Reason's nearest kin : philosophies of arithmetic from Kant to Carnap / Michael Potter.

Format:

Book

Author/Creator:

Potter, Michael D.

Language:

English

Subjects (All):

Arithmetic--Philosophy.

Arithmetic.

Mathematics--Philosophy.

Mathematics.

Physical Description:

x, 305 p.

Edition:

1st ed.

Other Title:

Philosophies of arithmetic from Kant to Carnap

Place of Publication:

Oxford ; New York : Oxford University Press, 2000.

Summary:

Reason's Nearest Kin is a critical examination of the most exciting period there has been in the philosophical study of the properties of the natural numbers, from the 1880s to the 1930s. Reassessing the brilliant innovations of Frege, Russell, Wittgenstein, and others, which transformed philosophy as well as our understanding of mathematics, Michael Potter places arithmetic at the interface between experience, language, thought, and the world.

Contents:

Intro

Contents

Introduction

0.1 Arithmetic

0.2 The a priori

0.3 Empiricism

0.4 Psychologism

0.5 Pure formalism

0.6 Trivial formalism

0.7 Reflexive formalism

0.8 Arithmetic and reason

1 Kant

1.1 Intuitions and concepts

1.2 Geometrical propositions

1.3 Arithmetical propositions

1.4 The Transcendental Deduction

1.5 Analytic and synthetic

1.6 The principle of analytic judgements

1.7 Geometry is not analytic

1.8 Arithmetic is not analytic

1.9 The principle of synthetic judgements

1.10 Geometry as synthetic

1.11 Arithmetic as synthetic

1.12 Arithmetic and sensibility

2 Grundlagen

2.1 Axiomatization

2.2 Arithmetic independent of sensibility

2.3 The Begriffsschrift

2.4 Frege's conception of analyticity

2.5 Numerically definite quantifiers

2.6 The numerical equivalence

2.7 Frege's explicit definition

2.8 The context principle again

2.9 The analyticity of the numerical equivalence

3 Dedekind

3.1 Dedekind's recursion theorem

3.2 Frege and Dedekind

3.3 Axiomatic structuralism

3.4 Existence

3.5 Uniqueness

3.6 Implicationism

3.7 Systems

3.8 Dedekind on existence

3.9 Dedekind on uniqueness

4 Frege's account of classes

4.1 The Julius Caesar problem yet again

4.2 The context principle in Grundgesetze

4.3 Russell's paradox

4.4 Numbers as concepts

4.5 The status of the numerical equivalence

5 Russell's account of classes

5.1 Propositions

5.2 The old theory of denoting

5.3 The new theory of denoting

5.4 The substitutional theory

5.5 Russell's propositional paradox

5.6 Frege's hierarchy of senses

5.7 Mathematical logic as based on the theory of types

5.8 Elementary propositions

5.9 The hierarchy of propositional functions in * 12.

5.10 The hierarchy of propositional functions in the Introduction

5.11 Typical ambiguity

5.12 Cumulative types

5.13 The hierarchy of classes

5.14 Numbers

5.15 The axiom of reducibility

5.16 Propositional functions and reducibility

5.17 The regressive method

5.18 The Introduction to Mathematical Philosophy

6 TheTractatus

6.1 Sign and symbol

6.2 The hierarchy of types

6.3 The doctrine of inexpressibility

6.4 Operations and functions

6.5 Sense

6.6 The rejection of class-theoretic foundations for mathematics

6.7 Number as the exponent of an operation

6.8 The adjectival strategy

6.9 Equations

6.10 Numerical identities

6.11 Generalization

6.12 The axiom of infinity

6.13 A transcendental argument

6.14 Another transcendental argument

7 The second edition of Principia

7.1 Logical atomism and empiricism

7.2 The hierarchy of propositional functions

7.3 Mathematical induction

7.4 The definition of identity

8 Ramsey

8.1 Propositions

8.2 Predicating functions

8.3 Extending Wittgenstein's account of identity

8.4 Propositional functions in extension

8.5 Wittgenstein's objections

8.6 The axiom of infinity

9 Hilbert's programme

9.1 Formal consistency

9.2 Real arithmetic

9.3 Schematic arithmetic

9.4 Ideal arithmetic

9.5 Metamathematics

9.6 Hilbert's programme

10 Gödel

10.1 Incompleteness

10.2 Formal theories

10.3 The unprovability of outer consistency

10.4 The demise of Hilbert's programme

10.5 The unprovability of consistency

10.6 Axiomatic formalism

11 Carnap

11.1 Language and symbolism

11.2 The rejection of the Tractatus

11.3 Conventionalism

11.4 Completeness

11.5 Consistency

11.6 Semantics

11.7 Pragmatics

Conclusion

Bibliography

Index

A

B

C

D

E

F

G

H

I

J

K.

L

M

N

O

P

Q

R

S

T

U

V

W

Z.

Notes:

Includes bibliographical references (p. [290]-298) and index.

Description based on metadata supplied by the publisher and other sources.

Description based on publisher supplied metadata and other sources.

ISBN:

9780191520228

0191520225

OCLC:

309340984