ISBN:
1283086107
,
9789400704312
,
9781283086103
Language:
English
Pages:
Online-Ressource
,
v.: digital
Edition:
Online-Ausg. Springer eBook Collection. Humanities, Social Science and Law Electronic reproduction; Available via World Wide Web
Series Statement:
The Western Ontario Series in Philosophy of Science 76
Keywords:
Mathematics
;
Logic
;
Science Philosophy
;
Logic, Symbolic and mathematical
Abstract:
" The book ""Foundational Theories of Classical and Constructive Mathematics"" is a book on the classical topic of foundations of mathematics. Its originality resides mainly in its treating at the same time foundations of classical and foundations of constructive mathematics. This confrontation of two kinds of foundations contributes to answering questions such as: Are foundations/foundational theories of classical mathematics of a different nature compared to those of constructive mathematics? Do they play the same role for the resp. mathematics? Are there connections between the two kinds of foundational theories? etc. The confrontation and comparison is often implicit and sometimes explicit. Its great advantage is to extend the traditional discussion of the foundations of mathematics and to render it at the same time more subtle and more differentiated. Another important aspect of the book is that some of its contributions are of a more philosophical, others of a more technical nature. This double face is emphasized, since foundations of mathematics is an eminent topic in the philosophy of mathematics: hence both sides of this discipline ought to be and are being paid due to. "
Description / Table of Contents:
Introduction; Giovanni Sommaruga; References; Part I Senses of `Foundations of Mathematics'; Foundational Frameworks; Geoffrey Hellman1; 1 Introduction: Questions of Justification and Rational Reconstruction (Between Hermeneutics and Cultural Revolution); 2 Desiderata; 3 Implications: Set Theory and Category Theory; 4 Modal-Structural Mathematics and Foundations; References; The Problem of Mathematical Objects; Bob Hale; 1 Parsons on Mathematical Intuition; 1.1 Intuition of and Intuition That; 1.2 Pure Abstract and Quasi-concrete Objects; 1.3 The Language of Stroke Strings; 2 Frege's Proof
Description / Table of Contents:
3 Dummett's Objections4 Dummett's Objection Refurbished; References; Set Theory as a Foundation; Penelope Maddy; References; Foundations: Structures, Sets, and Categories; Stewart Shapiro; 1 Ontology, Maybe Even Metaphysics; 2 Epistemology: What We Know and How We (Can) Know; 3 Organizing Things; References; Part II Foundations of Classical Mathematics; From Sets to Types, to Categories, to Sets; Steve Awodey; 1 Sets to Types; 1.1 IHOL; 1.2 Semantics; 2 Types to Categories; 2.1 Topoi; 2.2 Syntactic Topos; 3 Categories to Sets; 3.1 Category of Ideals; 3.2 Basic Intuitionistic Set Theory
Description / Table of Contents:
4 Composites4.1 Sets to Categories; 4.2 Types to Sets; 4.3 Categories to Types; 5 Conclusions; References; Enriched Stratified Systems for the Foundations of Category Theory; Solomon Feferman; 1 Introduction; 2 What the Various Proposals Do and Don't Do; 3 The System NFU With Stratified Pairing; 4 First-Order Structures in NFUP; 5 Meeting Requirements (R1) and (R2) in NFUP; 6 The Requirement (R3); Type-Shifting Problems in NFUP; 7 The Requirement (R3), Continued; Building in ZFC; 8 Cantorian Classes and Extension of NFU in ZFC; References; Recent Debate over Categorical Foundations
Description / Table of Contents:
Colin McLarty1 The Founding Ideas; 2 Feferman and Rao; 3 The Differences; References; Part III Between Foundations of Classical and Foundations of Constructive Mathematics; The Axiom of Choice in the Foundations of Mathematics; John L. Bell; References; Reflections on the Categorical Foundations of Mathematics; Joachim Lambek and Philip J. Scott; 1 Introduction; 2 Type Theory; 3 Elementary Toposes; 4 Comparing Type Theories and Toposes; 5 Models and Completeness; 6 Gödel's Incompleteness Theorem; 7 Reconciling Foundations; 7.1 Constructive Nominalism; 7.2 What Is the Category of Sets?
Description / Table of Contents:
8 What Is Truth?9 Continuously Variable Sets; 10 Some Intuitionistic Principles; 11 Concluding Remarks; References; Part IV Foundations of Constructive Mathematics; Local Constructive Set Theory and Inductive Definitions; Peter Aczel; 1 Introduction; 2 Inductive Definitions in CST; 2.1 Inductive Definitions in CZF; 2.2 Inductive Definitions in CZF+; 3 The Free Version of CST; 3.1 A Free Logic; 3.2 The Axiom System CZFf; 3.3 The Axiom Systems CZFf-, CZFfI and CZFf*; 4 Local Intuitionistic Zermelo Set Theory; 5 Some Axiom Systems for Local CST; 5.1 Many-Sorted Free Logic
Description / Table of Contents:
5.2 The Axiom System LCZFf-
Note:
Includes bibliographical references
,
Electronic reproduction; Available via World Wide Web
DOI:
10.1007/978-94-007-0431-2
URL:
Volltext
(lizenzpflichtig)
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