The man who crushed the mathematical dream
from Lars Syll
Gödel’s incompleteness theorems raise important questions about the foundations of mathematics.
The most important concerns the question of how to select the specific systems of axioms that mathematics are 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 unproved truths.
This, of course, ought to be of paramount interest for those mainstream economists who still adhere to the dream of constructing a 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.
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Real-World Economics Review Blog 
Most economists do not think like scientists. They have not studied scientific methodology.
Logical completeness is not a necessary condition of a functional system. In fact, Goedel’s theorem is a proof that to have a function (that is, to be useful in predicting, modifying or determining something) a system ‘must’ be incomplete, must have an ‘outside’ with respect to which it is intended to function. In other words, a function must quantify over something other than itself. Goedel’s theorem was a formal proof of incompleteness, but its conclusion was already widely accepted prior to Goedel, informally, as a matter of logical necessity, most notably by Tarski, to overcome semantic circularity (the liar paradox), and by Whitehead-Russell in their effort to solve the Russell’s paradox by means of the theory of types. On a more metaphysical note, incompleteness is a condition of ‘seeing’, of being a witness to something other that oneself, and therefore a condition of one’s own existence: that ‘x exists’ entails that x is in a causal relation with something other than itself, or that ‘x exists for y’. Taking this back to economics, it’s incompleteness is not a deficiency but rather a necessary condition of its usefulness, so any economist claiming to pursue a model of logical completeness is simply inconsistent, and the model is provably false.
“…a function must quantify over something other than itself. ” That should be sufficient to debunk the standard stupidity of treating money as a measure of value while charging a unit cost for its use!
* a unit cost in terms of itself
Not sure what you mean by “Charging a unit cost for its use”… interest? That’s a bit more complicated and possibly not something I would see refuted by the above.
On the other hand i do believe the above proves that money can’t be both a measure of value and be valuable itself. This is formally obvious: it violates the law of identity about value of money on account of definitional circularity (self inclusion).
Extremely important understanding. It means that what economics actually needs is a new philosophy. In fact it also signals that what is required is a new paradigm to be integrated into the present monetary and economic systems. The signature of paradigm changes has always been a swapping/inversion of position of dominance/primacy and also a thorough integration of opposites which re-integrates all of the components of the entire system. As the Copernican helio-centric paradigm change was a swapping/inversion of the position of the earth and the sun and the understanding that the sun was to be considered the primary astrological center it was a thorough integration of opposites and a re-integration of the entire system. As Finance is the current dominant/primary business model and its enforced paradigms for the individual are Only Via Debt/ Loan and For Enterprise, For Production Only, so direct monetary Gifting For Consumption is its opposite. Intelligently integrated and implemented into the digital money system this would end the curious and almost entirely overlooked dominance of Finance in what is supposedly a system that advocates free competitive markets, re-integrating and stabilizing both the micro and macro economy.
The question for a science, which economics should be, is whether a mathematical construction bears a useful resemblance to the observable world. This can only be decided by *observation*.
Going on about Godel’s theorem only continues the confusion that mathematics is science. It is not. It is a tool used in science.
Economics should strive toward scientific status, however, human beings and enterprises and their leadership not being altogether rational and ethical actors will always make any theory less than scientific. It’s just the nature of the system, the agents in it and hence the body of knowledge itself. Economic theory should strive most importantly and urgently for Wisdom. Science is a form of Wisdom actually, but an incomplete form. We need “the full armor” of Wisdom which includes science, philosophy and ethics, and we need an open mind to new ideas regarding all three.
Craig, I don’t disagree with your description of what economics needs to consider, but your picture of “science” seems to be that it deals only with inexorable clockwork mechanisms. Modern science includes complex self-organising systems that can embrace living systems without killing them. I’m not saying it’s easy, but it’s not excluded.
So when I say economics should be a science, I’m really only saying that it should look for the regularities discernible in observations and describe them. Even human affairs have some patterns, imperfect though they may be. Otherwise, why would historians bother?
