Hardy

On Four Quotes By G.H. Hardy

David Aldous

Dec 31, 2020 — 3 min read

occasionally let slip that I am an academic great-grandchild of G.H. Hardy, whose non-technical writings about mathematics, in particular A Mathematician’s Apology are often quoted in general discussions of the nature of mathematics. Here I give my personal thoughts related to four of these well known quotations. These of course reflect my own background, with a research career in theorem-proof mathematics (albeit in the field of probability, which Hardy had a rather low opinion of) but also with interests in “real data” mathematics and in exposition. Note that Hardy writes “men”, as was the custom at the time. What would have happened if, for instance, Ramanujan had turned out to be a woman, might make an interesting “alternate history” novel.

1. Exposition, Criticism and Appreciation Is Work For Second-Rate Minds

Hardy’s continuation on this theme strikes me as missing the point. Like Hardy, I have no doubt that novelists scorn literary critics, movie directors scorn movie reviewers, and so on. In an ideal world, creative works intended for a public audience would “speak for themselves” without need for extra commentary. But it’s not an ideal world, and extra commentary is often informative. Moreover mathematical research, especially in Hardy’s sense of “creative” theorem-proof mathematics, is not directed at a public audience. Popular exposition is beneficial both at an idealistic level (“education” as desirable in itself) and at a pragmatic level. Recall a line from The Right Stuff

You know what makes this bird [rocket] go up? FUNDING makes this bird go up.

Funding is what makes professional math research happen. Even if, like Hardy, you are doing only pure theory and are honest enough (or proud enough) to say so, letting the world know you exist is helpful. And in fact writing clear and engaging non-technical exposition is just as challenging as research.

2. A Young Man’s First Duty Is To Be Ambitious

Hardy writes in praise of ambition in general, then moves on to what he asserts are the three major motivations for research. In brief, they are
(1) intellectual curiosity, desire to know the truth.
(2) professional pride, anxiety to be satisfied with one’s performance.
(3) ambition, desire for reputation, and the position, even the power or the money, which it brings.

These quotes prompt two thoughts. We nowadays might interpret ambition as “ambition to be the best at something”. But this is like volunteering for a massively-negative-sum game — most such attempts are doomed to fail. Hardy is more nuanced, interpreting ambition as desire to create “something of permanent value”. I might just say “something worthwhile”, in that permanence can be over-rated.

My second thought is that rather than considering “motivation” it might be more helpful to consider “what is important for a satisfying career in mathematics.” (That is, instead of what drives you?, why not consider where are you going?). Here my own list below partially overlaps Hardy’s list above:
(1) Intellectual curiosity
(2) Intellectual discipline
(3) Talent
and I usually introduce my list by saying

Ability is overrated: raw mathematical talent is only the third most important factor for a satisfying career in mathematics.

Moreover there is always someone cleverer than you, but hopefully they are working on something harder or more significant than you are. In planning one’s research, I believe the key principle is “value added”:

Find the most worthwhile project that no-one else is both able and willing to tackle.

3. There Is No Permanent Place In This World For Ugly Mathematics

“Beauty is the first test”. Indeed, much has been written about the “beauty and elegance” of pure mathematics. I have nothing against beauty and elegance, but they strike me as part of a “small-scale” view — because it’s mostly individual theorems and proofs that are taken to be elegant. So my view is

One can be elegant only on a small stage. A Shakespearean sword fight can be elegant; a real battlefield is not.

4. I Hate Teaching (But I love lecturing)

(Disclaimer. Using this quote outside Oxbridge is rather disingenuous, because there Hardy’s ‘teaching’ presumably meant one-on-one meetings with undergraduates, not the contemporary meaning.)

I like repeating this Hardy quote to pure mathematicians, and follow up by saying that going through the traditional definition-theorem-proof-example format on a blackboard or electronic analog is lecturing, not teaching. The distinction is important. OK, so I can’t write a short definition of what teaching is, but it involves more interaction than does lecturing. An old joke says that a lecture is a means of transferring material from the instructor’s notes to the student’s notes without passing through the mind of either.