Dear Victor,

You make a good point about Gödel, distinguishing your work for his. I think the mistaken comparison comes form people who confuse the terms "formalism" or "formalist method" with the "axiomatic method." What I mean is that the formalists follow Descartes in claiming that a theory is a series of propositions in a long chain of reasoning. This is a sort of syllogistic method that transform ideas, and is different from the axiomatic one that tries to characterize the ideas. Gödel incompleteness theorem shows their impossibility as regarding the reduction of mathematics to logics. We find similar analysis with Arrow's impossibility theorem in economics, and some people have mistakenly associated it with Gödel's work. But Arrow corrected them by pointing out that it was started with Condorcet and not Gödel. I think you are seeking similar clarification.

I hold with others that the axiomatic method seeks the deep-lying intelligibility of the subject. It divides up difficult proposition and tries to solve them independently, isolate them, push their abstraction to the limit, and afterward recombine them to bring out their separateness and interactions. Surely, there are many such methods in economics. You mentioned Debrue's. There is also the Libertarian Action Axiom that I am sure you are familiar with. It seem you have done an appraisal of them in your book.

From what you have written (in the introduction) it is clear that those who invoke Godel's theorem in the context of your work do not understand Godel's theorem.

Many of your statistical assumptions can be disputed - and your work sounds very polemical.

Torkel Franzen's book on Gödel's Incompleteness Theorems provides an accessible critique of those who falsely relate Gödel's work to other fields. You may find it useful in such discussions. 

Simply you have to answer them that Godel's Theorem isn't about the fact that an axiomatic structure works (Math, for example, works pretty well...).

The only thing Godel Theorem states is that, among any number of axioms you choose to build a theory, one of them is instead a postulate, but one you couldn't prove.

So, like (for example) in Geometry we have the non-euclidean geometry (that implies the same axioms of ordinary geometry, but change the parallel postulate), the same should applies to your theory.

So, if someone uses Godel's Theorem to contradict you, they're making it wrong. They should, instead, find out what is, between your axioms, the one that can be changed resulting in a still functional theory.

But it's perfectly fine to create an axiomatic theory using a number of axioms and a single non-proven postulate taken - for convention - as true.

In my opinion it will be interesting for you to find out by yourself what of your axioms is a postulate instead, and try to find out what happens if you change it.

This will also allows you to answer more specifically to any objection, showing the REAL outcome if you properly applies Godel's Theorem.

The main parts of Torkel Franzen's book are available online, I have delivered talks on the book

Dear Victor,

it is true that many contexts in which one cites Gödel have nothing to do with Gödel. This comes mainly from the fact that they don't understand the result of Gödel.

Gödel says that there is no complete recursive axiomatization of number theory. Otherwise said, the theory of the structure (Z, +, -, *, 0, 1) is undecidable. It does not say that the theory does not exist, but just that there is no mechanical method to decide the truth of some formal sentence in this theory.

Parallel to Gödel, there is a theorem of Tarsky saying that the first order theory of (R, +, -, *, < , 0, 1) is decidable. There is decision algorithm for the real algebra, contrary to the number theory of integers!

Now about you. I have seen that you use real numbers, but also some other functions, like exponential and logarithm. Interesting enough, If we add to the real numbers the exponential function, it is not known if the theory is decidable or not. Wilkie and Macintyre have shown that decidability is more or less equivalent to an open problem of Shanuel about transcendental numbers. So a Gödel result for your frame of work does not exist - and also a Tarsky result does not exist. The problem is just open.

But every observer should remark that this whole discussion has nothing to do with your work.

1. You didn't want to do a decidable theory (with universal decision algorithm). You wanted to fix some axioms and to derive some sentences important in economy. This is very well and very legitimate.

2. Transcendental numbers and decision problems have nothing to do with your work, as like other number theoretic tasks. Your interest is mainly quantitative and related to the behaviour of some functions.

3. It is interesting if a fragment of the theory - which is relevant for economy - is decidable or not, but this is not the job of economists, just the job of some logicians or some philosophers.

