Saturday, March 21, 2015

On Mainstream Economists' Ignorance Of Real Analysis

"Logic sometimes makes monsters. Since half a century we have seen a crowd of bizarre functions which seem to try to resemble as little as possible the honest functions which serve some purpose. No longer continuity, or perhaps continuity, but no derivatives, etc. Nay more, from the logical point of view, it is these strange functions which are the most general, those one meets without seeking no longer appear except as particular cases. There remains for them only a small corner.

Heretofore when a new function was invented, it was for some practical end; to-day they are invented expressly to put at fault the reasonings of our fathers, and one never will get from them anything more than that." -- Henri Poincaré (1908, as quoted in Lakatos 1976, pp. 22-23).

Mainstream economists these days seem unwilling to accept claims about economics that are not backed up by mathematical models. (I think that views on mathematical formalism are pluralistic among non-mainstream economists. Mathematical models are just one of several approaches to acceptable claims about economics, and some non-mainstream economists are quite good at producing mathematical models.) Generally speaking, mainstream economists seem to me to reject norms common among mathematicians.

Anybody taking a standard undergraduate sequence in mathematics at a reasonably good university has an opportunity to be introduced to real analysis. Often, such a class is where the mathematician is introduced to a certain style of definitions and proofs, particularly epsilon-delta proofs. Besides this style, these classes teach a certain content, that is, the theory of limits, the differential calculus, and the integral calculus, from a rigorous standpoint. (I also draw on measure theory below, which, for me, was not taught at the undergraduate level.) In such a class, one should see various examples and purported counter-examples. The examples help the student to understand the range of behavior consistent with certain axioms. The supposed counter-examples help the student understand why theorems contain certain assumptions and why certain concepts useful for stating these assumptions were introduced into mathematics. Given an example inconsistent with the conclusion of a theorem, the student should identify a clause in the assumptions of the theorem that rules out the example.

To make my point, I'll list some examples. For my amusement, I'm not (initially) looking up anything for this post. Just as when someone criticizes somebody else's grammar, the probability approaches unity that they will make a typographic error, so I'll almost certainly be mistaken somewhere below. Does anybody have suggestions for additions to the following list of examples from real analysis?

  1. Define a function that is discontinuous at some point.
  2. Define a function that is continuous everywhere, but differentiable nowhere.
  3. Define a sequence of functions that converges pointwise, but is not uniformly convergent. (Or is it the other way 'round?)
  4. Define a function that is Lebesque integrable, but not Riemann integrable.
  5. Provide an example of a non-(Lebesque) measurable set.

The style of reasoning introduced in courses on real analysis has been important in economics since, at least, Debreu (1959). And economics provides many examples analogous to the answers to the above problems. Lexicographic preferences can provide an example of a complete order on a commodity space - that is, rational preferences - that cannot be represented by an utility function. Such preferences highlight the need for an assumption on the continuity of preferences, given that the commodity space is a continuum; "rationality" is not sufficient. Menu-dependent preferences suggest the possibility of specifying deeper structures that do and do not allow the construction of binary preference relation providing an order for a commodity space. I suppose the concept of hemi-continuity is proof generated in economics.

Sraffians have also provided many examples not consistent with outdated mainstream teaching. Ian Steedman's work, over the last quarter century, is particularly good on examples illustrating that the Cambridge critique is not exhausted by the possibilities highlighted by reswitching and capital-reversing. As of yet, economists have not specified any general assumption on production processes that rules out these sort of Sraffian examples and yields neoclassical conclusions. Yet many economists - who, I guess, treat their training in mathematics as a hazing ceremony for induction into the brotherhood of economists - proceed as if they have some such theorem.

Obviously, despite my generalization, some economists, both mainstream and non-mainstream understand and accept mathematical analysis. Maybe more mainstream economists understand than my generalization would suggest. The refusal I have seen of economists to accept their own logic may be the manifestation of anti-intellectualism and boundary-patrolling that I think is so common among properly socialized economists. The general public must not come to understand how vulnerable the conclusions of mainstream economists are to slight perturbations in model assumptions. Demonstrations of the failure of the logic in the teaching and public pronouncements of economists must be distracted in blather about credentials or (false?) irrelevancies about empirical results. What economists say in public and what they say in professional seminars need not be consistent. (This is not quite the right link from Dani Rodrik making his point.) I can easily be led to believe that explanation for some behavior I have seen is more a matter of the sociology of economics and less a lack of understanding of mathematics. So, in general, are economists still exhibiting a century-outdated attitude to mathematics?

