## Gödel and the limits of mathematics

from **Lars Syll**

*irrelevant model abstractions*

with no bridges to real-world economies

with no bridges to real-world economies

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.

The hole at the bottom of math is genuinely creative thought. Creative ideas are born in a world of ambiguity, contradiction, and paradox.

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Gödel invented a way of coding mathematical arguments into single integers with no loss of information to prove his incompleteness theorem—ironically transforming metamathematical statements into mathematics (arithmetic) to create a paradox to do so.

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Gödel’s insight is the truth of the non-algorithmic nature of mathematical insight. Human mind, its ability to engage in genuinely creative thinking, is non-algorithmic. We are, in other words, as Lars notes, not “nomological machines.”

… as Lars notes, notes