Lectures on the Philosophy of Mathematics

ISBN: 9780262362986 | Copyright 2020

Click here to preview

Instructor Requests

Digital Exam/Desk Copy Ancillaries
Expand/Collapse All
Preface (pg. xvii)
About the Author (pg. xix)
Numbers (pg. 1)
Numbers versus numerals (pg. 1)
Number systems (pg. 2)
Natural numbers (pg. 2)
Integers (pg. 3)
Rational numbers (pg. 3)
Incommensurable numbers (pg. 4)
An alternative geometric argument (pg. 5)
Platonism (pg. 6)
Plenitudinous platonism (pg. 7)
Logicism (pg. 7)
Equinumerosity (pg. 7)
The Cantor-Hume principle (pg. 8)
The Julius Caesar problem (pg. 10)
Numbers as equinumerosity classes (pg. 10)
Neologicism (pg. 11)
Interpreting arithmetic (pg. 12)
Numbers as equinumerosity classes (pg. 12)
Numbers as sets (pg. 12)
Numbers as primitives (pg. 14)
Numbers as morphisms (pg. 15)
Numbers as games (pg. 16)
Junk theorems (pg. 18)
Interpretation of theories (pg. 18)
What numbers could not be (pg. 19)
The epistemological problem (pg. 20)
Dedekind arithmetic (pg. 21)
Arithmetic categoricity (pg. 22)
Mathematical induction (pg. 23)
Fundamental theorem of arithmetic (pg. 24)
Infinitude of primes (pg. 26)
Structuralism (pg. 27)
Definability versus Leibnizian structure (pg. 28)
Role of identity in the formal language (pg. 29)
Isomorphism orbit (pg. 30)
Categoricity (pg. 31)
Structuralism in mathematical practice (pg. 32)
Eliminative structuralism (pg. 34)
Abstract structuralism (pg. 35)
What is a real number? (pg. 36)
Dedekind cuts (pg. 36)
Theft and honest toil (pg. 37)
Cauchy real numbers (pg. 38)
Real numbers as geometric continuum (pg. 38)
Categoricity for the real numbers (pg. 38)
Categoricity for the real continuum (pg. 40)
Transcendental numbers (pg. 42)
The transcendence game (pg. 42)
Complex numbers (pg. 43)
Platonism for complex numbers (pg. 44)
Categoricity for the complex field (pg. 44)
A complex challenge for structuralism? (pg. 45)
Structure as reduct of rigid structure (pg. 46)
Contemporary type theory (pg. 47)
More numbers (pg. 48)
What is a philosophy for? (pg. 48)
Finally, what is a number? (pg. 49)
Questions for further thought (pg. 49)
Further reading (pg. 51)
Credits (pg. 52)
Rigor (pg. 53)
Continuity (pg. 53)
Informal account of continuity (pg. 53)
The definition of continuity (pg. 55)
The continuity game (pg. 56)
Estimation in analysis (pg. 56)
Limits (pg. 57)
Instantaneous change (pg. 57)
Infinitesimals (pg. 58)
Modern definition of the derivative (pg. 59)
An enlarged vocabulary of concepts (pg. 59)
The least-upper-bound principle (pg. 61)
Consequences of completeness (pg. 62)
Continuous induction (pg. 63)
Indispensability of mathematics (pg. 64)
Science without numbers (pg. 65)
Fictionalism (pg. 67)
The theory/metatheory distinction (pg. 68)
Abstraction in the function concept (pg. 68)
The Devil's staircase (pg. 69)
Space-filling curves (pg. 70)
Conway base-13 function (pg. 71)
Infinitesimals revisited (pg. 74)
Nonstandard analysis and the hyperreal numbers (pg. 75)
Calculus in nonstandard analysis (pg. 76)
Classical model-construction perspective (pg. 77)
Axiomatic approach (pg. 78)
``The'' hyperreal numbers? (pg. 78)
Radical nonstandardness perspective (pg. 79)
Translating between nonstandard and classical perspectives (pg. 80)
Criticism of nonstandard analysis (pg. 81)
Questions for further thought (pg. 82)
Further reading (pg. 84)
Credits (pg. 85)
Infinity (pg. 87)
Hilbert's Grand Hotel (pg. 87)
Hilbert's bus (pg. 88)
Hilbert's train (pg. 88)
Countable sets (pg. 89)
Equinumerosity (pg. 90)
Hilbert's half-marathon (pg. 92)
Cantor's cruise ship (pg. 93)
Uncountability (pg. 93)
Cantor's original argument (pg. 95)
Mathematical cranks (pg. 96)
Cantor on transcendental numbers (pg. 96)
Constructive versus nonconstructive arguments (pg. 97)
On the number of subsets of a set (pg. 99)
On the number of infinities (pg. 99)
Russell on the number of propositions (pg. 100)
