m open to the cases for Wald, Friedman, Lucas and Harsanyi…) The bulk of the essay is a description of the development of general equilibrium, from the discovery of existence through to n-replica economies and Sonnenschein-Mantel-Debreu’s disappointing theorem. It is Part II that I find more interesting, though. Here, Debreu discusses why economics has been formalized mathematically (and this is a question which any economist ought be able to answer!)
He gives two main reasons. First, mathematics is an efficient language for unambiguous communication among economists and among other social scientists; we needn’t worry about technical terms being misunderstood when theorems are stated mathematically. Debreu studied mathematics under the influence of Hilbert and the Bourbaki school, so this argument seems taken directly from the analogous argument in math. Second, axiomatization provides “secure bases from which exploration could start in new dimensions” without requiring new researchers to question every statement made by previous researchers; as long as we agree on the axioms, the conclusions are deductions, and can be accepted. This is a very Kuhnian view of social science, with an axiomatic system playing the role of the field’s paradigm.
I agree that these reasons are at the heart of why economists formalize, but I do not think they are very compelling reasons. The problem lies at the heart of the difference between social science and mathematics. In an axiomatic mathematical system, the axioms are by assumption true – there is no sense in which an axiom can be false, since it is merely an abstract statement used to derive implications. That is, if I work on a program in Euclidean space, the very definition of that space means that I accept Euclid’s axioms. Social science is not this way. We know, pace the arguments of Lionel Robbins, that our assumptions are false, though we hope that they are in some sense “good enough” to derive implications (many economists will disagree with that last sentence, and instead argue that economics is a study of abstract relationships, and that deductions from axioms are merely mathematical deductions, but this “overformalization” has been countered many, many times – most famously, by von Neumann in an essay I may discuss on this site at a later date). If axioms are only approximate, though, as in the standard Humean problem of induction, we have no way of knowing whether conclusions will also be approximate; there is no “universal continuity”. I think economists would be better served to think of axiomatization as the formalization of analogies. That is, an axiomatic deduction when axioms are imprecise may tell us nothing about the real world, but it tells us as much as a qualitative analogy, and does so in a formal way that deemphasizes the rhetorical ability of the author. I am working on further results along these lines, so hopefully within the next year I’ll have an expansion of the above idea on these pages.
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.121.4168&rep=rep1&type=pdf (it is not obvious from the pdf, but this speech was transcribed in AER 74.3 (1984))