 Modern philosophy of mathematics starts with the Frege-Russell theory of numbers (Frege 1879, 1892, Goodstein 1957, see entry on  alternative axiomatic set theories)  in terms of sets. If we accept the notion of a class of objects as valid and fundamental, together with the notion of a one-to-one correspondence between classes of objects, then we can define numbers as sets of equinumerous classes. 