## Arrow-Debreu and the Bourbaki illusion of rigour

23 Jul, 2019 at 13:01 | Posted in Economics | 9 CommentsBy the time that we have arrived at the peak first climbed by Arrow and Debreu, the central question boils down to something rather simple. We can phrase the question in the context of an exchange economy, but producers can be, and are, incorporated in the model. There is a rather arid economic environment referred to as a purely competitive market in which individuals receive signals as to the prices of all goods. All the individuals have preferences over all bundles of goods. They also have endowments or incomes defined by the prices of the goods, and this determines what is feasible for them, and the set of feasible bundles constitutes their budget set. Choosing the best commodity bundle within their budget set determines their demand at each price vector. Under what assumptions on the preferences will there be at least one price vector that clears all markets, that is, an equilibrium? Put alternatively, can we find a price vector for which the excess demand for each good is zero? The question as to whether a mechanism exists to drive prices to the equilibrium has become secondary, and Herb Scarf’s famous example (1960) had already dealt that discussion a blow.

The warning bell was sounded by such authors as Donald Saari and Carl Simon (1978), whose work gave an indication, but one that has been somewhat overlooked, as to why the stability problem was basically unsolvable in the context of the general equilibrium model. The most destructive results were, of course, already there, those of Hugo Sonnenschein (1974), Rolf Mantel (1974), and Debreu (1974) himself. But those results show the model’s weakness, not where that weakness comes from. Nevertheless, the damage was done. What is particularly interesting about that episode is that it was scholars of the highest reputation in mathematical economics who brought the edifice down. This was not a revolt of the lower classes of economists complaining about the irrelevance of formalism in economics; this was a palace revolution.

Mainstream theoretical economics is still under the spell of the Bourbaki tradition in mathematics. Theoretical rigour is everything. Studying real-world economies and empirical corrobation/falsification of theories and models nothing. Separating questions of logic and empirical validity may — of course — help economists to focus on producing rigorous and elegant mathematical theorems that people like Lucas and Sargent consider as “progress in economic thinking.” To most other people, not being concerned with empirical evidence and model validation is a sign of social science becoming totally useless and irrelevant. Economic theories building on known to be ridiculously artificial assumptions without an explicit relationship with the real world is a dead end. That’s probably also the reason why Neo-Walrasian general equilibrium analysis today (at least outside Chicago) is considered a total waste of time. In the trade-off between relevance and rigour, priority should always be on the former when it comes to social science. The only thing followers of the Bourbaki tradition within economics — like Karl Menger, John von Neumann, Gerard Debreu, Robert Lucas and Thomas Sargent — has given us are irrelevant model abstractions with no bridges to real-world economies. It’s difficult to find a more poignant example of a total waste of time in science.

Applying an axiomatic hypothetico-deductive system to the real world can only be done by means of a mapping, which creates a model for the axiomatic system. These mappings then lead to assertions about the real world which require empirical verification. These assertions (which are proposed scientific laws) can NEVER be proven in the sense that mathematical theorems can be proven …

The scientific method arose as a rejection of the axiomatic method used by the Greeks for scientific methodology. It was this rejection of axiomatics and logical certainty in favour of empirical and observational approach which led to dramatic progress in science. However, this did involve giving up the certainties of mathematical argumentation and learning to live with the uncertainties of induction. Economists need to do the same – abandon current methodology borrowed from science and develop a new methodology suited for the study of human beings and societies.

Models may help us think through problems. But we should never forget that the formalism we use in our models is not self-evidently transportable to a largely unknown and uncertain reality. The tragedy with mainstream economic theory is that it thinks that the logic and mathematics used are sufficient for dealing with our real-world problems. They are not! Model deductions based on questionable assumptions can never be anything but pure exercises in hypothetical reasoning. And that kind of reasoning cannot establish the truth value of facts. Never has. Never will.

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I think there are two distinctive issues here though: one is the extent to which formalizing is helpful in the social sciences. IMO, beyond the level of formalization in Keynes and Marshall, I don’t think it’s very helpful at all. The other is the truth value of your assumptions, regardless of whether your use a formal semantic language or natural language. I think the latter is the bigger issue with respect to mainstream economics.

