If Mathematics is Informal, Then Perhaps we Should Accept That Economics Must be Informal Too

Discussions of formalism in economics are bedevilled by confusion over what the word means. Though finer distinctions may be possible, it will suffice to distinguish three meanings of the term, which I will refer to here as axiomatisation, mathematisation and a third category that can be termed methodological formalisation. Axiomatisation is perhaps the best understood term. It involves reducing a body of knowledge to a set of independent axioms, with all propositions being derived from those axioms using well-defined logical rules. If a statement is well-defined, it is in principle possible to state unambiguously whether or not it follows from the axioms or is inconsistent with them. Proof is simply a matter of applying logical rules. It is a mechanical process. It is the approach of Debreu (1986, 1991) and has, as Weintraub (1998) points out, roots in a specific view of mathematics.1 More general is mathematisation: the use of mathematical techniques (geometry, algebra, set theory, topology) in economic arguments. This definition is informal,2 but it has the advantage of corresponding exactly to what people have in mind when they talk of the mathematisation of economics in the last fifty to sixty years. Put crudely, we are referring to the rise in the proportion of articles in this JoURNAL and the AER that use algebra from no more than 10% in 1930 to around 75% in 1980 (Backhouse, forthcoming; Middleton, 1998). This change is significant in that it signals a different mode of arguing a different rhetoric. Note that these remarks apply equally to economic theory and empirical or econometric work. Finally, there is methodological formalisation, defined as the use of an agreed set of methods for the solution of certain types of problem. Economics has become methodologically more formal during the twentieth century in