On the number of possible committees (pg. 100)
The diary of Tristram Shandy (pg. 101)
The cartographer's paradox (pg. 101)
The Library of Babel (pg. 103)
On the number of possible books (pg. 103)
Beyond equinumerosity to the comparative size principle (pg. 104)
Place focus on reflexive preorders (pg. 106)
What is Cantor's continuum hypothesis? (pg. 107)
Transfinite cardinals—the alephs and the beths (pg. 109)
Lewis on the number of objects and properties (pg. 110)
Zeno's paradox (pg. 111)
Actual versus potential infinity (pg. 112)
How to count (pg. 112)
Questions for further thought (pg. 114)
Further reading (pg. 116)
Credits (pg. 117)
Geometry (pg. 119)
Geometric constructions (pg. 119)
Contemporary approach via symmetries (pg. 121)
Collapsible compasses (pg. 122)
Constructible points and the constructible plane (pg. 123)
Constructible numbers and the number line (pg. 125)
Nonconstructible numbers (pg. 126)
Doubling the cube (pg. 127)
Trisecting the angle (pg. 127)
Squaring the circle (pg. 128)
Circle-squarers and angle-trisectors (pg. 128)
Alternative tool sets (pg. 128)
Compass-only constructibility (pg. 129)
Straightedge-only constructibility (pg. 129)
Construction with a marked ruler (pg. 130)
Origami constructibility (pg. 130)
Spirograph constructibility (pg. 131)
The ontology of geometry (pg. 132)
The role of diagrams and figures (pg. 133)
Kant (pg. 133)
Hume on arithmetic reasoning over geometry (pg. 134)
Manders on diagrammatic proof (pg. 135)
Contemporary tools (pg. 135)
How to lie with figures (pg. 136)
Error and approximation in geometric construction (pg. 137)
Constructing a perspective chessboard (pg. 140)
Non-Euclidean geometry (pg. 142)
Spherical geometry (pg. 143)
Elliptical geometry (pg. 145)
Hyperbolic geometry (pg. 145)
Curvature of space (pg. 146)
Errors in Euclid? (pg. 147)
Implicit continuity assumptions (pg. 147)
The missing concept of ``between'' (pg. 148)
Hilbert's geometry (pg. 149)
Tarski's geometry (pg. 149)
Geometry and physical space (pg. 149)
Poincaré on the nature of geometry (pg. 151)
Tarski on the decidability of geometry (pg. 151)
Questions for further thought (pg. 153)
Further reading (pg. 154)
Credits (pg. 155)
Proof (pg. 157)
Syntax-semantics distinction (pg. 157)
Use/mention (pg. 158)
What is proof? (pg. 159)
Proof as dialogue (pg. 160)
Wittgenstein (pg. 161)
Thurston (pg. 161)
Formalization and mathematical error (pg. 162)
Formalization as a sharpening of mathematical ideas (pg. 163)
Mathematics does not take place in a formal language (pg. 163)
Voevodsky (pg. 165)
Proofs without words (pg. 165)
How to lie with figures (pg. 166)
Hard arguments versus soft (pg. 166)
Moral mathematical truth (pg. 167)
Formal proof and proof theory (pg. 169)
Soundness (pg. 170)
Completeness (pg. 170)
Compactness (pg. 171)
Verifiability (pg. 172)
Sound and verifiable, yet incomplete (pg. 172)
Complete and verifiable, yet unsound (pg. 173)
Sound and complete, yet unverifiable (pg. 173)
The empty structure (pg. 174)
Formal deduction examples (pg. 175)
The value of formal deduction (pg. 176)
Automated theorem proving and proof verification (pg. 177)
Four-color theorem (pg. 177)
Choice of formal system (pg. 178)
Completeness theorem (pg. 180)
Nonclassical logics (pg. 182)
Classical versus intuitionistic validity (pg. 183)
Informal versus formal use of ``constructive'' (pg. 185)
Epistemological intrusion into ontology (pg. 186)
No unbridgeable chasm (pg. 186)
Logical pluralism (pg. 187)
Classical and intuitionistic realms (pg. 187)
Conclusion (pg. 188)
Questions for further thought (pg. 188)
Further reading (pg. 190)
Credits (pg. 191)
Computability (pg. 193)
Primitive recursion (pg. 194)
Implementing logic in primitive recursion (pg. 195)
Diagonalizing out of primitive recursion (pg. 197)
The Ackermann function (pg. 198)
Turing on computability (pg. 200)
Turing machines (pg. 201)
