Arithmetic....

I am trying to understand the mathematical intuition. For now, I believe it is a priori, and it can be given a logical foundation, in a classical sense. Anyway, I do not know if it would be useful for mathematics when there is a non-classical logical formulation of it, from fuzzy logic etc.

What are numbers? I believe the intuition of 0 and 1 as numbers and the knowledge of them being different, and the concept of successor relation as the form of time (Kant) is enough for our mathematical intuition of numbers. Whether we intuit more is another question. This is the main idea of Frege's construction of logical system of arithmetic; he gives the formulation of numbers 0 and 1 logically, define successor relation and prove that 0 and 1 are different.

To define all cardinal numbers, he formulates the ancestral relation to define the concept "member of natural number series ending with n" where n+1 as a number belongs to. (Begriffsscrift)

Frege's logical system is ineffectual not because of the Russell Paradox which has been given a solution by Wright (Conception of Numbers as Objects). His system cannot distinguish objects from concepts in a unique way. He suggested that definite descriptions properly construed give classification for objects. However that is not the case, as the mystery of definite descriptions still cannot be solved and occupies a large literature in philosophy of language.