> I wonder what he would think if he could see the ways in which number theory, once often regarded as the purest of the branches of math, is now used in things

> I wonder what he would think if he could see the ways in which number theory, once often regarded as the purest of the branches of math, is now used in things like cryptography. I would say number theory proves his statement, although perhaps not his point. Applied math is useful for the applications that are known at the time of its creation, and it is likely that it will remain with that level of applicability in the future, although if the real world applications that it is used for fall out of favor we might find that the applied math decreases in importance, given its importance is in its applicability, and the applicability of things has an importance contingent on the importance of the thing that they are applied to. This is of course not 100% sure, as there can also arise new applications of things in the future. Pure math on the other hand, being not tethered to any particular application on the time of its creation, may find all sorts of applications in the future, pure math has as such infinite potential applicability waiting to be discovered and thus infinite potential usefulness, whereas applied math has limited known applicability and thus limited known usefulness.