4. A lot of undecidable and / or uncomplete theories work and are very important: the theory of groups, the theory of fields, the theory of analytic functions, the theory of number rings, etc. Nobody will say to a mathematician: do not consider Group Theory, because of Gödel. The answer will be: shut up and go to learn something.

I think that you can give exactly this answer to those who dismiss your work, specially without reading it and without reading Gödel. You may cite me!

 Dear Victor,

A simple way to grasp the meaning of Gödel's theorem is to digest the self-referential conundrum of "Everything I say is a lie". The theorem can be formulated by simple propositional logics containing so-called well-formulated formulas, wff's. The Gödel sentence then become something like "This theorem cannot be proved". 

The theorem, oversimplified, usually has two faces: There are always wff's in a consistent system that are not provable within the system and if the wff's include statements of its own completeness then it becomes inconsistent. This includes truths about natural numbers or simply about arithmetic.

If someone use Gödel to dismiss your theory ask them to cite a theory that is immune to "Gödel". In this vain it is not allowed to quote axiomatic theories based on higher order logics, because this is only a "cheating", since these logics, however interesting as a mathematical game purports to limit the axioms so as to avoid self-references.

I am not an economist so I cannot directly evaluate the soundness of your axiomatic theory, but I would not take a general Gödelian type criticism as the only reason to dismiss it.

The Godelian ideal of completeness is not the ideal of, and aim of, most economists and their work. In fact economists have become increasingly fascinated with inconsistencies and paradoxes. Even admitting  dualities. The point with Godel is that having a universal theory is tall order. Having an incomplete theory that works within a small confine of human reality is a more productive enterprise. While the pursuit of formal consistency is unattainable , the task of "theory building" becomes one of specifying conditions/premises under which, the theoretical edifice holds (i.e. optimal/equilibrium conditions). Sadly economics will have to grapple with changing premises and assumptions because our science is socio-psychological in nature (pure empiricists such as neuroeconomists may beg to differ). Given such an ontological constraint, a recursive, bottom- up approach, may be taken for being axiomatic. And in this way be on a higher order of completeness relative to an axiomatics based on pure logic.Godel is not the greatest opponent of the axiomatic approach but rather the empiricists. I wouldn't waste my time convincing them. Technical issues and the latest econometric tricks seem to be their preoccupation- an end in itself.Data is not truth. In the end both the theorists and empiricists have to be contented with an "estimation"-incompleteness.The question that begs an answer, "How incomplete?"

I think they jump when you use the term axiomatic.

Then by association they think about the axioms for integers, and from there the

incompletness predicted by Godel.

On the other hand this is what is original in your book, and if the axioms lead to

reasonable conclusions, you  should not mind too much about the criticism.

I think you could respond as follows.

Assume that your theory is subject to Gödel. That would mean there are statements in your theory that satisfy your Axioms, but which your Axioms cannot prove, nor disprove.

But, presumably, what you are interested in is statements that your Axioms do prove.   Gödel does not negate your theory.

Maybe another response you could give would be to ask them to explain Gödel to you.

Regards, Steve.

I agree with many of the above comments to the effect that Gödel is just a red herring in this context. A supplementary consideration might be to invoke a notion I would assume economists are familiar with, namely "satisficing". An axiomatic system need not attempt to capture everything, not even many things that would be easily capturable if one cared enough to capture them. One may simply want a system that is good enough for the purposes at hand. And such a pragmatic approach can itself be justified on economic grounds.

What is it about your work which triggers the "Gödel reaction"?   Are they referring to the Incompleteness Theorem, or the relatively trivial Completeness Theorem?  Maybe the dismissive person is just responding to the use of the words "Axiomatic" or "Theory". But I like Steve's suggestion best, just ask them to explain Gödel's Theorem to you.  No one who understands GT well enough to teach the mathematical details would play the challenge game, so the pretender will be in over his head as soon as you ask the question.

Dear Victor,  I have just realised there is something I should say.  My own subject is quantum theory.  And the work I have been doing for some years concerns logical independence in quantum theory. When spoken about in relation to Gödel, logical independence is often called mathematical undecidability.     Logical independence in quantum theory is answering questions that have been around in quantum theory for 90 years.  My point is that logical independence is not necessarily destructive; it can be exactly what a theory needs.