Partiality is inherent in computability (pg. 202)
Examples of Turing-machine programs (pg. 202)
Decidability versus enumerability (pg. 203)
Universal computer (pg. 204)
``Stronger'' Turing machines (pg. 205)
Other models of computatibility (pg. 206)
Computational power: Hierarchy or threshold? (pg. 207)
The hierarchy vision (pg. 207)
The threshold vision (pg. 207)
Which vision is correct? (pg. 207)
Church-Turing thesis (pg. 208)
Computation in the physical universe (pg. 209)
Undecidability (pg. 210)
The halting problem (pg. 210)
Other undecidable problems (pg. 211)
The tiling problem (pg. 211)
Computable decidability versus enumerability (pg. 212)
Computable numbers (pg. 212)
Oracle computation and the Turing degrees (pg. 214)
Complexity theory (pg. 215)
Feasibility as polynomial-time computation (pg. 217)
Worst-case versus average-case complexity (pg. 217)
The black-hole phenomenon (pg. 218)
Decidability versus verifiability (pg. 219)
Nondeterministic computation (pg. 219)
P versus NP (pg. 220)
Computational resiliency (pg. 221)
Questions for further thought (pg. 221)
Further reading (pg. 224)
Incompleteness (pg. 225)
The Hilbert program (pg. 226)
Formalism (pg. 226)
Life in the world imagined by Hilbert (pg. 228)
The alternative (pg. 228)
Which vision is correct? (pg. 228)
The first incompleteness theorem (pg. 229)
The first incompleteness theorem, via computability (pg. 229)
The Entscheidungsproblem (pg. 230)
Incompleteness, via diophantine equations (pg. 232)
Arithmetization (pg. 233)
First incompleteness theorem, via Gödel sentence (pg. 235)
Second incompleteness theorem (pg. 238)
Löb proof conditions (pg. 238)
Provability logic (pg. 240)
Gödel-Rosser incompleteness theorem (pg. 240)
Tarski's theorem on the nondefinability of truth (pg. 241)
Feferman theories (pg. 242)
Ubiquity of independence (pg. 243)
Tower of consistency strength (pg. 244)
Reverse mathematics (pg. 244)
Goodstein's theorem (pg. 247)
Löb's theorem (pg. 250)
Two kinds of undecidability (pg. 251)
Questions for further thought (pg. 252)
Further reading (pg. 253)
Set Theory (pg. 255)
Cantor-Bendixson theorem (pg. 256)
Set theory as a foundation of mathematics (pg. 258)
General comprehension principle (pg. 261)
Frege's Basic Law V (pg. 262)
Cumulative hierarchy (pg. 265)
Separation axiom (pg. 267)
Ill-founded hierarchies (pg. 268)
Impredicativity (pg. 269)
Extensionality (pg. 270)
Other axioms (pg. 271)
Replacement axiom (pg. 271)
The number of infinities (pg. 272)
The axiom of choice and the well-order theorem (pg. 274)
Paradoxical consequences of AC (pg. 276)
Paradox without AC (pg. 276)
Solovay's dream world for analysis (pg. 277)
Large cardinals (pg. 278)
Strong limit cardinals (pg. 279)
Regular cardinals (pg. 280)
Aleph-fixed-point cardinals (pg. 280)
Inaccessible and hyperinaccessible cardinals (pg. 281)
Linearity of the large cardinal hierarchy (pg. 283)
Large cardinals consequences down low (pg. 284)
Continuum hypothesis (pg. 284)
Pervasive independence phenomenon (pg. 285)
Universe view (pg. 286)
Categoricity and rigidity of the set-theoretic universe (pg. 286)
Criterion for new axioms (pg. 288)
Intrinsic justification (pg. 289)
Extrinsic justification (pg. 289)
What is an axiom? (pg. 290)
Does mathematics need new axioms? (pg. 292)
Absolutely undecidable questions (pg. 292)
Strong versus weak foundations (pg. 293)
Shelah (pg. 294)
Feferman (pg. 294)
Multiverse view (pg. 295)
Dream solution of the continuum hypothesis (pg. 295)
Analogy with geometry (pg. 296)
Pluralism as set-theoretic skepticism? (pg. 296)
Plural platonism (pg. 297)
Theory/metatheory interaction in set theory (pg. 297)
Summary (pg. 298)
Questions for further thought (pg. 298)
Further reading (pg. 300)
Credits (pg. 301)
Bibliography (pg. 303)
Notation Index (pg. 315)
Subject Index (pg. 319)
Go paperless today! Available online anytime, nothing to download or install.