On some quotes from G.H. Hardy

I 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 well known to mathematicians and often quoted in outside discussions of mathematics. Here I give my personal thoughts related to some of the 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.

"There is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, 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 pure theory and are honest enough (or proud enough) to say so, letting the world know you exist is helpful, and writing clear and engaging non-technical exposition is in fact as challenging as research.

"A man's first duty, a young man's at any rate, 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 smarter 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.

Beauty is the first test: there is no permanent place in this world for ugly mathematics.

Much has been written about the "beauty and elegance" of pure mathematics. I really have nothing against beauty and elegance, but they strike me as part of a "small-scale" view -- 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 is elegant; a real battlefield is not.

"I hate teaching [but] I love lecturing"

(Disclaimer. Using the 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, even though I can't write a short definition of what teaching is.