Applied vs Pure Mathematics: Is The Distinction Still Valid?
Written by Ben Watkins, who recently graduated from the University of Cambridge with a degree in Mathematics. He is staying on for a Masters at Cambridge to pursue mathematics further.
The dichotomy between pure and applied mathematics has been a central theme in the evolution of mathematical thought. Gauss once said, "The greatest thing is purely mathematical thinking: this is worth much more than the application of mathematics." G.H. Hardy also remarked, “But is not the position of an ordinary applied mathematician in some ways a little pathetic?” However, I believe that increasingly, in the modern day, the boundaries between these two domains are becoming blurred to the point where questions arise: is the distinction between pure and applied mathematics dead? And what does this mean for how we understand mathematics as a discipline?
To begin addressing these questions, we must provide some definitions for the terms 'pure' and 'applied'. Of course, there are some amusing ways to define them. I recall a particularly dry lecturer from my undergrad years who said, “Applied mathematics is where things work but we don’t know why. Pure mathematics is where things go wrong, but we know why.” Indeed, I would personally add that “Maths as an undergrad is where things go wrong and you don’t know why.”
However, I think the fairest way to define the distinction is whether the mathematics is carried out towards the end of creating real-world applications. It is ‘pure’ if it is not, and it is ‘applied’ if it is.
This seems simple enough, so why might one suggest that distinction is dead? The first and main point to mention is how frequently there is an interchange of mathematics between the two fields. For example, historically, ‘group theory’ was considered solely a branch of pure mathematics. It used to be called ‘modern algebra’ (which, as Tom Lehrer once pointed out, implies that we are now doing post-modern algebra). Group theory is the field of mathematics concerned with ‘symmetries’ in the most abstract sense. A symmetric object, such as a polygon, has associated with it a ‘group’ that represents its symmetries (for example, the dihedral group is the group that represents the polygon’s symmetries: its lines of mirror symmetry and its rotational symmetries). It was never originally considered for real-world applications. However, you’d be hard-pressed to find a modern physics paper that does not, in some way, discuss group theory; groups have become integral to our understanding of particles and fields. This is because, as it turns out, symmetries are central to the way the universe operates.
So the question is: is group theory still pure mathematics because it wasn’t originally created with real-world applications in mind? If not, at what point did group theory switch from being pure to being applied? I think the fairest answer is that the distinction between applied and pure mathematics isn’t particularly clean-cut.
There are also inverse cases of this adoption by the other field. For example, Fourier transforms were originally discovered to solve the heat equation, which dictates how heat distributes in an object. As such, by our definitions, one might call the field applied. However, nowadays, the study of Fourier analysis has also been adopted by analysts and treated in a way that real-world applications aren’t its end goal. As such, one might call the field pure. Again, the distinction between the two fields is quite murky.
Furthermore, it is also true that someone researching in a ‘pure’ field of mathematics might be doing it because they believe that, although something might not immediately have a real-world application, it might do so someday. For example, graph theory is the field of mathematics, typically taken to be a pure field, that looks at objects (called graphs) that consist of dots (vertices) connected by lines (edges). It turns out that all of AI and machine learning (in its current paradigm) work by weighting particular paths of a graph that represent something useful, except we tend to give the graph the special name ‘neural network’. For example, you might weight a path which represents a word that follows the last word written (this is how large language models work). I don’t think it is a particularly controversial claim that there are mathematicians who consider themselves ‘pure mathematicians’ working, knowing that their work might be beneficial in years to come even though it doesn’t have any particular use now. Even Terence Tao’s (one of the most famous ‘pure’ mathematicians) most cited paper is on the subject of compressed sensing, which is now incredibly useful to those who call themselves ‘applied mathematicians’. Equivalently, it is entirely reasonable that an applied mathematician might do maths in what is considered an applied field without knowing if it has any particularly crucial application or if it ever will.
It is probably worth qualifying my argument somewhat by stating that one can certainly see that even if it is a lame distinction, the distinction between pure and applied maths still has some pragmatic use. For example, we still organise our maths education at higher institutions by ‘pure’ and ‘applied’ to a great extent. However, when the politics of deciding where to invest money gets involved, this certainly isn’t always to the pure mathematician’s benefit!
Ultimately, I would argue that the actual state of affairs of ‘pure’ vs ‘applied’ mathematics is merely one in which some fields are just agreed upon to be considered ‘pure’ and some are just agreed upon to be considered ‘applied’, and that the reasons for these distinctions are typically historical, rather than representative of how much real-world use they have. Alternatively, you could also argue that this distinction of pure vs applied exists more as a spectrum on which fields can both move around and spread. Either way, I believe the distinction as traditionally described is certainly dead. Henri Poincare once said “Mathematics is the art of giving the same name to different things”, why then do we persist in giving different names to the same thing?
