The Identity of Numbers: From Arithmetic Invariants to Mathematical Structuralism

Abstract

What makes a number the same number across different representations and mathematical realizations? This paper examines numerical identity at the intersection of number theory and philosophy of mathematics. It distinguishes numeral representation, arithmetic structure, and mathematical ontology. The integer 30, for example, may be represented as 30, XXX, or 11110₂, while its arithmetic properties remain unchanged. Moving beyond notation, the paper considers isomorphic realizations of the natural numbers and shows explicitly that arithmetic properties such as primality, divisibility, and unique factorization are preserved under arithmetic isomorphism. This provides a precise mathematical basis for asking what, philosophically, constitutes numerical identity. Benacerraf's identification problem then shows why arithmetic does not privilege a particular set-theoretic realization of numbers. The paper argues that these mathematical facts strongly support methodological structuralism - the practice of identifying mathematical objects through structure-preserving relations - without thereby establishing metaphysical structuralism as the unique ontology of mathematics. Computer-assisted proof and artificial intelligence are considered as a contemporary extension of the same distinction between formal preservation and mathematical understanding. The resulting thesis is deliberately interdisciplinary: arithmetic invariance constrains philosophical accounts of number while leaving the ultimate ontology of mathematical objects open.