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?