Can you do math without knowing how to count? 
*

*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?
 A: This question, like many others, is based on a fundamental conceptual error. Any formal system whatsoever that humans can devise and use must be one with unambiguous rules. Therefore its theorems (i.e. statements that it allows you to deduce) must necessarily be a computably decidable set. But even basic logic relies crucially on the assumption that any finite symbol strings can be concatenated. Otherwise you cannot even have a rule that allows you to deduce "A∨B" from "A"!
Because of this, the answer to your questions are trivially "no" and "no".
A: 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.
A: "Are there mathematical theories that do not use intuitive integers?"
Geometry?  A lot of Euclid's Elements is done without numbers.
A: A "philosophy of math" tag would have been a good idea.
To answer your question, take a look at Hatry Field's "Science without Numbers".  Also, Edward Nelson has developed a theory of "proto-integers", which is discussed in "Diffusion, Quantum Theory, and Radically Elementary Mathematics" (MN-47).
In my opinion, most attempts at nominalism (be it about abstract objects, infinite sets, or numbers) spend a lot of energy, and for what in the end? Moreover, the approach invariably comes across as a cheap trick (to me).  This is my personal opinion, but I get the impression it is shared by many mathematicians.
I would quote Alonzo Church here, but the current climate does not allow for that anymore.
A: Here are two attempts to answer what I view is the underlying question.  The first attempt is taken from "Some Philosophical Prolegomena" (a section of the notes "The Axioms of Set Theory", by Tom Foster).

Many people come to set theory having been sold a story about its foundational significance; such people are often worried by apparent circularities such as...


Before we even reach set theory we have to have the language of first-order logic. Now the language of first-order logic is an inductively defined set and as such is the minimal [with respect to inclusion] set satisfying certain closure properties, and wasn’t it in order to clarify things like this (among others) that we needed set theory . . . ? And how can we talk about arities if we don’t already have arithmetic? And weren’t we supposed to get arithmetic from set theory?


...this doesn’t mean that set theory cannot serve as a foundation for Mathematics, but it does make the point that the whole foundation project is a bit more subtle than one might expect, and that the cirularities which launched this digression are not really pathologies, but a manifestation of the fact that life is complicated.

I have taken various liberties in selecting these quotations - you may want to consult the original text.
The second attempt is my own, more pessimistic, view.  I think that "meaning" (if it exists at all) comes from a very lengthy bootstrapping process; thus any attempt at mathematical foundations is hopeless.
A: Topology and group theory do not use the integers.
However, I'd say that groups are - one of many - generalisations of the integers.
Also set theory and as one poster has already pointed out, Euclidean geometry.
