Does Mathematical Possibility Imply Existence?

In the article I discuss the use of modal notions in philosophy of mathematics, in particular the (seemingly) uncontroversial “maximum principle”, according to which the mathematical universe is very rich – it contains “implementations” of all coherent mathematical notions. In particular, I discuss the more general problem of ontological reductions, and argue, that completeness theorems are not the appropriate formal paraphrase of this intuition. I also discuss Gödel’s, and Quine’s versions of mathematical realism in this context. Finally, I argue, that this issue is important both for the realism-antirealism debate in philosophy of mathematics, and for the problem of mathematical explanation.