It is now generally agreed that economic theory, combined with new data as well as historical, statistical and mathematical methods are necessary at the theoretical level, to formulate problems precisely, to draw conclusions from postulates and to gain insight into workings of complicated processes and, at the applied level, to measure variables, to estimates parameters and to organize the elaborate calculations involved in reaching empirical results.
Yes: it has been generally agreed since Koopmans' Three Essays in 1957. But I myself have found repeatedly that "general agreement" is, roughly half the time, a signal that people have not thought through the issue, and are mistaken about it, and don't want to hear anything more about it, and will get indignant and scornful if someone asks them to rethink it. The date of Koopmans' manifesto should make one wonder whether a methodological framework set up long, long before we knew what we were doing in either mathematical theory or in econometrics, and in contradiction to every other scientific program (physics, for example, which depends not at all on Bourbaki-style axiomatization or on statistical significance, even in fields like astronomy with observations similar to ours), is quite what we need now in 2011.
I've written a good deal on these matters, as in Knowledge and Persuasion in Economics, 1994, but have no illusions about how much my colleagues have seriously thought through what they are doing, or what influence my simple-minded doubts have had. My conclusion after looking into it with care is that the double claim that Koopmans made—that one branch of economics should produce theorems and the other test the implications with econometrics—sounded plausible in 1957 but has not succeeded. No wonder it hasn't, as I say, since no other science has proceeded in such a fashion.
The claim to get "insight" (the word economists always use without explaining what scientific use it would be) from an unbounded search through the hyperspace of postulates is unpersuasive. What insight does one get from the conclusion that if one postulates A one can deduce C, and if one postulate A' one can deduce C', and if one postulates A" one can deduce C", in an unbounded series? The unboundedness is plain, for example, in the theory of international trade, or in game theory: 150 solution concepts (and counting); the Folk Theorem implying that an infinite game has an infinite number of solutions. The way forward is economic thinking; ideas, sometimes verbal (Coase), sometimes mathematical (Samuelson), not diverted into pointless exercises of postulate and proof.
"Precision" and axiomatic consistency are very minor intellectual virtues. There was a famous exchange on the matter in 1939 between Wittgenstein and Turing. Turing the mathematician (he soon proved his worth in applied matters) kept challenging Wittgenstein the philosopher (whose scientific training was as an aeronautical engineer) on consistency and contradiction. Wittgenstein concluded:
Wittgenstein: Why are people afraid of contradictions? It is easy to understand why they should be afraid of contradictions, etc., outside mathematics. The question is: Why should they be afraid of contradictions inside mathematics? Turing says, 'Because something may go wrong with the application.' But nothing need go wrong. And if something does go wrong — if the bridge breaks down — then your mistake was of the kind of using a wrong natural law. ...
Turing: You cannot be confident about applying your calculus until you know that there are no hidden contradictions in it.
Wittgenstein: There seems to me an enormous mistake there. ... Suppose I convince Rhees of the paradox of the Liar, and he says, 'I lie, therefore I do not lie, therefore I lie and I do not lie, therefore we have a contradiction, therefore 2 x 2 = 369.' Well, we should not call this 'multiplication,' that is all...
Turing: Although you do not know that the bridge will fall if there are no contradictions, yet it is almost certain that if there are contradictions it will go wrong somewhere.
Wittgenstein: But nothing has ever gone wrong that way yet...Wittgenstein, Lectures on the Foundations of Mathematics, Cambridge 1939
University of Chicago Press, 1976, pp. 217-218
Mathematicians since Bolzano, Cauchy, and Abel have admired lack of contradiction extravagantly. Yet notice that calculus was used for a century and half with great scientific success before it was axiomatized and proven free of contradiction (at any rate of one accepts the Law of the Excluded Middle). We scientists, as against philosophers and theologians and mathematicians (all of whom in their own fields I sincerely admire), should care chiefly about relevance, richness, experiment, oomph. Existence theorems are not economic thinking. They are scientifically irrelevant ceremonies in the Hilbert Program—which didn't fully succeed even in mathematics.
The claim "at the applied level" to "estimate parameters" is equally unpersuasive. It's the other half of Koopmans' division of labor. I do not mean that empirical economics in studies of, say, national income or the market for autos has had no results (no more than I mean that the economic thinking we call theory has had no results—though never without confrontation with quantified facts or classifications does it bear scientific, as against philosophical, fruit). I'm not "against theory" or "against mathematics" or "against econometrics" or "against curve fitting" or "against statistics": or "against testing." I'm against the two detailed procedures that economists uniquely among serious scientists have adopted since Koopmans (existence theorems in theory and testing by statistical significance without a loss function in econometrics). I say that they have had no results.
I expect any economist of open mind will agree on statistical significance if she reads The Cult of Statistical Significance and on existence theorems and axiomatization if she reads the relevant chapters in Knowledge and Persuasion. A version of both points, directed at non-economists, is The Secret Sins of Economics, available by googling. The way forward in applied economics—which is all of economic science, as against economic philosophy—is measurement, experiment, history, thinking, graphing, simulation, and, sometimes, hyperplane fitting. "Testing" in the form recommended since Lawrence Klein has been phony, and has led nowhere. Ask yourself, oh economist: have tests of significance persuaded any economist of your acquaintance to change her mind on any subject of importance? Can't think of anyone? I'm not surprised!