- Title Pages
- Dedication
- Preface
- Introduction
- I.1 Descartes's first philosophy
- I.2 Neo‐cartesian skepticism
- I.3 Hume's naturalism
- I.4 Kant's transcendentalism
- I.5 Carnap's rational reconstruction
- I.6 Quine's naturalism
- I.7 Putnam's anti‐naturalism
- II.1 What's left to do?
- II.2 An illustration: truth and reference
- II.3 Reconfiguring the debate
- II.4 Disquotation
- II.5 Minimalism
- II.6 Correlation
- III.1 Naturalistic options
- III.2 Kant on logic
- III.3 Undoing the Copernican revolution<sup>1</sup>
- III.4 The logical structure of the world
- III.5 The logical structure of cognition
- III.6 The status of rudimentary logic
- III.7 From rudimentary to classical logic
- III.8 Caveats
- IV.1 Second philosophy of science
- IV.2 Mathematics in application
- IV.3 Second methodology of mathematics
- IV.4 Second philosophy of mathematics
- IV.5 Second metaphysics
- References
- Index

# Second methodology of mathematics

# Second methodology of mathematics

- Chapter:
- (p.344) IV.3 Second methodology of mathematics
- Source:
- Second Philosophy
- Author(s):
### Penelope Maddy (Contributor Webpage)

- Publisher:
- Oxford University Press

Given that natural science no longer dictates the course of mathematical development and that no curtailment of the free flowering of pure mathematics seems prudent, what are the appropriate methods to use? This chapter argues that those methods are not properly based in extra-mathematical metaphysics, that internal mathematical ends and values should carry the day. The various branches of mathematics aim at different goals, which explains why set theorists strive for a single unified theory of sets while geometers and algebraists embrace a variety of structures. Set theory's foundational role requires that its fundamental axioms form a unified list and that they be as generous as possible. These desiderata count against an axiom candidate like the Axiom of Constructibility (V=L).

*Keywords:*
abstract algebra, Axiom of Constructibility, geometry, mathematical methods, pure mathematics, set theory

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- Title Pages
- Dedication
- Preface
- Introduction
- I.1 Descartes's first philosophy
- I.2 Neo‐cartesian skepticism
- I.3 Hume's naturalism
- I.4 Kant's transcendentalism
- I.5 Carnap's rational reconstruction
- I.6 Quine's naturalism
- I.7 Putnam's anti‐naturalism
- II.1 What's left to do?
- II.2 An illustration: truth and reference
- II.3 Reconfiguring the debate
- II.4 Disquotation
- II.5 Minimalism
- II.6 Correlation
- III.1 Naturalistic options
- III.2 Kant on logic
- III.3 Undoing the Copernican revolution<sup>1</sup>
- III.4 The logical structure of the world
- III.5 The logical structure of cognition
- III.6 The status of rudimentary logic
- III.7 From rudimentary to classical logic
- III.8 Caveats
- IV.1 Second philosophy of science
- IV.2 Mathematics in application
- IV.3 Second methodology of mathematics
- IV.4 Second philosophy of mathematics
- IV.5 Second metaphysics
- References
- Index