## What are axiomatizations good for?

10 July, 2018 at 19:55 | Posted in Economics | 2 CommentsAxiomatic decision theory was pioneered in the early 20th century by Ramsey (1926) and de Finetti (1931,1937), and achieved remarkable success in shaping economic theory … A remarkable amount of economic research is now centered around axiomatic models of decision …

What have these axiomatizations done for us lately? What have we gained from them? Are they leading to advances in economic analysis, or are they perhaps attracting some of the best minds in the field to deal with difficult problems that are of little import? Why is it the case that in other sciences, such as psychology, biology, and chemistry, such axiomatic work is so rarely found? Are we devoting too much time for axiomatic derivations at the expense of developing theories that fit the data?

This paper addresses these questions … Section 4 provides our response, namely that axiomatic derivations are powerful rhetorical devices …

‘Powerful rhetorical devices’? What an impressive achievement indeed …

Some of us have for years been urging economists to pay attention to the ontological foundations of their assumptions and models. Sad to say, economists have not paid much attention — and so modern economics has become increasingly irrelevant to the understanding of the real world.

Within mainstream economics internal validity is still everything and external validity nothing. Why anyone should be interested in that kind of theories and models is beyond imagination. As long as mainstream economists do not come up with any export-licenses for their theories and models to the real world in which we live, they really should not be surprised if people say that this is not science, but autism!

Studying mathematics and logic is interesting and fun. It sharpens the mind. In pure mathematics and logic , we do not have to worry about external validity. But economics is not pure mathematics or logics. It’s about society. The real world. Forgetting that, economics is really in dire straits.

Mathematical axiomatic systems lead to analytic truths, which do not require empirical verification, since they are true by virtue of definitions and logic. It is a startling discovery of the twentieth century that sufficiently complex axiomatic systems are undecidable and incomplete. That is, the system of theorem and proof can never lead to ALL the true sentences about the system, and ALWAYS contain statements which are undecidable – their truth values cannot be determined by proof techniques. More relevant to our current purpose is that 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 …

Many more arguments can be given to explain the difference between analytic and synthetic truths, which corresponds to the difference between mathematical and scientific truths … 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.

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What I do has often been described as ‘analysis’, so I find the Carneades quote somewhat distracting, and tending to detract from Asad’s work by a kind of converse holo effect. Omit?

I wish that Asad had written “It was this rejection of [the Greek use of] axiomatics and logical certainty… ” and “giving up the certainties of [naive] mathematical argumentation “. Then I could agree with him whole heartedly.

Most hard science can be seen in terms of a search – at least – for axioms, but in a good sense, as ‘the core assertions of a subject that ought to be critiqued’, not ‘things that you mustn’t question’, or anything similar. To put it another way, you need to appreciate the difference in the role of axioms in pure and applied subjects. (Asad is not, I think, sufficiently clear on this. But this is not to criticise his main thrust. I just wish you’d humour me more!)

Comment by Dave Marsay— 10 July, 2018 #

I cannot say that I am much in sympathy with either the Carneades quotation or Asad Zaman in this case.

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The scientific method arose as a rejection of Scholasticism, or at least its most pretentitious forms emphasizing argument from authority. The ancient Greeks, of whom we know too little, should not be blamed for developments that occured during a revival of learning centuries later.

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Moreover, scientific method, far from rejecting logical analysis, embraces the practical application of logical reasoning — the supreme scientific presumption: the world is a logical place! That leads to the insight, if that be a proper term, that the world is driven and organized not by meaning but by functional relations, which is to say, systems constrained by logical possibility.

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Economists go wrong when they insist their theories can be mapped to the world and the theory tested, as Asad Zaman carelessly implies. This is silly. We must map the world using analytic theory as a toolkit, and test the world to see what the world is. Test the world! The factual truth is in the world and can only be approximated in systematic observation and measurement that makes use of logic to discover the nature of the world.

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We have a fine example of how this should work in that axiomatic-deductive masterpiece of Greeks, Euclid’s Geometry. Only a economic theorist could be obdurate enough to dismiss the results of Euclid’s theorems as a mere restatement of predicate meaning. No, Euclid explores the functional logic of space. Nor would any but a fool mistake a geometry for a map. But, a cartographer, surveyor, architect or geodesist must apply the logic of geometry to the task of modeling and mapping and measuring the world. This doesn’t lead, as one might suppose from the precedents of economics, to dogmatic assertions that the earth forms a perfect sphere. It results in carefully and precisely qualified measurments defined by operational models.

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Economists fail to operationalize properly and I would go farther and say they do not do axomatic-deductive analytic theory right either, the braying about “rigor” notwithstanding. I would not complain about axiomatic-deductive methods per se; the problem is not, in my view, that economists do analysis, but that they do it wrong. They confuse their

a prioritheories with statements about the world, thinking their armchair speculation makes assertions about the world. And, they refuse to remedy the many flaws in their theoretical logic, to pursue seriously an exploration of what is necessary and sufficient. The elementary textbook economic theory of production makes no acknowledgment of entropy or uncertainty, to take one consequential example. Serious analysis would never be satisfied to leave those lacunae at the core of theory, while preaching the necessity of true conclusions from true premises. That preaching is a false piety: conclusions drawn from incomplete premises cannot be logically true! If the functional system requires additional elements, these must be considered, before logical necessity can kick in. Continuing with example of the elementary theory of production: Output isnota function of factor inputs — 2 minutes of thought should confirm that — and without that relation, what does Samuelson’s theory of production consist of? An undefined and undefineable concept of “maximization”? This is not a case of premises that might be “true” in some conceptual sense, but a problem in logic not subjected to critical assessment. Axiomatic-deductive theory fails on its own terms — the economists doing it fail to do it right or seriously..

Axiomatic-deductive theory has great power, as geometry demonstrates. And, though geometry makes no use of measurement, measurement makes extensive use of geometry and geometry’s logic. Because the world is a logical place, exhibiting systems organized around functional relations.

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Economics, though, has been doing it wrong.

Comment by Bruce Wilder— 12 July, 2018 #