April 23, 2015

The insignificance of Gödel's theorem for economics

Comment on ‘Gödel’s theorems and the limits of reason’

Blog-Reference

In his Tribune article Asad Zaman writes: “Progress in science becomes possible only after we abandon the quest for logical certainty.”

True, of course, logic has limits. Yet, from this does not follow that inconsistency is acceptable. From Gödel's theorem in particular does not follow that the pluralism of false theories is acceptable. And not by any stretch of the imagination follows that inconsistent orthodox and heterodox economic models will ever be acceptable.

It still holds: science does not explain everything, but non-science explains nothing. Science is not logic alone and not fact alone: it is the seamless synthesis of both.

“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)

Zaman's understanding of the interaction of mathematics and physics is utterly confused. He writes:
“The axiomatic-deductive methodology of mathematics leads to logical certainty without requiring empirical confirmation — we do not assess the validity of the Pythagorean Theorem by drawing triangles and measuring their sides.” (see Tribune)

Exactly the opposite is true.

“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)

Or, with regard to Newton's axiomatization of physics:
“In the most fruitful applications of mathematics to the physical world, some nonmathematical axioms also enter. The Newtonian system of mathematical mechanics depends as much on the Newtonian laws of motion and gravitation as it does on the axioms of mathematics.” (Kline, 1981, p. 469)

In pure mathematics progress has not been and will not be achieved by abandoning the quest for logical certainty.

“In hindsight, the basic idea at the heart of the incompleteness theorem is rather simple. Gödel essentially constructed a formula that claims that it is unprovable in a given formal system. ... Thus there will always be at least one true but unprovable statement.” (Wikipedia)

This, though, is no hindrance at all for the advancement of economics because a ‘true but unprovable’ statement can be added as an additional axiom to the initially given formal system and thus make it more comprehensive. More comprehensive, to be precise, does not mean complete. Not complete, on the other hand, does not mean, that non-scientific approaches provide better results or any acceptable results at all.

The fundamental methodological problem of economics does not arise from Gödel's theorem but from the sad fact that economists fail already at simple logical tasks. It can be shown, for example, that the basic Keynesian formalism is logically defective (2011). Nonetheless, economists blindly apply it for more than 70 years.* This scientific incompetence, and not the Incompleteness Theorem, is the real problem of economics.

The manifest confusion of economists and the proto-scientific mess of contradictory and logically defective models cannot be justified with Gödel's theorem.

Egmont Kakarot-Handtke


References
Brown, K. (2011). Reflections on Relativity. Raleigh, NC: Lulu.com.
Kakarot-Handtke, E. (2011). Why Post Keynesianism is Not Yet a Science. SSRN Working Paper Series, 1966438: 1–20. URL
Klant, J. J. (1994). The Nature of Economic Thought. Aldershot, Brookfield, VT: Edward Elgar.
Kline, M. (1981). Mathematics and the Physical World. New York, NY: Dover.

*See also here