Name a famous mathematician. A live one, if you don’t mind. Who is the modern equivalent of Gauss, whose hometown Napoleon spared because “the foremost mathematician of all time lives there”? Perhaps this is unfair: we are unlikely ever to witness another person quite like Gauss in human history, but the point still stands.
And the point is not that mathematicians are no longer as famous as they once were. A quick survey of great mathematicians reveals a van Gogh-esque tendency not to be recognised in their time (except perhaps by other mathematicians and French generals). The point is rather this: mathematics today may seem parochial, or obscure, but it has the power to change history. This power is just hidden for now.
Being only at the very start of my Masters, this will be, to use a favourite mathematical term, a very “hand-wavy” post. But hopefully I will have said something by the end of it.
No one can imagine the world without Newton’s mathematics, which essentially marks the birth of analysis. Newtonian math put Neil Armstrong on the moon (as did Leibniz’s, but let’s not get into that now), and yet it was discovered almost 300 years before him. Perhaps the giant leap for mankind took place not in 1969, but in 1687.
The theory of prime numbers enable RSA encryption, relying heavily on work by Euler and others. Euler himself predated RSA by 200 years. Riemann supplied the mathematics crucial to relativity a comparatively short 50 years or so before Einstein needed it. Yet he, like the others, would not live to see it used in such a famous application.
It is not surprising, then, that no mathematician alive today can point to an application of homological algebra, topos theory or non-commutative geometry. Yet this puts mathematics at a sad disadvantage in an applications-obsessed world. Being applications-obsessed makes sense in a lot of ways. As one of my computational biology friends put it: “Would you rather cure cancer or prove the Riemann hypothesis?”.
Yet sacrificing mathematics for the sake of application seems a bit like chasing get-rich-quick schemes rather than investing. In essence, I believe neither in a demand for applications from all mathematicians, nor a complete indifference to application from mathematicians themselves. Just because we can’t see exactly how mathematics will be used in the future doesn’t mean we can’t guess at, speculate about and work on applications right now.
My field of interest is category theory, which has applications in the three fields of mathematics I just mentioned. But it also being used to uncover the dark secrets of quantum mechanics, model emergence in biological systems, improve software development and formalise ontology research. But often slowly, and not always in a way that provides improvement you can point to and say “there, isn’t that better?”.
The maths that the mathematician in the corner is doing may not be accessible, or even applicable, but I believe that as long as people keep trying to develop beautiful mathematics, and people (hopefully the same ones even) try and use it to improve every aspect of science, we will one day look back and thank him.