Gödel and the limits of mathematics

10 May, 2022 at 16:50 | Posted in Theory of Science & Methodology | 1 Comment

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Gödel’s incompleteness theorems raise important questions about the foundations of mathematics.

The most important concern is the question of how to select the specific systems of axioms that mathematics is supposed to be founded on. Gödel’s theorems irrevocably show that no matter what system is chosen, there will always have to be other axioms to prove previously unproven truths.

This, of course, ought to be of paramount interest for those mainstream economists who still adhere to the dream of constructing deductive-axiomatic economics with analytic truths that do not require empirical verification. Since Gödel showed that any complex axiomatic system is undecidable and incomplete, any such deductive-axiomatic economics will always consist of some undecidable statements. When not even being able to fulfil the dream of a complete and consistent axiomatic foundation for mathematics, it’s totally incomprehensible that some people still think that could be achieved for economics.

Separating questions of logic and empirical validity may — of course — help economists to focus on producing rigorous and elegant mathematical theorems that people like Lucas and Sargent consider “progress in economic thinking.” To most other people, not being concerned with empirical evidence and model validation is a sign of social science becoming totally useless and irrelevant. Economic theories building on known to be ridiculously artificial assumptions without an explicit relationship with the real world is a dead end. That’s probably also the reason why general equilibrium analysis today (at least outside Chicago) is considered a total waste of time. In the trade-off between relevance and rigour, priority should always be on the former when it comes to social science. The only thing followers of the Bourbaki tradition within economics — like Karl Menger, John von Neumann, Gerard Debreu, Robert Lucas, and Thomas Sargent — have given us are irrelevant model abstractions with no bridges to real-world economies. It’s difficult to find a more poignant example of intellectual resource waste in science.

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  1. 1.1.2 Gödel and Turing on Rationalistic Optimism
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    “Rationalistic optimism” is the view that there are no mathematical questions that the human mind is incapable of settling, in principle at any rate, even if this is not so in practice (due, say, to the occurrence of the heat-death of the universe). 5 In a striking observation about the implications of his incompleteness result, Gödel said:
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    My incompleteness theorem makes it likely that mind is not mechanical, or else mind cannot understand its own mechanism. If my result is taken together with the rationalistic attitude which Hilbert had and which was not refuted by my results, then [we can infer] the sharp result that mind is not mechanical. This is so, because, if the mind were a machine, there would, contrary to this rationalistic attitude, exist number-theoretic questions undecidable for the human mind (Gödel in Wang 1996, 186-187)
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    What Gödel calls Hilbert’s “rationalistic attitude” was summed up in Hilbert’s celebrated remark that “in mathematics there is no ignorabimus” — no mathematical question that in principle the mind is incapable of settling (Hilbert 1902, 445). Gödel gave no clear indication whether, or to what extent, he himself agreed with what he called Hilbert’s “rationalistic attitude” (a point to which we shall return in section 1.3). On the other hand, Turing’s criticism (in his letter to Newman) of the “extreme Hilbertian ”view is accompanied by what seems to be a cautious endorsement of the rationalistic attitude. The “sharp result” stated by Gödel seems in effect to be that there is no single machine equivalent to the mind (at any rate, no more is justified by the reasoning that Gödel presented) — and with this Turing was in agreement, as his letter makes clear. Incompleteness, if taken together with a Hilbertian optimism, excludes the extreme Hilbertian position that the “whole formal outfit” corresponds to some one fixed machine. (Copeland et. al., 2013, 5, Computability: Turing, Gödel, Church, and Beyond, The MIT Press. Kindle Edition.)
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    (Copeland, Jack B. Posy Carl J. and Shagrir Oron, Eds. Computability (Copeleand et. al., ed.) [Turing, Gödel, Church, and Beyond]. Cambridge, Massachusetts: MIT Press; 2013; p. 5. )

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    To raise the question of mind is almost anathema; it is easier to ignore than to openly address the fact that mind is real and transcends mathematical formalism (logic) and that it is the source of truly creative thinking.


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