TITLE: To what extent is it true that "number theory = mathematics"?
QUESTION [19 upvotes]: In a thought-provoking answer to this MO question, Kevin Buzzard
and several commentators have described a multitude of ways in which
number theory is related to other parts of mathematics. It seems that,
in practice, to know number theory you have to know all mathematics.
But what is "all mathematics"? The usual description is top-down -- that is, 
give a high-level theory, such as category theory, 
that includes nearly everything we currently consider to be important. 
Alas, there is no telling whether such a theory will continue to be a 
good description; category theory has only been around for a few decades.
Another way to describe "all mathematics" is from the bottom up -- give a 
basic form of mathematics that has always existed and which keeps growing 
and ramifying in all mathematical directions. Elementary number theory is 
very tempting bottom-up answer, because of the connections with other parts 
of mathematics  already noted, and because it will satisfy our non-mathematical
friends who think that mathematicians are people who are good with numbers.
So my basic questions are:

Is number theory a good bottom-up description of all mathematics? And
  if so, why?

Answers can be anything from general theories about the universality of
number theory to examples of unexpected appearance of number theory in other
branches of mathematics. And if you are not convinced that number theory rules:

Is there any good bottom-up description of all mathematics (one you
  can explain to a non-mathematical friend), and if so what?

REPLY [3 votes]: You really don't need PDE's and propagation of singularities (just to name an example) in order to do number theory. It simply never comes up. However PDE's are a part of mathematics, a big one at that. Therefore this disproves your equation (I'm a number theorist and in honesty I do find the equation mathematics = number theory borderline offensive).