Reading The Classics: Mathematics Vs. Economics

There’s been a fun (but silly) long-running debate on Twitter whether economists need to read canonical texts: Smith, Marx, Ricardo, Keynes, etc. What caught my eye is that a mainstream economist compared economics to mathematics — why don’t we learn calculus by studying the history of calculus? Why this is interesting is that is showed a lack of understanding of the situation in both mathematics and economics.

Please note that this article is a discussion of the philosophy of teaching at the university level, so do not expect any conclusions that will help make analysing bond markets easier. That said, there is an outline of a critique of the core methodological principles of neoclassical macro.

I will first start with mathematics.

The Eras of Mathematics

My academic background was as an (honours) electrical engineering undergraduate, with a minor in mathematics. I then ended up with a doctorate in control theory — an area of applied mathematics that is normally attached to engineering faculties. As such, I missed most of the undergraduate mathematics curriculum, but I did cover relevant courses, and hung out with mathematicians (my supervisor had a background in pure mathematics until he went to the dark applied side).

I took a course in the history of mathematics at McGill. My comments here reflect my hazy memory of that course — as well as an editorial bias that might horrify some historians, but probably reflects how many mathematical students would approach the topic. 

(I grabbed the cover of the book History of Mathematics (Amazon affiliate link) to give a pretty picture to distract from my text. The text looks interesting, and overlaps the topics in the course I took, although my course terminated at an earlier historical period. I have not read that book, but I would give 100% odds that it is a more reliable source about the history of mathematics than myself.)

I would divide mathematics into three eras.

1. Modern. Mathematics pursued in a fashion that is consistent with what is done in the present research literature.
2. Early Modern. Mathematics that is structured in a fashion similar to the present (although perhaps more “wordy”), but often uses proof techniques not used in the modern literature.
3. Pre-Modern. Any mathematics that a present-day math undergraduate would need a historian to explain the context.

Modern Mathematics

If we look at the output of modern mathematicians, we see two main classes of texts: articles, and long-form texts. Long form texts include textbooks, monographs, and theses (which is a monograph published at a University instead of publisher). Meanwhile, a “textbook” is just a monograph that has been optimised for teaching purposes.

When we look at monographs, a significant portion of the text is devoted to a survey of the field, as well as setting up notation (which is typically tied to the survey). The reason is straightforward: attempting to review a monograph that consists of purely new results would be a daunting task. It might happen in a brand new area of mathematics, but that is unusual (almost by definition).

We can then see why mathematical teaching does not consist of “reading canonical texts”: the underlying texts are a huge volume of journal articles, along with monographs that survey the journal literature. For reasons of time, teachers use the surveys — but students are expected to be able to go into the primary literature as needed.

(Jumping ahead, this is how many neoclassicals like to view economics.)

With respect to textbooks, within an area of specialisation, the textbooks generally evolve to meet teaching needs. However, the core contents are often relatively uniform. For example, both Rudin and Kolmogorov were popular textbooks for real analysis when I was student. Once you worked through one, you could cover the other without picking up much new content.

Early Modern Mathematics

For a philosopher of science, early modern mathematics (as I define it) is probably way more interesting. There were a lot of debates on how to prove results.

Unfortunately for the pre-moderns, the typical reason why a debate existed was that both sides were wrong. Over time, more robust proof techniques were developed, and the debates were ended.

A modern student of mathematics would be expected to be able to go through that literature, and very easily demonstrate the weaknesses (if not outright errors) of the old proof techniques.

This is not usually covered, although it can appear in surveys. For example, Rudin covers both the Riemann and Lebesgue definition of the integral, and explains the advantage of the Lebesgue formulation.

These debates are otherwise not covered for the basic reason that it is impossible to find properly-trained mathematicians who take the old proof techniques seriously. To what extent a debate exists, it would continue within “modern” mathematics.

Calculus

Calculus has its roots in the “early modern” period (as I define it). There were debates on how to do proofs. But, those debates were settled, and the modern proof techniques that settled the debate are the ones taught in courses.

Although those were important debates, they are not a major feature of university level teaching for an important reason: calculus is barely considered to be a university level subject. For example, in Quebec, local students have a 3 year university program, and for students in the mathematics/science streams, Calculus I (differentiation) and II (integration) are taught in CEGEP — a pre-university level of education. (Students from outside the province have an extra year, where they learn calculus.)

Calculus is a settled area of mathematics, far from the research frontier. The only pedagogical interest in teaching it is to find the way that best fits the backgrounds of the enrolled students. For students of Calculus I/II, teaching them invalid proof techniques is distracting and not helpful.

Pre-Modern Mathematics

What about the canonical texts of pre-modern mathematics? Well, here is my list of such canonical texts.

• Euclid’s Elements.

(Disclaimer: my list is meant to capture canonical texts in terms of contributions for university level mathematics, and not texts that matter a lot for history. For example, Arab mathematicians were critical for preserving the existence of mathematics and would thus appear in a course on mathematical history. I explain later why they missed the list above.)

