Roads to Infinity

by John Stillwell

Winner of a CHOICE Outstanding Academic Title Award for 2011!

This book offers an introduction to modern ideas about infinity and their implications for mathematics. It unifies ideas from set theory and mathematical logic, and traces their effects on mainstream mathematical topics of today, such as number theory and combinatorics. The treatment is historical and partly informal, but with due attention to the subtleties of the subject. Ideas are shown to evolve from natural mathematical questions about the nature of infinity and the nature of proof, set against a background of broader questions and developments in mathematics. A particular aim of the book is to acknowledge some important but neglected figures in the history of infinity, such as Post and Gentzen, alongside the recognized giants Cantor and Gödel.

Genres: Mathematics, Philosophy, Science, Logic, Nonfiction

216 pages, Hardcover

First published July 12, 2010

About the author

John Colin Stillwell (born 1942) is an Australian mathematician on the faculties of the University of San Francisco and Monash University

Reviews

[Chelsea M]

Having finished the "prequel", Yearning for the Impossible: The Surprising Truths of Mathematics, I was looking for a similar experience from this book.

However, this was written for a significantly higher level than the previous book, and I, a student in her last year of high school studying a stage 1 university mathematics course, was not able to follow much of the content.

I think I have been mistaken in my impression that this book is a sequel to "Yearning". A positive side of this is that it's completely possible to read this without reading "Yearning" first, so if you found that "Yearning" was a little too simple or basic, this may be the book for you.

[Bob · Rating 4 out of 5]

A tough subject to understand; this book helped me come closer to figuring it out. On infinity, I'm still plenty lost but at least now I know what "countably infinite" means. At least, sort of.

I will probably go through this book again at some point on the future.

[Liquidlasagna · Rating 5 out of 5]

In 1963, Edwin E. Moise published Elementary Geometry from an Advanced Standpoint and his book became a classic. … [this book] deserves the same outcome. One of the most enjoyable features is Stillwell’s use of techniques of logic and set theory to solve real mathematical problems … Another enjoyable feature is Stillwell’s uniform coverage of unprovability, undecidability and non-computability... suitable for self-study... it is excellent background material for computer scientists and mathematicians in other fields. The historical notes alone are worth perusing by anyone who is interested in the development of mathematical ideas.
Phill Schultz, Gazette of the Australian Mathematical Society

[an] excellent book....the investment the reader makes―be he an intellectually curious adult or a math grad student with extra time on her hands―pays off with an increased understanding of the fascinating world of mathematical logic. The author’s thorough, well-researched historical comments are particularly valuable, as well as the philosophical quotations from the important players in this game. There is a very complete bibliography. What the reader might appreciate most is the ability of the author to share his deep insights into what is important and what it all means in the most profound sense...it is clear that the book received excellent proofreading before publication.
Mathematical Reviews

I love reading anything by John Stillwell. If you've ever been tantalized by the puzzles of infinity, set theory, and logic, and want to understand what's really going on, this is the book for you. It's an exceptionally fine piece of mathematical exposition.
Steven Strogatz, Cornell University

[Stefan · Rating 4 out of 5]

I agree with a previous reviewer that this books is written at a much higher level than the work it cites as a prequel, "Yearning for the Impossible..." It is a brief but exceedingly well written look at some areas of modern mathematics that may not otherwise get the coverage they deserve. I am not a mathematician but have an extensive mathematics background. Some of these topics were just barely touched upon in my advanced coursework and I never quite grasped them until I read this. To be honest, I still don't feel comfortable with some of the material presented but that is no fault of the author. It simply would take more time and work than I have at the moment to repeat the proofs myself and put the ideas into practice until they truly clicked. It was a mind-expanding journey that I expect to return to again. It also helped to give me background context for exploring other works on infinity by Woodin and others.