Platonism is an influential and well-established conception of mathematics and mathematical practice, which states that the axioms of mathematics ''refer'' to an independently existing realm of mathematical entities.

In particular, set-theoretic platonism entails the view that the axioms of set theory refer to an independently existing realm of objects (the ''universe of sets'') and that, given any reasonably formulated set-theoretic statement φ, φ has a ''definite'' truth-value.

In my talk, I will address the question of whether there any sufficient grounds to advocate platonism in view of contemporary set theory. In particular, the ''truth-value indeterminacy'' connected to the independence phenomenon in ZFC and its extensions does not seem to validate platonists’ epistemic optimism concerning our ability to have access to ''unique'' truths.

I will be particularly concerned with Gödel’s platonism and its ontological and epistemological implications, but I also intend to pay attention to several alternative formulations which are deemed to belong to the big family of realism in mathematics. Naturalism, full-blooded platonism, extreme platonism, conceptual realism will also be taken into account. My aim is to show that they may not perform better than Gödel’s conception.