Founding Mathematics on Semantic Conventions (Synthese Library, 446)

✍ Scribed by Casper Storm Hansen

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 259
• Category: Library

✦ Synopsis

This book presents a new nominalistic philosophy of mathematics: semantic conventionalism. Its central thesis is that mathematics should be founded on the human ability to create language – and specifically, the ability to institute conventions for the truth conditions of sentences.

This philosophical stance leads to an alternative way of practicing mathematics: instead of “building” objects out of sets, a mathematician should introduce new syntactical sentence types, together with their truth conditions, as he or she develops a theory.

Semantic conventionalism is justified first through criticism of Cantorian set theory, intuitionism, logicism, and predicativism; then on its own terms; and finally, exemplified by a detailed reconstruction of arithmetic and real analysis.

Also included is a simple solution to the liar paradox and the other paradoxes that have traditionally been recognized as semantic. And since it is argued that mathematics is semantics, thissolution also applies to Russell’s paradox and the other mathematical paradoxes of self-reference.

In addition to philosophers who care about the metaphysics and epistemology of mathematics or the paradoxes of self-reference, this book should appeal to mathematicians interested in alternative approaches.

✦ Table of Contents

Founding Mathematics on Semantic Conventions
Preface
Contents
1 Introduction
1.1 Overview and Guide to Partial Reading
2 Classical Mathematics and Plenitudinous Combinatorialism
2.1 Large Cardinal Axioms and Theorems of Arithmetic
2.2 Transfinite Ordinals
2.3 Transfinite Cardinals
2.4 The Continuum Hypothesis
3 Intuitionism and Choice Sequences
3.1 General Introduction
3.2 Brouwer on Freely Proceeding Choice Sequences
3.3 Constitution of Free Choice Sequences
3.4 Evaluation of Brouwer's Claim
3.5 Verificationism and Intuitionistic Logic
4 From Logicism to Predicativism
4.1 Frege
4.2 Russell
4.3 Weyl
4.4 Weyl's Failure to Include All Real Numbers
5 Conventional Truth
5.1 The Obvious Solution to the Liar Paradox
5.2 Conventional Truth Conditions
5.3 The Dogma
5.4 Possible Language Conventions
5.5 T-schemas and Expressive Strength
5.6 Dialectical Situation
5.7 The View from Nowhere
5.8 Comparison with Chihara's Position
5.9 Revenge
6 Semantic Conventionalism for Mathematics
6.1 Needs Assessment
6.2 Simple Arithmetic as a Conventional Language
6.3 Quine's Anti-Conventionalism
6.4 Rule-Following
6.5 Choice of Logic
7 A Convention for a Type-free Language
7.1 The Kripke Convention and Its Shortcomings
7.2 Reformulating the Kripke Convention
7.2.1 Collapsing Truth and Satisfaction of View-From-Nowhere Truth Conditions
7.2.2 Kleenification
7.2.3 Kripke Recursion
7.3 Adding a Conditional with Supervaluational Semantics
7.3.1 Supervaluation over All Possibilities
7.3.2 View-From-Nowhere Truth Conditions for the Strong Conditional
7.3.3 If the Supervaluation Criterion is Not Satisfied
7.3.4 Ensuring Quantification over All Possibilities in the Presence of Supervaluation
7.3.5 Iteration of the Strong Conditional
7.3.6 Summary
7.4 Denoting Terms for Applied Mathematics
7.5 Meta-Theorems
8 Basic Mathematics
8.1 Logic
8.2 Natural Numbers
8.3 Integers
8.4 Rational Numbers
8.5 Classicality So Far
8.6 Classes
8.7 An Example of Applied Mathematics
9 Real Analysis
9.1 Functions
9.2 Real Numbers
9.3 Exponentiation
9.4 Completeness
9.5 Suprema, Infima, and Roots
9.6 Continuity
9.7 Operations on Functions
9.8 Differentiation
9.8.1 Calculating Derivatives
9.8.2 Uniform Differentiability
9.9 Integration
9.10 Unbounded Intervals and Piecewise Continuity
9.11 Completifications of Functions Generalized
9.12 Another Example of Applied Mathematics
9.13 Diagonalization
10 Possibility
10.1 All Possible Real Numbers
10.2 Modal Metaphysics
10.3 Conclusion
References
Index of Symbols
General Index