The fatal flaw of mathematics

21 Nov, 2021 at 18:16 | Posted in Theory of Science & Methodology | 7 Comments

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

The most important concerns the question of how to select the specific systems of axioms that mathematics are 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 unproved truths.

This, of course, ought to be of paramount interest for those mainstream economists who still adhere to the dream of constructing a 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 — has given us are irrelevant model abstractions with no bridges to real-world economies. It’s difficult to find a more poignant example of an intellectual resource waste in science.

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  1. “It’s difficult to find a more poignant example of an intellectual resource waste” than philosophers trying to suggest that difficult puzzles in the fundamentals of pure mathematics are “fatal” or relevant in any practical way for applied sciences.
    Applied economists do not “adhere to the dream of constructing a deductive-axiomatic economics with analytic truths that do not require empirical verification”.

    • 《fatal” or relevant in any practical way for applied sciences.》
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      Have you ever had a computer hang due to the Halting Problem?

  2. 《If it is deductive certainty you are after, rather than the ampliative and defeasible reasoning in inference to the best explanation — well, then get into math or logic, not science.》
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    https://larspsyll.wordpress.com/2021/11/05/social-mechanisms-and-inference-to-the-best-explanation/
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    《Since Gödel showed that any complex axiomatic system is undecidable and incomplete, any such deductive-axiomatic economics will always consist of some undecidable statements.》
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    Are these two propositions incompatible? If you are after deductive certainty, will undecidable propositions still frustrate you?
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    Bruce said: “Go, look and see. Then measure.”
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    How can you measure when so much information (accounting books) is private and the owners want to keep it that way?
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    Henry said: “So how does one do economics?”
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    Move on to finance, already?

  3. ” It’s difficult to find a more poignant example of an intellectual resource waste in science.”
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    So how does one do economics?

    • Go, look and see. Then measure.
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      I know, wild and crazy.

    • Then what?
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      Past is not prologue. (See Lars’ “Rethinking Economics”).
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      The mere act of taking a measurement invalidates it – it’s no longer reliable as we cannot have any means of knowing whether the forces that determined it will apply in the next instance – this is the curse of ever extant uncertainty.

  4. Roger Penrose agrees. https://wordpress.com/post/motanomics.com/1659


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