I endorse Composition as Identity, broadly and loosely understood as the thesis that a composite whole is nothing over and above its parts, and the parts nothing over and above the whole. Thus, given an object, x, composed of n proper parts, y1, ..., yn, I feel the tension between my Quinean heart and its Lewisian counterpart. I feel the tension between my obligation to countenance n+1 things, x and the y’s, each of which is a distinct portion of reality, (...) and my inclination to count just 1 thing, x, or just n things, the y’s, the former encompassing the same amount of reality as the latter. This paper is an attempt to reconstruct this tension and to explain it away without forgoing the intimate link between counting and countenancing. (shrink)

In this chapter, I examine Lewis's ideas about ontological innocence, ontological commitment and double-counting, in his discussion of composition as identity in Parts of Classes. I attempt to understand these primarily as epistemic or methodological claims: how far can we get down this route without adopting radical metaphysical theses about composition as identity?

The connection between whole and part is intimate: not only can we share the same space, but I’m incapable of leaving my parts behind; settle the nonmereological facts and you thereby settle what is a part of what; wholes don’t seem to be an additional ontological commitment over their parts. Composition as identity promises to explain this intimacy. But it threatens to make the connection too intimate, for surely the parts could have made a different whole and the whole have (...) had different parts. In this paper I attempt to offer an account of parthood that is intimate enough but not too intimate: the parts generate the whole, but they are not themselves the whole. (shrink)

The book begins with an overview that introduces the Theorem and the issues surrounding it, and explores how the essays that follow contribute to our understanding of those issues.

Consider the kinds of macroscopic concrete objects that common sense and the sciences allege to exist: tables, raindrops, tectonic plates, galaxies, and the rest. Are there any such things? Opinions differ. Ontological liberals say they do; ontological radicals say they don't. Liberalism seems favored by its plausible acquiescence to the dictates of common sense abetted by science; radicalism by its ontological parsimony. Priority theorists claim we can have the virtues of both views. They hold that tables, raindrops, etc., exist, but (...) they aren't fundamental. The ontological liberal's ontology provides the correct inventory of existent individuals. The ontological radical's more restricted ontology provides the correct inventory of fundamental individuals. The priority theorist claims that the derivative individuals are "no addition in being" to the fundamental ones, so we can have our cake and eat it too. It would be nice if priority theorists were right. In this paper I argue, with regret, that they are not. One upshot is that the sort of explanations which underwrite the priority theorist's distinction between fundamental and derivative individuals do not mitigate our ontological commitments. Another is that we still have to choose between the charms of liberalism and radicalism. (shrink)

I examine the link between extensionality principles of classical mereology and the anti‐symmetry of parthood. Varzi's most recent defence of extensionality depends crucially on assuming anti‐symmetry. I examine the notions of proper parthood, weak supplementation and non‐well‐foundedness. By rejecting anti‐symmetry, the anti‐extensionalist has a unified, independently grounded response to Varzi's arguments. I give a formal construction of a non‐extensional mereology in which anti‐symmetry fails. If the notion of ‘mereological equivalence’ is made explicit, this non‐anti‐symmetric mereology recaptures all of the structure (...) of classical mereology. (shrink)

This essay interprets Butler’s distinction between identity in the loose and popular sense and in the strict and philosophical sense. Suppose there are different standards for counting the same things. Then what are two distinct things counting strictly may be one and the same thing counting loosely. Within a given standard identity is one-one. But across standards it is many-one. An alternative interpretation using the parts-whole relation fails, because that relation should be understood as many-one identity. Another alternative making identity (...) relative to sort fails, because whole and parts can be of the same sort. (shrink)

§ i. After deserting for a time the old Euclidean standards of rigour, mathematics is now returning to them, and even making efforts to go beyond them. ...

