Non-well-founded set theories allow set-theoretic exotica that standard ZFC will not allow, such as a set that has itself as its sole member. We can distinguish plenitudinous non-well-founded set theories, such as Boffa set theory, that allow infinitely many such sets, from restrictive theories, such as Finsler-Aczel or AFA, that allow exactly one. Plenitudinous non-well-founded set theories face a puzzle: nothing seems to explain the identity or distinctness of various of the sets they countenance. In this paper I aim to (...) sharpen this puzzle, make clear who it does and does not apply to and, ultimately, to argue in favor of a plenitudinous theory like Boffa. (shrink)

Gideon Rosen, Brian Leiter, and Catarina Dutilh Novaes raise deep questions about the arguments in Morality and Mathematics (M&M). Their objections bear on practical deliberation, the formulation of mathematical pluralism, the problem of universals, the argument from moral disagreement, moral ‘perception’, the contingency of our mathematical practices, and the purpose of proof. In this response, I address their objections, and the broader issues that they raise.

A basic way of evaluating metaphysical theories is to ask whether they give satisfying answers to the questions they set out to resolve. I propose an account of “third-order” virtue that tells us what it takes for certain kinds of metaphysical theories to do so. We should think of these theories as recipes. I identify three good-making features of recipes and show that they translate to third-order theoretical virtues. I apply the view to two theories—mereological universalism and plenitudinous platonism—and draw (...) out their third-order virtues and vices. One lesson is that there is an important difference between essentially and non-essentially third-order vicious theories. I also argue that if a theory is essentially third-order vicious, it cannot be assessed for more standard “second-order” theoretical virtues and vices, like parsimony. This motivates the idea that third-order virtues are distinct from second-order ones. Finally, I suggest that the relationship between truth, progress, and third-order virtue is more complex than it seems. (shrink)

Strict finitism posits a largest natural number. The view is usually thought to be objectionably arbitrary. After all, there seems to be no apparent reason as to why the natural numbers should ‘stop’ at a specific point and not a bit later on the natural line. Drawing on how arguments from arbitrariness are employed in mereology, I propose several ways of understanding this objection against strict finitism. No matter how it is understood, I argue that it is always found wanting.