Jack Himelright

Advocates of the explanatory indispensability argument for platonism say two things. First, we should believe in the parts of our best scientific theories that are explanatory. Second, mathematical objects play an explanatory role within those theories. I give a two-part response. I start by using a Bayesian framework to argue that the standards many have proposed must be met to show that mathematical objects are dispensable are too demanding. In particular, nominalistic theories may be more probable than platonistic ones even if they are extremely complicated by comparison. This is true even if there are genuine cases of mathematical explanation in science. The point made here is a matter of principle, holding regardless of how one assesses nominalistic theories already on offer. I then examine my recent nominalization of second-order impure set theory in light of the correct, laxer standards. I make a tentative case that my nominalistic theory meets those standards, which would undermine the explanatory indispensability argument. While this case is provisional, I aim to bring attention to my nominalization and others in light of the revised standards for demonstrating dispensability.

In this paper, we will introduce a novel argument (the “Region Argument”) that objects do not have frame-independent shapes in special relativity. The Region Argument lacks vulnerabilities present in David Chalmers' argument for that conclusion based on length contraction. We then examine how views on persistence interact with the Region Argument. We argue that this argument and standard four-dimensionalist assumptions entail that nothing in a relativistic world has any shape, not even stages or the regions occupied by them. We also argue that endurantists have viable ways of preserving shape despite the Region Argument. The upshot of these arguments is that contrary to conventional wisdom, considerations about shape in relativity support endurantism rather than four-dimensionalism. We conclude by examining the implications of our discussion for the debate over Edenic shapes, noting that endurantists have a satisfying response to skeptical arguments about Edenic shapes similar to the one they have against the Region Argument.

In this paper, I develop a "safety result" for applied mathematics. I show that whenever a theory in natural science entails some non-mathematical conclusion via an application of mathematics, there is a counterpart theory that carries no commitment to mathematical objects, entails the same conclusion, and the claims of which are true if the claims of the original theory are "correct": roughly, true given the assumption that mathematical objects exist. The framework used for proving the safety result has some advantages over existing nominalistic accounts of applied mathematics. It also provides a nominalistic account of pure mathematics.

Nominalists are confronted with a grave difficulty: if abstract objects do not exist, what explains the success of theories that invoke them? In this paper, I make headway on this problem. I develop a formal language in which certain platonistic claims about properties and certain nominalistic claims can be expressed, develop a formal language in which only certain nominalistic claims can be expressed, describe a function mapping sentences of the first language to sentences of the second language, and prove some facts about that function and facts about some sound logics for those languages. In doing so, I prove that, given some plausible metaphysical assumptions, a large class of sentences about properties of concrete objects are “safe” on nominalistic grounds. Whenever some true sentences about concrete objects and some sentences belonging to this class that are true according to platonists collectively entail a conclusion about concrete objects, some nominalistically acceptable sentences are true and entail the same conclusion. Because the proof can itself be formulated without abstract objects, it provides a nominalistic explanation of the success of theories that invoke properties of concrete objects.

In this paper, I argue that all expressions for abstract objects are meaningless. My argument closely follows David Lewis’ argument against the intelligibility of certain theories of possible worlds, but modifies it in order to yield a general conclusion about language pertaining to abstract objects. If my Lewisian argument is sound, not only can we not know that abstract objects exist, we cannot even refer to or think about them. However, while the Lewisian argument strongly motivates nominalism, it also undermines certain nominalist theories.

In this paper, I argue that for the purposes of ordinary reasoning, sentences about properties of concrete objects can be replaced with sentences concerning how things in our universe would be related to inscriptions were there a pluriverse. Speaking loosely, pluriverses are composites of universes that collectively realize every way a universe could possibly be. As such, pluriverses exhaust all possible meanings that inscriptions could take. Moreover, because universes necessarily do not influence one another, our universe would not be any different intrinsically if there were a pluriverse. These two facts enable anti-realists about abstract objects to replace, e.g. talk of anatomical features with talk of the inscriptions concerning anatomical structure that would exist were there a pluriverse. The availability of such replacements enables anti-realists to carry out essential ordinary reasoning without referring to properties, thereby making room for a consistent anti-realist worldview. The inscriptions of the would-be pluriverse are so numerous and varied that sentences about them can play the roles in ordinary reasoning served by simple sentences about properties of concrete objects.