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 "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.
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.
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".