For some time now I have wondered if chaotic systems might be explained by logical independence.  Is economics chaotic?  Is chaos consistent with your Axioms, even if maybe your Axioms do not derive it?

Logical independence of the irrationals is what I'm thinking of.

regards, Steve.

Thank you everyone!  All of your comments were wise, but let me specifically reply to Steve Faulkner and Federico Danelli.

Faulkner and others have pointed out that I need only be interested in the statements that my axioms do prove; Gödel’s Theorem isn't about the fact that an axiomatic structure works.  This is welcome news and basically what I was trying to say when I responded to critics by comparing my theory to Leonhard Euler founding ballistics where nothing but the wildest guesswork had existed before. 

I make the comparison to ballistics here:

https://www.researchgate.net/publication/280948689_Critique_of_Tony_Lawson_on_Neoclassical_Economics

Axiom #1:       Constant atmospheric density from the ground to the apogee.

Axiom #2:       Drag is proportional everywhere to the square of speed.

Axiom #3:       Gravity is everywhere pointed downwards; e.g. the Earth is flat.

In point of fact, exactly none of Euler’s axioms are universally true.  The beauty of the axiomatic method is that it can always be extended, as modern artillerists did when their guns became powerful enough to fire projectiles with a muzzle speed over 240 m/s into the stratosphere at targets over the horizon.  But the fact the Euler’s work was in need of extension when applied to long-range artillery is no recommendation to stand calmly on the receiving end of low-velocity trench mortar fire from a kilometer away, even though it is guided only by Euler’s original theory.

But you are right, Faulkner; I have needlessly allowed economists to put me on the defensive.  Like in a court of law, the burden of proof is on the accuser.  I have dodged their criticism because – I will be the first to admit – I know little about number theory.  I am sure that I remember writing analysis on my résumé.  But, if economists know so much about number theory, then let them explain Gödel to me!

In the meantime, I purchased a copy of Torkel Franzen's book on Gödel's Incompleteness Theorems and will read it as soon as it arrives in the mail.

Note:  This is in reply to Faulkner’s first comment.  Regarding his recent question, I will have to think about that before replying.

Regarding Danelli’s suggestion that I find out by myself which of my axioms is a postulate and try to find out what happens if I change it, I have done some work along these lines. 

Note:  In my own writing I use axiom and postulate as synonyms, though I understand what Danelli means by the distinction between Euclid’s intuitive first four axioms and his more dubious fifth postulate.

Over the years a number of people have questioned my second axiom, though I thought their counter-examples were a bit contrived.  But just this year I reformulated this axiom in a way that states only the minimum needed to prove my theory while dispelling all counter-examples, even the most contrived.

In my 1999 book I said that marginal (diminishing) utility is negative monotonic; that is, u’(s) < 0.  Now I say only that the tail need be negative monotonic; that is, there exists an s* such that, for all s > s*, u’(s) < 0.  This is how axiom 2ii will read in future editions.

But the axiom that is most like the Parallel Postulate is my third one, regarding proportionate effect.  Many people have noted that it is considerably more complicated and less intuitive than the others and that it stands apart from them in the same way that Euclid’s fifth postulate stands apart from the first four.  In my Simplified Exposition (p. 10) I write:

“The axiom that first-unit demand conforms to the characteristics of proportionate effect is a generalization of calculating interest as a percentage of the amount owed.  In fact, this is how people have calculated interest throughout recorded history, although economics having always been a soft science, they never asked for proof.  Perhaps the value of money decays harmonically over time or in another way besides exponentially?”

https://www.researchgate.net/publication/270687116_AXIOMATIC_THEORY_of_ECONOMICS

Appendix A of my book is titled Alternative Distributions for First-Unit Demand and here I endeavor to be my own Lobachevski by offering an alternative economics based on the harmonic or some other decay function for the value of money to be received in the future.  I write:

“The present value of money to be received a year from now is currently calculated by multiplying the amount by a constant near unity called the common ratio.  Under the proposed system, one would invert, add a constant, and invert again.”

