In arithmetic, if only because many of its methods and concepts originated in India, it has been the tradition to reason less strictly than in geometry, ...

Are there objects that are “thin” in the sense that their existence does not make a substantial demand on the world? Frege famously thought so. He claimed that the equinumerosity of the knives and the forks suffices for there to be objects such as the number of knives and the number of forks, and for these objects to be identical. The idea of thin objects holds great philosophical promise but has proved hard to explicate. This book attempts to develop the (...) needed explanations by drawing on some Fregean ideas. First, to be an object is to be a possible referent of a singular term. Second, singular reference can be achieved by providing a criterion of identity for the would-be referent. The second idea enables a form of easy reference and thus, via the first idea, also a form of easy being. Paradox is avoided by imposing a predicativity restriction on the criteria of identity. But the abstraction based on a criterion of identity may result in an expanded domain. By iterating such expansions, a powerful account of dynamic abstraction is developed. (shrink)

This paper is a reply to George Boolos's three papers (Boolos (1987a, 1987b, 1990a)) concerned with the status of Hume's Principle. Five independent worries of Boolos concerning the status of Hume's Principle as an analytic truth are identified and discussed. Firstly, the ontogical concern about the commitments of Hume's Principle. Secondly, whether Hume's Principle is in fact consistent and whether the commitment to the universal number by adopting Hume's Principle might be problematic. Also the so-called `surplus content' worry is discussed, (...) which points out that the conceptual resources to grasp Hume's Principle vastly outstrip the conceptual resources employed in arithmetical reasoning. And lastly whether Hume's Principle is in bad company with other unsuccessful implicit definitions. In the last section, an account towards our entitlement to Hume&'s Principle is sketched. (shrink)

One recent `neologicist' claim is that what has come to be known as "Frege's Theorem"–the result that Hume's Principle, plus second-order logic, suffices for a proof of the Dedekind-Peano postulate–reinstates Frege's contention that arithmetic is analytic. This claim naturally depends upon the analyticity of Hume's Principle itself. The present paper reviews five misgivings that developed in various of George Boolos's writings. It observes that each of them really concerns not `analyticity' but either the truth of Hume's Principle or our entitlement (...) to accept it and reviews possible neologicist replies. A two-part Appendix explores recent developments of the fifth of Boolos's objections–the problem of Bad Company–and outlines a proof of the principle $N^q$, an important part of the defense of the claim that what follows from Hume's Principle is not merely a theory which allows of interpretation as arithmetic but arithmetic itself. (shrink)

Neo-Fregeans need their stipulation of Hume's Principle — $NxFx=NxGx \leftrightarrow \exists R (Fx \,1\hbox {-}1_R\, Gx)$ — to do two things. First, it must implicitly define the term-forming operator ‘Nx…x…’, and second it must guarantee that Hume's Principle as a whole is true. I distinguish two senses in which the neo-Fregeans might ‘stipulate’ Hume's Principle, and argue that while one sort of stipulation fixes a meaning for ‘Nx…x…’ and the other guarantees the truth of Hume's Principle, neither does both.

This comprehensive volume gives a panorama of the best current work in this lively field, through twenty specially written essays by the leading figures in the field. All essays deal with foundational issues, from the nature of mathematical knowledge and mathematical existence to logical consequence, abstraction, and the notions of set and natural number. The contributors also represent and criticize a variety of prominent approaches to the philosophy of mathematics, including platonism, realism, nomalism, constructivism, and formalism.

By a `denoting phrase' I mean a phrase such as any one of the following: a man, some man, any man, every man, all men, the present King of England, the present King of France, the center of mass of the solar system at the first instant of the twentieth century, the revolution of the earth round the sun, the revolution of the sun round the earth. Thus a phrase is denoting solely in virtue of its form. We may distinguish (...) three cases: (1) A phrase may be denoting, and yet not denote anything; e.g., `the present King of France'. (2) A phrase may denote one definite object; e.g., `the present King of England' denotes a certain man. (3) A phrase may denote ambiguously; e.g. `a man' denotes not many men, but an ambiguous man. The interpretation of such phrases is a matter of considerably difficulty; indeed, it is very hard to frame any theory not susceptible of formal refutation. All the difficulties with which I am acquainted are met, so far as I can discover, by the theory which I am about to explain. (shrink)

