Inspired by Cantor's Theorem (CT), orthodoxy takes infinities to come in different sizes. The orthodox view has had enormous influence in mathematics, philosophy, and science. We will defend the contrary view---Countablism---according to which, necessarily, every infinite collection (set or plurality) is countable. We first argue that the potentialist or modal strategy for treating Russell's Paradox, first proposed by Parsons (2000) and developed by Linnebo (2010, 2013) and Linnebo and Shapiro (2019), should also be applied to CT, in a way that (...) vindicates Countabilism. Our discussion dovetails with recent independently developed treatments of CT in Meadows (2015), Pruss (2020), and Scambler (2021), aimed at establishing the mathematical viability, and therefore epistemic possibility, of Countabilism. Unlike these authors, our goal isn't to vindicate the mathematical underpinnings of Countabilism. Rather, we aim to argue that, given that Countabilism is mathematically viable, Countabilism should moreover be regarded as true. After clarifying the modal content of Countabilism, we canvas some of Countabilism's many positive implications, including that Countabilism provides the best account of the pervasive independence phenomena in set theory, and that Countabilism has the power to defuse several persistent puzzles and paradoxes found in physics and metaphysics. We conclude that in light of its theoretical and explanatory advantages, Countabilism is more likely true than not. (shrink)

Zeno argued that since at any instant an arrow does not change its location, the arrow does not move at any time, and hence motion is impossible. I discuss the following three views that one could take in view of Zeno's argument:(i) the "at-at" theory, according to which there is no such thing as instantaneous velocity, while motion in the sense of the occupation of different locations at different times is possible,(ii) the "impetus" theory, according to which instantaneous velocities do (...) exist but these are only contingently and causally related to the temporal developments of positions,(iii) the "no instants" theory, according to which instants in time do not exist, and hence instantaneous velocities do not exist, while motion, in the sense of different areas occupied during different time intervals, is possible.I argue that, despite the fact that there have been interesting and relevant developments in mathematics and physics since the time of Zeno, each of these views still has serious drawbacks. (shrink)

The aim of this paper is to show that topology has a bearing on<br><br>combinatorial theories of possibility. The approach developed in this article is “mapping account” considering combinatorial worlds as mappings from individuals to properties. Topological structures are used to define constraints on the mappings thereby characterizing the “really possible” combinations. The mapping approach avoids the well-known incompatibility problems. Moreover, it is compatible with atomistic as well as with non-atomistic ontologies.It helps to elucidate the positions of logical atomism and monism (...) with theaid of topological separation axioms. (shrink)

Could space consist entirely of extended regions, without any regions shaped like points, lines, or surfaces? Peter Forrest and Frank Arntzenius have independently raised a paradox of size for space like this, drawing on a construction of Cantor’s. I present a new version of this argument and explore possible lines of response.

I argue that it may well be the case that space and time do not consist of points, indeed that they have no smallest parts. I examine two different approaches to such pointless spaces : a topological approach and a measure theoretic approach. I argue in favor of the measure theoretic approach.

Doing Worlds with Words throws light on the problem of meaning as the meeting point of linguistics, logic and philosophy, and critically assesses the possibilities and limitations of elucidating the nature of meaning by means of formal logic, model theory and model-theoretical semantics. The main thrust of the book is to show that it is misguided to understand model theory metaphysically and so to try to base formal semantics on something like formal metaphysics; rather, the book states that model theory (...) and similar tools of the analysis of language should be understood as capturing the semantically relevant, especially inferential, structure of language. From this vantage point, the reader gains a new light on many of the traditional concepts and problems of logic and philosophy of language, such as meaning, reference, truth and the nature of formal logic. (shrink)

"Accessibility" is a crucial concept of possible worlds semantics. The simplest approach to accessibility is the "magical theory" that construes this relation as analogous to spatial or temporal relations. In this paper I give a nonmagical structural account of the accessibility relation that can be used to give a necessitarian account of kinds and laws. Laws are characterized in a structural way as stable invariants of the world's gestalt. Finally, I point out how the structural approach can be embedded in (...) a general representational theory of modality. (shrink)

Say that space is ‘gunky’ if every part of space has a proper part. Traditional theories of gunk, dating back to the work of Whitehead in the early part of last century, modeled space in the Boolean algebra of regular closed subsets of Euclidean space. More recently a complaint was brought against that tradition in Arntzenius and Russell : Lebesgue measure is not even finitely additive over the algebra, and there is no countably additive measure on the algebra. Arntzenius advocated (...) modeling gunk in measure algebras instead—in particular, in the algebra of Borel subsets of Euclidean space, modulo sets of Lebesgue measure zero. But while this algebra carries a natural, countably additive measure, it has some unattractive topological features. In this paper, we show how to construct a model of gunk that has both nice rudimentary measure-theoretic and topological properties. We then show that in modeling gunk in this way we can distinguish between finite dimensions, and that nothing in lost in terms of our ability to identify points as locations in space. (shrink)

This paper is concerned with Wittgenstein's early doctrine of the independence of elementary propositions. Using the notion of a free generator for a logical calculus–a concept we claim was anticipated by Wittgenstein–we show precisely why certain difficulties associated with his doctrine cannot be overcome. We then show that Russell's version of logical atomism–with independent particulars instead of elementary propositions–avoids the same difficulties.

