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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