This will be a talk for the Wijsgerig Festival DRIFT 2019, held in Amsterdam May 11, 2019. The theme of the conference is: Ontology.

**Abstract.** What does it mean to make existence assertions in mathematics?

Is there a mathematical universe, perhaps an ideal mathematical reality, that the assertions are about? Is there possibly more than one such universe? Does every mathematical assertion ultimately have a definitive truth value? I shall lay out some of the back-and-forth in what is currently a vigorous debate taking place in the philosophy of set theory concerning pluralism in the set-theoretic foundations, concerning whether there is just one set-theoretic universe underlying our mathematical claims or whether there is a diversity of possible set-theoretic worlds.

An interesting item in this topic is synthetic differential geometry, which works beautifully, but only when the underlying logic does not have the law of the excluded middle.