Introduction to 'Belief and Probability,' issue of Philosophical Studies, January 1995. Remarks on contents, including van Fraassen, 'Belief and the Problem of Ulysses and the Sirens,' Skyrms, 'Strict Coherence, Sigma Coherence and the Metaphysics of Quantity,' Jeffrey, 'Probability Reparation,' de Finetti, 'The Logic of Probability,' Kolmogorov, 'Complete Metric Boolean Algebras,' 4 more.

In a recent paper, Belot argues that Bayesians are epistemologically flawed because they believe with probability 1 that they will learn the truth about observational propositions in the limit. While Belot’s considerations suggest that this result should be interpreted with some care, the concerns he raises can largely be defused by putting convergence to the truth in the context of learning from an arbitrarily large but finite number of observations.

This paper forms part of a wider campaign: to deny pointillisme. That is the doctrine that a physical theory's fundamental quantities are defined at points of space or of spacetime, and represent intrinsic properties of such points or point-sized objects located there; so that properties of spatial or spatiotemporal regions and their material contents are determined by the point-by-point facts. More specifically, this paper argues against pointillisme about the structure of space and-or spacetime itself, especially a paper by Bricker (1993). (...) A companion paper argues against pointillisme in mechanics, especially about velocity; it focusses on Tooley, Robinson and Lewis. To avoid technicalities, I conduct the argument almost entirely in the context of ``Newtonian'' ideas about space and time. But both the debate and my arguments carry over to relativistic, and even quantum, physics. (shrink)

We think of a boundary whenever we think of an entity demarcated from its surroundings. There is a boundary (a line) separating Maryland and Pennsylvania. There is a boundary (a circle) isolating the interior of a disc from its exterior. There is a boundary (a surface) enclosing the bulk of this apple. Sometimes the exact location of a boundary is unclear or otherwise controversial (as when you try to trace out the margins of Mount Everest, or even the boundary of (...) your own body). Sometimes the boundary lies skew to any physical discontinuity or qualitative differentiation (as with the border of Wyoming, or the boundary between the upper and lower halves of a homogeneous sphere). But whether sharp or blurry, natural or artificial, for every object there appears to be a boundary that marks it off from the rest of the world. Events, too, have boundaries — at least temporal boundaries. Our lives are bounded by our births and by our deaths; the soccer game began at 3pm sharp and ended with the referee's final whistle at 4:45pm. It is sometimes suggested that even abstract entities, such as concepts or sets, have boundaries of their own, and Wittgenstein could emphatically proclaim that the boundaries of our language are the boundaries of our world. Whether all this boundary talk is coherent, however, and whether it reflects the structure of the world or simply the organizing activity of our mind, are matters of deep philosophical controversy. (shrink)

An object is T-gunky just in case all its parts (i) have proper parts and (ii) are of non-zero measure in every spatial dimension. I show that a recent argument due to Hud Hudson—though not intended as a threat to gunk—bears on the possibility of T-gunky material objects in non-gunky space. I then show that the friend of T-gunk can circumvent Hudson’s argument without abandoning pointy space or standard mereology. What is needed is a novel conception of the relation of (...) exact occupation of pointy regions by T-gunky objects. With the new conception in hand, the friend of material T-gunk need not look to theories of gunky spacetime in order to reconcile the possibility of material gunk with mathematical physics. (shrink)

The volume includes twenty-five research papers presented as gifts to John L. Bell to celebrate his 60th birthday by colleagues, former students, friends and admirers. Like Bell’s own work, the contributions cross boundaries into several inter-related fields. The contributions are new work by highly respected figures, several of whom are among the key figures in their fields. Some examples: in foundations of maths and logic ; analytical philosophy, philosophy of science, philosophy of mathematics and decision theory and foundations of economics. (...) Most articles are contributions to current philosophical debates, but contributions also include some new mathematical results, important historical surveys, and a translation by Wilfrid Hodges of a key work of arabic logic. (shrink)

It is widely accepted that continuity is the most essential characteristic of motion; the motion of macroscopic objects is apparently continuous, and classical mechanics, which describes such motion, is also based on the assumption of continuous motion. But is motion really continuous in reality? In this paper, I will try to answer this question through a new analysis of the cause of motion. It has been argued that the standard velocity in classical mechanics cannot fulfill the causal role required for (...) explaining continuous motion in a deterministic way. However, there is a hot debate over the solution to this causal explanation problem. The existing solutions, i.e. the "at-at" theory, the impetus view and the "no instants" view, have serious drawbacks. After reviewing their drawbacks and presenting my own objections, I propose a new solution to this causal explanation problem. It is argued that the relativity of motion or the equivalence between motion and rest demands that motion has no deterministic cause, and thus no causal explanation is needed for motion. Based on this result, I further argue that the uncaused motion is not deterministic and continuous but essentially random and discontinuous. Moreover, objects have a potential that determines the probabilities of their future positions. This suggests a possible connection with quantum mechanics. Lastly, I give a brief explanation for the apparent inconsistency between the suggested potential theory of motion and the appearance of continuous motion in the macroscopic world. (shrink)