Objectivity, Realism, and Proof

The essays collected here  explore  the semantic and epistemic problems raised by different kinds of mathematical objects, by their characterization in terms of axiomatic theories, and by the objectivity of both pure and applied mathematics. They investigate controversial aspects of contemporary theories such as neo-logicist abstractionism, structuralism, or multiversism about sets, by discussing different conceptions of mathematical realism and rival relativistic views on the mathematical universe.  They consider fundamental philosophical notions such as set, cardinal number, truth, ground,  finiteness and infinity, examining  how their informal conceptions can best be captured in formal theories.