By a `denoting phrase' I mean a phrase such as any one of the following: a man, some man, any man, every man, all men, the present King of England, the present King of France, the center of mass of the solar system at the first instant of the twentieth century, the revolution of the earth round the sun, the revolution of the sun round the earth. Thus a phrase is denoting solely in virtue of its form. We may distinguish (...) three cases: (1) A phrase may be denoting, and yet not denote anything; e.g., `the present King of France'. (2) A phrase may denote one definite object; e.g., `the present King of England' denotes a certain man. (3) A phrase may denote ambiguously; e.g. `a man' denotes not many men, but an ambiguous man. The interpretation of such phrases is a matter of considerably difficulty; indeed, it is very hard to frame any theory not susceptible of formal refutation. All the difficulties with which I am acquainted are met, so far as I can discover, by the theory which I am about to explain. (shrink)

This article is a critical notice of Bob Hale and Crispin Wright's *The Reason's Proper Study* (OUP). It focuses particularly on their attempts (crucial to their neo-logicist project) to say what a singular term is. I identify problems for their account but include some constructive suggestions about how it might be improved.

Syntactic Reductionism, as understood here, is the view that the ‘logical forms’ of sentences in which reference to abstract objects appears to be made are misleading so that, on analysis, we can see that no expressions which even purport to refer to abstract objects are present in such sentences. After exploring the motivation for such a view, and arguing that no previous argument against it succeeds, sentences involving generalized quantifiers, such as ‘most’, are examined. It is then argued, on this (...) basis, that Syntactic Reductionism is untenable. (shrink)

Frege, famously, held that there is a close connection between our concept of cardinal number and the notion of one-one correspondence, a connection enshrined in Hume's Principle. Husserl, and later Parsons, objected that there is no such close connection, that our most primitive conception of cardinality arises from our grasp of the practice of counting. Some empirical work on children's development of a concept of number has sometimes been thought to point in the same direction. I argue, however, that Frege (...) was close to right, that our concept of cardinal number is closely connected with a notion like that of one-one correspondence, a more primitive notion we might call just as many. (shrink)

Philosophers of mathematics agree that the only interpretation of arithmetic that takes that discourse at 'face value' is one on which the expressions 'N', '0', '1', '+', and 'x' are treated as proper names. I argue that the interpretation on which these expressions are treated as akin to free variables has an equal claim to be the default interpretation of arithmetic. I show that no purely syntactic test can distinguish proper names from free variables, and I observe that any semantic (...) test that can must beg the question. I draw the same conclusion concerning areas of mathematics beyond arithmetic. This paper is a greatly extended version of my response to Stewart Shapiro's paper in the conference 'Structuralism in physics and mathematics' held in Bristol on 2–3 December, 2006. (shrink)

This work is the long awaited sequel to the author’s classic Frege: Philosophy of Language. But it is not exactly what the author originally planned. He tells us that when he resumed work on the book in the summer of 1989, after a long interruption, he decided to start afresh. The resulting work followed a different plan from the original drafts. The reader does not know what was lost by their abandonment, but clearly much was gained: The present work may (...) be the most compactly organized of Dummett’s philosophical books, and its value as commentary on Frege is enhanced by its being organized as a study of Die Grundlagen der Arithmetik and selected parts of Grundgesetze. Moreover, since Dummett has been thinking and writing about Frege for forty years, we have a single, mature chronological stage of his thought. (shrink)

Mathematical realism is the doctrine that mathematical objects really exist, that mathematical statements are either determinately true or determinately false, and that the accepted mathematical axioms are predominantly true. A realist understanding of set theory has it that when the sentences of the language of set theory are understood in their standard meaning, each sentence has a determinate truth value, so that there is a fact of the matter whether the cardinality of the continuum is א2 or whether there are (...) measurable cardinals, whether or not those facts are knowable by us. (shrink)

That reference is inscrutable is demonstrated, it is argued, not only by W. V. Quine's arguments but by Peter Unger's "Problem of the Many." Applied to our own language, this is a paradoxical result, since nothing could be more obvious to speakers of English than that, when they use the word "rabbit," they are talking about rabbits. The solution to this paradox is to take a disquotational view of reference for one's own language, so that "When I use 'rabbit,' I (...) refer to rabbits" is made true by the meaning of the word "refer." The reference relation is extended to other languages by translation. The explanation for this peculiarly egocentric conception of semantics-questions of others' meanings are settled by asking what I mean by words of my language-is to be found in our practice of predicting and explaining other people's behavior by empathetic identification. I understand other people's behavior by asking what I would do in their place. (shrink)

Much of The Reason’s Proper Study is devoted to defending the claim that simply by stipulating an abstraction principle for the “number-of” functor, we can simultaneously fix a meaning for this functor and acquire epistemic entitlement to the stipulated principle. In this paper, I argue that the semantic and epistemological principles Hale and Wright offer in defense of this claim may be too strong for their purposes. For if these principles are correct, it is hard to see why they do (...) not justify platonist strategies that are not in any way “neo-Fregean,” e.g. strategies that treat “the number of Fs” as a Russellian definite description rather than a singular term, or employ axioms that do not have the form of abstraction principles. (shrink)

Frege suggests that criteria of identity should play a central role in the explanation of reference, especially to abstract objects. This paper develops a precise model of how we can come to refer to a particular kind of abstract object, namely, abstract letter types. It is argued that the resulting abstract referents are ‘metaphysically lightweight’.

