Versatility

One major advantage of mathematics is its versatility. Learning the same mathematics often enables you to make contributions to a variety of different fields. Indeed, there is a long history of mathematical ideas crossing disciplinary boundaries.

There are plenty of instances where “pure” math found application in the sciences: non-Euclidean geometry as the basis for Einstein’s general relativity, complex numbers in Schrödinger’s wave function, abstract algebra as the cornerstone of modern cryptography. The examples I have in mind are more expansive yet, for they deal not with mathematics branching outwards to other fields, but with fields connecting to one another by means of mathematics.

I remember the following experience quite vividly. In the middle of a topology lecture, Professor Frederic Schuller provides a completely unexpected example. In all disciplines, we have theories, and we often claim that our new theories are better than the old ones. In order for us to defend this claim, we must be able to compare theories. However, in the general case, it is not possible to define a transitive quality metric to compare any entities!¹ We may then qualify our claim, saying that at least the theories are “com[ing] closer to the truth”. But to verify this we require a “topology on the space of all theories”! Again this is something that we cannot begin to conceive. Here, in the middle of an otherwise abstract lecture, we have a piece of wisdom that applies universally because it deals with the very definition of what it means to compare things.

Another excellent example is between physics and economics. The researcher Ole Peters was familiar with the physical models describing the behaviour of gas particles. He then discovered that these equations were the same as the ones governing statistical expectation in economics! In doing so, he discovered a flawed assumption made by most economists, and published an important paper on the topic.² This is now his main area of study.

In finance, too, there is a surprising degree of crossover from physics. Louis Bachelier’s Brownian motion equations, used in finance to model random price fluctuations, were also developed independently by Albert Einstein, who used it as empirical evidence for the kinetic theory of gases. Even the famous Black-Scholes equation is just a variant of the heat equation from physics.

A more philosophical example is to be found in Douglas Hofstadter’s book Gödel, Escher, Bach. Hofstadter makes connections between mathematical logic and a variety of subjects including art, computer science, molecular biology, neuroscience. The fundamental idea is that the particular type of self-referential structure present in Gödel’s famous incompleteness proofs is also present in many other unexpected places. In keeping with the tradition of abstraction in mathematics, Hofstadter extracts the core structures of these seemingly separate fields and shows how they in fact share certain commonalities by constructing an extended analogy.³

My favourite example, however, is that of chaos theory, which deals with the behaviour of systems that are highly sensitive to their initial conditions (which we call nonlinear systems). Fundamentally, chaos theory is about things that change in complex ways. It should be no surprise that humans devote a lot of energy studying things that change in complex ways, but it may be surprising just how many disciplines converged on the same mathematical techniques to do so. In his book Chaos, James Gleick cites examples from cardiology, ecology, economics, information theory, meteorology, microbiology, and several branches of physics, to name just a few.

Surely, then, mathematics is one of the most versatile of all disciplines. With the rapid progress of technology, there are more applications of mathematics than ever before.⁴ And as we continue to innovate, we will surely find new areas of mathematics altogether and applications which we had never imagined.