Introduction
1: 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 Russell
3: 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
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.
Mancosu's book is packed with new ideas, novel perspectives, and
important insights, offering the reader a thorough and exciting
examination.... The book should be required reading for anyone
interested in the history and foundations of mathematics.
*Ray T. Cook and Michael Calasso, Philosophia Mathematica*
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*
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*
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*
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