Not really. There is actually three types of science. Orthodox science, “good” open minded science and scientific breakthroughs which are generally characterized by a re- integration of a scientific viewpoint that is aided and abetted by an aspect or aspects of consciousness. The most recent of which was Einstein’s imagination/visualization of a man leaping out of a window and sliding down space-time. Physics has been struggling for almost 100 years to reconcile quantum realities with empirical particles when they would probably be smarter to realize that the quantum universe where particles continually come into being and then go back out of existence is perfectly reflective of the actions of human consciousness reaching out and experiencing the temporal universe and then withdrawing from it and in so doing that part of the temporal universe…for them and in that moment….goes out of existence for them. This does not invalidate the reality of the physical/temporal universe, but it does mean that our consciousness and its actions are reflective of quantum reality and are very likely a factor that rigid and arrogant orthodox scientific empiricism/positivism unconsciously uses to impede scientific breakthrough. Economics would undoubtedly scientifically benefit from the inclusion of consciousness and its many reflectivities in considering new policies.
Oh and I’ve written about this several times on this site.
Mathematics is a social construction. A product of culture. It is a system of theorems and axioms. Sometimes useful for such tasks as building, navigation, medicine, war, and of course commerce. It is not a pathway to truth, to significance, or, as Newton and Copernicus believed to God. This is how economists should use mathematics. It is how all scientists use mathematics.
Economists don’t need to abandon mathematics but they do need to stop using it to fake a world that operates according to one or another, or some group of mathematics theorems and axioms. No world operates this way. The physical universe forces the re-write of mathematical proofs and theorems. Why would anyone believe that the same would not be the case for the social and cultural worlds of human collective life?
It saddens me that contributors to the Real Word Economic Review are still failing to distinguish (a) Reality from (b) References to it, (c) the Form and/or Grammar of the references and (d) such Facts as we have been able to manuFACTure that are (e) logically True and hence worth referencing – i.e, in which relationships and actions expressible in the Form of our References to Reality correspond to constructed expressions and actual reality.
Following up Chomsky’s struggle to understand how children are able to learn different languages, the European-“manufactured” Algol68 I experimented with for twenty years made these distinction half a century ago, though of course mighty Americans rejected it as “not invented here”, cornering the market with a simplified version, ‘C’, which simplified away the philosophical point of it. My simplest version of Godel is thus that if new forms of Reality emerge from evolution, Formal languages need to be able to evolve too. Algol68 made that possible, e.g. allowing logical definition of pictures as well as numerical processing.
Professors are quite rightly in the business of helping new generations to learn to think, but when the problems are economic and the philosophical answers are already available, they need to acquaint themselves with them and start from an explanation of the answers so the students can move on with better understanding to the work they have seen needs doing.
Of the responses above, I agree with Craig on the importance of what Michael is saying (though Michael might find Kant a better starting point than Tarski), and emphasis the importance of what Craig is saying about inversion as a sign of Copernican-type mental revolution. Such a revolution in economics needs to parallel the one that took place in electric theory, when sub-atomic physics revealed that battery polarity premised on electrons being countable and hence numerically positive was actually the wrong way round. Money is considered valuable because it is countable, but actually it it is not a thing but a reference which can take many forms, and in Reality it represents debt.
I do wish Ken would get his head round what I am saying here, for he’s going off half-cock precisely because he hasn’t yet done so. I’m sure he COULD say it far better than I can.
Click to access How%20to%20Define%20Money.pdf
The idea that interest is the only reason for the instability of the macro-economy is false and confuses stocks with flows. However, as it is a cost it is part of the actual problem, especially with modern technologically advanced economies, which is that the ratio of the rate of flow of total individual incomes is less than the rate of flow of total costs and so results in a continual increase to the lower bound of price. (This is not marginalism which is also false, but rather an inherent cost inflationary condition of advanced capital intensive economies which have ever increasing fixed asset costs.)
Currently the only way to keep the economy from continually sinking is to continuously borrow which only re-inforces the business model of Finance’s already dominant position over every other business model and probably 95% of the general populace and doesn’t solve the problem because their enforced monetary paradigms of Debt, Loan and For Production ONLY, simply enforces a flow of additional costs in an already inherently cost inflationary system. The remedy is integrating a new monetary paradigm of Gifting into the system directly to the individual via a universal dividend and reciprocally back to enterprise after they have gifted the consumer a discount to their prices thus effectively integrating what is considered impossible by economists who forget that the money and pricing system is digital, price deflation into profit making systems.