The Law of Proportionate Effect (Theorem 1) states that when every individual changes each day by a small and random proportion, they will eventually all be logrithmico-normally distributed.  In the appendix I prove what I call the Law of Harmonic Effect which states that when every individual changes each day by inverting, adding a small and random amount and then inverting again, they will eventually all be reciprico-normally distributed.  While interesting in its own right, I then prove that this newly-defined distribution does not work for my purpose.

“To replace our third axiom with the assertion that first-unit demand conforms to the characteristics of harmonic effect would imply that the present value of money held at interest is always infinite.”

I then define what I call paranormal distributions, of which the lognormal distribution is (so far) the only example; these are the distributions that will work for the purpose of founding an economic theory.

Next I make a mistake:  I derive what I call the root-reciprico-normal distribution and claim that it is paranormal, but that it is not a viable candidate for founding an economic theory because it implies an almost god-like concern for the future.

“However, it is interesting that alternative systems exist which contain no internal contradictions but which are still very different from human economies.  Perhaps the inhabitants of another planet organize their economy in this or another way and consider their calculation of interest to be as natural as we consider multiplication by a constant.”

I call this a mistake because I have since realized that the definition of paranormal distributions must be further restricted for them to actually serve as the foundation for an economic theory and that the root-reciprico-normal distribution, while close, does not meet these more stringent requirements.

So I failed to be my own Lobachevski.  But perhaps one of you guys here at Research Gate can be Lobachevski to my Euclid.  I do not mind.  It was narrow-minded of old-time geometers to admit no alternative to Euclid; interest in geometry, including Euclidean, increased tenfold after it was realized that there was a second and then a third possible geometry.

Research Critique of Tony Lawson on Neoclassical Economics

Chapter Simplified Exposition of Axiomatic Economics

Alexander,

I was intrigued by your unorthodox answer to Victor as I have also been by your publications. Unfortunately I have to leave this thread, as other threads, as someone, who I know, is constantly down voting my postings as soon as I mention quantum mechanics and Gödel.

Best regards

erkki 

 Aleksandar,  I am not sure what to understand from what you are saying.  Are you saying that logical independence should be kept out of quantum theory? 

Steve

Aleksander,

As said I am intrigued by your understanding, however I do not fully agree with your conclusions. "Gödel" is not unknown in physics or chemical physics. It is for instance mentioned and applied in Linear Algebra for Quantum Theory (Wiley, 1998) by P.O. Löwdin and properly arranged in an axiomatic derivation of quantum theory. 

One way to "tackle" the conundrum would be to formulate the Gödel theorem(s) with respect to quantum logic rather than propositional logic. 

In light of this discussion, you might be interested to see the the following paper. In that paper, the axioms are The Field Axioms, which is a first order theory and is subject to Gödel.   That is to say, axioms in that theory are not a collection of stated principles, adopted for the theory.

https://www.researchgate.net/publication/282647562_Logical_independence_inherent_in_Elementary_Algebra_seen_in_context_of_quantum_randomness

Article Logical independence inherent in Elementary Algebra seen in ...

 Aleksander,

I am simply thinking about Gödel's paradox formulated  in propositional logics. In quantum logics the conundrum goes away.

http://www.researchgate.net/publication/282647562_Logical_independence_inherent_in_Elementary_Algebra_seen_in_context_of_quantum_randomness

Excellent work!  It helped very much to clarify what Dr. Faulkner is saying in this thread.

Abstract:

“As opposed to the classical logic of true and false, when elementary logic is treated as a formal axiomatised system, formulae in that algebra are either provable, disprovable or otherwise, logically independent of axioms.  This logical independence is well known to Mathematical Logic.  The intention here is to cover the subject in a way accessible to physicists.  This work is part of a project researching logical independence in quantum mathematics, for the purpose of advancing a complete theory of quantum randomness.”

I have written on this same subject though without reference to quantum randomness, proposing a three-term system of formal logic, the three terms being “possible,” “impossible” and “maybe.”  This latter term seems very similar to Faulkner’s excluded middle, though not quite the same thing, and about all of logic, not just mathematical logic.

