Why does economics escape Godel's theorems?

Question

I've seen many professors say that Godel's incompleteness theorems don't apply to economics. Of course I've seen others like Yanis Varoufakis who has on record said that many economics papers defy the basic principles of logic, while also saying that empirical papers can reach the complete opposite conclusion with the same data; both greater and less bargaining power of labor unions can reach the same equilibrium of better profits.

If you look at Godel's incompleteness theorem...

https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-080-great-ideas-in-theoretical-computer-science-spring-2008/lecture-notes/lec6.pdf

"For any fixed formal system of logic F, if the system is sound and computable, then there exist true statements about the integers that are not provable within the system, F"

1. Is economics a "formal system of logic"? If not, then is it okay to use illogical reasoning in economics? Even if we accept that some reasoning may be intuitive, is the mathematics underlying much of economic theory not susceptible, as a formal system of logic, to Godel's incompleteness theorem? If mathematics is incomplete, then any economic theory that has the pretense of rigor based on mathematics, cannot stand the test of Godel's theorems.

2. If economics is a formal system of logic, then is it sound and computable? I'm guessing all the mathematic formulations of constrained optimization, comparative statics, and causal analysis in econometrics is a "sound and computable" system of applied math, in which case it strives to be a "formal system of logic that is sound and computable". If not, then its effort to be such a field is counterintuitive or even self-defeating in some sense.

3. If economics is formal but not a system, then what is it? Is it a set of competing theories which are themselves systems? If not, then does it house within every theorem or principle a sort of incompleteness where we can relax the requirement for a set of theories to work like a coherent system?

Am I making sense here?

Answer 1

The Incompleteness Theorems apply to computable, first-order, deductive systems. That means that there must be both a computable set of axioms and a computable inference system. In other words, you must be able to write a computer program that can answer the following question: Given a finite sequence of sentences, is it the case that each statement is either an axiom or follows inferentially from previous statements?

Furthermore, the Incompleteness Theorems require that the system be able to interpret Peano arithmetic (i.e., talk about the non-negative integers with plus and times).

Given these constraints, I'm sure it wouldn't be hard to construct such a first-order, axiomatic system for economics (the Austrian concept of praxeology would be a good starting point). Now, would such a system be able to interpret Peano arithmetic? If so, then all the Incompleteness Theorems would tell you is that there would be certain statements (specifically, about integers) that could neither be proved nor refuted. Would such statements be relevant to someone studying economic theory? Godel doesn't tell us. The system would only be incomplete because it could talk about integers and the integers are what creates the incompleteness. The fact that the system could also talk about economics would be merely incidental.

I do think that Godel's theorems (and not just the ones about incompleteness; keep in mind that his dissertation was proving the Completeness Theorem) are fascinating. However, I fear that many people exaggerate their significance beyond their original bounds and try to make some grand philosophical epiphany out of them.

Answer 2

Every science using mathematical reasoning is in some sense subject to Goedel's first incompleteness Theorem, but in a rather trivial sense. This didn't diminish the success of, e.g., physics, and it won't impact economics at all. So yes, in some sense economics is "incomplete", but that's for sure the least of its problems.

Answer 3

Kurt Gödels Incompleteness Theorem is the negative answer to the quest of the mathematician Davild Hilbert in the early 20th century to find a set of complete and consistent axioms upon which to build the whole of mathematics. It turns out that it is not possible to find such a set. Any set of axioms which is complete will lead to inconsistencies; and every set of axioms that avoids inconsistencies will be incomplete.

This is it, basically. It is a highly technical mathematical proof which has very little impact on any real world application. It decided a long discussion between mathematicians, solving an until then unknown question which is mainly interesting for mathematicians: it tells them that it makes no sense to spend decades of your life searching for such a set of axioms. It also ended (or began, depending on your viewpoint, the "foundational crisis of mathematics"), which is a very interesting topic for another question.

There are also very interesting, closely related proofs that show that it is impossible to define truth inside a formal system (Tarski's Undefinability Theorem) and impossible to decide whether a given program with ever halt for a given input (Alan Turing's Halting Problem regarding algorithms/computation).

To answer your question: I am not familiar with formal definitions of the term "economics", but the term "formal system" (which the Incompleteness Theorem talks about) has a formal definition. So if you have a definition of "economics" which is based on a set of axioms and rules how to infer statements from those axioms (and other statements), and if it contains at least the integer numbers, then the IC applies.

Every practical field "escapes" the theorems anyways because none of these theorems tells anything about the usability of said formal systems, logics, or algorithms. In practice it does not matter whatsoever.

N.B., as the comments mention, things gets more ugly soon if you dig deeper; i.e. from these kinds of theorems you can deduce that it is impossible to do things in, say, software developement which would be very practical indeed, in this respect it is not really fair to say that all of them are just "theoretical".

Answer 4

Godel's incompleteness says that there is a dichotomy: A set of axioms is either complete or consistent, but not both.

If it is complete then you can prove any theorem from it but it will be inconsistent i.e. there will be paradoxes hidden somewhere.

If it is consistent there won't be paradoxes hidden in it but it will be incomplete i.e some theorems are not provable.

Is economics a "formal system of logic"?

It's a collection of empirical facts and a set of mathematical models build upon those facts. It might be a "formal system of logic" but that would be by accident.

In economics proving a theorem is not the criteria for whether the model is bad or good. What we really care about is whether the model is falsifiable and able to make predictions (we also care about the reproducibility of experiments).

Suppose that I came up with a new idea (theorem) and I want to know whether it's true: I don't really need to prove it, I only need to test it empirically then add the result to my list of known facts. If the result agrees with an already existing model then good for the model, it passed another test. If the result disagrees with an already existing model then bad for the model, it needs to be discarded or reformulated.

