There are three types of mathematician: those who can count and those who can't.
It takes a few seconds to get this joke the first time you hear it, and every once and a while when I tell it to a class one student will raise a hand and asked for the 'third type'. Its a good joke because it challenges assumptions, here the assumption is that mathematics is concerned with arithmetic, which is just a minor branch of number theory.
At the end of June, before I went on holiday, I had been thinking about the role of mathematics in democracy. This was prompted by an invitation from a Norwegian mathematician to do some work in the topic. I contacted @BrendanLarvor asking who were the main scholars in the area. He pointed me to a recent paper that discusses a key issue. Many people see the role of mathematics in democracy as educating the public so that they can do their own calculations. Citizens are able to calculate the cost/benefit of Brexit, for example. But calculation is not what mathematics is concerned with. The paper is not an easy read but highlights the awareness amongst (some) mathematicians that mathematics is not straightforward. In particular the phrase "the role of mathematics in formatting the world as we experience it" resonated with me as being a key issue.
This led me to think about the role of mathematics in defining how the disciplines of finance and economics are arranged. A consequence of this was I invited people to answer a short survey on "Does a mathematical proof enhance a financial theory?". The survey was widely distributed (via @MarkThoma amongst others) but only elicited seven responses. The results are here (the survey is still open, btw). I was disappointed that only seven people seemed to share my interest to the degree that they would spend a little time answering the question. I concluded that either people were disinterested (possibly because they thought the question was trivial, like "Does water flow downhill") or that they did not understand the question (they do not feel confident about what is meant by a 'mathematical theory').
On returning from holiday, I noticed that @freakonometrics had retweeted an article from aeon about how "By fetishising mathematical models, economists turned economics into a highly paid pseudoscience"
"By fetishising mathematical models, economists turned economics into a highly paid pseudoscience" https://t.co/2ODpHZg1It by @Naseeoh— Arthur Charpentier (@freakonometrics) July 24, 2017
and then @rethinkecon sent out this
— Rethinking Economics (@rethinkecon) July 26, 2017
I have the opinion that almost all of the criticism of the use of mathematics in economics stems from a lack of understanding of what mathematics is, reflecting a general ignorance in economics that has led to the failure of mathematics in economics. To get an idea of my frustration consider the following argument about journalism. One might observe that there are many more photographs in newspapers today than there were 100 or so years ago. Using the argument that the problems of economics are in its use of mathematics is rather like saying the problems of contemporary journalism is down to photography.
The starting point of understanding the role of mathematics in finance and economics is to appreciate what mathematics is concerned with. Mathematics is concerned with identifying relations between objects: bigger smaller, to the left/right, symmetry, before/after and so forth. Top class mathematical research is concerned with discovering new ways of representing how things are related. More every-day research shows that A=B or how you go from A to B. Once the mathematicians have done their work, of "formatting the world as we experience it" by identifying how we see relations between objects, others then get on and do things. Mercator figured out how to make maps - a mathematical operation - sailors then used the maps and in the process forgot that what they were doing was using mathematics.
Mathematicians rely on other disciplines providing problems, mathematics, whatever the caricature of a mathematician dealing with abstract ideals will say. Mathematics then figures out a way of looking at the problem - the relations between its components - so that a solution can be found. The caricature of the mathematician is explained by how mathematics is presented. Rather than starting with the problem and then breaking it down into its components, mathematics is presented back to front. It starts with the components and then shows how these combine to deliver the observed phenomena. This 'back-to-front' approach originates in Euclid. The theorems at the end of Euclid's Elements were all well known hundreds, if not thousands, of years before he wrote The Elements around 300 BCE.
Euclid's approach is useful in that it identifies the essential elements of a theorem, these elements can be the used to construct novel theorems by combining them in innovative ways; think of a mathematical assumption as a chemical element and a theorem as a useful molecule. However there are a number of problems resulting from the way mathematics is presented. One effect is encapsulated in Kant's argument that synthetic a priori knowledge was possible. Kant used the example of Euclid to argue that it was because he had assumed Euclid had deduced the theorems from first principles. This is significant in that this fallacious argument was a foundation of Kant's rejection of Hume's claim that a necessary cause of an effect could never be identified. Another effect is it provides a model for a powerful rhetorical form that is persuasive, it was used in particular by Hobbes and Spinoza while Aquinas' writing has been compared to mathematics. Today 'mathematical' proofs that 1=2 are commonplace. More significantly this 'mathematical' approach was used by Hobbes to argue that if a highwayman offered you the choice of 'your money or your life' and you handed over your money, you were giving consent. It is not easy to discern flaws in these 'mathematical' arguments, and this is the day to day job of research mathematicians (a social scientists once Tweeted they had had a productive day, reviewing three papers: as a mathematician it will take me a week of hard graft to review a 10 page paper).
