Critiques
Hardy's opinions were heavily influenced by the academic culture of the universities of Cambridge and Oxford between World War I and World War II.
Some of Hardy's examples seem unfortunate in retrospect. For example, he writes, "No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years." Since then, the application of relativity was part of the development of nuclear weapons, while number theory figures prominently in public-key cryptography. However, Hardy's more prominent examples of elegant mathematical discoveries with no use (proofs of the infinity of primes and of the irrationality of the square root of two) still hold up.
The applicability of a mathematical concept is not the reason that Hardy considered applied mathematics somehow inferior to pure mathematics, though; it is the simplicity and prosiness that belongs to applied mathematics that led him to describe them as he did.
He considers that Rolle's theorem for example, though it is of some importance for calculus, cannot be compared to the elegance and preeminence of the mathematics produced by Leonhard Euler or Évariste Galois and other pure mathematicians.
Read more about this topic: A Mathematician's Apology