Penelope Maddy has uploaded a recent paper, Set-Theoretic Foundations, to her page on academia.edu (I needed to download the paper to get a readable version).
The paper shows Maddy’s usual admirable virtues of great clarity. In the first main section, she says something about the various senses in which set theory might be said to “provide a foundation for classical mathematics” (and in which it can’t be said to do so). The second section considers whether, and in what senses, category theory can be said to provide an alternative/better foundation. The third main section — and this makes the paper apt for a volume of essays celebrating Hugh Woodin, for it is on a topic he has views on! — is about the idea that we should think in terms of a multiverse of set universes, rather than a single universe in which (in some sense) all mathematics can live. Her ambitions are modest, however. Maddy writes, sensibly enough,
I won’t pretend to sort out all these complex and contentious matters, but I do hope to compile a few relevant observations that might help bring illumination somewhat closer to hand.
And, yes, I did find a number of Maddy’s remarks clarifying and illuminating. So I’ll spread the word about her paper here.