The paper formulates and proves a strengthening of ‘Frege’s Theorem’, which states that axioms for second-order arithmetic are derivable in second-order logic from Hume’s Principle, which itself says that the number of Fs is the same as the number ofGs just in case the Fs and Gs are equinumerous. The improvement consists in restricting this claim to finite concepts, so that nothing is claimed about the circumstances under which infinite concepts have the same number. ‘Finite Hume’s Principle’ also suffices for (...) the derivation of axioms for arithmetic and, indeed, is equivalent to a version of them, in the presence of Frege’s definitions of the primitive expressions of the language of arithmetic. The philosophical significance of this result is also discussed. (shrink)

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.

In “Double Vision Two Questions about the Neo-Fregean Programme”, John MacFarlane’s raises two main questions: (1) Why is it so important to neo-Fregeans to treat expressions of the form ‘the number of Fs’ as a species of singular term? What would be lost, if anything, if they were analysed instead as a type of quantifier-phrase, as on Russell’s Theory of Definite Descriptions? and (2) Granting—at least for the sake of argument—that Hume’s Principle may be used as a means of implicitly (...) defining the number operator, what advantage, if any, does adopting this course possess over a direct stipulation of the Dedekind-Peano axioms? This paper attempts to answer them. In response to the first, we spell out the links between the recognition of numerical terms as vehicles of singular reference and the conception of numbers as possible objects of singular, or object-directed, thought, and the role of the acknowledgement of numbers as objects in the neo-Fregean attempt to justify the basic laws of arithmetic. In response to the second, we argue that the crucial issue concerns the capacity of either stipulation—of Hume’s Principle, or of the Dedekind-Peano axioms—to found knowledge of the principles involved, and that in this regard there are crucial differences which explain why the former stipulation can, but the latter cannot, play the required foundational role. (shrink)

Michael Dummett mounts, in Frege: Philosophy of Mathematics, a concerted attack on the attempt, led by Crispin Wright, to salvage defensible versions of Frege's platonism and logicism in which Frege's criterion of numerical identity plays a leading role. I discern four main strands in this attack—that Wright's solution to the Caesar problem fails; that explaining number words contextually cannot justify treating them as enjoying robust reference; that Wright has no effective counter to ontological reductionism; and that the attempt is vitiated (...) by the unavoidable impredicativity of its leading principle—and argue that none of them succeeds. (shrink)

This paper discusses Michael Dummett’s criticism of the Neo-Fregean concep- tion of the context principle. I will present four arguments by Dummett that purport to show that the context principle is incompatible with platonism. I discuss and ultimately reject each argument. I will close this paper by identifying what I take to be a deep rooted tension in the Neo-Fregean project which might have motivated Dummett’s opposition to the Neo-Fregean use of the context principle. I argue that this tension does (...) give rise to a legitimate concern, yet it does not affect the Neo-Fregean conception of the context principle. (shrink)

SummaryA dilemma concerning reference is posed: on the one hand it seems essential, if we are to give an account of truth, to first give an account of reference. On the other hand, reference is more remote than truth from the evidence in behavior on which a radical theory of language must depend, since words refer only in the context of sentences, and it is sentences which are needed to promote human purposes. The solution which is proposed is to treat (...) reference as a theoretical construct whose sole function is to serve a theory of truth. Since more than one relation between words and objects will serve a theory of truth equally well, this amounts to giving up the concept of reference as basic. (shrink)

SummaryA dilemma concerning reference is posed: on the one hand it seems essential, if we are to give an account of truth, to first give an account of reference. On the other hand, reference is more remote than truth from the evidence in behavior on which a radical theory of language must depend, since words refer only in the context of sentences, and it is sentences which are needed to promote human purposes. The solution which is proposed is to treat (...) reference as a theoretical construct whose sole function is to serve a theory of truth. Since more than one relation between words and objects will serve a theory of truth equally well, this amounts to giving up the concept of reference as basic. (shrink)

This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments for arithmetic and set theory.