# Abstraction and Infinity

## Paolo Mancosu

### Abstract

The book provides an original investigation of historical and systematic aspects of the notions of abstraction and infinity and their interaction. The notion of abstraction in question is that related to the use of abstraction principles in neo-logicism. The most familiar abstraction principle in this context is Hume’s Principle. Hume’s Principle says 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. Chapter 1 shows that abstraction principles were quite widespread in the mathematical practice that preceded Fr ... More

The book provides an original investigation of historical and systematic aspects of the notions of abstraction and infinity and their interaction. The notion of abstraction in question is that related to the use of abstraction principles in neo-logicism. The most familiar abstraction principle in this context is Hume’s Principle. Hume’s Principle says 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. Chapter 1 shows that abstraction principles were quite widespread in the mathematical practice that preceded Frege’s discussion of them. The second chapter provides the first contextual analysis of Frege’s discussion of abstraction principles in section 64 of the Grundlagen; the second part investigates the foundational reflection on abstraction principles in the Peanosets not by using school and Russell. Chapter 3 discusses a novel approach to measuring the size of infinite sets known as the theory of numerosities. This theory assigns numerosities to infinite sets not by using one–one correspondence but by preserving the part–whole principle, namely the principle according to which if a set A is strictly included in a set B, then the numerosity of A is strictly smaller than the numerosity of B. Mancosu shows how this new development leads to deep mathematical, historical, and philosophical problems. Chapter 4 brings the previous strands together by offering some surprising novel perspectives on neo-logicism.

*Keywords: *
Abstraction,
Infinity,
Neo-logicism,
Frege,
philosophy of mathematics,
Hume’s Principle

### Bibliographic Information

Print publication date: 2016 |
Print ISBN-13: 9780198746829 |

Published to Oxford Scholarship Online: January 2017 |
DOI:10.1093/acprof:oso/9780198746829.001.0001 |