Tuesday, July 23, 2019

Lars P. Syll — Arrow-Debreu and the Bourbaki illusion of rigour

It's about mathematical economics and its limitations. Don't let the title scare you off. Not at all wonkish (no math), although it helps if have some background in the controversy.

Basically, it's Plato (formalism) versus Aristotle (empiricism). Most mathematicians today are Platonists, while most scientists are Aristotelians. But that is another story.

Lars P. Syll’s Blog
Arrow-Debreu and the Bourbaki illusion of rigour
Lars P. Syll | Professor, Malmo University


AXEC / E.K-H said...


Incompetence — the original sin in economics
Comment Lars Syll on ‘Deductivism — the original sin in economics’

Tony Lawson summarizes: “This emergence of the axiomatic method removed at a stroke various hitherto insurmountable constraints facing those who would mathematise the discipline of economics.”

This is not quite accurate, the mathematization of economics started before Hilbert’s general axiomatization of mathematics: “… economists have long been mathematicians without being aware of the fact. The unfortunate result is that they have generally been bad mathematicians, and their works must fall. Hence the explicit recognition of the mathematical character of the science was an almost necessary condition of any real improvement of the theory.” (Jevons, 1871, PS. 13)

The axiomatization of economics started even earlier “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)

Where Tony Lawson is right is that economists messed the whole thing up, and this resulted in the ongoing ridiculous mathiness discussion.

The key of formalization, axiomatics, or mathiness is “Formal axiomatic systems must be interpreted in some domain … to become an empirical science.” (Boylan and O'Gorman, 1995, p. 198)

Now, Debreu did explicitly the very opposite, that is, he disconnected “Allegiance to rigor dictates the axiomatic form of the analysis where the theory, in the strict sense, is logically entirely disconnected from its interpretations.” (Debreu, 1959, p. x)

It is well-known from where Debreu got his methodological inspiration “It seems clear that Debreu intended his Theory of Value to serve as the direct analogue of Bourbaki’s Theory of Sets, right down to the title.” (Weintraub, 2002, p. 121)

Now, it is pretty obvious that Debreu missed the crucial point of Bourbaki’s axiomatics. “From the axiomatic point of view, mathematics appears thus as a storehouse of abstract forms — the mathematical structures; and it so happens — without our knowing why — that certain aspects of empirical reality fit themselves into these forms, as if through a kind of preadaptation. … It is only in this sense of the word ‘form’ that one can call the axiomatic method a ‘formalism’.” (Bourbaki, 2005, p. 1276)

It was quite clear to Bourbaki that NOT ALL mathematical structures incorporate ‘certain aspect of empirical reality’, which means, that there is a “… whole crop of monster-structures, entirely without application” (Bourbaki, 2005, p. 1275, fn. 9).

Hence, Debreu’s axiomatization of Walrasian General Equilibrium is a monster-structure that is due to Debreu’s misunderstanding of Bourbaki. It is NOT the axiomatic-deductive method that is wrong, it is neoclassical economics that is wrong.

See part 2

AXEC / E.K-H said...

Part 2

Generally, it can be said that the so-called social scientists cannot get their head around the essentials of the scientific method. Tony Lawson is one of them. It is quite obvious that no genuine scientist ever abused the axiomatic-deductive method by declaring green-cheese assumptions like constrained optimization and equilibrium as axioms. In marked contrast, economists never had any scruples to make fools of themselves: “most of what I and many others do is sorta-kinda neoclassical because it takes the maximization-and-equilibrium world as a starting point” (Krugman).

Einstein explained in a short paragraph what the so-called social scientists do not understand until this very day “The basic concepts and laws which are not logically further reducible constitute the indispensable and not rationally deducible part of the theory [= axioms]. It can scarcely be denied that the supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience.” (Einstein, 1934, p. 165)

Note that Einstein treats axioms and experience as two sides of the same coin — just like Bourbaki — no disconnect here!

The fact of the matter is that both orthodox and heterodox economists have not been very skilled users of the scientific method up to now. This explains why economics is a failed science for more than 200 years.

Egmont Kakarot-Handtke

Bourbaki, N. (2005). The Architecture of Mathematics. In W. Ewald (Ed.), From
Kant to Hilbert. A Source Book in the Foundations of Mathematics, Volume II,
pages 1265–1276. Oxford, New York, NY: Oxford University Press. (1948).
Boylan, T. A., and O’Gorman, P. F. (1995). Beyond Rhetoric and Realism in Economics.
Towards a Reformulation of Economic Methodology. London: Routledge.
Debreu, G. (1959). Theory of Value. An Axiomatic Analysis of Economic Equilibrium.
New Haven, London: Yale University Press.
Einstein, A. (1934). On the Method of Theoretical Physics. Philosophy of Science,
1(2): 163–169. URL http://www.jstor.org/stable/184387.
Jevons, W. S. (1871). The Theory of Political Economy. Library of Economics and
Liberty. URL http://www.econlib.org/library/YPDBooks/Jevons/jvnPE.html.
Schumpeter, J. A. (1994). History of Economic Analysis. New York, NY: Oxford
University Press.
Weintraub, E. R. (2002). How Economics Became a Mathematical Science. Durham,
NC, London: Duke University Press.

Calgacus said...

Have to say I mostly agree with Egmont Kakarot-Handtke here. Syll calls Karl Menger and John von Neumann "followers of the Bourbaki tradition within economics" - More than a little ridiculous. Syll's repetitious diatribes against axiomatic-deductive thought miss the mark. The logical faintness of these damns amounts to praise of neoclassical economics. So they have nothing to worry about from this kind of both overblown and inadequate critique.

AXEC / E.K-H said...


You say: “The logical faintness of these damns amounts to praise of neoclassical economics. So they have nothing to worry about from this kind of both overblown and inadequate critique.”

Frank Hahn was always very happy with the incompetent critics of the neoclassical research program: “The enemies, on the other hand, have proved curiously ineffective and they have very often aimed their arrows at the wrong targets.”

See also
The stupidity of Heterodoxy is the life insurance of Orthodoxy

Egmont Kakarot-Handtke

Matt Franko said...

"economists have not been very skilled users of the scientific method"

yo they are not trained in that methodology... you can't hold them responsible for poor conduct of a methodology they have never been trained in...