Oliver Marshall

Saul Kripke’s analysis of the concept of the natural numbers that we are taught in school yields a novel and axiomatically economical way of representing arithmetic in standard set theory—one that helps to answer Benacerraf’s objection from extraneous content as well as Wittgenstein’s objection from unsurveyability. After describing Kripke’s proposal in some detail, we examine it in the light of work by Quine, Steiner, Parsons, Boolos and Burgess. Although the primary aim of this paper is to present and explicate Kripke’s view, we conclude by discussing some of the issues that are faced by Kripke’s proposal, so that the reader can get a sense of the geography of these issues.

Alonzo Church (1903–1995) was a renowned mathematical logician, philosophical logician, philosopher, teacher and editor. He was one of the founders of the discipline of mathematical logic as it developed after Cantor, Frege and Russell. He was also one of the principal founders of the Association for Symbolic Logic and the Journal of Symbolic Logic. The list of his students, mathematical and philosophical, is striking as it contains the names of renowned logicians and philosophers. In this article, we focus primarily on Church’s philosophical contributions. (For an account of his life and academic history see the Introduction to The Collected Works of Alonzo Church (2019).) However, we also discuss his mathematical results when it is desirable to do so in order to pursue a philosophical issue.

Marcus Giaquinto claims that finite cardinal numbers are sensible properties, and that the smallest ones are known by acquaintance. In this paper I compare Giaquinto’s epistemology to the Russellian one with which it invites comparison, before showing how it is subject to a version of Jody Azzouni’s “epistemic role” objection. Then I argue that the source of this problem is Giaquinto’s misconception that numbers, like quantities, are sensible properties. Finally, I offer a sketch of a theory of how we grasp finite cardinals on the assumption that they are not sensible, and show why this is not subject to the epistemic role objection.

David Liebesman argues that we never count by identity. He generalizes from an argument that we don't do so with sentences indicating fractions, or with measurement sentences on their supposed count readings. In response, I argue that measurement sentences aren't covered by the thesis that we count by identity, in part because they don't have count readings. Then I use the data to which Liebesman appeals, in his argument that we don't count by identity using measurement sentences, in order to rebut his argument that we don't count by identity using sentences indicating fractions.

ABSTRACT I argue against both neuropsychological and cognitive accounts of our grasp of numbers. I show that despite the points of divergence between these two accounts, they face analogous problems. Both presuppose too much about what they purport to explain to be informative, and also characterize our grasp of numbers in a way that is absurd in the light of what we already know from the point of view of mathematical practice. Then I offer a positive methodological proposal about the role that cognitive science should play in the philosophy of mathematics.