“The Impossible” and “Infinity”: Two Outstanding Books on Math

When I was at MathFest in Pittsburgh this summer, I bought a pile of math books.  It’s taken a lot of bus rides to get through them all, but now that I’ve read them all, there are several that I want to recommend.

My favorites from the bunch are two books on mathematics for a general audience by John Stillwell.  Stillwell is best known, I believe, for his excellent tome Mathematics and its History, which is an outstanding textbook for a course (or two or three) in the history of mathematics, expertly blending mathematical content, biographical information, and insight into the historical progression.  I can’t say enough good things about that text — it’s one of my favorite books — but it’s not something that a nonmathematician is likely to buy (because it’s so big and correspondingly expensive).

These two books, though, are short, affordable, accessible, and beautifully written.  Each tells a story of mathematicians dealing with a certain big theme, with a logical progression of increasingly sophisticated and deep mathematics.

The first is Yearning for the Impossible: The Surprising Truths of Mathematics (ISBN 156881254X).  Here the theme is the ongoing evolution of mathematics in response to questioning the impossibility of certain ideas.  There is no real solution to $x^2=-1$, so we could just throw up our hands and say “It’s impossible!”, but the results are much more interesting if we ask “Is there some other sense in which it is possible?”  This book, just over 200 pages, is really remarkable for the number of disparate and sophisticated ideas it manages to introduce to a general audience.  Let me illustrate this by simply listing the chapter headings.

1. The Irrational
2. The Imaginary
3. The Horizon
4. The Infinitesimal
5. Curved Space
6. The Fourth Dimension
7. The Ideal
8. Periodic Space
9. The Infinite

The follow-up is Roads to Infinity: The Mathematics of Truth and Proof (ISBN 1568814666).  The title makes it sound very profound, and it is.  This book goes deep into various notions of infinity, including a friendly but surprisingly thorough treatment of ordinal and cardinal numbers.  Godel’s Theorem(s) made accessible without being dumbed down.  Good stuff.

Here’s the takeaway.  These books are beautiful, they, make me happy, you should go buy them and read them.  Someday I want to be able to write like that.

P.S.

If anyone local wants to borrow any of the books mentioned, just drop me a line or stop by my (Ann Arbor) office.  I have a 2nd edition and a 3rd edition of Mathematics and its History, though I plan to award the 2nd ed to a worthy student at semester’s end.