I find this interesting Craig:
“The remedy is integrating a new monetary paradigm of Gifting into the system directly to the individual via a universal dividend and reciprocally back to enterprise after they have gifted the consumer a discount to their prices thus effectively integrating what is considered impossible by economists who forget that the money and pricing system is digital, price deflation into profit making systems.”
But not sure what you mean by “after they have gifted the consumer a discount to their prices”? How is this discount given if not voluntarily?
I have argue in issue 70 of RWER for a “gift” linked to individual contribution to the real GDP, not a universal gift. I had technical reasons for that but in the context of what we are discussing here the key advantage of that approach is incentivisation of constructive, economic activity as opposed to incentivising leisure. I still believe that universal gifting is likely to seriously affect the taxable capacity by many individuals reducing work hours or stepping out of the workforce altogether.
Marc, I appreciate your brave attempt to define money, but I don’t accept your conclusion that money is a measure. It is more like an estimate; and surely the point of both these is these is that it is an estimate or measure of something, which in the case of ‘value’ is itself, for valuation is estimating, while as a measure of its own value money is notoriously elastic (hence Sawyyer’s ‘topological’, quantitatively useful only in an ordinal sense, as an indicator of more or less).
Hence, anyway, the interest of Godel, and the point of the Algol68 scientific language: to enable scientists to define terms unambiguiously (http://www.eah-jena.de/~kleine/history/languages/Algol68R-UserGuide.pdf). You will find it instructive to try and define money taking account of the reference levels, structures (e.g. sets) and procedures of this: data being assigned to and referred to by objects, objects to types (which include sets or structures of objects) and types to open sets of procedures (e.g. such as can be legitimately carried out on whole numbers: type INT) which are already defined by the time of use. On the different forms of money, try comparing them with water, which remains water whether solid ice, liquid, gas or actively ionic, each of which has different properties and is useful for different tasks.
Michael, I too think Craig is not solving the problem by going for price cutting rather than gifting basic income, but your aims and its problems can be met by supplementing gifting of basic income with the incentive of prizes for good work.
The problem with paying employees is that there are more employees than there is constructive economic activity for them, so we have the current ludicrous situation of making up useless work (including money-making and the bureaucratic management of profit-making and taxation) using vast quantities of resources which are either irreplaceable or not being replaced. As the money in this system turns out to represent debt, there is the more economic alternative of providing basic income to both individuals and enterprises as credit (this becoming debt only insofar as it is used) and writing off what people need for the work of looking after and educating themselves and doing such work as is needed of them in the community, incentivising competitive development of skills, ideas and quality with fixed (so computable) rather than piecework-related prizes. Those running the show can still get all they need, but the incentive is to keep down their debt, not to show off their earnings.
Dave,
I’m actually advocating both a universal dividend and retail discount
«no matter what system is chosen, there will always have to be other axioms to prove previously unproved truths»
Oh please this is a decades old huge misunderstanding: Gödel two theorems don’t state anything like that, or anything of wider philosophical importance, they are just narrow technicalities, that say more or less that a mathematical system cannot be used as its own own proof theory to prove that all its own true theorems are provable in it (completeness) and that none of its own false theorems are provable in it (consistency). It is a technicality of interest only for the historical purpose of discussing Hilbert’s programme, and in particular Hilbert’s hope that finitary (in the induction sense, not the size of proof sense) proofs of arithmetic would be possible.
The two subsequent theorems by Gentzen are far more interesting on the topic of the difference between logical provability and logical truth, even if still fairly technical, and they much generalize Gödel two theorems by showing that to prove the completeness and consistency of a mathematical system is possible but only using a system with strictly higher order induction. That has nothing to do with «other axioms to prove previously unproved truths» even if there is a vague similarity.
The Gentzen theorems while quite interesting pose no wider philosophical questions or have any implications like «such deductive-axiomatic economics will always consist of some undecidable statements».