At the time when I wrote about the logical foundations of economics I was informed by Willard Quine’s work and had little further knowledge of the literature of mathematical logic, specifically that of Gödel.  While I later studied (at the undergraduate level) quantum mechanics and was quite successful at it, it did not inspire me to edit the early chapters of my book about the foundations of economics, whose later chapters about economic theory I was working on at the time.  I never specifically studied the work of Gödel.

Thus, a comparison of my three-term system of logic and Faulkner’s work does not belong in this thread about Gödel and so I have started a new question where we can discuss this. 

https://www.researchgate.net/post/How_does_my_three-term_system_of_formal_logic_compare_to_Steve_Faulkners_work

Article Logical independence inherent in Elementary Algebra seen in ...

Dear Victor,

The only reason I can think of for taking such invocation seriously is if you feel the grounds for dismissing your work deserve to be addressed, perhaps because they raise issues that might enrich your thesis and/or perspective.

Now, it is highly unlikely that such invocation has an unequivocal perspective of what Goedel actually achieved in his seminal 1931 paper, say Go31, on formally undecidable arithmetical propositions.

As I show in the paper cited below, Goedel's remarkable achievement in Go31 was to constructively define an arithmetical expression R(x) such that (proofs are given in the paper):

(i) given any natural number n, there is always some algorithm T(n) that depends upon n and will compute R(m) = 1 (which, by definition, means that R(m) is true) for all 0 < m < n+1;

(ii) there is no single algorithm T that is independent of n and will compute R(n) = 1 for all n > 0.

However, what Goedel explicitly claimed in Go31 was that:

(iii) every axiomatic system which proves the PA axioms---subsuming the standard first-order logic FOL---is omega-consistent;

which implies what Goedel considered as his remarkable achievement in Go31 (leading to his two famous Undecidability Theorems), namely that:

(iv) every such axiomatic system contains a formal expression G which is formally undecidable since neither G nor ~G can be proven from the axioms of the system.

Now, presuming your work must implicitly assume that the PA axioms hold for the arithmetical functions and relations that you must need to consider in any axiomatic theory of economics, it can be naively argued that (iii) and (iv) must also apply to your axiomatic system.

However, you may safely assume, at the moment, that anyone who invokes Goedel---whether naively or not---in order to dismiss your work is not only unaware of the implications of (i) and (ii), but also unaware that---although Goedel's formal argumentation in Go31 for (iv) is valid---his two famous Undecidability Theorems hold vacuously---because Goedel's claim (iii) is false, since it can be shown that (see paper cited below):

(v) the first-order Peano Arithmetic PA is not omega-consistent;

and that:

(vi) there are no formally undecidable arithmetical propositions.

So, perhaps the best advice for now---in order to sift out the irrelevant criticisms so you may focus on the substantial ones---would be to refer your Goedel-invoking critics to the following paper which contains formal proofs of (i), (ii), (v) and (vi); and which is due to appear in the December 2016 issue of 'Cognitive Systems Research' (freely downloadable from the link below till 8th May 2016):

Title: The Truth Assignments That Differentiate Human Reasoning From Mechanistic Reasoning: The Evidence-Based Argument for Lucas' Goedelian Thesis.

CSR paper: http://authors.elsevier.com/a/1SkGv4xrDw50dQ

Kind regards,

Bhup

Your Axiomatic theory of economias has the same problems as set theory. The consistency of your theory can be proved as  I  did with set theory. Your theory can't be simultaneusly consistent and complete. If, for instance you did the proof of consistency, then your theory is incomplete. Set theory is incomplete, which means that your axioms can't prove or refute some propositions, like the axiom of choice and the continuum hypothesis, in set theory.

Its easier to dismiss than to verify and make something work.

Though Gödel prided himself in discovering a priori truths, in reality its the pragmatics of that truth that will win the day.

Article Methodology Doesn't Matter, but the History of Thought Might

If we compare axiomatics to a "constitution", only time will tell whether a priori truths will catch up with practice:

https://www.quora.com/What-was-the-flaw-Kurt-G%C3%B6del-discovered-in-the-US-constitution-that-would-allow-conversion-to-a-dictatorship