Wednesday, August 6, 2014

Addendum to SCIOD 4: Liar's Paradox

Gödel's Theorem illustrated with regard to Say's Law and the Theory of the Lump of Labour:
  1. "...a fallacy long recognized by economists as being especially pernicious. It is based upon the old exploded 'lump of labor theory'—the theory that there is a limited amount of work to be done. The truth is, of course, that every worker turning out a salable product thereby automatically generates a demand for other products, and thus sets others to work."

  2. "The theory of a wage-fund and that of a 'lump of labor' naturally cohere, and there seems to be as much truth in one as in the other."

  3. "Mill's successors rejected his wages-fund theory but overlooked the fact that Mill's refutation of Malthus depended on it." -- Keynes

  4. "Of all the opinions advanced by able and ingenious men, which I have ever met with, the opinion of M. Say, which states that, Un produit consommé ou detruit est un débouché fermé appears to me to be the most directly opposed to just theory, and the most uniformly contradicted by experience." -- Malthus
Daniel R. Fusfeld (1980) "The Conceptual Framework of Modern Economics."  Journal of Economic Issues, 14, 1, pp. 7-8:
Gödel's theorem offers a profound challenge to the theory of knowledge on which logical empiricism is based. Unprovable propositions compromise the completeness of theoretical models. The results depend on the assumptions made about the unprovable propositions, and since those assumptions cannot be proven to be correct (or incorrect), the results must also be unprovable. If investigators try to avoid assumptions by using casual empiricism as the foundation for their unprovable propositions, they will then find themselves testing their hypotheses with the empirical data from which their casual empiricism was drawn, and the argument becomes circular. The only alternative is to leave the entire system open and unprovable through pure assumption or faith. The knowledge derived from formal axiomatic models must be imperfect, subjective, problematic.  
Gödel's theorem has had little impact on economic method, however, and has not been discussed by economists [Sandwichman: prior to Mirowski]. They have always used the ceteris paribus assumption to close their models. The embarrassing presence of an undecidable proposition can be avoided by inserting at any convenient point in the chain of deductive reasoning the simple assumption that "everything else remains the same." This procedure closes the model at that point and allows the remaining axiomatic model to be logically complete -- qualified, of course, by the "as if" assumption made when "everything else remains the same." This procedure was traditional in economics long before the epistemological problems raised by Gödel's theorem were recognized. Economists have been able to proceed as if Gödel had never lived.
"Gödel's Theorem seems to me to prove that Mechanism is false, that is, that minds cannot be explained as machines." J. R. Lucas (1961) "Minds, Machines and Gödel." Philosophy, 36, 137.

5 comments: said...


You have now gotten over your head citing Godel's Theorem (which one???). Not a good approach.

So, this is family for me. My old man proved the now textbook version of the incompleteness theorm. After decades I was finally pressured into officially commenting on all this, which I did so in "On the Foundations of Mathematical Economics." It was pubbed in a math/cs journal and can be easily tracked down by its title.

I do not disagree with your main point in this point, but please do not join the long list or ignorami who spout off about Godel inappropiatel, and, sorry, this post does not cut the mustard.

Sandwichman said...

Let me get this straight, Barkley.

1. You don't disagree with my main point.

2. I am "over my head."

First, let me say that you are right that I am "over my head" with regard to Godel's theorems in the sense that this is not my area of expertise and thus I am relying on what other people have written, which could be wrong or which I might misinterpret.

I am NOT over my head with regard to lump of labor and Say's Law.

But that aside, what I would ask of you is, instead of pulling rank and giving no substantive critique of how I am "over my head" that you actually point out how the post "doesn't cut the mustard." By all means, don't leave out why you don't disagree with my main point.

Sandwichman said...

By the way, in my background research on this, I did have a look at TWO papers you wrote. I don't recall if one of them was "On the Foundations of Mathematical Economics" but I'll check back and see. The two papers that I had a look at only briefly mentioned Godel's Theorem(s) and didn't really go into any detail.

Sandwichman said...

Nope. "On the Foundations of Mathematical Economics" wasn't one of the articles by you I consulted. It was published in New Mathematics and Natural Computation in 2012. Not a journal my university library gives me access to. So why don't you just tell me where I've fucked up?

Sandwichman said...

ABSTRACT: Kumaraswamy Vela Velupillai presents a constructivist perspective on the foundations of mathematical economics, praising the views of Feynman in developing path integrals and Dirac in developing the delta function. He sees their approach as consistent with the Bishop constructive mathematics and considers its view on the Bolzano-Weierstrass, Hahn-Banach, and intermediate value theorems, and then the implications of these arguments for such "crown jewels" of mathematical economics as the existence of general equilibrium and the second welfare theorem. He also relates these ideas to the weakening of certain assumptions to allow for more general results as shown by Rosser in his extension of Gödel's incompleteness theorem in his opening section. This paper considers these arguments in reverse order, moving from the matters of economics applications to the broader issue of constructivist mathematics, concluding by considering the views of Rosser on these matters, drawing both on his writings and on personal conversations with him.

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