Comment by Citizen Rat— 23 Jul, 2019 #

Lars, I think you’ve pointed the finger at Bourbaki before. Among mathematicians and physicists they used to be notorious for failing to provide any sort of logical basis for mathematical physics, and based on a quick scan of the web this still seems to be the case, e.g. according to https://en.wikipedia.org/wiki/Nicolas_Bourbaki “Anything connected with mathematical physics is totally absent from Bourbaki’s text”. This seems not to be an oversight, as I understand it they at least recognized the “well-known serious difficulties with stochastic processes.” (https://web.archive.org/web/20070304000104/http://www.math.tu-dresden.de/~pos_iv/Abstracts/koenig_abstract/index.html ).

My understanding is further, as wikipedia says, that to some extent Bourbaki was overtaken by category theory, which I have always seen as an attempt to distinguish between ‘proper’ mathematics and what Keynes called ‘pseudo-mathematics’ and which applies to mainstream economists views of stochastic processes.

But maybe you have some particular (obscure?) work by Bourbaki in mind? If so, please let me know and I’d be delighted to distance myself from them.

(On a slightly pedantic note, if von Neumann was a follower of Bourbaki, did he crack the problem of time-travel?)

Comment by Dave Marsay— 24 Jul, 2019 #

Dave, let me be very clear on this. I have no problem at all with the Bourbaki project within mathematics per se. The problem is when you transport it into science. It is one thing to separate rigour and precision from empirical corrobation/falsification/ in mathematics, and a totally different thing to try and do the same for (especially social) sciences. That is exactly — in line with the Bourbaki thinking — the wrong-headed tradition that Karl Menger started (and von Neumann, Wald and Debreu followed up on) with his article “Weitere Bemerkungen zu den Ertragsgesetzen” in ZfN 1936:

“Die Bemerkungen über Ertragsgesetze, welche wir in dieser und der vorangehenden Abhandlung gemacht haben, dürfen in methodischer Hinsicht vielleicht den Anspruch erheben, zum ersten Male in der Ökonomie die Frage nach den gegenseitigen logischen Beziehungen zwischen Aussagen von der Frage nach der empirischen Gültigkeit der Aussagen klar getrennt zu haben. Welche der verschiedenen Formen der Ertragsgesetze und sonstigen in diesen Bemerkungen behandelten Aussagen empirisch gelten oder nicht gelten, das sind Fragen, die lediglich empirisch durch Beobachtungen entschieden werden können. Wie die verschiedenen Aussagen miteinander zusammenhängen, welche aus welchen folgen oder nicht folgen, das sind logisch-mathematische Fragen, die mit der Erfahrung nichts zu tun haben. Die Wichtigkeit der empirischen Fragen für die Ökonomie ist so klar, dab sie kaum eines Hinwelses bedarf. Dass in diesen Bemerkungen die formalen Fragen so ausführlich behandelt wurden, hat folgende Gründe: Erstens, dass die Ökonomen selbst sie angeschnitten, zugleich aber in formal unzurichender Weise behandelt haben, während aus diesen Bemerkungen klar werden sollte, dass und wie sie einer völlig strengen Behandlung durchaus fähig sind. Zweitens, dass die formalen Fragen bisher meist gar nicht klar als solche hingestellt, sondern vielfach mit den empirischen Fragen verquickt, jedenfalls aber nicht als das behandelt wurden, was sie sind, nämlich als Fragen, ob diese und jene genau zu präzisierenden Aussagen diese und jene anderen Aussagen implizieren oder nicht implizieren. Drittens, dass ganz analoge Verhältnisse in mehreren Teilen der Ökonomie vorliegen, so dass die Methoden dieser Bemerkungen mehrfach anwendbar sind.”

Comment by Lars Syll— 25 Jul, 2019 #

Lars, my German is not up to your quote. Even my Bourbaki relies on translations into English. If what you are saying is that some mathematicians have said things about the social sciences that are well beyond their competance, then I have to agree. Mathematicians have to take much of the share of the blame. So, from a social science perspective, it seems reasonable to say that from a Wittgenstinian perspective, mathematics is hopelessly corrupt.

