Comment on ‘It’s all over — Gödel’s incompleteness theorems’

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“But lots of people have misused Gödel's theory.” (Tony Mann, Gödel's incompleteness theorem, youtube, 5:01)

There have always been lots of people who had considerable trouble with the clear-cut truth claim of mathematics. For one reason or another they prefer the realm of vagueness, ambivalence, indeterminism, fogginess, wish-wash, inconclusiveness, twilight, uncertainty, where “... nothing is clear and everything is possible.” (Keynes, 1973, p. 292)

For these people, who are at least mildly anti-scientific, science itself offered two god-sents: Heisenberg's uncertainty principle and Gödel's incompleteness theorem. These masterpieces of pure logic seemed to establish an ecological niche where true and false overlap, where contradictions can peacefully coexist, and where opinion and knowledge are equal. Tortured souls hailed von Neumann's “It's all over” as the end of their nightmare and the licence for anything-goes.

What von Neumann, who was for a time member of the most ambitious program in mathematics, meant was that Hilbert's goal of a complete proof of all mathematical truths was unattainable. What Gödel had shown was that there are true propositions that cannot be proved within a given consistent formal system. This was by no stretch of imagination the end of mathematics because there are many, many proposition that can be proved. And this invalidated none of the propositions that had already been proved since the days of Euclid. So Pythagoras's theorem and 2+2=4 still stand after Gödel.

Gödel had used Hilbert's axiomatic-deductive method to demonstrate that the axiomatic-deductive method has limits. To conclude from this demonstration that the method is inapplicable in economics is a gross logical blunder of the math-Luddites.

To tell the logically handicapped majority of economists of the limits of logic and mathematics is to warn a snail that it will encounter an absolute limit when it approaches the speed of light.

“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)

The problem of economics is not that there is one true but unprovable statement. To the contrary, the problem is that there is not one true and provable statement.

The curious thing is that there are so many economists who are quite content with this deplorable state of affairs.*

Egmont Kakarot-Handtke

References

Keynes, J. M. (1973). The General Theory of Employment Interest and Money. The Collected Writings of John Maynard Keynes Vol. VII. London, Basingstoke: Macmillan.

*For details see also here and here