Abstraction and Infinity by Paolo Mancosu
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Two central topics of this book are definitions by abstraction and the concept of the size of infinite sets.
When I was a student, I was under the impression that most of modern mathematics was about equivalence relations and quotients. Cantor’s cardinal and ordinal numbers, quotient topological spaces, quotient groups and rings, isomorphism types of algebraic structures, Turing degrees, germs of analytic functions: everywhere you looked there was an equivalence relation giving rise to a new concept that was more abstract than the one one started with. It was all solidly grounded in axiomatic set theory that we all learned and accepted as the foundation of mathematics. However, once one steps outside the realm of set theory, the status of definitions by abstraction becomes less clear.
In the first chapter, Definition by Abstraction from Euclid to Frege (and Beyond), Mancosu shows how abstraction principles were used in geometry, number theory, and algebra long before their formal...