1. Are there mathematical theories that do not use intuitive integers? [That is, do not use integers to write statements.]
2. Can you propose a theory that describes natural integers, without using intuitive integers to state axioms, definitions, and propositions?



Can you do math without knowing how to count?
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.


