Lectures on the Philosophy of Mathematics

Description

An introduction to the philosophy of mathematics grounded in mathematics and motivated by mathematical inquiry and practice. In this book, Joel David Hamkins offers an introduction to the philosophy of mathematics that is grounded in mathematics and motivated by mathematical inquiry and practice. He treats philosophical issues as they arise organically in mathematics, discussing such topics as platonism, realism, logicism, structuralism, formalism, infinity, and intuitionism in mathematical contexts. He organizes the book by mathematical themes--numbers, rigor, geometry, proof, computability, incompleteness, and set theory--that give rise again and again to philosophical considerations. Title Page Copyright Dedication Table of Contents Preface About the Author 1. Numbers 1.1. Numbers versus numerals 1.2. Number systems Natural numbers Integers Rational numbers 1.3. Incommensurable numbers An alternative geometric argument 1.4. Platonism Plenitudinous platonism 1.5. Logicism Equinumerosity The Cantor-Hume principle The Julius Caesar problem Numbers as equinumerosity classes Neologicism 1.6. Interpreting arithmetic Numbers as equinumerosity classes Numbers as sets Numbers as primitives Numbers as morphisms Numbers as games Junk theorems Interpretation of theories 1.7. What numbers could not be The epistemological problem 1.8. Dedekind arithmetic Arithmetic categoricity 1.9. Mathematical induction Fundamental theorem of arithmetic Infinitude of primes 1.10. Structuralism Definability versus Leibnizian structure Role of identity in the formal language Isomorphism orbit Categoricity Structuralism in mathematical practice Eliminative structuralism Abstract structuralism 1.11. What is a real number? Dedekind cuts Theft and honest toil Cauchy real numbers Real numbers as geometric continuum Categoricity for the real numbers Categoricity for the real continuum 1.12. Transcendental numbers The transcendence game 1.13. Complex numbers Platonism for complex numbers Categoricity for the complex field A complex challenge for structuralism? Structure as reduct of rigid structure 1.14. Contemporary type theory 1.15. More numbers 1.16. What is a philosophy for? 1.17. Finally, what is a number? Questions for further thought Further reading Credits 2. Rigor 2.1. Continuity Informal account of continuity The definition of continuity The continuity game Estimation in analysis Limits 2.2. Instantaneous change Infinitesimals Modern definition of the derivative 2.3. An enlarged vocabulary of concepts 2.4. The least-upper-bound principle Consequences of completeness Continuous induction 2.5. Indispensability of mathematics Science without numbers Fictionalism The theory/metatheory distinction 2.6. Abstraction in the function concept The Devil’s staircase Space-filling curves Conway base-13 function 2.7. Infinitesimals revisited Nonstandard analysis and the hyperreal numbers Calculus in nonstandard analysis Classical model-construction perspective Axiomatic approach “The” hyperreal numbers? Radical nonstandardness perspective Translating between nonstandard and classical perspectives Criticism of nonstandard analysis Questions for further thought Further reading Credits 3. Infinity 3.1. Hilbert’s Grand Hotel Hilbert’s bus Hilbert’s train 3.2. Countable sets 3.3. Equinumerosity 3.4. Hilbert’s half-marathon Cantor’s cruise ship 3.5. Uncountability Cantor’s original argument Mathematical cranks 3.6. Cantor on transcendental numbers Constructive versus nonconstructive arguments 3.7. On the number of subsets of a set On the number of infinities Russell on the number of propositions On the number of possible committees The diary of Tristram Shandy The cartographer’s paradox The Library of Babel On the number of possible books 3.8. Beyond equinumerosity to the comparative size principle Place focus on reflexive preorders 3.9. What is Cantor’s continuum hypothesis? 3.10. Transfinite cardinals—the alephs and the beths Lewis on the number of objects and properties 3.11. Zeno’s paradox Actual versus potential infinity 3.12. How to count Questions for further thought Further reading Credits 4. Geometry 4.1. Geometric constructions Contemporary approach via symmetries Collapsible compasses Constructible points and the constructible plane Constructible numbers and the number line 4.2. Nonconstructible numbers Doubling the cube Trisecting the angle Squaring the circle Circle-squarers and angle-trisectors 4.3. Alternative tool sets Compass-only constructibility Straightedge-only constructibility Construction with a marked ruler Origami constructibility Spirograph constructibility 4.4. The ontology of geometry 4.5. The role of diagrams and figures Kant Hume on