This book does three main things. First, it defends mathematical platonism against the main objections to that view (most notably, the epistemological objection and the multiple-reductions objection). Second, it defends anti-platonism (in particular, fictionalism) against the main objections to that view (most notably, the Quine-Putnam indispensability objection and the objection from objectivity). Third, it argues that there is no fact of the matter whether abstract mathematical objects exist and, hence, no fact of the matter whether platonism or anti-platonism is true.

Naive set theory, as found in Frege and Russell, is almost universally believed to have been shown to be false by the set-theoretic paradoxes. The standard response has been to rank sets into one or other hierarchy. However it is extremely difficult to characterise the nature of any such hierarchy without falling into antinomies as severe as the set-theoretic paradoxes themselves. Various attempts to surmount this problem are examined and criticised. It is argued that the rejection of naive set theory (...) inevitably leads one into a severe scepticism with regard to the feasibility of giving a systematic semantics for set theory. It is further argued that this is not just a problem for philosophers of mathematics. Semantic scepticism in set theory will almost inevitably spill over into total pessimism regarding the prospects for an explanatory theory of language and meaning in general. The conclusion is that those who wish to avoid such intellectual defeatism need to look seriously at the possibility that it is the logic used in the derivation of the paradoxes, and not the naive set theory itself, which is at fault. (shrink)

CZF is an intuitionistic set theory that does not contain Power Set, substituting instead a weaker version, Subset Collection. In this paper a Kripke model of CZF is presented in which Power Set is false. In addition, another Kripke model is presented of CZF with Subset Collection replaced by Exponentiation, in which Subset Collection fails.

George Boolos has argued that the iterative conception of set justifies most, but not all, the ZFC axioms, and that a second conception of set, the Frege-von Neumann conception (FN), justifies the remaining axioms. This article challenges Boolos's claim that FN does better than the iterative conception at justifying the axioms in question.

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)

It is commonly assumed that classical logic is the embodiment of a realist ontology. In “Sets and Semantics”, however, Jonathan Lear challenged this assumption in the particular case of set theory, arguing that even if one is a set-theoretic Platonist, due attention to a special feature of set theory leads to the conclusion that the correct logic for it is intuitionistic. The feature of set theory Lear appeals to is the open-endedness of the concept of set. This article advances reasons (...) internal to Lear’s account to show that his argument for intuitionistic set theory is unsuccessful. (shrink)

Some philosophers have argued that the open-endedness of the set concept has revisionary consequences for the semantics and logic of set theory. I consider (several variants of) an argument for this claim, premissed on the view that quantification in mathematics cannot outrun our conceptual abilities. The argument urges a non-standard semantics for set theory that allegedly sanctions a non-classical logic. I show that the views about quantification the argument relies on turn out to sanction a classical semantics and logic after (...) all. More generally, this article constitutes a case study in whether the need to account for conceptual progress can ever motivate a revision of semantics or logic. I end by expressing skepticism about the prospects of a so-called non-proof-based justification for this kind of revisionism about set theory. (shrink)

Mark Balaguer has written a provocative and original book. The book is as ambitious as a work of philosophy of mathematics could be. It defends both of the dominant views concerning the ontology of mathematics, Platonism and Anti-Platonism, and then closes with an argument that there is no fact of the matter which is right.

to constructive systems is significant for contemporary metaphysics. However, many are surprised by these results, having learned that the Axiom of Choice (AC) is constructively valid. Indeed, even among specialists there were, until recently, reasons for puzzlement-rival versions of Intuitionistic Type Theory, one where (AC) is valid, another where it implies classical logic. This paper accessibly explains the situation, puts the issues in a broader setting by considering other choice principles, and draws philosophical morals for the understanding of quantification, choice (...) principles, and the prospects for constructivism. (shrink)