# Modes of Referring and the Problem of Universals: An Essay in Metaphysics

By D. S. Shwayder | Go to book overview

CHAPTER IV
THE NATURAL NUMBERS

PART 1. WHAT IS A NATURAL NUMBER?
1. A philosophical analysis of numbers. -- I wish to use the schemas of chapter ii to illuminate some metaphysical puzzles over the nature of natural numbers. Such puzzles have muddled the clearest minds and set powerful intellects wandering upon the most inauspacious and dubious philosophical ways.What will follow bears exclusively on metaphysical puzzles, and has nothing to do with problems in mathematics; it deals rather with the philosophy of mathematics. I give no directions for proving theorems in number theory; say nothing about how best to organize that subject; offer no axioms or definitions of mathematical objects. My aim is to trace out the logical surroundings of the concept of a natural number. Those who are devoted to Peano's Axioms, in any one of their many forms, may regard what follows as an attempt to account for our happening to hit upon just those axioms, or a near equivalent. But if you do choose to regard my efforts in that way, I must insist that this work has even less to do with psychology than with mathematics. I shall seek to disclose the logical character of the natural numbers as understood and employed by the average thirteen-year-old who no longer thinks in terms of apples and bananas when making and stating the results of his arithmetical calculations.2. Some uses of numerals. -- It is imperative that we be clear, even from the outset, that we commonly employ numerals in a variety of related though differing uses. Our chief task here, as with property words, is to uncover these differences and relations. Let us for the moment suppose that we are employing the familiar system of Arabic numerals. These numerals are commonly used in at least the following eight ways.
 1 They are used to count with, as when we count off the students in a room. I shall call numerals so used counting-numerals. (It is of some importance that most of us first learn to count with words like "one," "two," and only later come to employ Arabic numerals for the job. But certainly Arabic numerals can be so employed.) 2 They are used to identify objects by number, as referring expressions, a kind commonly employed to refer to rooms in a hotel or men

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Modes of Referring and the Problem of Universals: An Essay in Metaphysics

• Title Page iii
• Acknowledgments vii
• Contents ix
• Introduction 1
• Chapter I - Prolegomena to the Theory Of Referring 5
• Chapter II - Modes of Referring 37
• Chapter III - Properties 72
• Chapter IV - The Natural Numbers 116
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