As Caldwell notes, Hayek initially thought the dividing line between possible and impossible positivism lay in the distinction between natural sciences and social sciences, but by the 1950s he had come to understand that the issue was really one of complexity. A positivist, predictive science is possible only for phenomena, whether human or natural, that are relatively simple—particle physics, for example. One can never fully model and predict complex phenomena such as the spontaneous orders produced by the interactions of simpler agents. These orders include the human brain, whose higher functions cannot possibly be inferred from its physical substratum, as well as ecosystems and, of course, markets, cultures, and other human institutions.
...Thus, the highly mathematical and ahistorical turn that academic economics has taken in recent years would have been, for Hayek, as much an abuse of reason as the socialist planning of earlier generations.
In discussing the role of math and econometrics, I think that the fundamental issue is how to evaluate an economic argument. How do we decide that on e person's paper is valid and worth publishing, while another person's is not?
If we believe that mathematics can establish the logical validity of an argument, and if we believe that econometrics can establish the empirical validity of an argument, then we have an approach to evaluating papers in economics. Otherwise, we do not appear to have clear criteria.
I believe that economic papers ought to be logical and scientific in spirit, with arguments expressed as precisely as possible. They ought to make falsifiable predictions. However, I believe that math and econometrics are neither necessary nor sufficient for establishing the validity of an economic argument. Thus, to borrow a Hayekian phrase, there is a "fatal conceit" to the sort of technical emphasis that was dominant when I was in graduate school.
For Discussion. If the mathematical approach to economics were eliminated, what would take its place?