Gödel, Tarski and the Lure of Natural Language: Logical Entanglement, Formalism Freeness
Cambridge University Press, 12/17/2020
EAN 9781107012578, ISBN10: 1107012570
Hardcover, 220 pages, 22.9 x 15.2 x 1.9 cm
Language: English
Originally published in English
Is mathematics 'entangled' with its various formalisations? Or are the central concepts of mathematics largely insensitive to formalisation, or 'formalism free'? What is the semantic point of view and how is it implemented in foundational practice? Does a given semantic framework always have an implicit syntax? Inspired by what she calls the 'natural language moves' of Gödel and Tarski, Juliette Kennedy considers what roles the concepts of 'entanglement' and 'formalism freeness' play in a range of logical settings, from computability and set theory to model theory and second order logic, to logicality, developing an entirely original philosophy of mathematics along the way. The treatment is historically, logically and set-theoretically rich, and topics such as naturalism and foundations receive their due, but now with a new twist.
1. Introduction
1.1 The Syntax/Semantics Distinction
1.2 Our Logical Pluralism
1.3 Formal vs Linguistic Semantics
2. Formalism Freeness and Entanglement
Definitions
2.1 Precedents
2.2 Entanglement and Formalism Freeness
Varieties
2.3 A Simple Preference for Semantic Methods?
3. Computability
the Primary Example
3.1 On Adequacy
3.2 Different Notions of Computability Emerge in the 1930s
3.3 The 'Scope Problem'
3.4 Turing's Analysis of Computability
3.5 Gödel's Reaction to Turing's Work at the Time
3.6 Coda
a Word About Deviant Encodings
4. Gödel and Formalism Independence
4.1 Gödel on Formalism
4.2 Episodes of Formalism Independence in Gödel's Writings
4.3 Gödel's Princeton Bicentennial Lecture
4.4 Implementation
4.5 Logical Autonomy?
5. Tarski and 'the Mathematical'
5.1 'The Mathematical', Definable Sets of Reals, and Naïve Set Theory
5.2 Tarski's Naturalism
5.3 Squeezing First Order Definability
5.4 Tarski and Logicality
5.5 In Sum
Parataxis
5.6 Coda
an Improvement of McGee's Theorem
6. Model Theoretic Aspects
6.1 Abstract Elementary Classes
6.2 Patchwork Foundations, On-Again-Off-Again-Sim and Implicit Syntax
6.3 Implicit Syntax, Implicit Logic
6.4 A Remark on Set Theory
6.5 Symbiosis
6.6 Coda
Symbiosis in Detail
7. On the Side of Natural Language.