Beyond the axioms: The question of objectivity in mathematics (Q2765285)

The author distinguishes in the philosophy of mathematics realism from super-realism. Realism boils down to Hilbert's view that the demands of mathematical existence and truth are met by axiomatization of logic and the cumulative hierarchy of sets, plus meta-mathematical arguments. Super-realism implies the existence of an external criterion of mathematical existence and truth. This implies that it makes sense to question the validity of mathematics as a whole, because the existing axiomatization may not correctly represent mathematical reality. In the present paper the author rejects super-realism and defends the realism represented by Hilbert by discussing several objections against it. Tait views super-realist statements like ``mathematical objects are immutable, outside space and time, uncreated'' as merely grammatical statements, meaning that spatial and temporal properties cannot be applied meaningfully to mathematical objects. As for objections against realism he deals with (in-)consistency and (in-)completeness of axiomatized theories and with undecidable propositions. Post-modern relativism is rejected with the argument that although other very different practices of mathematics may be possible, that supposition does not make our own practice invalid.