Lectures on the philosophy of mathematics / Joel David Hamkins.

Format:

Book

Author/Creator:

Hamkins, Joel David, author.

Language:

English

Subjects (All):

Mathematics--Philosophy.

Mathematics.

Physical Description:

xviii, 329 pages : illustrations (chiefly color) ; 23 cm

Place of Publication:

Cambridge, Massachusetts : The MIT Press, [2020]

Summary:

"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. Hamkins shows, for example, how number systems set the stage for discussions of such philosophical issues as platonism, logicism, and the nature of abstraction. Consideration of the rise of rigor in the calculus leads to a discussion of whether the indispensability of mathematics in science offers grounds for mathematical truth. Sophisticated technical developments in set theory give rise to a necessary engagement with deep philosophical concerns, including the criteria for new mathematical axioms. Throughout, Hamkins offers a clear and engaging exposition that is both accessible and sophisticated, intended for readers whose mathematical backgrounds range from novice to expert"-- Provided by publisher.

Contents:

Machine generated contents note: 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 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

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

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

7. Incompleteness

7.1. The Hilbert program

Formalism

Life in the world imagined by Hilbert

The alternative

7.2. The first incompleteness theorem

The first incompleteness theorem, via computability

The Entscheidungsproblem

Incompleteness, via diophantine equations

Arithmetization

First incompleteness theorem, via Godel sentence

7.3. Second incompleteness theorem

Lob proof conditions

Provability logic

7.4. Godel-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. Lob's theorem

7.11. Two kinds of undecidability

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

III-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

Credits.

Notes:

Includes bibliographical references and indexes.

ISBN:

9780262542234

0262542234

OCLC:

1154072847

Publisher Number:

99987600193