Three Infinities in Mathematics

This book is intended as conceptual introduction to three mathematical theories arising from the attempt to understand (actual) infinity. Though the treatment does not avoid technical aspects, it tries to explain their origins and purposes. The book is divided into three parts, each composed of two chapters and a few appendices. Each part presents in detail one of three moments in which mathematicians made a concerted effort to control the idea of infinity: Projective Geometry, Infinitesimal Calculus, and Set Theory. The text includes exercises and philosophical and/or historical remarks. It is written both for non-mathematicians interested in understanding the most relevant ideas that govern these theories, and for professional mathematicians interested in exploring the possibility of a conceptually-engaged presentation of mathematical contents. The book can be used for advanced undergraduate students, as well as for beginning graduate students.

Introduction. - Part 1. Infinity in Projective Geometry. - The Geometry of the Painters. - Appendices to Chapter 1. Projective Geometry. - Part 2. Infinitesimal Analysis. - Inklings of Infinity in Greek Mathematics. - Appendices to Chapter 3. - The Birth and Development of Calculus. - Appendices to Chapter 4. - Part 3. Set Theory and Infinity. - Naive Set Theory. - Appendice to Chapter 5. - Axiomatic Set Theory. - Appendices to Chapter 6. - Conclusions: Three Infinities or Just One? .