Nate Silver’s questionable foray into predicting World Cup results got me thinking about the limitations of maths in economics (and the social sciences in general). I generally stay out of this discussion because it’s completely overdone, but I’d like to rebut a popular defence of mathematics in economics that I don’t often see challenged. It goes something like this:
Everyone has assumptions implicit in the way they view the world. Mathematics allows economists to state our assumptions clearly and make sure our conclusions follow from our premises so we can avoid fuzzy thinking.
I do not believe this argument stands on its own terms. A fuzzy concept does not become any less fuzzy when you attach an algebraic label to it and stick it into an equation with other fuzzy concepts to which you’ve attached algebraic labels (a commenter on Noah Smith’s blog provided a great example of this by mathematising Freud’s Oedipus complex and pointing out it was still nonsense). Similarly, absurd assumptions do not become any less absurd when they are stated clearly and transparently, and especially not when any actual criticism of these assumptions is brushed off the grounds that “all models are simplifications“.
Furthermore, I’m not convinced that using mathematics actually brings implicit assumptions out into the open. I can’t count the amount of times that I’ve seen people invoke demand-supply without understanding that it is built on the assumption of perfect competition (and refusing to acknowledge this point when challenged). The social world is inescapably complex, so there are an overwhelming variety of assumptions built into any type of model, theory or argument that tries to understand it. These assumptions generally remain unstated until somebody who is thinking about an issue – with or without mathematics – comes along and points out their importance.
For example, consider Michael Sandel’s point that economic theory assumes the value or characteristics of commodities are independent of their price and sale, and once you realise this is unrealistic (for example with sex), you come to different conclusions about markets. Or Robert Prasch’s point that economic theory assumes there is a price at which all commodities will be preferred to one another, which implies that at some price you’d substitute beer for your dying sister’s healthcare*. Or William Lazonick’s point that economic theory presumes labour productivity to be innate and transferable, whereas many organisations these days benefit from moulding their employees’ skills to be organisation specific. I could go on, but the point is that economic theory remains full of implicit assumptions. Understanding and modifying these is a neverending battle that mathematics does not come close to solving.
Let me stress that I am not arguing against the use of mathematics; I’m arguing against using gratuitous, bad mathematics as a substitute for interesting and relevant thinking. If we wish to use mathematics properly, it is not enough to express properties algebraically; we have to define the units in which these properties are measured. No matter how logical mathematics makes your theory appear, if the properties of key parameters are poorly defined, they will not balance mathematically and the theory will be logical nonsense. Furthermore, it has to be demonstrated that the maths is used to come to new, falsifiable conclusions, rather than rationalising things we already know. Finally, it should never be presumed that stating a theory mathematically somehow guards that theory against fuzzy thinking, poor logic or unstated assumptions. There is no reason to believe it is a priori desirable to use mathematics to state a theory or explore an issue, as some economists seem to think.