Further thoughts on mathematical formalism in Economics

In a recent entry,  I offered a brief, and consequently simplified explanation of why formalism in mathematics and logic per se is not the cause of the perceived (accurately in my view) lack of realism in mainstream economics. This is not to say, however, that the emphasis in mainstream economics on formal mathematical is particularly useful. Coincidentally, an economics journal called The Review of Economics and Economic Methodology is planning on what looks to be a quite interesting issue on the role of Mathematics in Economics. Alas, the deadline for submission is most likely shorter than I could meet and it’s a distraction from my main project. That noted, it’s a good opportunity to at least attempt a further analysis.

In this entry, I will expand on my argument that formal mathematics brings little to the social sciences and also introduce a second point. Multiple people who criticize mathematical formalism in economics, perhaps the most prominent being Tony Lawson have argued that the use of the covering law model of scientific explanation is both ontologically flawed, as it rests on the assumption of “event regularities” and that it’s presence in economics is a strong contributor to the emphasis in economics on mathematical formalism. I will argue in contrast that though the covering law model does not comprise the entirety of scientific explanation, it is both useful and indispensable for explanation in the social sciences. For the most part, mainstream economists have not generally couched their explanations in the form of theories articulated via the structure of the covering law model of explanation. And that, as I argue, is to be lamented.

In the interests of attribution and acknowledgement of intellectual debt, where as my previous post on this topic was perhaps a bit heavier on the reliance of analytical philosophy for my arguments than I am fully comfortable with, in this post, a substantial part of my argument owes much to the contributions of  Susan Haack.

Part I: Clarifying a previous point

Firstly, I want to expand on and clarify my previous argument.  Any formal system that is applied to the study of  the natural or social sciences is a dual system: It is composed of the rules of the mathematics that are used, whether the mathematics used are set theory, linear algebra, calculus or Euclidean geometry and all the assumptions, definitions, etc. that attempt to define the entity being studied. In The Theory of Value for example, Gerard Debreu devotes his first chapter to explaining the math, and then in subsequent chapters defines and applies the economic assumptions. It is the latter chapters, not the first chapter, that present the problem. In principle, Debreu could have employed different economic assumptions and arrived at very different economic conclusions, and still couched his proof in the form of set theory and topography. Debreu’s irrealism is not a product or consequence of set theory and topography.

Part II: Does formalism bring anything to the social sciences?

Nevertheless, there is still the question of what is actually to be gained by this level of formalism in the natural and social sciences. In principle, the move towards formal systems, as opposed to informal systems,  is supposed to bring clarity and an end to ambiguity (for a quick, useful distinction, see this ).  The contributions of formal mathematics to physics are clear. If like me, you are neither a physicist nor a mathametician, I recommend Walter Isaacson’s biography of Einstein as one very accessible source of the relationship between Hilbert’s mathematics and Einstein’s formulation of the Theory of General Relativity.  The removal of ambiguity, subtlety and nuance are to some extent, virtues in physics.

What do we gain from this in the social sciences? IMO, very little that requires the substitution of formal, symbolic languages for the use of natural language. There are degrees of formalization. In   The General Theory   Keynes employs some of the method of Russell and the early Wittgenstein, both of whom were associates of Keynes, in defining his terms clearly and precisely, and occasionally employing symbols, rather than prose to make his arguments. But the main arguments in The General Theory are expressed via natural language. It would, IMO, be possible to fully formalize the The General Theory but, what would be lost would be the exceptions, qualifications, subtleties and nuances that Keynes expresses via natural language. In other words, a little formalization goes a long way. Because the social sciences are dealing with conscious, purposive agents who create institutions, the reduction of an argument in economics to steps that a computer program can follow may actually cloud, rather than clarify the argument. Nevertheless, the judicious application of the techniques of formal semantic analysis, can be useful, provided we do not get too caught up in it. This puts the burden of proof on the formalizers to demonstrate the usefulness of the formalization, rather than on those who express the argument in natural language.

Part III: An initial defense of the covering law model in the social sciences

This now brings me to the use of covering law models of explanation. A covering law model simply put has the logical form of   modus ponens .  In sciences where general laws apply, such as physics, given that the relevant phenomena are in actuality accurately described by the law, the law applies with iron necessity. There is an interesting issue in that for the most part, the logical positivists, especially Carnap and Hempel were nominalists. Part of Lawson’s objection is that Lawson is a realist and he thinks that all sciences, including the social sciences, should address real properties of the phenomena being studied: I agree in principle, thought not necessarily in full with Lawson’s version of realism. In so far as I understand Lawson’s criticism of the covering law model, it is that it relies on the conjunction of event regularities, rather than on real properties. But I think that a careful reading of Hempel shows that a covering law model can and often does describe actual properties of the phenomena being studied, which does raise some interesting questions about Hempel’s commitment to nominalism. Any explanation couched in the form of the covering law seeks to discern between accidental and essential conjunctions and explanation in terms of the mechanism of causality between X and Y. And that noted, I am curious as to how Lawson thinks we can reason without modus ponens.

Lawson’s second objection to the covering law model, especially in the social sciences, centers on the Critical Realist view of open systems.  I agree that socio cultural systems are open and evolving, and that even within a given system, patterns and habits may change so that forecasts based on time series will be unreliable. For example, a rate of unemployment of 4% in the 1960’s may have been sufficient to trigger an increase in real wages, thus leading to an acceleration of inflation, but insufficient in 2019. So Hempel’s use of the probabilistic version of the covering law is at least partially defeated. But does this mean that there is no relationship between the rate of unemployment and the ability of workers to bargain for a wage increase, and that all other things held equal, an increase in real wages will be passed on in the form of higher prices, if there is sufficient aggregate demand? I think not. In other words, the statement: for each and every developed capitalist economy there exists a level of aggregate demand that will be sufficient to lead to an acceleration of the level of unemployment, absent specific policy interventions, can be taken as a true description of the actual properties of  a developed, capitalist economy. What is uncertain is where that level of unemployment is, so the concept of a “natural rate” of unemployment at best obscures the existence of the reserve amy of the unemployed. In sum, the existence of open systems does not prevent us from having degrees of warrant of belief in the statement of a general principle.

In my view, the task of the social sciences, including economics, should be to determine the level of warrant for the applicability of general principles that can explain a wide range of phenomena. No single principle can explain the totality of a socio-cultural system, let alone of all human endeavors in every time and place. Different principles will have varying degrees of warrant and degrees of applicability. So we cannot rely on modus ponens alone, to do all the work for us. But on the other hand, I don’t see how we can escape it.

This now brings me finally to the problem(s) in economics. Formal models, of the kind economists use, are not in actuality general principle covering law explanations. In contrast, they are more often caricatures of specific features of the economy, whose conclusions are then overgeneralized far beyond their range of relevance.



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