Answers
  1. This is an easy question. For amusement, I'll name a function that exhibits a discontinuity of the second kind, if I correctly remember the terminology. Consider the limit of the following function of the reals as x approaches zero: f(x) = sin(1/x), if x ≠ 0; 0, if x = 0.
  2. Various space filling curves provide examples. I think both Hilbert and Sierpinski provide examples.
  3. I'm vague on this one, but consider the Fourier series for a square wave, where the value of the square wave at points of discontinuity is the midpoint of the left-hand and right-hand limits. I think mathematicians greeted Fourier's work on functions that were only piecewise continuous with some degree of incredulity.
  4. f(x) = 0, for x rational; 1 for x irrational.
  5. Consider a decomposition of the real numbers between zero and unity, inclusive, into equivalence classes. For this example, two real numbers in the range are considered equivalent if the difference between them, modulo one, more or less, is a rational number. The axiom of choice allows one to select a real number in each equivalence class. Take the union, with the index set for the union formed by the choice from each equivalence class. The index set contains an infinite number of elements, and the union is the desired closed interval. Furthermore, each equivalence class can be put into a one-to-one correspondence with any other equivalence class. Thus, the measure of each equivalence class must be the same. And these measures must add up to one, since that is the Lebesque measure of the closed interval. But assigning a measure of zero to each equivalence class will not do, and the sum over equivalence claess for any finite measure would be positive infinity. So any equivalence class formed in this way in non-measurable.
References
  • Gerard Debreu (1959). Theory of Value: An Axiomatic Analysis of Economic Equilibrium. John Wiley & Sons.
  • Imre Lakatos (1976). Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge University Press.
  • Walter Rudin (1976). Principles of Mathematical Analysis, Third edition. McGraw-Hill.

6 comments:

Brian Romanchuk said...

In your second example, you have "differential" where it should be "differentiable".

One thing that surprised me when I took a look at DSGE models was the description of Calvo pricing. The text said that "firms have an index i, where i lies in the interval [0,1]", and it integrated over that interval to get the aggregate.

It took me awhile for me to believe they were serious. I thought that anyone who was writing down all this heavy math knew that the interval [0,1] is uncountably infinite, and so could never be the limit of models where you have a finite number of firms. This was a situation where firms were a point in a continuum. How was this supposed to be microfounded?

Robert Vienneau said...

Thanks, fixed.

Steve Keen has trouble with that kind of assumption, too. I think you can see the first example in papers (e.g., Markets with a Continuum of Traders) by Robert Aumann in the 1960s.

Nicholas Georgescu-Roegen had similar objections. See his paper "Methods in Economic Science" (Journal of Economic Issues, V. 13 (1979): pp. 317-328

Anonymous said...

Some blogger attacks Marxist theory, what do you think about this ??:
http://socialdemocracy21stcentury.blogspot.com/

Robert Vienneau said...

I'm not making much progress in revising my empirical demonstration of the labor theory of price. There's all sorts of discussions I have not had time to get round to commenting on - his earlier take on post modernism and discussions of Krugman's non-Keynesianism.

Unlearningecon said...

One of the escapes I've seen from critiques like Brian's (and especially Keen's) is that although the model doesn't make sense mathematically, you can assume that firms act 'as if' they don't have an impact on price (in Keen's case) or, presumably 'as if' the economy is represented by a continuum of firms (in Brian's case).

Basically, you can always seem to escape an internal critique by inventing a new assumption in economics, coherent or not.

Unlearningecon said...

Just to add another small rant which leads on (hopefully) from your post.

One of the problem in economics is how maths is taught. Students are only given training in maths insofar as it helps them to deal with particular economic models, rather than being given a firm grounding in maths and then shown how some of this is used in economics. So you will know constrained optimisation, both dynamic and static, but will not have much of an understanding of what Hamiltonians and Lagrangians are actually supposed to do, and how they relate to the real world in detail. (Nothing like this: https://www.youtube.com/watch?v=xz7jLnWcxMs&list=PLTAnYeruJqLeW0sK3Gzwh2VxAPEhI8RWc , for example)

The result - at least in my experience - is that there are a lot of holes in the students' knowledge, and the type of deep understanding demonstrated by the critical thinking questions in this post is often lacking. Professors tend to have further training in maths but again this is often limited and instrumental. It's therefore no wonder you get complete misapplications like Brian's above.