The Art of the Infinite : The Pleasures of Mathematics (2003)

I worked for over four years as the archivist for a mathematics collection, even though the last math class I took was pre-calculus as a junior in high school and I did not consider myself mathy in the least. But something strange happened as I got further enmeshed in my work. As I browsed through higher-level mathematical journals and attended annual mathematics conferences and wrote articles for mathematics newsletters and helped mathematical historians do their research I began to relax into the idea of mathematics. Not the actual math itself, but the ideas behind the math. Why it was interesting and creative. What it was good for. All that stuff.

And I began to really like it -- although every time I told someone where I worked I still felt like I had to defend it. "It's not that bad!" "It's not as boring as it sounds!" (I actually still have to do that with my current job archiving a religious collection...) Still, very few people really seemed to understand that these mathematicians and the work that they do were different than the drudgery of high-school algebra. If I could make all those mathematics-doubters read The Art of the Infinite : The Pleasures of Mathematics by Robert Kaplan and Ellen Kaplan (2003), I think they could start to see my point.

Of course, I could also get you doubters to start with Robert Kaplan's actually pretty popular (for math writing) book, The Nothing that Is: A Natural History of Zero (1999), which I have also read and which is really excellent. It is much shorter than The Art of the Infinite and involves less equations, so it might be a little more approachable as a first dip into popular mathematics reading.

Yes, The Art of the Infinite has a lot of equations, schematics, graphs, and geometric projections (although the really intricate ones are kindly set aside in the Appendix). But don't let that scare you. Remember, we are relaxing into the idea of really understanding how cool upper-level mathematics is -- don't tense up at all those Greek symbols and acute angles, just let them wash over you and the Kaplan's will lead you through some pretty amazing mathematical concepts one step at a time. Along the way, you will get a taste of the major mathematical figures: from Pythagoras to Cantor -- all nicely illustrated by Ellen Kaplan, who also hand-draws all the mathematical figures in the book.

I wouldn't claim to have understood (or really thought out) every proof in this book, but I feel like I got enough of a taste to understand what was going on every step of the way and why it was interesting. I feel like I really understand the definitions and distinctions between Real, Natural, Rational and Imaginary numbers. I have a huge appreciation for the imagination and creativity of mathematicians who attack the mind-blowing complications of infinite numbers and come up with something that can be wrangled into an equation and applied to any number at all. This isn't the kind of thing I usually read about, especially since I left the math archives behind nearly two years ago, and it got me a little pumped up to dip back into that world again.

Okay, here is just one of the mind-blowing things the book got me thinking of: think about all the counting numbers. 1, 2, 3, 4, 5.... They are infinite, right? Just pick the highest one you can think of, and you can always add one more. But then think about the rationals (which are basically fractions). Between 1 and 2 on the number line, there are an infinite number of rational numbers. You can always divide the piece of pie one more time, add another number to a decimal, etc. But if there are an infinite number of rationals between each pair of natural numbers (which are also infinite), the infinity of the rationals is more infinite than the infinity of the naturals.

Just think on the awesomeness of that for awhile. No wonder some mathematicians go crazy and try to prove the existence of God. This is some heady and cool stuff...