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Maths and economics

from Lars Syll

Many American undergraduates in Economics interested in doing a Ph.D. are surprised to learn that the first year of an Econ Ph.D. feels much more like entering a Ph.D. in solving mathematical models by hand than it does with learning economics. Typically, there is very little reading or writing involved, but loads and loads of fast algebra is required. Why is it like this? …

One reason to use math is that it is easy to use math to trick people. Often, if you make your assumptions in plain English, they will sound ridiculous. But if you couch them in terms of equations, integrals, and matrices, they will appear more sophisticated, and the unrealism of the assumptions may not be obvious, even to people with Ph.D.’s from places like Harvard and Stanford, or to editors at top theory journals such as Econometrica …

Given the importance of signaling in all walks of life, and given the power of math, not just to illuminate and to signal, but also to trick, confuse, and bewilder, it thus makes perfect sense that roughly 99% of the core training in an economics Ph.D. is in fact in math rather than economics.

Douglas L. Campbell

Indeed.

No, there is nothing wrong with mathematics per se.

No, there is nothing wrong with applying mathematics to economics.

Mathematics is one valuable tool among other valuable tools for understanding and explaining things in economics.

What is, however, totally wrong, are the utterly simplistic beliefs that

• “math is the only valid tool”

• “math is always and everywhere self-evidently applicable”

• “math is all that really counts”

• “if it’s not in math, it’s not really economics”

• “almost everything can be adequately understood and analyzed with math”

Mainstream economists have always wanted to use their hammer, and so have decided to pretend that the world looks like a nail. Pretending that uncertainty can be reduced to risk and that all activities, relations, processes and events can be adequately converted to pure numbers, have only contributed to making economics irrelevant and powerless when confronting real-world financial crises and economic havoc.

How do we put an end to this intellectual cataclysm? How do we re-establish credence and trust in economics as a science? Five changes are absolutely decisive.

(1) Stop pretending that we have exact and rigorous answers on everything. Because we don’t. We build models and theories and tell people that we can calculate and foresee the future. But we do this based on mathematical and statistical assumptions that often have little or nothing to do with reality. By pretending that there is no really important difference between model and reality we lull people into thinking that we have things under control. We haven’t! This false feeling of security was one of the factors that contributed to the financial crisis of 2008.

(2) Stop the childish and exaggerated belief in mathematics giving answers to important economic questions. Mathematics gives exact answers to exact questions. But the relevant and interesting questions we face in the economic realm are rarely of that kind. Questions like “Is 2 + 2 = 4?” are never posed in real economies. Instead of a fundamentally misplaced reliance on abstract mathematical-deductive-axiomatic models having anything of substance to contribute to our knowledge of real economies, it would be far better if we pursued “thicker” models and relevant empirical studies and observations.

(3) Stop pretending that there are laws in economics. There are no universal laws in economics. Economies are not like planetary systems or physics labs. The most we can aspire to in real economies is establishing possible tendencies with varying degrees of generalizability.

(4) Stop treating other social sciences as poor relations. Economics has long suffered from hubris. A more broad-minded and multifarious science would enrich today’s altogether too autistic economics.

(5) Stop building models and making forecasts of the future based on totally unreal micro-founded macro models with intertemporally optimizing robot-like representative actors equipped with rational expectations. This is pure nonsense. We have to build our models on assumptions that are not so blatantly in contradiction to reality. Assuming that people are green and come from Mars is not a good – not even as a ‘successive approximation’ – modelling strategy.

1. April 5, 2020 at 11:02 am

The Douglas Campbell quote is spot on, but Lars, despite my having repeatedly pointed this out to you, there are other types of maths besides the symbolic ones he is talking about, in particular the iconic geometry which Descartes turned into algebraic short-hand. That doesn’t even involve numbers, just similarities and symmetries. You need to realise that our brains don’t just provide algebraic hammers and sequential language, they provide imaginative mirrors and complex outlines in what Kenneth Boulding in “The Image” called iconic language, in terms of which economics needs not numerical answers like GDP but a map of which way money and other traffic flows, paying particular attention to the possibilities open to us at cross roads. Try rethinking your prohibitions with that in mind.

