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.
