From David Mumford’s May 2013 interview in SIAM News:
The applied mathematician has the difficult job of looking at a problem in context with no explicit mathematics and trying to see what kinds of mathematical ideas are under the surface that could clarify the situation. I think the most successful applied mathematicians are those who look in both directions, at the science and the math.
You can’t become too attached to one way of looking at things. Applied math has always rejuvenated pure, and theorems in pure math can unexpectedly lead to new tools with vast applications.
Emphasis added. I wish Mumford had said “at the problem and at the math” because not all applications of math are scientific.