Overview

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 that preceded 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 the theory 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.

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9780198746829

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Reviews

This is a very rich and rewarding book. It provides not merely a much-needed, detailed account of the history of so called definitions by abstraction or abstraction principles, but also advances the current philosophical debate about the foundational status of such principles. These themes are presented in a beautifully clear prose which makes this book, quite simply, a joy to read - not something that is easily accomplished by a book in the history and philosophy of mathematics ... I highly recommend Mancosu's book to philosophers and mathematicians interested in the philosophy or the history of mathematics and logic. ... Mancosu not only proves to be one of the great detectives of the history of mathematical practice, but shows us how an historical approach to mathematical practice can, and in this case, does successfully move forward our current debates in the philosophy of mathematics. * Notre Dame Philosophical Reviews * Paolo Mancosu's Abstraction and Infinity expertly straddles the history and philosophy of mathematics. ... Mancosu's work is well written and has been well received by those involved with neo-logicism. But it also offers readers not engaged in such debates an opportunity to reflect on the fact that of two intuitively true principles for comparing finite collections (one-to-one correspondence and part-whole inclusion), the historic choice of equinumerosity by Cantor for investigating infinite collections was and is not as cut-and-dried as some believe. And, of course, the book provides a valuable scholarly account of the history of definition by abstraction, the use of equivalence relations, and notions of infinity. * Calvin Jongsma, Mathematical Association of America Reviews * Mancosu's book is valuable for its original insights into abstraction principles and its exceptionally scholarly presentation of the history of both abstraction and infinity. His facility with the history of mathematics is unparalleled and this work is rich in sources, often in original translations of German, Latin, or Italian that are otherwise unavailable in English. * Paolo Mancosu, MathSciNet *

This is a very rich and rewarding book. It provides not merely a much-needed, detailed account of the history of so called definitions by abstraction or abstraction principles, but also advances the current philosophical debate about the foundational status of such principles. These themes are presented in a beautifully clear prose which makes this book, quite simply, a joy to read - not something that is easily accomplished by a book in the history and philosophy of mathematics ... I highly recommend Mancosu's book to philosophers and mathematicians interested in the philosophy or the history of mathematics and logic. ... Mancosu not only proves to be one of the great detectives of the history of mathematical practice, but shows us how an historical approach to mathematical practice can, and in this case, does successfully move forward our current debates in the philosophy of mathematics. * Notre Dame Philosophical Reviews *

This is a very rich and rewarding book. It provides not merely a much-needed, detailed account of the history of so called definitions by abstraction or abstraction principles, but also advances the current philosophical debate about the foundational status of such principles. These themes are presented in a beautifully clear prose which makes this book, quite simply, a joy to read - not something that is easily accomplished by a book in the history and philosophy of mathematics ... I highly recommend Mancosu's book to philosophers and mathematicians interested in the philosophy or the history of mathematics and logic. ... Mancosu not only proves to be one of the great detectives of the history of mathematical practice, but shows us how an historical approach to mathematical practice can, and in this case, does successfully move forward our current debates in the philosophy of mathematics. * Notre Dame Philosophical Reviews * Paolo Mancosu's Abstraction and Infinity expertly straddles the history and philosophy of mathematics. ... Mancosu's work is well written and has been well received by those involved with neo-logicism. But it also offers readers not engaged in such debates an opportunity to reflect on the fact that of two intuitively true principles for comparing finite collections (one-to-one correspondence and part-whole inclusion), the historic choice of equinumerosity by Cantor for investigating infinite collections was and is not as cut-and-dried as some believe. And, of course, the book provides a valuable scholarly account of the history of definition by abstraction, the use of equivalence relations, and notions of infinity. * Calvin Jongsma, Mathematical Association of America Reviews *

Author Information

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). During his career he has taught at Stanford, Oxford, and Yale. He has been a fellow of the Humboldt Stiftung, the Wissenschaftskolleg zu Berlin, the Institute for Advanced Study in Princeton, and the Institut d'Études Avancées in Paris. He has received grants from the Guggenheim Foundation, the NSF, and the CNRS.

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