My skill in mathematics pales in comparison to Hardy's, but here goes anyway.
Essentially, pure mathematics is more 'important' (if such a thing can be said) than applied mathematics, because the research conducted in the latter depends on the former. For example, cryptographers would not have a job were it not for the pure mathematicians who discovered the basis of the field in the 1970s. However, Hardy's comments about the utility of mathematics are outdated due to the importance of number theory in the modern landscape (cryptography, the Internet, etc.
). It is lamentable that he did not live to see the fruition of his work in that domain, but the fact remains that pure mathematics has become very
real and important to today's world. Goodreads would almost surely not exist were it not for this research.
I was quite puzzled to see Hardy label Pythagoras' theorem concerning the irrationality of the square root of 2 more 'deep' than Euclid's proof of the infinitude of the primes. This is a curious choice on Hardy's part, as I would wager that Hardy dealt with primes far more often than with irrationals. That is of course speculation on my part, but it does lead the reader to question how one seminal proof can be more 'deep' than another. It is also curious to read Hardy's thoughts on the importance of mathematics as it relates to other domains like the arts. He states, for example, that poetry is more important than cricket, but fails to justify this assertion in any meaningfully rigorous way. One might have expected more formality from a pure mathematician.A Mathematician's Apology
is worth reading in order to better understand the mind of a mathematician, although I might take some parts with a grain of salt.