A Mathematician’s Utility
Over the holidays, I re-read Hardy’s book A Mathematician’s Apology. Briefly put, it is a justification—if not a defense—of (pure) mathematics, written at the end of Hardy’s life. He gives a description of mathematical beauty that is accessible to the layperson and sheds some light on the quirky world of mathematics. Yet, there is an underlying theme of great sadness, if not depression in the book, as well as an arrogance when it comes to “denouncing” whole fields of mathematics. It is this arrogance that I object to, and in the following, I shall attempt to present a different view. To some extent, this is of course presumptuous of me: I am not an eminent mathematician as Hardy; but I have the feeling that he would enjoy hearing dissenting opinions.
How mathematical research works
I do not want to get bogged down in deep discussions about whether mathematics is discovered or invented. I merely want to comment on one theme that is common in mathematics:
mathematicians research for the purpose of having a copious supply of research for the future
This means that, when not being driven by practical constraints, many mathematicians are fully content in following their own research down a rabbit hole. They find joy in playing with definitions and objects, and are not afraid of being dumbfounded. Often, interesting things do crop up only by asking simple questions. For example, the notion of a cohomology structure can to some extent be seen as a natural answer to the question “What happens if we change the direction of all arrows of an exact sequence?”—the remarkable thing is the fact that this actually leads to something meaningful! There are more examples of this form (and of course I do not want to imply that cohomology is trivial):
By asking which polynomials do not have roots in the same field, a hierarchy of fields was established, i.e. rational numbers, real numbers, complex numbers.
Hyperbolic geometry arose after the parallel postulate from Euclidean geometry was dropped and replaced with something else.
Category theory was in part driven by the realization that to understand mathematical structures, one has to understand the processes that preserve said structure and, moreover, there are constructions that apply to more than one type of object.
But enough of this—I want to make the point that mathematical research is often, to the eye of the observer, somewhat solipsistic, and this is not a bad thing. Hardy would not object, I would wager here.
The utility of mathematical research
But this is where I take my leave from Hardy. He writes:
The ‘real’ mathematics of the ‘real’ mathematicians, the mathematics of Fermat and Euler and Gauss and Abel and Riemann, is almost wholly ‘useless’ (and this is as true of ‘applied’ as of ‘pure’ mathematics). It is not possible to justify the life of any genuine professional mathematician on the ground of the ‘utility’ of his work.
I agree with the last sentence—the life of no person should be measured by their utility, regardless of what they are doing or not doing. However, I strongly disagree with Hardy’s attempt at “gatekeeping”, as we would call it, for “real” mathematicians. I also disagree with his hard dichotomy of “applied” and “pure” mathematics. If anything, this should be a spectrum, depending on how far away your research is from a practical problem in the real world. In that sense, the researchers working on, say, algebraic number theory, may still label themselves to be purer than a statistician who works with—gasp—clinicians in a hospital.
But even if you feel that you need to this for some reason, the utility of a mathematical theory is a function of time: given enough time, practical applications might be found for any area of mathematics. Here are just some examples:
Number theory, one of the subject in which Hardy arguably excelled, is now commonly used in, among other things, cryptographic systems. RSA, PGP, etc. all make use of deep theorems in this field.
Relativity and quantum mechanics, which Hardy called “at present at any rate, almost as useless as the theory of numbers” revolutionized our modern communication systems.
Algebraic topology has seen applications in computational topology, a whole new field that aims to solve complex data analysis methods.
Category theory sees more and more applications in systems biology, IT security, and data science. There are workshops on applied category theory now.
This demonstrates that even abstract and seemingly obscure fields of mathematics might be used in time. Hardy seems to believe that these uses only occur for “dull” mathematics, though:
No one foresaw the applications of matrices and groups and other purely mathematical theories to modern physics, and it may be that some of the ‘highbrow’ applied mathematics will become ‘useful’ in as unexpected a way; but the evidence so far points to the conclusion that, in one subject as in the other, it is what is commonplace and dull that counts for practical life.
And again, I emphatically disagree: beauty is in the eye of the beholder, and mathematics does not lose an iota of its lustre just because there are now more applications of it.
I understand Hardy’s sentiment though: having seen how “dull” mathematics was used in the world wars, I can see the pacifist Hardy being relieved that his subject is safe, for the time being, from being used to bring misery into the world. Yet, as I outlined above, utility is a function of time. So even though a specific piece of research seems innocuous for the time being, this may change—not necessarily tomorrow, but maybe five years from now, or in another political climate.
So, what is a way out of that dilemma? Ignoring the bad aspects of mathematical research by only focusing on abstract nonsense will only work for a brief period of time—and this strategy strikes me as somewhat cowardly. It is also exacerbated by the fact that future uses of a theory, an algorithm, or something else are extremely hard to predict. An equally foolish thing would be to either shoulder the full responsibility for research or try to diffuse it among society. I admit that I do not have a straightforward or convincing answer to this issue. In the words of E. O. Wilson:
The real problem of humanity is the following: we have paleolithic emotions; medieval institutions; and god-like technology.
We should thus keep our eyes open towards potential misuses of technology and try to involve governing bodies as well as the public if possible. In machine learning, the Montréal Declaration for a Responsible Development of Artificial Intelligence is one example of how such an involvement might look like. While this is of course not directly applicable to many areas of mathematics, it can serve as an inspiration.
Happy proofs, until next time!