Widespread interest in Frege's general philosophical writings is, relatively speaking, a fairly recent phenomenon. But it is only very recently that his philosophy of mathematics has begun to attract the attention it now enjoys. This interest has been elicited by the discovery of the remarkable mathematical properties of Frege's contextual definition of number and of the unique character of his proposals for a theory of the real numbers. This collection of essays addresses three main developments in recent work on Frege's (...) philosophy of mathematics: the emerging interest in the intellectual background to his logicism; the rediscovery of Frege's theorem; and the reevaluation of the mathematical content of The Basic Laws of Arithmetic. Each essay attempts a sympathetic, if not uncritical, reconstruction, evaluation, or extension of a facet of Frege's theory of arithmetic. Together they form an accessible and authoritative introduction to aspects of Frege's thought that have, until now, been largely missed by the philosophical community. (shrink)

Neo-Fregeans argue that substantial mathematics can be derived from a priori abstraction principles, Hume's Principle connecting numerical identities with one:one correspondences being a prominent example. The embarrassment of riches objection is that there is a plurality of consistent but pairwise inconsistent abstraction principles, thus not all consistent abstractions can be true. This paper considers and criticizes various further criteria on acceptable abstractions proposed by Wright settling on another one—stability—as the best bet for neo-Fregeans. However, an analogue of the embarrassment of (...) riches objection resurfaces in the metatheory and I conclude by arguing that the neo-Fregean program, at least insofar as it includes a platonistic ontology, is fatally wounded by it. (shrink)

Say that some things form a set just in case there is a set whose members are precisely the things in question. For instance, all the inhabitants of New York form a set. So do all the stars in the universe. And so do all the natural numbers. Under what conditions do some things form a set?

A brief, non-technical introduction to technical and philosophical aspects of Frege's philosophy of arithmetic. The exposition focuses on Frege's Theorem, which states that the axioms of arithmetic are provable, in second-order logic, from a single non-logical axiom, "Hume's Principle", which itself is: The number of Fs is the same as the number of Gs if, and only if, the Fs and Gs are in one-one correspondence.

Introduction to a special issue based on a summer school on set theory and high-order logic.

Gideon Rosen and Robert Schwartzkopff have independently suggested (variants of) the following claim, which is a varian of Hume's Principle: -/- When the number of Fs is identical to the number of Gs, this fact is grounded by the fact that there is a one-to-one correspondence between the Fs and Gs. -/- My paper is a detailed critique of the proposal. I don't find any decisive refutation of the proposal. At the same time, it has some consequences which many will (...) find objectionable. (shrink)

This collection of essays is the first of its kind to focus on the relationship between composition and identity. Twelve original articles--written by internationally renowned scholars and rising stars in the field--argue for and against the controversial doctrine that composition is identity.--Provided by publisher.

Science Without Numbers caused a stir in 1980, with its bold nominalist approach to the philosophy of mathematics and science. It has been unavailable for twenty years and is now reissued in a revised edition with a substantial new preface presenting the author's current views and responses to the issues raised in subsequent debate.

Our conception of logical space is the set of distinctions we use to navigate the world. Agustn Rayo argues that this is shaped by acceptance or rejection of 'just is'-statements: e.g. 'to be composed of water just is to be composed of H2O'. He offers a novel conception of metaphysical possibility, and a new trivialist philosophy of mathematics.

Region R Question: How many objects — entities, things — are contained in R? Ignore the empty space. Our question might better be put, 'How many material objects does R contain?' Let's stipulate that A, B and C are metaphysical atoms: absolutely simple entities with no parts whatsoever besides themselves. So you don't have to worry about counting a particle's top half and bottom half as different objects. Perhaps they are 'point-particles', with no length, width or breadth. Perhaps they are (...) extended in space without possessing spatial parts (if that is possible). Never mind. We stipulate that A, B and C are perfectly simple. We also stipulate that they are connected as follows. A and B are stuck together in such a way that when a force is applied to one of them, they move together 'as a unit'. Moreover, the two of them together exhibit behavior that neither would exhibit on its own — Perhaps they emit a certain sound, or glow in the dark — whereas C is.. (shrink)