In theCritique of Pure Reason,Kant argues for two principles that concern magnitudes. The first is the principle that ‘All intuitions are extensive magnitudes,’ which appears in the Axioms of Intuition ; the second is the principle that ‘In all appearances the real, which is an object of sensation, has an intensive magnitude, that is, a degree,’ which appears in the Anticipations of Perception. A circle drawn in geometry and the space occupied by an object such as a book are paradigm (...) examples of extensive magnitudes, while the intensity of a light is a paradigm example of an intensive magnitude. These principles justify and explain the possibility of applying mathematics to objects of experience. The Axioms principle also explains the possibility of any mathematical cognition at all. (shrink)

Intuitionism’s disagreement with classical logic is standardly based on its specific understanding of truth. But different intuitionists have actually explicated the notion of truth in fundamentally different ways. These are considered systematically and separately, and evaluated critically. It is argued that each account faces difficult problems. They all either have implausible consequences or are viciously circular.

This note examines the mereological component of Geoffrey Hellman's most recent version of modal structuralism. There are plausible forms of agnosticism that benefit only a little from Hellman's mereological turn.

William Tait's standing in the philosophy of mathematics hardly needs to be argued for; for this reason the appearance of this collection is especially welcome. As noted in his Preface, the essays in this book ‘span the years 1981–2002’. The years given are evidently those of publication. One essay was not previously published in its present form, but it is a reworking of papers published during that period. The Introduction, one appendix, and some notes are new. Many of the essays (...) will be familiar to the readers of this journal; indeed two first appeared here.It should be no surprise to those who know Tait's work that this is a very rich collection, with contributions on a wide variety of issues, both systematic and historical. That poses a problem for a reviewer, because a serious discussion of even all the major issues would be beyond the scope of a review.Before the essays in this collection, Tait's publications were almost all in mathematical logic, especially proof theory. He was a leading actor in the work on the revised Hilbert program stimulated after the war by Georg Kreisel, Kurt Schütte, and Gaisi Takeuti. Tait thus has an experience in mathematical research that is rare among those whose careers have been institutionally in philosophy. In the 1970s Tait became disillusioned with the proof-theoretic program on which he had worked. However, he draws on that background in several of these essays, and he has continued to work on mathematical questions. In particular, the set-theoretic project reflected in essay 6 is as much mathematical as philosophical.It is not easy to characterize Tait's philosophical viewpoint briefly. Both the title of the book and some of its content mark him as a rationalist. His hero in the earlier history …. (shrink)

It is sometimes argued that mathematical knowledge must be a priori, since mathematical truths are necessary, and experience tells us only what is true, not what must be true. This argument can be undermined either by showing that experience can yield knowledge of the necessity of some truths, or by arguing that mathematical theorems are contingent. Recent work by Albert Casullo and Timothy Williamson argues (or can be used to argue) the first of these lines; W. V. Quine and Hartry (...) Field take the latter line. I defend a version of the argument against these, and other objections. (shrink)

In this article, I present a novel account of a priori warrant, which I then use to examine the relationship between a priori and a posteriori warrant in mathematics. According to this account of a priori warrant, the reason that a posteriori warrant is subordinate to a priori warrant in mathematics is because processes that produce a priori warrant are reliable independent of the contexts in which they are used, whereas this is not true for processes that produce a posteriori (...) warrant. Following this, I show how this difference explains why a priori warranting processes, such as proof (and mathematical intuition) can, and a posteriori warranting processes cannot, definitively establish mathematical results. I then show why, in the event of a conflict between a priori and a posteriori warranting processes, the former undermine, and cannot be undermined by, the latter. Finally, I explain the connection between apriority and mathematical necessity, but do so in a way that allows for the existence of contingent a priori knowledge. (shrink)

Consider one of several things. Is the one thing necessarily one of the several? This key question in the modal logic of plurals is clarified. Some defenses of an affirmative answer are developed and compared. Various remarks are made about the broader philosophical significance of the question.