Many mathematicians, when asked to justify their research on abstract structures, will give some form of the argument from future utility. “Sure, semiperfect Andret complexes have no application right now” they might say,1 “but history is littered with instances of pure mathematics developed for its own sake turning out unexpectedly to be the perfect tool to solve some real-world problem”.
And the thing is they’re not wrong. The most famous example, the one you’re most likely to have quoted at you, is Bernhard Riemann’s work on curved spaces in general dimension (known now as Riemmanian geometry). Others before Riemann had figured out the details of the mathematics of curved surfaces such as a sphere or hyperboloid; this work was by no means easy, but it was at least motivated by considering surfaces, such as soap bubbles or arched roofs, that are immersed in our three-dimensional world. Riemann’s imaginative leap to conceive of spaces of three dimensions and more that are also intrinsically curved in the same ways surfaces can be curved was mathematics in its purest form. But some 60 years later, Einstein used exactly the ideas of Riemann and those following him to describe gravity in general relativity as a curvature of space and time. The theory matched reality beautifully, successfully predicting the bending of starlight during a solar eclipse and resolving nagging problems with the Newtonian account of gravity, such as the advance of the perihelion of Mercury.
But this is not the only example: time and again something developed for the sake of it turns out to be the right tool for some piece of science. The representation theory of groups explains the vibrations of certain molecules in terms of their symmetries. Analysis of finite geometries is key for developing optimal error-correcting codes. Number theory, either elementary (modular arithmetic) or more sophisticated (elliptic curves over finite fields), is the foundation of all modern cryptography. This is despite G. H. Hardy’s confident assertions in A Mathematician’s Apology that the theory of numbers is the Queen of Mathematics, the purest of the pure, and could never affect the world either positively or negatively, unlike the cruder sciences. This does give the Apology a certain bathos.
I’ve never really believed this defense of pure mathematics though, and I don’t think many of those who profess it do either, really. The amount of maths that turns out to be useful is indeed surprisingly large, but the total of all mathematics is many time larger. Most of it does not and will not ever be applied outside of mathematics itself (and a non-trivial proportion won’t even be used inside maths).
I was in the audience for a panel discussion at the weekend on “Is God a Mathematician?” (I know, horrible title/topic, but the discussion was good), and it was refreshing to hear Kevin Buzzard say outright
I can understand people’s desire to believe and promote the eventually-applicable view. The teasing hope that understanding derived categories of Calabi-Yau manifolds will eventually lead us to the correct form of string theory certainly helps raise grant money. But mathematicians, for the most part, aren’t motivated by the promise of future applications—they do it for the fun of it, to satisfy their own curiosity. Utility, if it ever comes, is a bonus. In this sense pure mathematics is much more like an art or the humanities than a science. I don’t say this to demean maths (or the arts and humanities either), but we should be honest about why we do things and who they’re for. We know deep down that pure mathematicians’ universities pay them to teach and tolerate their research so far as it raises their department’s reputation, and we should acknowledge that. And then, because we are happy with this arrangement and want it to continue, we should extend the same privilege to those working in the humanities and arts and anybody else who wants to create for a living.
Related reading: The Ideal Mathematician by Philip J. David & Reuben Hersh.
(“beyond their obvious importance for the characteristic zero case of the endomorphism tower collapse conjecture”) ↩︎
Quoted from memory. Not his precise words but close, and (I believe) faithful to his meaning. ↩︎
Buzzard is refreshingly forthright in general; in answer to a question about whether absolute infinity causes a fundamental incompatibility between idealised mathematics and the real world: “Oh, infinity? Yeah we just made it up.” ↩︎