The effect in economics is most clearly seen in Friedman's argument, in the Methodology of Positive Economics, that the validity of an economic theorem should not rest on the realism of its assumptions. I will not dismiss Friedman as the arch-priest of neo-liberalism as I think the argument he makes has some merits (he focuses on the empirical outcome and would normally be regarded as 'anti mathematiciastion'). The attitude he shares with most economists, along with Kant, Hobbes and Spinoza, is that a 'mathematical' argument flows from assumptions to conclusions. A mathematician approach would be to try and tease out the correct assumptions from the observed behaviour. I would prefer the problem to be re-cast as "By fetishising synthetic a priori knowledge, economists turned economics into a highly paid pseudoscience".
The next question is why do economists do this. The answer is rooted in the observation that the 'mathematical' approach is powerful rhetorically: you can use it to convince everyone of almost anything, providing you can make the chain of arguments tricky enough to follow. From a philosophical perspective, Kant distinguished the ‘lower faculties’, such as mathematics, that would consider matters of pure reason independently of the concerns of the state from the ‘higher faculties’, engineering, jurisprudence, medicine and theology, were concerned with matters of authority and would be regulated and monitored by the state. If economics is mathematical it should inform the state, not be directed by the state, if it is not then it will have the same status (and funding) as theology (and, one would suppose, other modern social and human sciences).
More practical motivations were characterised by Frank Knight, who, around 1920, felt that economics had split into two strands. There was a mathematical science, which studied closed systems based on distorting assumptions, and a descriptive science, which could deduce nothing. Economics needed to take a middle path that was both realistic and informative. However, before the Second World War, most economists doubted the usefulness of mathematics in addressing problems involving radical uncertainty and human volition, such as the economy. These attitudes changed when it was seen that mathematics had transformed how the war, a similarly uncertain and human activity, had been fought; operations research, cryptography, supporting the physics of radar and weapons. Based on this experience and government faith in mathematics, economics began presenting itself as a mathematical science after the war. Two publications of 1944 led this transformation: The Probability Approach in Econometrics by Trygve Håvelmo and The Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern.
Håvelmo argued also that if economics wanted to be taken as seriously as physics, chemistry and biology, it needed to employ probability because that was the way that opinions were expressed in science. He believed that if this was done, economics would make new insights, just as physicists and biologists had. He also observed that the natural sciences had found a perspective on nature that made it appear to follow stable laws. The goal of The Probability Approach in Econometrics was to present how this could be realised. Morgenstern began The Theory of Games, like Håvelmo, with an argument for the use of mathematics in economics and explained that what was required was the careful definition of terms, a pre-requisite of mathematics but lacking in economics. To this end, von Neumann started with the axioms of utility that had been at the core of Carl Menger’s, unmathematical, economics.
When Håvelmo was awarded the Nobel Prize for Economics in 1989 he reflected that his aspirations for introducing mathematics to economics had not been met. He identified the primary issue as being that the economic models that ‘econometricians’ had been trying to apply to the data were probably wrong. More fundamentally, economics never generated new mathematics ‒ ways of seeing relationships ‒ in the way that the physical sciences had stimulated developments in mathematics. Economists had simply adapted concepts from other fields to their own devices.
To my mind, Håvelmo captures why mathematics is not unreasonable effective in economics. It is because economists use mathematics as 'part of the plumbing', a rhetorical tool to convince an audience of an argument. The Unreasonable Effectiveness of Mathematics in the Natural Sciences is founded on the fact that the natural science use mathematics to figure out relationships. The one exception to this rule (that I am aware of) in modern economics is the Fundamental Theorem of Asset Pricing, formulated by Harrison, Kreps and Pliska around 1980 (I dismiss game theory as this was originated in the early 1700s). The FTAP is analogous to the Mercator projection, it describes the basis on which models (maps) are made that guide probationers (navigators). “A market admits no arbitrage, if and only if, the market has a martingale measure” establishes a relationship.
Once mathematics has delivered ways of identifying relations in physics, 'invariants' can be identified, such as momentum, energy or the speed of light (Noether's Theorem is critical here). Physical theories are then tested on the basis of whether or not they adhere to a particular conservation law. Because economics is disinterested in using mathematics to identify relationships it has been unable to accomplish the next step of discovering invariants. It has tried, notably by sometimes hoping 'money' is an economic invariant.
In writing Ethics in Quantitative Finance (the points made here are expanded upon in the book) one of my aims was to think of finance as a mathematician. That is to consider the fundamental relationship, as expressed in the FTAP, and then think about what this implies as to the fundamental invariant. My conclusion was that reciprocity - and equality between what is given and received - is the invariant and I explore why this might be so. The hope is that finance and economics can actually achieve something useful for the wider community.