We live in a mathematical universe! There’s so much talk these day about the need for new economic thinking, and so little action. The problem about the role of mathematics in economics is a key stumbling block here, I think, and many would agree to this. But let me try to put the problem under a new angle that is somewhat different than the usual. Every economist educated from a reputable university would recognise being told in introductory courses that modern economic theory is based on consistent mathematical methodology. The idea taught is that mathematics provides the basis for stable axioms and deducible theorems, and economics is then merely the empirical application to a complicated world. The truth of economics is supposed to come from mathematics understood as a consistent logical procedure in a line of argumentation. The principle of falsification, the modus tollens proposition heavily propagated by good old Karl Popper about 80 years ago that has formed the philosophical basis for positivist science ever since, is itself such a mathematical argument. It claims that science can be axiomatised, that is, organised according to logically binding principles. The presumption then is that math is a basis for rational thought, a method that deals with logic only, not reality. The mathematician Gregory Chaitin has challenged this idea over the last 40 years or so. I think economists need to listen to what he says (see link below). According to him mathematics has its own problems. It is supposed to possess absolute truth, while empirical science such as economics doesn’t get to that. This is wrong, he says. Mathematics does not have absolute truth. No mathematical theory can give you the truth and just the truth. There will always be truths that cannot be reached, and there are mathematical theorems that prove this. What we therefore need to recognise is that we can have relatively simple laws (such as equilibrium of supply and demand in economics, if I may add) and still be confronted with a very complex mathematical world. Chaitin thinks that mathematics can be very helpful in physics, but that it is more difficult with biology, or economics. Why is that? Pure mathematics is provably infinitely complicated, he says. It’s impossible to have a complete mathematics, a theory of everything. That is then less of a problem for a relatively simple system such as physics. It’s a bigger problem for complicated systems such as biology or economics. But we cannot even realise that this is a problem at all if mathematics is merely seen as a convenient method, a way of methodologically approaching the world. Mathematics would then merely describe a limited world of consistent logic. In sum, we are led by Chaitin to the insight that math is about some real objects in the world. Logical relations exist! He then builds this insight into a theory of information. According to Chaitin, information comes from the difference between description and theory. There is no information in pure replication, that is, description, but there is also no information without observation and data. Information is the extent to which one through computation can compress an algorithm that writes the data to an algorithm that writes the theory. There is a lower limit to this called computational complexity. If we then add the idea that macroeconomics is part of such a mathematical universe, information theory allows us, I believe, to escape Popper’s fallacy of deducting a set of laws from axioms and then impose these on observations of a thing called the economy. This is difficult but nevertheless very rewarding stuff for economists. Economics should start researching some new economic thinking along these lines. Economic theory has since at least the 1960s acknowledged that information guides the decision-making of people. But the focus in economic theory became for some reason optimization instead of decision-making. The focus should have been on whether and how equations can be solved. But this has not been discussed except at the fringes of mainstream. Information is regarded as data assembled conveniently by a benign data collector for input in a model; the nature of the model is external to the nature of the data; the model is build on consistent math; expectations then solve the model – so said Muth and Lucas. Information is seen as observer relative, which means that there is no complexity limit. In sum, if you ask me, a philosophy of mathematics associated with a concept of information is strongly warranted but strangely wanting in modern economics. It’s one of the really great puzzles why economics uses the mathematical technology that it does. Mathematics is, I would say, a language with which you reach conclusions and produce arguments that you cannot by purely verbal means. But once you’ve reached these conclusions it is always possible retroactively to explain a result in verbal terms. Math is a forum for thinking both the possible and the impossible. It could make us realise that an economy is a complexity, that is, a constraint on the computation of mathematical expressions. There’s a problem of computation and solvability that needs to be approached both in mathematical and economic terms. It is not satisfactory merely to assume that equilibrium exists on the basis of some proof given 63 years ago.
Posted on: Thu, 18 Dec 2014 16:32:23 +0000
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