Lectures on the Philosophy of Mathematics by Joel David Hamkins -- Read -- Imperial Library of Trantor

Index

Cover 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

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