PS A very little noticed detail about Gödel and Gentzen’s theorems is that they are about whether consistency and completeness of a mathematical system can be proven, not whether the mathematical system is in fact complete and consistent.
Yes, Goedel’s proof works only for a formal, narrow system, precisely because it would not work, by its own premise, for all of reality. All formal theorems operate on a very limited basis, but that does not preclude broader philosophical implications that are both relevant and logically consistent.
If truth must be knowable as truth in order to be meaningful, and therefore be truth per se, then provability is a necessary and sufficient condition of truth. If this is correct, then the theorem implies that truth (not just provability) and logical completeness are mutually exclusive.
The proof of course was only a formality confirming something already known, at least since Aristotle’s formulation of the law of identity: that identity cannot be a proper part of itself because that would imply difference of itself from itself, therefore non-identity, therefore untrue identity.
Consequence: totality is untrue; truth is essentially incomplete.
Conversely, we can talk about a ‘formal’ system as an identity, we can maintain its truth only because its formality and systematicity is defined from the outside, in relation to external identities or some meta-language, and is as such logically incomplete.
We never needed Goedel for that, but his rigour is certainly appreciated.
«Yes, Goedel’s proof works only for a formal, narrow system, precisely because it would not work, by its own premise, for all of reality.»
«Consequence: totality is untrue; truth is essentially incomplete.»
That is random gibberish, it has nothing to do with the narrow ambitions of Hilbert’s programme, and the meaning and implications of Gdel’s two theorems in relation to Hillbert’s programme.
Unfortunately those two theorems attract wild meaningless speculation by those that like to string important-sounding words together.
it is usually a good test of the understanding of Gödel two minor theorems whether the much more interesting Gentzen extensions of those two theorems is mentioned.
«truth (not just provability) and logical completeness are mutually exclusive»
That’s more gibberish, and there is an interesting detail: the consistency (“truth”) and completeness (“provability”) of the theory of arithmetic have both been proven decades ago. How has that been possible if they are mutually exclusive?
Blissex et al: If you haven’t read it, you might like Torkel Franzen’s Gödel’s Theorem: An Incomplete Guide to Its Use and Abuse here is a positive review. Though I don’t think he would have agreed with all of Blissex’s judgments above.
I am not a logician but i studied some –including both goedel and Gentzen (i used to have a kind of informal proof of one of godel’s theorems in from 3 to one lines –basically a restatement of liar’s paradox) A famous book by Trostky’s secreatary collects many famous papers —from frege to godel–and there are later ones on Emil Post, etc.–like Godel somewhat eccentric.
regarding the view that Kowalik writes ‘random gibberish’, one should read some of Emil Post ‘s stuff–who is famous for ‘post’s problem’ (a very nice one—i think some have shown that can be solved in 10-15 lines). Some of Post reads like ‘random gibberish’ or ‘poetry’ but i personally find it somewhat stimulating and coherent–its saying the same thing but in a different way, and while imprecise may be one way to get nonexperts somewhat of an understaning or the concept. (People do this with relativity theory all the time — this gives some physicists (who i view as being into ‘rigor (mortis)’
heart attacks, but some others (eg VI Arnold of KAM (ergodic) theory say actually that’s fine , and also that’s how he was educated early on to develop an interest in physics.
I don’t quite remeber the references or even the exact results, but i’ve seen quite a few applications of godel’s theorem , undecidability and computability results in econ. The math is basically all the same. Very abstract, mostly useuelss except philosophically, but i think maybe economists should spend more time on this rather than pretend they are solving economic problems—eg unemployment. One could have full employment tomorrow if we gave everyone a UBI or an MMT government job and put them to work solving math problems which interest them or their community.
(eg some unemployed person could manage my dwindling bank account–write a budget while still possible. ).
The Lowenheim Skolem theorem i think is as interesting as goedel, gentzen, post etc. (later stuff by Paul Cohen on AoC). At present i think its still an open question whether any of this will ever be applicable (T H hardy thought his number theory was ‘useless art’ but he was wrong. Philosopher Badiou thinks transfinite arithmatic applies to the economic system of trotskyism). On Va Voir. I think it may be a conveiant notation. ) (see S Feferman).
