Abstraction and Infinity by Paolo MancosuAbstraction and Infinity by Paolo Mancosu

Abstraction and Infinity

byPaolo Mancosu

Hardcover | January 14, 2017

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Paolo Mancosu provides an original investigation of historical and systematic aspects of the notions of abstraction and infinity and their interaction. A familiar way of introducing concepts in mathematics rests on so-called definitions by abstraction. An example of this is Hume's Principle,which introduces the concept of number by stating that two concepts have the same number if and only if the objects falling under each one of them can be put in one-one correspondence. This principle is at the core of neo-logicism. In the first two chapters of the book, Mancosu provides a historical analysis of the mathematical uses and foundational discussion of definitions by abstraction up to Frege, Peano, and Russell. Chapter one shows that abstraction principles were quite widespread in the mathematical practice thatpreceded Frege's discussion of them and the second chapter provides the first contextual analysis of Frege's discussion of abstraction principles in section 64 of the Grundlagen. In the second part of the book, Mancosu discusses a novel approach to measuring the size of infinite sets known as thetheory of numerosities and shows how this new development leads to deep mathematical, historical, and philosophical problems. The final chapter of the book explore how this theory of numerosities can be exploited to provide surprisingly novel perspectives on neo-logicism.
Paolo Mancosu is Willis S. and Marion Slusser Professor of Philosophy at the University of California at Berkeley. He is the author of numerous articles and books in logic and philosophy of mathematics. He is also the author of Inside the Zhivago Storm: The editorial adventures of Pasternak's masterpiece (Feltrinelli, Milan, 2013). Du...
Title:Abstraction and InfinityFormat:HardcoverDimensions:240 pages, 9.21 × 6.02 × 0 inPublished:January 14, 2017Publisher:Oxford University PressLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0198746822

ISBN - 13:9780198746829

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Table of Contents

Introduction1. The mathematical practice of definitions by abstraction from Euclid to Frege (and beyond)2. The logical and philosophical reflection on definitions by abstraction: From Frege to the Peano school and Russell3. Measuring the size of infnite collections of natural numbers: Was Cantor's theory of infinite number inevitable?4. In good company? On Hume's Principle and the assignment of numbers to infinite concepts