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Author Benis-Sinaceur, Hourya, author

Title Functions and generality of logic : reflections on Dedekind's and Frege's logicisms / Hourya Benis-Sinaceur, Marco Panza, Gabriel Sandu
Published Cham : Springer, 2015
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Description 1 online resource
Series Logic, epistemology and the unity of science, 2214-9783 ; volume 37
Logic, epistemology and the unity of science ; v. 37
Contents 880-01 Chapter 1: Is Dedekind a logicist?; Hourya Benis Sinaceur -- Chapter 2: Functions and Expressions; Marco Panza -- Chapter 3: Frege, Russell, Ramsey on arbitrary functions; Gabriel Sandu
880-01/(S Machine generated contents note: 1. Is Dedekind a LogicistWhy Does Such a Question Arise/ Hourya Benis-Sinaceur -- 1.1. Introduction -- 1.2. Logicist Thesis -- 1.2.1. "New Logic" -- 1.2.2. Logicist Foundations of Mathematics -- 1.3. Similar Claims, Different Fundamental Conceptions -- 1.3.1. Reason Versus Intuition and the Foundations of Arithmetic -- 1.3.2. Pure Thought, Objectivity, Logic, Proof -- 1.3.3. More on Inference: Truths and Logical Truths -- 1.4. On Definitions -- 1.4.1. Dedekind's Definition by Axioms -- 1.4.2. Frege's Ontological Conception of Definitions of Objects -- 1.4.3. Frege's Epistemology -- 1.4.4. Dedekind's Treppen-Verstand and Stuckeweise Definitions -- 1.4.5. Frege's Criticism of Dedekind's Stuckweise and Creative Definitions -- 1.4.6. Frege's Technical Conception of Definitions -- 1.4.7. Frege's and Dedekind's Philosophical Assumptions -- 1.5. System and Abbildung: Structuralism and/or Logicism -- 1.5.1. Concept -- 1.5.2. System and Abbildung: the Search of Generality -- 1.5.3. Dedekind's Chains and Frege's Following in a φ-Sequence -- 1.6. Conclusions -- 2. From Lagrange to Frege: Functions and Expressions / Marco Panza -- 2.1. Introduction -- 2.2. Lagrange's Notion of a Function -- 2.3. Arbitrary Functions and the Arithmetisation of Analysis -- 2.4. Functions in Frege's Grundgesetze -- 2.4.1. Elucidating the Notion of a Function -- 2.4.2. How (First-Level) Functions Work in the Begriffsschrift -- 2.4.3. (First-level) Functions and Names of Functions -- 2.4.4. Compositionality of Functions, Higher-Level Functions, and the Notion of an Arbitrary Function -- 2.5. Concluding Remarks -- 3. Frege, Russell, Ramsey and the Notion of an Arbitrary Function / Gabriel Sandu -- 3.1. Background -- 3.2. Standard versus Non-standard Distinction and Dedekind Theorem -- 3.3. Isomorphism Theorem -- 3.4. Ramsey's Notion of a Predicative Function in "Foundations of Mathematics" -- 3.5. Ramsey's Reduction of Type (2) -- 3.5.1. Logical Necessity versus Analytical Necessity -- 3.5.2. Ramsey's Propositional Functions in Extension -- 3.5.3. Sullivan's Objection to the Notion of Propositional Function in Extension: Containment -- 3.5.4. Substitution -- 3.5.5. Arbitrary Functions -- 3.6. Conclusion
Summary This book examines three connected aspects of Frege's logicism: the differences between Dedekind's and Frege's interpretation of the term 'logic' and related terms and reflects on Frege's notion of function, comparing its understanding and the role it played in Frege's and Lagrange's foundational programs. It concludes with an examination of the notion of arbitrary function, taking into account Frege's, Ramsey's and Russell's view on the subject. Composed of three chapters, this book sheds light on important aspects of Dedekind's and Frege's logicisms. The first chapter explains how, although he shares Frege's aim at substituting logical standards of rigor to intuitive imports from spatio-temporal experience into the deductive presentation of arithmetic, Dedekind had a different goal and used or invented different tools. The chapter highlights basic dissimilarities between Dedekind's and Frege's actual ways of doing and thinking. The second chapter reflects on Frege's notion of a function, in comparison with the notions endorsed by Lagrange and the followers of the program of arithmetization of analysis. It remarks that the foundational programs pursued by Lagrange and Frege are crucially different and based on a different idea of what the foundations of mathematics should be like. However, despite this contrast, the notion of function plays similar roles in the two programs, and this chapter emphasizes the similarities. The third chapter traces the development of thinking about Frege's program in the foundations of mathematics, and includes comparisons of Frege's, Russell's and Ramsey's views. The chapter discusses earlier papers written by Hintikka, Sandu, Demopoulos and Trueman. Although the chapter's main focus is on the notion of arbitrary correlation, it starts out by discussing some aspects of the connection between this notion and Dedekind Theorem
Bibliography Includes bibliographical references
Notes Online resource; title from PDF title page (Ebsco, viewed June 29, 2015)
Subject Cantor, Georg, 1845-1918.
Frege, Gottlob, 1848-1925.
Algebraic functions.
Algebraic logic.
Logic, Symbolic and mathematical.
Form Electronic book
Author Panza, Marco, author
Sandu, Gabriel, author
ISBN 3319171089 (print)
3319171097 (electronic bk.)
9783319171081 (print)
9783319171098 (electronic bk.)