2. April 20, 2020 at 2:29 am

Mathematics is a cultural invention. No one expresses this more clearly than Leslie White in The Locus of Mathematical Reality: An Anthropological Footnote.

We return now, in conclusion, to some of the observations of G. H. Hardy,
to show that his conception of mathematical reality and mathematical behavior
is consistent with the culture theory that we have presented here and is, in fact, explained by it.

“I believe that mathematical reality lies outside us,” he says. If by “us” he means “us mathematicians individually,” he is quite right. They do lie outside each one of us; they are a part of the culture into which we are born. Hardy feels that “in some sense, mathematical truth is part of objective reality,” (my emphasis, L.A.W.). But he also distinguishes “mathematical reality” from “physical reality,” and insists that “pure geometries are not pictures … [of] the spatio-temporal reality of the physical world.” What then is the nature of mathematical reality? Hardy declares that “there is no sort of agreement . . . among either mathematicians or philosophers” on this point. Our interpretation provides the solution. Mathematics does have objective reality. And this reality, as Hardy insists, is not the reality of the physical world. But there is no mystery about it. Its reality is cultural: the sort of reality possessed by a code of etiquette, traffic regulations, the rules of baseball, the English language or rules of grammar.
Thus we see that there is no mystery about mathematical reality. We need not search for mathematical “truths” in the divine mind or in the structure of the universe. Mathematics is a kind of primate behavior as languages, musical systems and penal codes are. Mathematical concepts are man-made just as ethical values, traffic rules, and bird cages are man-made. But this does not invalidate the belief that mathematical propositions lie outside us and have an objective reality. They do lie outside us. They existed before we were born.
As we grow up we find them in the world about us. But this objectivity exists only for the individual. The locus of mathematical reality is cultural tradition, i.e., the continuum of symbolic behavior.

For example, most cultures have concerns about the distribution of resources, the choice of laws and government structures, the conduct of religious and governmental affairs, assessing the number, health, and work of the culture’s members, etc. Mathematics is constructed to play a part in each of these areas of concern, and many others. Since the concerns vary in form and importance from one time and place to another, so mathematics does also. Mathematics is, thus culturally specific. Or, at least it was before cultures began sharing with and learning from one another. Chinese mathematics 1000 years ago was noticeably different from European mathematics. Today the two have developed a shared common core. Much the same is the case with Christian and Islamic mathematics. But they still have noticeable differences.

Contemporary economists work hard to invent mathematics to fit their theoretical creations. Economists use mathematics as a language or code to articulate the properties of these creations. These creations are often hard to grasp, conceptually and materially, and actors face difficulties in trying to understand them in a stable and useful manner. This is where mathematics comes in. Sometimes, these creations’ properties are expressed using mathematics, especially-but not exclusively the mathematics of partial differential equations such as the Black-Scholes equation. At the same time, however these mathematical expressions of economists’ creations also mystify them, as few everyday people have a background in such forms of mathematics.

But more than this, economists are obsessed with mathematics. The obsession is, as Gillian Tett describes it, comical. The obsession with mathematics did not just affect academic economics; it spread into finance too. Just before Lucas was developing his vision of economies based on “rational expectations,” Harry Markowitz, another economist, created the so-called capital asset pricing model, which tried to measure the risk of assets (and thus their supposed price) with statistics. This approach used a set of models that had been created by Kenneth Arrow and Gerard Debreu and became very influential. “[The Arrow-Debreu] mathematics was seized on as a road map towards the utopia of complete and efficient markets,” as Bill Janeway, an American venture capitalist and economist, observes. (The Silo Effect, The Peril of Expertise and the Promise of Breaking Down Barriers)
—-After nearly destroying the wellbeing of most humans twice in less than 25 years, the model remains relied upon. Economists are obsessed. The rest of us are crazy to continue listening to them.

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