January 19, 2016

Axiomatics — the heterodox bugbear

Comment on Lars Syll on ‘Axiomatics — the economics fetish’


Why can heterodox economists not get their heads around the methodological fact that axiomatization and empiricism are not mutually exclusive but the two sides of the coin called theory?

“In Einstein's words, geometry constituted one of the oldest physical theories. In the preface to his Principia Newton treats geometry as a branch of mechanics, i.e. as a branch of physics: Therefore geometry is founded in mechanical practice and is nothing but that part of universal mechanics which accurately proposes and demonstrates the art of measuring.” (Zahar, 1980, p. 3)

That axioms are strictly empirical was obvious to the economics founding fathers: “... in Mill's view, the axioms of Euclidian geometry are inductive.” (Whitaker, 1975, p. 1038)

The original commonsensical realism of axioms got out of sight in the process of ever-increasing refinement, precision, idealization, and the ever-increasing requirement of logical consistency.

Historically, axioms as the basis of subsequent deduction are themselves arrived at by intuitive induction. They are neither plucked out of thin air nor arbitrary. A direct proof of the fact that axioms are empirical is that Gauss set out to test them “One of the most famous stories about Gauss depicts him measuring the angles of the great triangle formed by the mountain peaks of Hohenhagen, Inselberg, and Brocken for evidence that the geometry of space is non-Euclidean.” (Brown, 2011, p. 565)

Axioms are ‘realistic’ but in a highly refined form derived from the deeper reality: “... the repugnance which it [the axiomatic method] still meets here and there can only be explained by the natural difficulty of the mind to admit, in dealing with a concrete problem, that a form of intuition, which is not suggested directly by the given elements (and which often can be arrived at only by a higher and frequently difficult stage of abstraction), can turn out to be equally fruitful.” (Bourbaki, 2005, p. 1275)

Axiomatization is not an easy thing as more than 2500 years of history since Euclid testify and it is quite natural to get it wrong. This, clearly, is not a fault of the method but of the user: “My opinion continues to be that axiomatics, like every other tool of science, is no better than its user, and not all users are skilled.” (Clower, 1995, p. 308)

It is correct of Heterodoxy to insist that something went wrong with Debreu’s axiomatization of general equilibrium (Ackerman et al., 2004). The obvious error/mistake is that there exists no such thing as an equilibrium in the economy. It is simply an inexcusable methodological blunder to axiomatize a nonentity. As Clower said, not all users are skilled.

It is correct of Heterodoxy to reject the neo-Walrasian axioms. But to reject axiomatization as a method is stupid and ultimately self-defeating. Axiomatics is not a fetish but an indispensable tool to secure logical consistency. The classical economists understood this well.

“To Senior belongs the signal honor of having been the first to make the attempt to state, consciously and explicitly, the postulates that are necessary and sufficient in order to build up … that little analytic apparatus commonly known as economic theory, or to put it differently, to provide for it an axiomatic basis.” (Schumpeter, 1994, p. 575)

Because neither Senior nor Debreu nor Heterodoxy has been successful, this task is still unfinished.#1 And this is why economics is, until this day, not much more than storytelling, ad hocery, and persistent confusion at the proto-scientific level.

Egmont Kakarot-Handtke

Ackerman, F., and Nadal, A. (Eds.) (2004). Still Dead After All These Years: Interpreting the Failure of General Equilibrium Theory. London, New York: Routledge.
Bourbaki, N. (2005). The Architecture of Mathematics. In W. Ewald (Ed.), From Kant to Hilbert. A Source Book in the Foundations of Mathematics, Vol. II, 1265–1276. Oxford, New York: Oxford University Press. (1948).
Brown, K. (2011). Reflections on Relativity. Raleigh: Lulu.com.
Clower, R. W. (1995). Axiomatics in Economics. Southern Economic Journal, 62(2): 307–319. URL
Schumpeter, J. A. (1994). History of Economic Analysis. New York: Oxford University Press.
Whitaker, J. K. (1975). John Stuart Mill’s Methodology. Chicago Journals, 83(5): 1033–1050. URL
Zahar, E. (1980). Einstein, Meyerson and the Role of Mathematics in Physical Discovery. The British Journal for the Philosophy of Science, 31(1): 1–43. URL

#1 Towards the true economic theory
Related 'Lars Syll creatively destructs Wren-Lewis' and 'Axiomatized nonentities and the failure of methodologists'. For details of the big picture see cross-references Axiomatisation and cross-references Paradigm shift.

Immediately following Economists’ proto-scientific methodology.