On purity

The estimable John von Neumann once argued that mathematicians were always in danger of straying too far from reality and becoming irrelevant. But G. H. Hardy could not disagree more: true mathematics is useless, uselessness is a positive good because useful mathematics could be misused, and it is only true mathematics that can be beautiful.

Since then, many a mathematician has followed in the footsteps of A Mathematician’s Apology. Many mathematicians I met take pride in being “useless”. Many only do so humorously, but there are still plenty who justify it as Hardy do: that applied mathematics can be abused, or that or it is just not pretty enough for their tastes.

Genuinely useless labor can be possibly moral when it is undertaken while the world burns, as our does. Certainly, most mathematics is not genuinely useless, because it has aesthetic value if nothing else. For example, I enjoy learning about large cardinals, and so their study is not genuinely useless, even though I doubt that the upper reaches of the higher infinite will never find application. All mathematics is applied mathematics if one enjoys mathematics and the purpose of applied science is to reduce human suffering. But there is value in keeping the number of brilliant minds working on set theory somewhat small, because every one comes at a high opportunity cost. Arguing on a less cosmic scale, university spending that goes to mathematics is university spending that cannot go to, say, cutting tuition, so must be spent wisely.

Chauvinism is hardly restricted to pure mathematicians. For example, I once had a run-in with an applied mathematician who took an interest in my past work in mathematical epidemiology, but when I proposed to generalize the mathematical ideas to more general results about ODE derived from certain graphs, she immediately became dismissive, considering the project a waste of time because I was assuming that biologists would find the work useful without knowing enough biology to verify this myself. She is certainly not the only applied mathematician I have met who holds such views.

This brings me to the true reason I reject Hardyism: it overlooks the work of countless mathematicians who are interested in nontrivial mathematics with nontrivial applications. Some examples I’ve had the opportunity to see up-close and personal: the work of Naveen Vaidya on the dynamics of HIV models (of which my work I mentioned above is a faint shadow of), the work of Maciej Zworski on quantum scattering with applications in MEMS, the work of Marc Rieffel on the foundations of string theory. Of note, none of these examples are mathematical statistics (though Dr. Vaidya certainly has also worked on the statistics and data-fitting of such models), though many people are happy to equate mathematical statistics and applied mathematics.

Science operates incrementally. It is hard to imagine the work of my mentors, Drs. Vaidya and Zworski, going very far if a deep theory of ordinary differential equations had not developed over the 19th and 20th centuries, much of which was quite pure, abstract, and originally developed solely for “aesthetic” reasons, and much of which was and is useless if not viewed in the broader context of developing the theory of dynamical systems. Would we have such a good understanding of the dynamics of differential equations today if we had not wanted to study such silly toy models as the Smale horseshoe?

Even the large cardinal hierarchy, whose upper reaches I previously mocked, has its place: inaccessible cardinals seem to make the lives of category theorists easier, allowing them to focus on algebra and applications rather than worrying about the sizes of limits — and if the work of John Carlos Baez and Blake Pollard takes off, category theory will some day be crucial to the theory of dynamical systems. So I can foresee a possible future wherein I am holding models of diseases in one hand and a universe-sized category in the other.

I do not claim that particular future is inevitable, or even terribly likely. But that it is possible is reason enough for us mathematicians to not so quickly write off each others’ work, and is reason enough for us mathematicians to stop boasting about being useless.