Peter D. Lax (one of the greatest mathematicians of our time) in his acceptance speech of the Abel Prize:
“Traditionally mathematics is divided into two kinds: pure and applied. The relation of the two is delicate. The great applied mathematician Joe Keller’s definition is: pure mathematics is a branch of applied mathematics. He meant that mathematics, beginning with Newton, was originally concerned with answering question[s] in physics, it is only later that the tools and concepts used were elaborated into theories that took on lives of their own. It was remarked by von Neumann that after a while abstract mathematics needs to be invigorated by the injections of new empirical material, like a new scientific theory, new experimental facts, or numerical studies.”
Peter D. Lax, Abel Prize acceptance speech, 2005 Abel Prize Multimedia
See also, an Interview with Peter D. Lax, Notices of the AMS, 53 (2), February 2006, 223–229 and his SIAM Review article The Flowering of Applied Mathematics in America, SIAM Rev. 31 (4), 1989, 533–541.