On Confusion About Gödel's Theorem, Including In Austrian Economics

1.0 Introduction

Gödel's theorems are often invoked in non-mathematical contexts, sometimes in a very imprecise fashion (Franzen 2005, Raatikaine 2007, Jaimungal). You should be skeptical of what many say about Gödel, including of what I say.

In this post, I look at two examples. One is of a confused economist of the Austrian school. The other is of Wittgenstein, who has been ably defended on this point.

2.0 A Claim About the Economic Calculation Problem

Nguyen (2024) says that somehow Gödel's theorem supports the claim that centralized economic planning is, in principle, impossible. Nguyen is sympathetic to the claim that our minds (brains?) transcend the capabilities of all formal systems. I find this claim dubious, but I have not read Penrose. For purposes of argument, Nguyen assumes, through most of this paper, that the central planner has the information von Mises grants them. Nguyen deliberately puts aside Hayek's concerns about non-articulated, distributed, tacit knowledge.

I have demonstrated that von Mises' argument is invalid. I find I am not original. Cockshott (2010) has done the same. Both of us put forth a linear programming formulation that does not require prices of intermediate goods as part of the data. We differ in our specifications of the planner's objective function. Neither of us are echoing Lange and Lerner's formulations of general equilibrium. As such, I do not see any issues are raised for us by the non-computability of utility functions, preference relations, or general equilibria in which ratios of marginal utilities enter the system of equations.

A polynomial-time algorithm exists for solving linear programs. Linear programs are not undecidable. So Gödel's theorem and issues arising from Turing's work do not seem to have any purchase here.

Furthermore, in practice, only a finite number of numbers would be used in drawing up plans. Real numbers would be approximated, if you can call it that, by IEEE Std. 754. This standard defines floating-point numbers, in single and double precision formats. Your computer does not only fail to represent the full range of real numbers. It also only represents a finite number of integers in words, typically of 32 or 64 bits.

I do not mean to suggest that many practical issues do not arise with central planning Nor am I advocating such. I continue to maintain that von Mises' argument is invalid. And I find Nguyen's paper to be one of those mystical, imprecise invocations of Gödel.

3.0 A Notorious Paragraph from Wittgenstein

Here is Wittgenstein from notes not published until after his death:

"8. I imagine someone asking my advice; he says: 'I have constructed a proposition (I will use "P" to designate it) in Russell's symbolism, and by means of certain definitions and transformations it can be so interpreted that it says: "P is not provable in Russell's system". Must I not say tha this proposition on the one hand is true, and on the other hand unprovable? For suppose it were false; then it is true that it is provable. that surely cannot be! And if it is proved, then it is proved that it is not provable. Thus it can only be true, but unprovable.'

Just as we ask, '"Provable" in what system?,' so we must also ask, '"True" in what system?' 'True in Russell's system' means, as was said, proved in Russell's system, and 'false in Russell's system' means the opposite has been proved in Russell's system. - Now what does your 'suppose it is false' mean? In the Russell sense it means, 'suppose the opposite is proved in Russell's system'; if that is your assumption you will now presumably give up the interpretation that it is unprovable. And by 'this interpretation' I understand the translation into this English sentence. — If you assume that the proposition is provable in Russell's system, that means it is true in the Russell sense, and the interpretation 'P is not provable' again has to be given up. If you assume that the proposition is true in the Russell sense, the same thing follows. Further: if the proposition is supposed to be false in some other than the Russell sense, then it does not contradict this for it to be proved in Russell's system. (What is called 'losing' in chess may constitute winning in another game.)" -- Wittgenstein (1978: Part I, Appendix III)

Wittgenstein seems to equate provability in PM with being true in PM. Is not part of Gödel's point to separate these concepts? Furthermore, Wittgenstein seems to be writing only about the heuristic argument given at the start of Gödel's paper. Do his remarks make sense of Gödel numbering, (primitive) recursive functions, and so on?

As I understand it, Gödel's proof of his incompleteness theorem is a conventional proof. The proof is an argument to convince you that a sequence of statements exists that follow one another, by conventional deduction rules. And the conclusion is a theorem about natural numbers.

Gödel's proof is about the syntactic manipulation of strings of symbols by formal rules. Wittgenstein questions interpretations, I guess, of Godel numbers as the statements or sequence of statements that map into them. Floyd and Putnam (2000) justify Wittgenstein in a way that draws on the distinction between consistency and ω-consistency.

J. B. Rosser replaced ω-consistency with consistency in Gödel's proof. A theory is ω-inconsistent if one can show, for some proposition p:

• not p(0), not p(1), not p(2), ..., and
• There exists x such that p(x).

I guess that a non-standard model can be ω-inconsistent, where x is not a natural number, somehow. Suppose PM is ω-inconsistent. And suppose the negation of the Gödel sentence 'P' is true. Then the Godel number for the proof of this proposition could be a non-standard natural number. Wittgenstein writes about interpretations, but does not mention ω-consistency. I suppose he could have known about the Löwenheim-Skolem theorem.

I am sympathetic to being suspicious of English-language interpretations of syntactical manipulations. I am not so sympathetic to how lightly Wittgenstein treats possible contradictions in mathematics. I am sympathetic to the idea that mathematical logic, set theory, and model theory, for example, just provides more maths. One does not need this math to justify what humans have been doing for millennia. Do mathematical propositions have meaning before proofs are found? As I understand Wittgenstein, he says not. Proofs draw connections and give the proved proposition a meaning.

But I am also aware that even if I can echo out some propositions from these fields, I am no expert in them.

References

• Cockshott, W. Paul. 2010. Von Mises, Kantorovich and in-natura calculation. European Journal of Economics and Economic Policies 7(1): 167-199.
• Dawson, Ryan. 2015. Leaving Mathematics As It Is: Wittgenstein's Later Philosophy of Mathematics. University of East Anglica. PhD Thesis.
• Floyd, Juliet and Hilary Putnam. 2000. A note on Wittgenstein’s 'notorious paragraph' about the Gödel theorem. Journal of Philosophy. 97(11): 624-632.
• Franzen, Torkel. 2005. Gödel's Theorem: An Incomplete Guide to Its Use and Abuse. A. K. Peters.
• Jaimungal, Curt. Theory of Everything: a podcast episode.
• Nguyen, Hai-Trieu. 2024. The incompleteness of central planning. Quarterly Journal of Austrian Economics. 27(4).
• Raatikaine, Panu. 2007. Review of Franzen. Notices of the AMS. 54(3): 380-383.
• Standford Encylopedia of Philosophy, Wittgenstein's philosophy of mathematics.
• Wittgenstein, Ludwig. 1978. Remarks on the Foundation of Mathematics, revised edition. MIT Press.