Chapter 16 of Debunking Economics is a short comment on the use of mathematics in economics. Keen offers a defence of maths itself, suggesting that it is neoclassical economist’s misuse of the tool, rather than the tool itself that has caused the problems in economics today. He compares it to the story of a king who hears an awful tune played on the piano, and proceeds to shoot the piano.
Keen first recaps on some of the mathematical mistakes he has discussed throughout the book, such as the problems with demand and supply curves. I won’t go over these again here – that would be a summary of a summary – but will instead briefly note a couple of general problems with economist’s use of mathematics.
First, it seems economists are not ready to acknowledge the limits of mathematics: mathematicians have known for some time that some equations simply cannot be solved, or are incredibly difficult. Since economists are often dedicated to proving the existence of an equilibrium, they have to stick to overly simplistic analysis, where equations can definitely be solved. This causes them to rely overly on linear models.
Second, Keen makes a pithy mathematical observation about emergent properties and reductionism. Reductionism can be characterised as reducing something down to its component parts. However, if these component parts are multiplied together – rather than added – as you aggregate up, you will see a substantial change in behaviour at the aggregate level. Hence, reductionism has clear and obvious limitations.
Overall, I agree with Keen that mathematics is useful in economics. Jevons put it most accurately when he said “[economics] must be mathematical, simply because it deals with quantities.” However, this shouldn’t mean quantifying things with erroneous measures – such as capital - just for the sake of mathematics. Equations have to have clearly defined parameters, can only be considered as good as their assumptions, and may not have clear implications. Such is the nature of modelling complex systems.
Update: I was going to leave this out for fear of digressing, but a couple of the comments reminded me of a quote Keen used to end the last chapter:
The real problem with my proposal for the future of economics departments is that current economics and finance students typically do not know enough mathematics to understand (a) what econophysicists are doing, or (b) to evaluate the neo-classical model (known in the trade as ‘The Citadel’) critically enough to see, as Alan Kirman  put it, that ‘No amount of attention to the walls will prevent The Citadel from being empty’. I therefore suggest that the economists revise their curriculum and require that the following topics be taught: calculus through the advanced level, ordinary differential equations (including advanced), partial differential equations (including Green functions), classical mechanics through modern nonlinear dynamics, statistical physics, stochastic processes (including solving Smoluchowski-Fokker-Planck equations), computer programming (C, Pascal, etc.) and, for complexity, cell biology. Time for such classes can be obtained in part by eliminating micro- and macro-economics classes from the curriculum. The students will then face a much harder curriculum, and those who survive will come out ahead. So might society as a whole.
This is from the (econo)physicist Joseph McCauley. It’s an interesting reversal of roles for economists, who often label critics as mathematically illiterate. Having said that, I think McCauley’s attitude shares some of the same characteristics that I hate to see in economists.