The tendency to conflate science and mathematics in economics

The discipline of economics is frequently criticised for using too much mathematics, for giving precedence and undue reverence to highly mathematical models, and for looking down on social sciences that don’t utilise mathematics as it does. I’d like to chime in to this debate. There are some compelling arguments on both sides. I’ll try to be brief. 

First, in favour of mathematics, let’s start with a comment from a lecturer at the ANU:

‘The other day in The Canberra Times some guy was complaining that “there are too many equations in economics”. That’s ridiculous. Can you imagine someone saying “There are too many equations in air traffic control…we would get better outcomes if we used fewer equations in air traffic control.”? You can’t, because it’s ridiculous.’

There are numerous situations in research and practice where you need numbers to get a deep understanding of something. You can’t determine the estimated time of arrival of a plane without mathematics. Similarly, you can’t give a compelling estimate of how much more money Australia’s tertiary education sector ‘ought’ to receive without a mathematical model of things like the costs and benefits of tertiary education compared to other potential recipients of government money or untaxed income.

You also need maths to engage in the empirical verification of hypotheses. This is very important. Without tests we have no knowledge, and without knowledge might makes right or rhetoric makes right, neither of which is a great outcome. Verification requires tests, and good tests require data—reams of it. Statistical tools become a necessity, and statistics is a branch of mathematics. This is especially true in the social sciences, where the inability to bring about controlled conditions in a lab means large samples and advanced identification techniques are critical for getting precise results. The process of verification is expedited if hypotheses arrive at the statistician’s table already in a mathematical form, so economists tend to put ideas into such forms early in their lifecycles.

However, while empirical verification of hypotheses is the essence of the scientific method (see Karl Popper) and maths is required for empirics, economists sometimes skip a step and think the essence of science is just maths. Thus calculus, set theory and the other branches of mathematics frequently employed by economics are sometimes utilised seemingly for their own sake.

One example passed to me from a friend: his lecturer spent nearly half an hour proving that utility curves are thin, because otherwise they various arcane mathematical assumptions required for sketching them graphically would be violated. Nobody cares about this except mathematicians. It isn’t required to tell a compelling story, to articulate the theory of consumer choice, or to test that theory.

Let’s look at another example from development theory. The Harrod-Domar model is constructed from three equations that describe the natural growth rate, the warranted growth rate and the actual growth rate. Once thus equipped, it discusses what happens when the warranted growth rate exceeds the natural growth rate, and vice versa. Seems fine, but the problem is that the math does nothing for the model. Indeed, it actually makes it harder to understand. The insight or hypothesis in the model is that sustained growth requires harmony between the growth of capital and the growth of the labour force that uses it (and their skill level). If capital abundance grows slower than labour abundance then structural unemployment emerges, as it does in most developing countries. If capital grows faster than labour then liquidity becomes abundant and the price of labour is bid up, as we see in many developed countries. I think there is probably some way to improve this story with math, but the maths in the Harrod-Domar model doesn’t make it clearer or more testable.

Now admittedly, sometimes putting things into a mathematical form provides important insights. For example, the basic microeconomic model of individual utility maximisation allows for the development of an increasingly complicated graphical representation of the economy that provides insights into why markets clear, why prices move the way they do and why pareto efficiency emerges, among many other things, that would be very difficult to articulate with words. Indeed, the core of microeconomics is marginal theory, which is fundamentally inextricable from calculus.

In cases where mathematisation doesn’t initially lead to new insights it might still lead eventually to a statistically testable hypothesis. Arguably, hypotheses expressed in largely non-mathematical forms can also give rise to forms accessible to statistics, but more mathematically articulated models tend to garner more precise answers from the data. For example, if you ask people ‘do you hate your job’, you can only get a binary answer. If you ask people instead to rate their job satisfaction on a scale of 1-10 and define <5 as being ‘dissatisfied’ you get a richer answer to your question.

Mathematics is also the language of logic, and stating hypotheses in a logically manipulable form is critical to the development of theory. But here it must be remembered that putting a theory into a mathematical form rather than sticking to linguistic logic frequently involves a simplification. For example, trade models certainly show, convincingly, that openness to trade should lead to utility gains for all involved. But they do not account for political or institutional factors, among other things, arguably because these factors are too hard to crowbar into the model. As a result, implementers of the ‘Washington consensus’ of trade-for-development overlooked several critical issues, such as the fact that American and European markets remained closed to African agricultural exports, and that corrupt elites in African nations were likely to funnel proceeds away from development to foreign accounts in order to enrich themselves. These issues were not lost on development practitioners, but they couldn’t be captured in models because they were too amorphous and therefore not amenable to a simple mathematical form.

This is not to say that simplification should always be avoided, or that economics does a bad job of it. Simplification often allows for certain key issues to float to the surface of an investigation. Economics is also unusual in that it explicitly states its simplifying assumptions while other disciplines have a tendency to act as though they haven’t made any assumptions (such as liberal international relations theory). This is especially important with regards to norms. Maths cannot engage in normative discussions. This is a weakness of sorts, but it also means economists must be very explicit when introducing a normative bias in favour of one outcome or another.    

What all these arguments add up to is simply a word of caution and a request for a bit more modesty. In recent years economics has gotten a little bit carried away with the mathematics. It has frequently been guilty of conflating mathematics with science. It has betrayed a failure to understand the different between induction, which is what science does, and deduction, which is what math does. Its emphasis on mathematical models has made it much less accessible to a lay audience at a time when that audience has been increasingly affected by the arguments of economists in the policy realm. Simplification has given rise to some spectacular failures on the part of the discipline, the most obvious being the self-correcting market hypothesis, which is great for making mathematical models work but far from reality. Finally, economics’ love affair with mathematics has occasionally resulted in a hubristic dismissal of other disciplines, something we don’t want in this era of complexity.

Economics should not abandon mathematics or its push to have other social sciences more fundamentally integrate mathematical techniques into their own methods. Rather, it should become more cogent of the strengths and weaknesses of mathematics compared to other analytical frameworks. As is so often the case some wider training in the basics of the theory of knowledge would be useful. They might then not conflate numbers with science or simplification with insight and instead simply go about humbly testing their clearly stated hypotheses—more than enough to set economics apart from other disciplines at the present time, at least in Australia.