Date of Degree

2-2016

Document Type

Dissertation

Degree Name

Ph.D.

Program

Philosophy

Advisor(s)

Gary Ostertag

Saul Kripke

Committee Members

Saul Kripke

Nathan Salmon

Jesse Prinz

Michael Levin

Subject Categories

Epistemology | History of Philosophy | Logic and Foundations of Mathematics | Philosophy of Language | Philosophy of Mind

Keywords

Number sense, Cardinal, Counting, Acquisition, Paradox of analysis, Set theory

Abstract

Saul Kripke once remarked to me that natural numbers cannot be posits inferred from their indispensability to science, since we’ve always had them. This left me wondering whether numbers are objects of Russellian acquaintance, or accessible by analysis, being implied by known general principles about how to reason correctly, or both. To answer this question, I discuss some recent (and not so recent) work on our concepts of number and of particular numbers, by leading psychologists and philosophers. Special attention is paid to Kripke’s theory that numbers possess structural features of the numerical systems that stand for them, and to the relation between his proposal about numbers and his doctrine that there are contingent truths known a priori. My own proposal, to which Kripke is sympathetic, is that numbers are properties of sets. I argue for this by showing the extent to which it can avoid the problems that plague the various views under discussion, including the problems raised by Kripke against Frege. I also argue that while the terms ‘the number of F’s’, ‘natural number’ and ‘0’, ‘1’, ‘2’ etc. are partially understood by the folk, they can only be fully understood by reflection and analysis, including reflection on how to reason correctly. In this last respect my thesis is a retreat position from logicism. I also show how it dovetails with an account of how numbers are actually grasped in practice, via numerical systems, and in virtue of a certain structural affinity between a geometric pattern that we grasp intuitively, and our fully analyzed concepts of numbers. I argue that none of this involves acquaintance with numbers.

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