Proofs are still useful in economics (and all empirical sciences) because you still need to know whether the model is really what you meant:

Did you divide by zero somewhere? Did you assume a number was always positive when in reality it can be negative? Did you miss some parameter and now the model doesn't work in all circumstances you thought it would?

Answer 5

while also saying that empirical papers can reach the complete opposite conclusion with the same data

From a mathematical point of view, if your assumptions and logic lead to a contradiction, it means that, by reductio ad absurdum, there is a mistake somewhere.

You cannot use the corresponding theory anywhere, because it contains a contradiction, which means that false can be derived from it, and that this theory could be used to prove anything.

This is obviously a much larger problem (as mentioned by @VARulle) than having questions that cannot be answered (as stated by Gödel's incompleteness theorems).

Basically, if a contradiction appears somewhere, the whole theory needs to be rebuilt from scratch, and every assumption and logical step needs to be checked.

Answer 6

Godel's theorems are about self-referencing statements of the form "This statement is unprovable by X" where 'X' is some system or mechanism for proving things. 'X' is usually some defined system of logic - a set of axioms and rules for proofs - but it doesn't have to be.

The basic idea is that if the statement is provable by X, then X can prove statements that are false, and so X is inconsistent. If X is consistent (only proves true statements) then the statement is unprovable by X, and therefore true. Godel's achievement was to encode both the statement and the proof system using only the methods of basic arithmetic applied to the positive whole numbers, thus showing (roughly speaking) that not every true arithmetical property of whole numbers can be proven.

Taken literally, it obviously doesn't apply to other systems like economics because economics isn't arithmetic. (Although arguably economics contains arithmetic...) But we can talk about a statement in the same spirit as Godel's theorems by asking whether "This statement cannot be proven in economics" is true or not?

If economics could prove it, economics would have proven something that is false, and so would be inconsistent. If economics cannot prove the statement, then the statement is true.

Various opinions could be had on that matter. Some would say economics is inconsistent ("empirical papers can reach the complete opposite conclusion with the same data"). Some would say it is incapable of providing proofs - only of inductively justifying increased or decreased belief. Some would ask "Which version of 'economics' are you talking about?" because there are many. Get any three economists together and you'll get four different opinions! And of course, some would say "Sure, but that's not really Godel's theorem."

What I don't think anyone would argue is to say that economics is both consistent (only proves true things) and can prove that the statement above can't be proved. Economists aren't that crazy! So in this sense, my view is that Godel's theorem applies just as powerfully to economics, too. It doesn't have the same significance, though, because nobody ever believed economics was complete and consistent, whereas lots of people held that belief about mathematics!

Answer 7

I would say you are getting into identifying a big issue in applied epistemology more generally namely that cross domain communication and knowledge transfer can be very hard which is to say that asking those trained in economics about the applicability of a mathematical proof to their domain is likely to yield unsatisfactory answers to the same degree that say, asking neurologists about the impact of some aspect of quantum physics is applicable to their understanding of neurological processes.

Sometimes even more worryingly the concepts may have been independently considered in two fields and while they conceptually discuss the same issues they may use entirely different vocabularies to describe this same concept. This occurs even amongst comparatively closely related fields of study like electrical and mechanical engineering. A particular example is that of control theory that spans these two fields and for various historical reasons have taken curious diverging paths at times.

That is not to say that the connections don't exist but the answers that make these questions interesting are not easy to come by as the concepts in each field can take significant study within the respective fields to even master and truly appreciate thus limiting the time available for experts from either field to really be able to discover the ideas from fields outside of their own even if there is considerable overlap never mind to meaningfully translate between the vocabularies to engage with those outside their field.

Is there a connection between the broad class of related ideas represented by the likes of Gödel's incompleteness theorems, Tarski's undefinability theorem, Church's answer to Hilbert's Entscheidungsproblem and finally and maybe more famously the Halting Problem, and more broadly all of human epistemology? I can't claim to be an expert in any of these specific proofs but I have to say there seems at least to me an intuitive sense in which this connection is unavoidable. They seem to point to some significant limits to what can and cannot be known and what it even means to know anything at all. Is the connection sometimes maybe overhyped and sensationalized a little by certain groups? Again I think the answer is yes but I think the resultant hesitancy in fields outside of those where these proofs first originated to investigate their consequences on their own fields is also a mistake as these are extremely significant and rigorous findings.

If highly simplified and formal systems are plagued with inherent contradictions, and computational limits then it does not bode well for larger more complex systems systems of knowledge. Sometimes practitioners unfamiliar with these proofs defend against this issue when first exposed to these ideas by making claims that these theoretical shortcomings do not translate to any limitations in practice but I would say that at the least computer science strongly refutes this proposition as the implications of the halting problem and other computability problems are quickly and trivially practical not least to massive economic consequence.

As for how these ideas have been addressed within economics I think there does feel like there are some similarities between the class of proofs previously listed and the Mises-Hayek economic calculation problem though I'm not familiar with any rigorous attempts to connect these ideas despite their passing similarities which as I initially mentioned seems related to the the difficulties encountered in sharing knowledge across domains.

I would therefore say that any attempt to reject the impact of this class of proofs on other fields is horribly misguided and smacks of significant doses of ignorance, but this must be weighed against the reality that in many practical fields the ability to make positive progress is not impeded by their applicability.

Not knowing and stated more strongly maybe never actually being able to know does not mean that significant value and progress cannot be achieved. After all, we aren't comparing the correctness and consistency of our knowledge systems to some system created by some perfect oracle somewhere but merely to those created by other humans similarly limited by the implications of these theories.