But it seems to me that if one accepts Keynes’ distinction between mathematics proper and pseudo-mathematics (or similar distinctions by Bourbaki or category theorists) then one can identify a sub-set of what social scientists regard as ‘mathematics’ that is highly critical of the pseudo variety, and is not corrupt and which may be useful.

Von Neumann keeps coming up in these discussions. He was an American so maybe I could simply agree with what I think is your view, that he was deeply deluded. On the other hand the work that he co-authored with Morgenstern seems to me to include all appropriate caveats, so the problem may not be that Von Neumann was an idiot, but that his work has been widely misinterpreted, like that of Boole, Bayes, Keynes, Russell and Turing. Either way, von Neumann’s work still seems to me very important, as long as you focus on the mathematics as such and regard all the expressions of opinion critically, as you do.

Maybe you have some ‘naughty’ quotes from von Neumann that I have overlooked? Alternatively, what logic or mathematics do you consider ‘proper’?

Comment by Dave Marsay— 25 Jul, 2019 #

Dave, just a few short comments.

1The vNM game theory book was a blend of vN’s math and M’s economics.

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2 vN did not really write anything (and absolutely not his ‘growth theory’) that can be considered ‘economic’ in any substantial meaning.

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3 Logic and mathematics per se is NOT the problem. I spent almost ten years when younger (back in the 80s and 90s) studying mathematics and mathematical logic. It was great. And very satisfying — at least until a friend of mine in a heated discussion told me that “you always have precise answers to questions, but they are always such, from the real-world perspective, insignificant and rather irrelevant questions!” Although I at the time wasn’t exactly überjoyed and didn’t want to admit he was right, the truth of what he said slowly dawned on me. After a couple of months I — painfully — accepted I had to redirect my intellectual endeavours into other paths if I wanted to have something relevant to say as a social scientist and economist!

Comment by Lars Syll— 25 Jul, 2019 #

I’m a bit boggled, I struggle to say something helpful, but I blunder on:

What kind of questions was your friend asking?

My own examples:

If asked ‘what is the probability of Heads for a fair coin’ I can give a precise answer. If asked ‘what is the probabity of heads’ for a real coin the only correct, unconditional and precise answer I can think of is ‘I don’t know’. I was taught to say ‘with the usual assumptions ….’ but then people ask about the assumptions, and then things are very far from ‘precise’. How would you answer a question about a particular real coin?

Another example, before 2007 I often used to say that ‘with the usual assumptions’ the probability of a crash would remain miniscule unless and until it actually happened. I might add that if one weakened the assumptions it wasn’t that the probability would increase, but that (as for Bourbaki) there would be no possible logical interpretation of ‘numerical probability’. This answer was, I think, correct and in a sense ‘precise’, but generally not acceptable: surely it was my job to quantify the uncertainty? How would you put it?

Comment by Dave Marsay— 27 Jul, 2019 #

“….. surely it was my job to quantify the uncertainty? ”

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What’s the point of quantifying uncertainty?

Comment by Henry Rech— 28 Jul, 2019 #

How would you answer a question about a particular real coin?.

Of course, you would investigate the particular real coin.

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In preparation for investigating the particular coin, you might also investigate the concept of a “fair coin”. Understanding the concept of a fair coin could be helpful in devising tests of whether a particular coin was “fair” to some measurable degree.

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I am not sure why economists should struggle so much with the essential distinction between working out conceptual logic on the one hand and systematically observing and measuring the particular and actual.

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Working out the concept of market equilibrium does not create an actual market or a particular equilibrium. Applying the logic of market equilibrium to testing whether a particular exchange relationship qualifies as a market or can arrive at a stable equilibrium in price might constitute an interesting empirical research project. It would certainly be an unusual project for a mainstream economist.

Comment by Bruce Wilder— 28 Jul, 2019 #

The persistent long term violation of Covered Interest Parity in currency swap markets proves prices are arbitrary.

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Fair pricing formulas explicitly use no arbitrage assumptions; if significant arbitrage is observed in the voluminous international currency swap market, fair future prices become arbitrary.

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Fischer Black was right!

Comment by Robert S Mitchell— 30 Jul, 2019 #