Understanding Arithmetic through Definitions

154 Pages·2013·0.61 MB·English

UCLA
UCLA Electronic Theses and Dissertations
Title
Understanding Arithmetic through Definitions
Permalink
https://escholarship.org/uc/item/5xr5x4f1
Author
Nutting, Eileen Susanna
Publication Date
2013

Peer reviewed|Thesis/dissertation
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University of California
UNIVERSITY OF CALIFORNIA
Los Angeles
Understanding Arithmetic through Definitions
A dissertation submitted in partial satisfaction of the
requirements for the degree Doctor of Philosophy
in Philosophy
by
Eileen Susanna Nutting
2013
(cid:13)c Copyright by
Eileen Susanna Nutting
2013
ABSTRACT OF THE DISSERTATION
Understanding Arithmetic through Definitions
by
Eileen Susanna Nutting
Doctor of Philosophy in Philosophy
University of California, Los Angeles, 2013
Professor Donald A. Martin, Chair
Identifying the source of our mathematical knowledge is an old and challenging prob-
lem. In order to avoid postulating cognitive faculties that seem entirely mysterious, such
as mathematical intuition, some philosophers have attempted to explain our mathematical
knowledge by claiming that it is grounded in definitions, stipulations, or postulations. In this
dissertation, I argue against a range of views of this sort in the domain of arithmetic. I ar-
gue that none of these views can account for our mathematical knowledge unless we already
understand arithmetical structure when we introduce the relevant definitions, stipulations,
or postulations. An adequate account of mathematical knowledge must explain how we have
this antecedent cognitive grip on the structure of arithmetic. The views under consideration,
I argue, are inadequate because they fail to explain this antecedent cognitive grip.
The dissertation begins with a discussion of the problem of mathematical knowl-
edge. In the process, I introduce several desiderata for theories of mathematical knowledge.
Definition-based accounts are attractive because they fare well on many of these desiderata.
ii
In the rest of the dissertation, I argue against three such accounts. First, it could not be
the case that all of our mathematical knowledge is grounded in axiomatic definitions; we
have arithmetical knowledge that could not emerge from axioms. Second, it could not be
the case that our ability to think about the natural numbers is grounded in a stipulation of
Hume’s Principle as an implicit definition; such a stipulation could not uniquely fix referents
for number-phrases, and ancient mathematicians could not have stipulated it in a suitable
way. And third, our knowledge of basic arithmetical truths is not entirely grounded in the
standard simple proofs of those truths, which rely on explicit definitions of the form ‘4=3+1’.
While these proofs surely contribute to the security of our arithmetical knowledge, we can
only understand these proofs if we already understand the structure of arithmetic. So, our
most basic arithmetical knowledge is cognitively prior to these explicit definitions.
iii
The dissertation of Eileen Susanna Nutting is approved.
John P. Carriero
Yiannis N. Moschovakis
Terence Dwight Parsons
Donald A. Martin, Committee Chair
University of California, Los Angeles
2013
iv
In memory of Matt Garber,
who first encouraged me to pursue philosophy
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Contents
1 The Problem of Mathematical Knowledge 1
1.1 Lessons from the Meno . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 The Obvious Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 The Insights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Mathematical Reasoning and Hilbert’s Geometry . . . . . . . . . . . . . . . 6
1.2.1 Constructive and Non-Constructive Geometry . . . . . . . . . . . . . 6
1.2.2 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.3 Implicit Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Benacerraf Problems: Truth and Knowledge . . . . . . . . . . . . . . . . . . 13
1.3.1 Sacrificing Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.2 Benacerraf’s ‘Combinatorial’ Classification . . . . . . . . . . . . . . . 20
1.3.3 The Epistemic Appeal . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.4 The Problem of Combinatorial Knowledge . . . . . . . . . . . . . . . . . . . 30
1.4.1 Hilbert and Antecedent Knowledge . . . . . . . . . . . . . . . . . . . 30
1.4.2 What Follows: An Overview . . . . . . . . . . . . . . . . . . . . . . . 32
2 Axiomatic Definitions of N 36
2.1 First-Order Peano Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.1.1 Uniqueness and Isomorphism . . . . . . . . . . . . . . . . . . . . . . 39
2.1.2 Standard and Nonstandard Models . . . . . . . . . . . . . . . . . . . 41
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2.1.3 Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.1.4 First-Order Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.2 Second-Order Peano Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.2.1 Second-Order Categoricity . . . . . . . . . . . . . . . . . . . . . . . . 47
2.2.2 Potential Limitations of Second-Order Quantification . . . . . . . . . 49
2.2.3 Understanding through Axioms . . . . . . . . . . . . . . . . . . . . . 50
2.3 To Distinguish Standard Models . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.3.1 Traditional and Intuitive Kinds of Finitude and Infinity . . . . . . . . 53
2.3.2 Technical Definitions of Finitude . . . . . . . . . . . . . . . . . . . . 58
2.3.3 Taking Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.3.4 The Least Number Principle . . . . . . . . . . . . . . . . . . . . . . . 65
2.3.5 Descending Progressions . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.4 Resisting Relativism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.5 Appendix I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.6 Appendix II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3 On Interaction Problems and Hume’s Principle 72
3.1 Interaction Problems for Platonism . . . . . . . . . . . . . . . . . . . . . . . 72
3.1.1 Descriptions, Interaction, and Object-Directed Thought . . . . . . . . 74
3.2 The Fregean Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.2.1 Abstraction Principles: Hume’s Principle and Direction Equivalence . 79
3.2.2 Implicit Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.3 Identifying Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.3.1 Fixing Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.3.2 Distinguishing Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.4 The Ancients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.4.1 The Possibility of an Alternative Method of Engagement . . . . . . . 94
3.4.2 Continuity of Object-Directed Thought . . . . . . . . . . . . . . . . . 98
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3.4.3 The Very Idea of Implicit or Innate Stipulation . . . . . . . . . . . . 100
3.5 Final Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4 2+2=4: A Basic Arithmetical Proof 103
4.1 Leibniz’s Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2 Leibniz’s Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.2.1 Defining Number-Objects . . . . . . . . . . . . . . . . . . . . . . . . 107
4.2.2 Defining Numerals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.2.3 Compositional Number-Expressions . . . . . . . . . . . . . . . . . . . 110
4.3 The Meaning of ‘a+b’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.3.1 Collections, Combinations, and Sums . . . . . . . . . . . . . . . . . . 112
4.3.2 An Extensional Function . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.3.3 Fregean Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.4 The Successor Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.4.1 Accounting for Successor . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.4.2 A Primitive Successor Function . . . . . . . . . . . . . . . . . . . . . 132
4.5 Final Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
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