Beau Mount

There is extensive debate among contemporary philosophers about the possibility of absolutely unrestricted quantification; thus far, the debate been almost entirely logically and metaphysically focused. We argue for a third axis of evaluation: the epistemological. We defend absolutism on epistemological grounds, by showing that one prominent and attractive alternative to absolutism—schematism—is epistemologically unacceptable. First, we spell out and motivate an epistemological desideratum for theories of generality, a desideratum which is easily satisfied by absolutists. Second, we consider five ways the schematist might satisfy this desideratum. We argue that none of the five ways is successful.

This is the transcript of a conversation between the participants of theWoodin, W. Hugh Arctic Set Theory Workshop VI and Hugh Woodin. The conversation took place on February 24, 2023 at the Biological Research Station of the University of Helsinki in Kilpisjärvi, Finland, and was moderated by Juliette Kennedy. The transcript was created by Beau Madison Mount and is based on detailed notes taken by him during the evening. Questions were asked by David Asperó (DA), Douglas Blue (DB), Juliette Kennedy (JK), Rahman Mohammadpour (RM), Jeffrey Schatz (JS), Corey Switzer (CS), Jouko Väänänen (JV) and Bartosz Wcisło (BW). Unidentified audience members are denoted AM. * * * indicates recording unintelligible or missing.

This paper examines the logical and metaphysical consequences of denying Leibniz's Law, the principle that if t1= t2, then φ(t1) if and only if φ(t2). Recently, Caie, Goodman, and Lederman (2020) and Bacon and Russell (2019) have proposed sophisticated logical systems permitting violations of Leibniz's Law. We show that their systems conflict with widely held, attractive principles concerning the metaphysics of individuals. Only by adopting a highly revisionary picture, on which there is no finest-grained equivalence relation, can a well-motivated metaphysics for rejecting Leibniz's Law be developed. We sketch one such picture—a metaphysics of stuff. Stuff ontologies can be initially motivated through ordinary language: stuff stands to mass nouns as objects stand to count nouns. The stuff ontology we propose takes stuff to be fundamental and views the world as composed of an infinite descending hierarchy of kinds and portions of stuff. We defend the coherence of this picture and offer a model theory demonstrating that it can be consistently formalized.

Vann McGee has argued that, given certain background assumptions and an ought-implies-can thesis about norms of rationality, Bayesianism conflicts globally with computationalism due to the fact that Robinson arithmetic is essentially undecidable. I show how to sharpen McGee's result using an additional fact from recursion theory—the existence of a computable sequence of computable reals with an uncomputable limit. In conjunction with the countable additivity requirement on probabilities, such a sequence can be used to construct a specific proposition to which Bayesianism requires an agent to assign uncomputable credence—yielding a local conflict with computationalism.

I extend theorems due to Roy Cook on third- and higher-order versions of abstraction principles and discuss the philosophical importance of results of this type. Cook demonstrated that the satisfiability of certain higher-order analogues of Hume's Principle is independent of ZFC. I show that similar analogues of Boolos's new v and Cook's own ordinal abstraction principle soap are not satisfiable at all. I argue, however, that these results do not tell significantly against the second-order versions of these principles

I develop a novel argument against the claim that ordinals are sets. In contrast to Benacerraf’s antireductionist argument, I make no use of covert epistemic assumptions. Instead, my argument uses considerations of ontological dependence. I draw on the datum that sets depend immediately and asymmetrically on their elements and argue that this datum is incompatible with reductionism, given plausible assumptions about the dependence profile of ordinals. In addition, I show that a structurally similar argument can be made against the claim that cardinals are sets.

Recent work on formal theories of truth has revived an approach, due originally to Tarski, on which syntax and truth theories are sharply distinguished—‘disentangled’—from mathematical base theories. In this paper, we defend a novel philosophical constraint on disentangled theories. We argue that these theories must be epistemically stable: they must possess an intrinsic motivation justifying no strictly stronger theory. In a disentangled setting, even if the base and the syntax theory are individually stable, they may be jointly unstable. We contend that this flaw afflicts many proposals discussed in the literature; we defend a new, stable disentangled theory, double second-order arithmetic.