Until relatively recently, Euclid was taught. The catch was that it was taught to school children (such as generations of English schoolboys). Although not assigned, I read the Elements in either junior high or high school.

(Update: Alex Douglas pointed out mathematical logic is another key area of modern mathematics. I forgot about logic as a stand-alone subject (oops), my excuse is that I implicitly used Euclid as a key example of the logical structure of proofs — which is one reason why it made the cut. I am unsure what other texts might be needed to cover logic, particularly for cultures outside the mathematical traditions that explicitly followed Euclid.)

What about everything else? Well, we need to look at what pre-modern mathematics consisted of. Firstly, people lumped a lot of things in with “mathematics” that mathematicians have dumped on other faculties, such as applied sciences and mysticism about numbers. What about the rest? We see the following areas of interest.

1. Plane geometry. Any culture that had access to the Elements took it seriously, and you could use the plane geometry either in it or as taught in modern high schools to solve the problems that the ancients faced.
2. Arithmetic. This is taught in modern primary schools. The best way to teach arithmetic depends upon your numbering system, and even modern schools keep experimenting.
3. Word problems. These can be solved by using high school algebra. The rise of modern notation is probably where you can draw a line between “pre-modern mathematics” and the rest.

This is reflected in how my history of mathematics course was taught. We got a small potted history in lectures, but for testing purposes, we were expected to solve problems using the mathematical techniques available to a given culture. Although a fun challenge, this is more a form of entertainment than serious mathematics. Solving a problem that you already know the answer to (courtesy of knowing modern techniques) is not that big a deal.

This ease would not hold for the earlier mathematicians. It took an effort to translate other cultures’ mathematics to their own conventions, and would be shocked by the coursework by a modern mathematics faculty.

Finally, my extremely limited list of “canonical” texts reflects that I am looking at this from the perspective of undergraduate level mathematics. As I argue, most of the (non-mystical) content would now be classified as material taught before the university level.  

Economics

The Twitter debate about reading the classics largely consisted of a fiesta of name-calling and strawman attacks. In other words, the kind of entertainment that I go onto Twitter for. If we wanted to be serious, we realise that there are three separate questions that are being mangled together.

• What do we expect an undergraduate in a field to know?
• What should academic researchers in a field know?
• What is the best way to teach an undergraduate (possibly graduate) students?

That is, we need to differentiate between what is considered required knowledge for an undergraduate versus researchers. In engineering, a knowledge of project finance is a requirement. I am out of the loop, but I believe that any Canadian faculty without such a requirement would face accreditation issues. This is despite the fact that it is exceedingly unlikely that faculty at engineering departments will publish research in the area of project finance.

I do not claim to be trained as an economist, and I leave it to the reader to decide whether a person can leave a university with an undergraduate degree in economics without knowledge of the thought of Smith, Ricardo, Marx, and Keynes (etc.).

At the graduate level? The reality is that students can parachute into an economics graduate program without an economics undergraduate degree. Since post-grad degree programs are already far too long in most places, you cannot shoehorn in an undergraduate degree into the program. That said, one does not normally crow about ignorance of a subject that is taught to undergraduates in their field. (My supervisor had an obligation to tutor undergraduates, and so he often left meetings with me to return home to study the next lecture of thermodynamics so that he could answer question. This is how academics should work.)

Finally, what is the best teaching technique? Speaking as someone who barely read any authors before Keynes, I am in the camp that secondary sources are probably the most time efficient teaching technique. That said, there is a catch: the secondary sources need to take the primary sources seriously. It should be possible for the reader of the secondary source to read the primary and go to almost any passage and understand what the original author is going on about.

Dunking on the Mainstream

With that out of the way, let’s have some fun and dunk on the mainstream’s positions in this area. More accurately, I will relay some critiques that I am aware of, without claiming that they are true.

• We cannot compare many areas of economics — such as my only area of interest, macroeconomics — to calculus, since the questions are not settled. The research frontier has not completely moved away from what Keynes was writing about.
• To what extent a mainstream academic argues that undergraduates do not need to be aware of foundational topics in their field since they are not current topics of research is out of line with accepted practice in every other university department that I have had contact with.
• Heterodox authors claim that neoclassical summaries of economic debates are botched because they do not take critiques of underlying assumptions seriously. Debates that neoclassicals consider “settled” are still disputed.
• Related to the previous: the survey undertaken neoclassicals of the literature is obviously biased. The case in point is MMT: from the perspective of a mainstream academic, MMT did not exist until the articles dunking on it by neoclassicals appeared. We have the obviously dysfunctional situation of earlier neoclassical articles on MMT critiquing it without citing any journal articles, with later neoclassical articles begrudgingly citing them since MMT has refused to disappear off the radar screen. I have encountered barriers to citation in my old area of academia, but nothing even slightly close to this behaviour.

Most of the discussion in this article could be viewed as academic mud wrestling, which not everyone is interested in. However, the latter point does affect the output of the neoclassical research program, and thus is of interest to anyone with an interest in macroeconomic theory.

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(c) Brian Romanchuk 2021