Why is, for example, the theory of ordinary differential equations considered less pure than number theory?

Question

This is a bit of a soft question, but why is it that number theory is considered the most "pure" branch of mathematics, and something like, say, the theory of differential equations, considered less "pure"? In my view, since real numbers (and hence real functions) can be derived from the set of natural numbers (via the well-known process of constructing the set of integers from the set of naturals, constructing the set of rationals from the set of integers, and finally constructing the set of reals from the set of rationals), the theory of differential equations is no less and no more "pure" than number theory. I am basically interested in the historical/sociological reason why certain subfields of even so-called "pure" mathematics are themselves considered more or less "pure" than other subfields.

Answer 1

There are various connotations & meanings of the relevant words. Here I am using a Dictionary to get the meanings of the Words.

Partial list [ WordWeb ] :

PURE 1 : Concerned with theory and data rather than practice

PURE 2 : Free of extraneous elements of any kind

APPLIED 3 : Concerned with concrete problems or data rather than with fundamental principles

APPLIED 4 : Put into service to work for a particular purpose or use as designed

In Mathematics , generally we use meanings "PURE 1" & "APLLIED 3" (or "APLLIED 4") not "PURE 2"

In that way , we have PURE Number theory or PURE Calculus or PURE theory of ODE or PURE Combinatorics , which may then have usage in the Sciences where it may become APPLIED ODE or APPLIED Combinatorics.