September 2, 2015

At last, mathiness problem settled

Comment on ‘Economics — a rogue branch of applied mathematics’

Blog-Reference

Noah Smith writes: “Traditionally, economists have put the facts in a subordinate role and theory in the driver’s seat. Plausible-sounding theories are believed to be true unless proven false, while empirical facts are often dismissed if they don’t make sense in the context of leading theories. This isn’t a problem with math ...” (See intro)

Exactly so. The problem has never been math but that economists have never grasped what science is all about: “Research is in fact a continuous discussion of the consistency of theories: formal consistency insofar as the discussion relates to the logical cohesion of what is asserted in joint theories; material consistency insofar as the agreement of observations with theories is concerned.” (Klant, 1994, p. 31)

The problem can be exactly located here: “As with any Lakatosian research program, the neo-Walrasian program is characterized by its hard core, heuristics, and protective belts. Without asserting that the following characterization is definitive, I have argued that the program is organized around the following propositions: HC1 economic agents have preferences over outcomes; HC2 agents individually optimize subject to constraints; HC3 agent choice is manifest in interrelated markets; HC4 agents have full relevant knowledge; HC5 observable outcomes are coordinated, and must be discussed with reference to equilibrium states.
By definition, the hard-core propositions are taken to be true and irrefutable by those who adhere to the program. "Taken to be true" means that the hard-core functions like axioms for a geometry, maintained for the duration of study of that geometry.” (Weintraub, 1985, p. 147)

The material refutation consists in the refutation of these behavioral axioms, that is, HC1 is vacuous and HC2 to HC5 lack material consistency. This suffices to put an end to the traditional research program.

The formal — i.e. mathiness — refutation consists in: “Thus not all axiomatic theories need to be phrased in terms of set theory but much more conveniently and intelligibly rather in terms of some advanced mathematical structures.” (Schmiechen, 2009, p. 367)

Solution of the mathiness problem: economics has to move from Debreu's set theoretical approach to an advanced formal structure (2014).

Genuine scientists always understood — and economists never got it — that a theory consists of two vital elements. “A scientific deductive system (“scientific theory”) is a set of propositions in which each proposition is either one of a set of initial propositions ... or a deduced proposition ... which is deduced from the set of initial propositions according to logico-mathematical principles of deduction, and in which some (or all) of the propositions of the system are propositions exclusively about observable concepts (properties or relations) and are directly testable against experience.” (Braithwaite, 1959, p. 429)

Note that the hard core propositions HC1 to HC5 lack observable concepts. This has always been the pivotal problem, not math — or the underlying axiomatic-deductive method — per se.

Egmont Kakarot-Handtke


References
Braithwaite, R. B. (1959). Axiomatizing a Scientific System by Axioms in the Form of Identifications. In L. Henkin, P. Suppes, and A. Tarski (Eds.), The Axiomatic Method, pages 429–453. Amsterdam: North-Holland.
Kakarot-Handtke, E. (2014). Objective Principles of Economics. SSRN Working Paper Series, 2418851: 1–19. URL
Klant, J. J. (1994). The Nature of Economic Thought. Aldershot, Brookfield, VT: Edward Elgar.
Schmiechen, M. (2009). Newton’s Principia and Related ‘Principles’ Revisited, volume 1. Norderstedt: Books on Demand, 2nd edition. URL
Weintraub, E. R. (1985). Joan Robinson’s Critique of Equilibrium: An Appraisal. American Economic Review, Papers and Proceedings, 75(2): 146–149. URL