Mart, good thoughts. For any of this to work, however, there has to be a community in which the work can be done and for which it is done. Neoliberals (current version fascists) have spent quite a bit of time and money the last 50 years destroying both the notion of community and its day-to-day functioning.
It was interesting to see the problem referred back to Aristotle and sounding very like Russell’s problem with self-reference, which was resolved in the Algol68 multiple-reference interpretation of his theory of types. An identity is a relation of reference to an object (which may itself be a form), but that makes an identity’s relation to itself a reference to a reference to the form of a reference relationship.
“If truth must be knowable as truth in order to be meaningful, and therefore be truth per se, then provability is a necessary and sufficient condition of truth.”
But it isn’t. Meaning isn’t about reference to reality, it is about relationships between forms. Truth is a reference to a definition of a relationship in formal logic which may or may not be true of the form of a relationship between a statement and what it refers to. If ‘dog’ refers to a reference to the form of particular object, itself referred to as X, then dereferencing, “X is a dog” is meaningful even if X is in fact a cat.
Glad, then, to see we agree on the “Consequence” here. “My simplest version of Godel is thus that if new forms of Reality emerge from evolution, Formal languages need to be able to evolve too”.
Yes, the theory of types was no doubt a major step towards development of modern programming logic.
“that makes an identity’s relation to itself a reference to a reference to the form of a reference relationship.”
You lost me a bit here. Should I understand that “the form of a reference rlationship” is defined by infinitely regressive reference relationship? I would then have to agree since self-reference can be regarded as a case of infinite regress… this is particularly evident in the liar paradox:
If we were to understand “This sentence is false” as a complete sentence of a language L (with no reference to a metalanguage) we should be able to substitute the same sentence for the word ‘sentence’, resulting in this:
“This (this ( this ( this … *inf*) is false.”
This is clearly an incomplete (infinitely regressive) sentence since the “this” reference is never able to reference to an object, but only to a next higher instance of the same form of referring.
I didn’t say that “meaningful” must be true, but only that “truth” must be meaningful, otherwise it would be a mystery in what sense it could count as true. And if a meaningless truth did have sense then it would be meaningful (because of its sense), therefore a contradiction.
“Meaning isn’t about reference to reality, it is about relationships between forms.”
It depends what is meant by reality. If reality meant only something that is objectively binding (independent of contingent and variable desires) then certain relationships between forms could be all the reality there is for any observer, but reality nonetheless.
If I understand you correctly, you are affirming a position sometimes referred to as ‘relational ontology’: that relations are fundamental while objects/forms/identities are derived or constructed from relations. This is commonly rejected because implies infinite regress (to locate an object), but I think it is still the most consistent position ever presented. The circularity is not vicious, because it is mediated via other relations, like a dictionary. Every word is defined only in terms of other words but indirectly, and so the complex system of overlapping relations it generates (language) is stable despite not being circular or definitionally fixed. This also satisfies the incompleteness/truth thesis: circularity ensures incompleteness, and as long it is not vicious it remains consistent.
… despite being circular but not definitionally fixed, I meant above.
“the consistency (“truth”) and completeness (“provability”) of the theory of arithmetic have both been proven decades ago. How has that been possible if they are mutually exclusive?”
Completeness is Not provability. But I get what you mean. The point is that the theory of arithmetic is not complete: the proof of consistency does not (and cannot) obtain from the theory itself, according to Goedel. Gentzen proof is meta-arithmetic as it is said to require accepatance of transfinite induction (whatever that is). In that sense, it can be taken as a demonstration of Peano arithmetic incompleteness.
One cannot prove that A is an A without some normative framework that is already accepted as true, the proof of which requires a meta-normative framework. This clearly leads to infinite regress unless interrupted by a decision to take some framework as true without a proof: therefore incompleteness.
For example, we cannot formally prove that the law of identity or non-contradiction is true, we simply assume this as a matter of practical necessity based on what kind of beings we already are. Consistency of our meaning relies on these laws, but they are not provable any more than ‘correcteness’ of our being.