Founding Mathematics on Semantic Conventions

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, this solution 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.

Auteur
Casper Storm Hansen is an Associate Professor at the Institute of Philosophy, Chinese Academy of Sciences, and has a background in both philosophy and mathematics from the universities of Copenhagen, Amsterdam, and Aberdeen. In addition to the philosophy of mathematics and the semantic paradoxes, he works on formal epistemology, decision theory, and formal semantics.

Contenu

1. Introduction1.1 Overview and Guide to Partial Reading
2. Classical Mathematics and Plenitudinous Combinatorialism2.1 Large Cardinal Axioms and Theorems of Arithmetic2.2 Transfinite Ordinals2.3 Transfinite Cardinals2.4 The Continuum Hypothesis
   3 Intuitionism and Choice Sequences3.1 General Introduction3.2 Brouwer on Freely Proceeding Choice Sequences3.3 Constitution of Free Choice Sequences3.4 Evaluation of Brouwer's Claim3.5 Verificationism and Intuitionistic Logic
3. From Logicism to Predicativism4.1 Frege4.2 Russell4.3 Weyl4.4 Weyl's Failure to Include All Real Numbers
4. Conventional Truth5.1 The Obvious Solution to the Liar Paradox5.2 Conventional Truth Conditions5.3 The Dogma5.4 Possible Language Conventions5.5 T-schemas and Expressive Strength5.6 Dialectical Situation5.7 The View from Nowhere5.8 Comparison with Chihara's Position5.9 Revenge
5. Semantic Conventionalism for Mathematics6.1 Needs Assessment6.2 Simple Arithmetic as a Conventional Language6.3 Quine's Anti-Conventionalism6.4 Rule-Following6.5 Choice of Logic
6. A Convention for a Type-free Language7.1 The Kripke Convention and Its Shortcomings7.2 Reformulating the Kripke Convention7.3 Adding a Conditional with Supervaluational Semantics7.4 Denoting Terms for Applied Mathematics7.5 Meta-Theorems
7. Basic Mathematics8.1 Logic8.2 Natural Numbers8.3 Integers8.4 Rational Numbers8.5 Classicality So Far8.6 Classes8.7 An Example of Applied Mathematics
8. Real Analysis9.1 Functions9.2 Real Numbers9.3 Exponentiation9.4 Completeness9.5 Suprema, Infima, and Roots9.6 Continuity9.7 Operations on Function9.8 Differentiation9.9 Integration9.10 Unbounded Intervals and Piecewise Continuity9.11 Completifications of Functions Generalized9.12 Another Example of Applied Mathematics9.13 Diagonalization
9. Possibility10.1 All Possible Real Numbers10.2 Modal Metaphysics10.3 Conclusion
   ReferencesIndex of symbolsGeneral index