arithmetic reasoning over geometry Manders on diagrammatic proof Contemporary tools How to lie with figures Error and approximation in geometric construction Constructing a perspective chessboard 4.6. Non-Euclidean geometry Spherical geometry Elliptical geometry Hyperbolic geometry Curvature of space 4.7. Errors in Euclid? Implicit continuity assumptions The missing concept of “between” Hilbert’s geometry Tarski’s geometry 4.8. Geometry and physical space 4.9. Poincaré on the nature of geometry 4.10. Tarski on the decidability of geometry Questions for further thought Further reading Credits 5. Proof 5.1. Syntax-semantics distinction Use/mention 5.2. What is proof? Proof as dialogue Wittgenstein Thurston Formalization and mathematical error Formalization as a sharpening of mathematical ideas Mathematics does not take place in a formal language Voevodsky Proofs without words How to lie with figures Hard arguments versus soft Moral mathematical truth 5.3. Formal proof and proof theory Soundness Completeness Compactness Verifiability Sound and verifiable, yet incomplete Complete and verifiable, yet unsound Sound and complete, yet unverifiable The empty structure Formal deduction examples The value of formal deduction 5.4. Automated theorem proving and proof verification Four-color theorem Choice of formal system 5.5. Completeness theorem 5.6. Nonclassical logics Classical versus intuitionistic validity Informal versus formal use of “constructive” Epistemological intrusion into ontology No unbridgeable chasm Logical pluralism Classical and intuitionistic realms 5.7. Conclusion Questions for further thought Further reading Credits 6. Computability 6.1. Primitive recursion Implementing logic in primitive recursion Diagonalizing out of primitive recursion The Ackermann function 6.2. Turing on computability Turing machines Partiality is inherent in computability Examples of Turing-machine programs Decidability versus enumerability Universal computer “Stronger” Turing machines Other models of computatibility 6.3. Computational power: Hierarchy or threshold? The hierarchy vision The threshold vision Which vision is correct? 6.4. Church-Turing thesis Computation in the physical universe 6.5. Undecidability The halting problem Other undecidable problems The tiling problem Computable decidability versus enumerability 6.6. Computable numbers 6.7. Oracle computation and the Turing degrees 6.8. Complexity theory Feasibility as polynomial-time computation Worst-case versus average-case complexity The black-hole phenomenon Decidability versus verifiability Nondeterministic computation P versus NP Computational resiliency Questions for further thought Further reading 7. Incompleteness 7.1. The Hilbert program Formalism Life in the world imagined by Hilbert The alternative Which vision is correct? 7.2. The first incompleteness theorem The first incompleteness theorem, via computability The Entscheidungsproblem Incompleteness, via diophantine equations Arithmetization First incompleteness theorem, via Gödel sentence 7.3. Second incompleteness theorem Löb proof conditions Provability logic 7.4. Gödel-Rosser incompleteness theorem 7.5. Tarski’s theorem on the nondefinability of truth 7.6. Feferman theories 7.7. Ubiquity of independence Tower of consistency strength 7.8. Reverse mathematics 7.9. Goodstein’s theorem 7.10. Löb’s theorem 7.11. Two kinds of undecidability Questions for further thought Further reading 8. Set Theory 8.1. Cantor-Bendixson theorem 8.2. Set theory as a foundation of mathematics 8.3. General comprehension principle Frege’s Basic Law V 8.4. Cumulative hierarchy 8.5. Separation axiom Ill-founded hierarchies Impredicativity 8.6. Extensionality Other axioms 8.7. Replacement axiom The number of infinities 8.8. The axiom of choice and the well-order theorem Paradoxical consequences of AC Paradox without AC Solovay’s dream world for analysis 8.9. Large cardinals Strong limit cardinals Regular cardinals Aleph-fixed-point cardinals Inaccessible and hyperinaccessible cardinals Linearity of the large cardinal hierarchy Large cardinals consequences down low 8.10. Continuum hypothesis Pervasive independence phenomenon 8.11. Universe view Categoricity and rigidity of the set-theoretic universe 8.12. Criterion for new axioms Intrinsic justification Extrinsic justification What is an axiom? 8.13. Does mathematics need new axioms? Absolutely undecidable questions Strong versus weak foundations Shelah Feferman 8.14. Multiverse view Dream solution of the continuum hypothesis Analogy with geometry Pluralism as set-theoretic skepticism? Plural platonism Theory/metatheory interaction in set theory Summary Questions for further thought Further reading Credits Bibliography Notation Index Subject Index