Consequently, structuralism maintains that mathematical objects do not possess any intrinsic properties but are defined by their external relations in a system.
Other examples of mathematical objects might include lines and planes in geometry, or elements and operations in abstract algebra.
[1] Structuralism in the philosophy of mathematics is particularly associated with Paul Benacerraf, Geoffrey Hellman, Michael Resnik, Stewart Shapiro and James Franklin.
[2] He developed an argument against the ontology of set-theoretic Platonism, which is now historically referred to as Benacerraf's identification problem.
Benacerraf noted that there are elementarily equivalent, set-theoretic ways of relating natural numbers to pure sets.
Consequently, Benacerraf inferred that this set-theoretic falsehood demonstrates it is impossible for there to be any Platonic method of reducing numbers to sets that reveals any abstract objects.
[3] The fundamental epistemological problem thus arises for the Platonist to offer a plausible account of how a mathematician with a limited, empirical mind is capable of accurately accessing mind-independent, world-independent, eternal truths.