Abstraction and Infinity.

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Cover; Abstraction and Infinity; Copyright; Dedication; Contents; Introduction; Abstraction; Infinity; Abstraction and Infinity; Acknowledgements; 1: The mathematical practice of definitions by abstraction from Euclid to Frege (and beyond); 1.1 Introduction; 1.2 Equivalence relations, invariants, and definitions by abstraction; 1.3 Mathematical practice and definitions by abstraction in classical geometry; 1.4 Definitions by abstraction in number theory, number systems, geometry, and set theory during the XIXth century; 1.4.1 Number theory; 1.4.2 Systems of Numbers and abstraction principles

1.4.3 Complex numbers and geometrical calculus1.4.4 SetTheory; 1.5 Conclusion; 2: The logical and philosophical reflection on definitions by abstraction: From Frege to the Peano school and Russell; 2.1 Frege's Grundlagen, section ; 2.1.1 The Grassmannian influence on Frege: Abstraction principles in geometry; 2.1.2 The proper conceptual order and Frege's criticism of the definition of parallels in terms of directions; 2.1.3 Aprioricity claims for the concept of direction: Schlömilch's Geometrie des Maasses; 2.1.4 The debate over Schlömilch's theory of directions

2.2 The logical discussion on definitions by abstraction2.2.1 Peano and his school; 2.2.2 Russell and Couturat; 2.2.3 Padoa on definitions by abstraction and further developments; 2.3 Conclusion; 2.4 Appendix; 3: Measuring the size of infinite collections of natural numbers: Was Cantor's theory of infinite number inevitable?; 3.1 Introduction; 3.2 Paradoxes of the infinite up to the middle ages; 3.3 Galileo and Leibniz; 3.4 Emmanuel Maignan; 3.5 Bolzano and Cantor; 3.6 Contemporary mathematical approaches tomeasuring the size of countably infinite sets

3.6.1 Katz's "Sets and their Sizes" (1981)3.6.2 A theory of numerosities; 3.7 Philosophical remarks; 3.7.1 An historiographical lesson; 3.7.2 Gödel's claim that Cantor's theory of size for infinite sets is inevitable; 3.7.3 Generalization, explanation, fruitfulness; 3.8 Conclusion; 4: In good company? On Hume's Principle and the assignment of numbers to infinite concepts; 4.1 Introduction; 4.2 Neo-logicism and Hume's Principle; 4.3 Numerosity functions: Schröder, Peano, and Bolzano; 4.4 A plethora of good abstractions; 4.5 Neo-logicism and Finite Hume's Principle

4.6 The 'good company' objection as a generalization of Heck's argument4.7 HP's good companions and the problem of cross-sortal identity; 4.8 Conclusion; 4.9 Appendix 1; 4.10 Appendix 2 ; Bibliography; Name Index

Mancosu offers an original investigation of key notions in mathematics: abstraction and infinity, and their interaction. He gives a historical analysis of the theorizing of definitions by abstraction, and explores a novel approach to measuring the size of infinite sets, showing how this leads to deep mathematical and philosophical problems.

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