The most commonly used/accepted foundation for mathematics is set theory, specifically ZFC. Relations are modeled as sets [of pairs, which are in turn modeled as sets].

A logician / formalist would argue that mathematics is principally (entirely?) about proving derivations from axioms - theorems. A game of logic with finite strings of symbols drawn from a finite alphabet.

An intuitionist might argue that there is something more behind this, and we are describing some deeper truth with this symbolic logic.