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March 05, 2005
Mathematical Constructivism and Proofs in Economics
Steve Han asks me what I think of Mathematical Constructivism, described in this Wikipedia entry. As the entry describes well, there are four parts to Mathematical Constructivism: (1) if you are proving that a class of things exists, you have to actually find one example in the class that does exist, (2) if you are proving that a property is true for every element of a class of things, you have to provide an algorithm that will show it is true for every individual element, (3) the concept of infinity can't be used, and (4) you can't use proofs by contradiction (something which flows from principles 1 and 2).I remember first hearing of this in Herb Scarf's math econ class at Yale around 1978. Brouwer was a constructivist, and Scarf was happy that in the Scarf Algorithm for finding fixed points he had found a constructive proof of Brouwer's Fixed Point Algorithm.
I use proofs by contradiction all the time, and do believe that they indeed lead me to truth. That is because I would define a true statement as one on which we can rely under all circumstances and to which everyone would agree if they thought hard and long enough. I do not, however, think that proofs by contradiction lead me to understanding. I can rely on my proved theorem, but I do not necessarily understand it, by which I mean that I cannot generalize it further or extend it to even slightly different premises (unless I can be assured that the contradiction carries over to those different premises).
Thus, knowledge attained via proofs by contradiction is useful but unsatisfactory. It is okay for theorems that one uses as a technical tool, but not for ultimate conclusions. (Maybe I'm caught in a contradiction here-- how can I rely on an ultimate conclusion reached by a technical tool that I don't understand?)
Take, for example, the question of existence of a Nash equilibrium in a particular game-- say, the Battle of the Sexes. Here are three ways I could prove that an equilibrium exists:
2. I could show that this game (with mixed strategies included) fits the premises of the standard theorem about when equilibrium exists-- that the game's payoffs are closed and bounded, that the payoff functions are continuous in the strategies, and so forth.
3. I could in effect re-prove an existence theorem by showing that a contradiction arises if no equilibrium exists. (Is it true that whenever a Proposition is true, one can find a proof of it by contradiction? Probably.)
1. I find a strategy profile that fits the definition of equilibrium-- that is, such that neither the Man nor the Woman would unilaterally deviate. One such strategy profile is that both players go to the Prizefight. The Man would not deviate, because he likes the Prizefight better than the Ballet and he likes being with the Woman. The Woman would not deviate, because although she likes the Ballet better, it is more important to attend the same event as the Man.
For this simple situation, method (1) is easiest and best. I not only achieve my goal of proving existence, but I end up with an example of it too. I understand the situation much better than if I had used (2) or (3).
More often, method (1) is not easiest, because it is hard to find an example but easy to show that the conditions of some other theorem apply, method (2). But method (1) still yields the best understanding, I think.
In economic modelling, understanding is particularly important. Our models are always special cases anyway, and we want to extend the mathematical conclusion to the real world. Thus what we call "intuition" is crucial. It is not enough to do the proof: the economist must make the reader understand why the general idea behind the proposition is true. The proof is there mainly to provide a check on the intuition, because purely verbal explanations are more likely to be flawed.
An example is the Walras-Arrow-Debreu proposition that there exists a set of prices which equate supply and demand. The premises of the theorem never apply exactly, but frequently apply approximately. The proof is a useful achievement, but what is most important is the idea that the Invisible Hand works, an idea as old as Adam Smith, but one which in the form he presented it would leave any careful reader uncomfortable.
Posted by erasmuse at March 5, 2005 09:27 AM
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