The Philosophy of Mathematics Today

By Matthias Schirn | Go to book overview

3
Do We Have a Determinate Conception of
Finiteness and Natural Number?

HARTRY FIELD

I'm going to begin with a rather familiar worry, about whether we have a determinate understanding of second-order quantification. Then I will consider an analogous worry about whether we have a determinate understanding of the notion of finiteness. At first blush the case for the second worry seems quite analogous to the case for the first; but for various reasons doubts about the determinacy of the notion of finiteness may seem harder to take seriously than doubts about the determinacy of second-order quantification. I'll be considering a number of different reactions to this situation.


1. INTRODUCTION

1A. First-order set theory

I'll start off the discussion not with second-order quantification, but with the notion of set; for the moment I'll tacitly take our logic to be first-order. How determinate is our understanding of the notion of set?

I think that familiar considerations against the determinacy of our notion of 'set' are fairly strong. I'll remind you of two of them. The first concerns very central set-theoretic questions such as the size of the continuum, that are unsettled not only by the standard axioms of set theory but by any axioms that anyone has been able to think of that are intuitively compelling. The continuum example seems to me an especially striking one because it isn't simply that there is an undecidable 'yes'/'no' question, of whether the

____________________
This paper is a shortened version of "'Are Our Logical and Mathematical Concepts Highly Indeterminate?'", which appeared in Midwest Studies in Philosophy, 19. I distributed a first draft of that paper at the Munich conference, and gave a talk based on it that roughly corresponds to the shortened version here. I received extremely helpful comments on the early draft from many of the conference participants, especially Penelope Maddy, Stewart Shapiro, and Bill Tait; outside of the conference, comments by Shaughan Lavine and Vann McGee also led to improvements. I am grateful to Midwest Studies for allowing me to reprint the paper in shorter form.

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