TITLE: Can you do math without knowing how to count?
QUESTION [10 upvotes]: Are there mathematical theories that do not use intuitive integers? [That is, do not use integers to write statements.]

Can you propose a theory that describes natural integers, without using intuitive integers to state axioms, definitions, and propositions?

REPLY [3 votes]: There is a famous anecdot concerning Grothendieck and a certain integer. In the sixties, Alexander Grothendieck was developping a complex algebro-arithmetic framework in order to prove a set of conjectures due to Weil concerning varieties over finite fields. One motivation was to get a better grasp of integer solutions of polynomial equations using reduction mod p.
The lectures given by Grothendieck were very abstract and one of the auditor asked for an example. "Sure," said Grothendieck, "let's take some prime number p, for example 57". And so now 57 is known as the Grothendieck prime.
In modern algebraic geometry, an prime integer is a maximal ideal in the ring $\bf Z$. I don't remember what $\bf Z$ is exactly. Some algebraic geometer may come forward and correct me if I am wrong but I think that this is the terminal object in the category of scheme. Or maybe the antediluvian one.
Another fashionable topic at the moment is the field with one element. The number 1 of course should not be understood in its intuitive meaning. All jokes taken aside, some parts of arithmetics nowadays are pretty sophisticated and do not rest on an intuitive notion of integers.
6 for example should be interpreted as a global section of the canonical sheaf of $spec({\bf Z})$, that is as a kind of function that associates to each point p in $spec({\bf Z})$ (a prime number actually) an element of ${\bf Z} / p{\bf Z}$ (the stalk of the sheaf at p) obtained by reducing 6 mod p. This allows to draw a parallel with the polynomial ring ${\bf C}[X]$, which happens to be a ring of functions over the complex line, and use idea coming from analytic geometry to study integers.
At that point, you should be confused enough to understand that integers are not what they used to be.