Hello, my name is BlackKnight, and despite all evidence to the contrary I am not quite nerdy enough. I would like to learn more about math.
One thing I’ve noticed of late is that math seems to be a diverse field. There are more branches of math than I’ve ever heard of, I’m sure. I’m wondering if there are any good books giving an overview of math - what branches of mathematics there are, how they relate to each other, etc. In short a “birds eye view” of math, without necessarily focusing deeply on any one area. From there, I could decide if any particular area looked interesting and understandable enough to persue or whether I should just give up and watch cartoons or something.
My background: I’ve had the normal arithmetic, geometry, algebra, calculus stuff in school. I’ve also studied a bit of logic, theory of computation, and a tiny bit of set theory. I’m not afraid of equations (or even inequalities :D) so long as everything is fairly well defined and explained in advance. In other words, it’s ok to recommend math books with actual math in them. If I bite off more than my mind can chew I’ll be happier than if I bite off less than enough to feed it.
I don’t know of any books like you’re looking for, but I can think of a couple good sources for that sort of info: the Wikipedia Mathematics Portal (in particular, the list of topics), and the Mathematical Atlas. Have a look around those, and if you see any particularly interesting topic, someone can probably recommend a book on it.
Concepts of Modern Mathematics by Ian Stewart might be the kind of thing you’re looking for.
If you’re at all interested in pursuing math as a career, or are just intrigued by the people who do, his Letters to a Young Mathematician is also a good read.
All the books I’ve listed are written (extremely well) at the popular level and are easy to read compared to a textbook. I’m sure there are more technical, textbooky overviews of math out there, but no specific one comes to mind to recommend.
This isn’t really an overview, but the most fascinating book I’ve ever read is about math and philosophy and intelligence and wordplay and a bunch of other things… Godel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter
I strongly second this, especially given the few areas of higher math you say you’ve been exposed to, and thus presumably have interests around (logic, computation, set theory).
Much as I like Asimov’s F&SF columns (and I’ve read all 399), I have to point out that numbers are not math. Most modern math never gets within a country mile (GQ in-joke) of an actual number. (Except for number theory, to be sure. And even there…)
Asimov never wrote much of anything that could be called math. Math history, maybe, but not math itself.
It’s so hard to get actual math out of this, though. I read it immediately after taking classes in logic and computation theory, and there were still a few times where I couldn’t figure out exactly what he meant in a given analogy. Still, it’s an entertaining book.
It’s true that the unconventionalness of the writing style can make it difficult to extract the mathematics underneath. But the OP seems to be looking particularly for books to help him decide which areas of math would be most interesting to him, and for this, I think, GEB is particularly well suited.
I read Four Colors Suffice by Robin Wilson some time ago, and thought it was an excellent general exploration of a math topic you may have heard of (a famous topological problem) but don’t really know much about. The historical development of the problem, an ingenious but ultimately flawed proof, the blind alleys, and the philosophical implications of the final, computer-based solution are explained with enough detail that you can follow the math and understand the proofs (at least in the early chapters). I highly recommend it if you are looking for something to prod your mathematical talent without getting overwhelmed by specialization.
I’ve decided to order two books from Amazon:
“Mathematics for the Nonmathematician”, which Larry Borgia recommended
and
“Concepts of Modern Mathematics” which Thudlow Boink and Exapno Mapcase simul-recommended
“Godel, Escher, Bach: An Eternal Golden Braid” is an excellent book, but it’s not quite what I was looking for. Plus, I’ve already read it a couple of times. (I’ve got the 20th anniversary edition on the shelf next to me.) I’m currently reading “I am a Strange Loop”, Hofstadter’s “sequel” to GEB. It’s not quite as playful as GEB, but it’s definitely whimsical and thought-provoking. Anyone who liked GEB would probably like “I am a Strange Loop”. And if you like both of them, look for another book by Hofstadter, “Le Ton Beau de Marot”.
The Knot Book by Colin Adams is, according to my son the topologist, a very good book on knot theory. If you’re looking for an introduction to the ideas of topology, you can’t go wrong with knot theory. It’s the study of infinitely stretchy loops, tied up in knots in space. The Knot Book starts with basics and explains how studying strings tied in knots relates to deeper questions in mathematics.
There are also chapters on DNA and biological applications, as well as the Poincare Conjecture which, if you read the right news, has been in the news a lot lately.
For an amateur, knot theory is a good place to start. It’s both a subject of current interesting research and also surprisingly accessible. The basic results are a little over a century old and, even now, there are theorems being proved which are very geometric and visual.
(I’ve got the 20th anniversary edition on the shelf next to me.) I’m currently reading “I am a Strange Loop”, Hofstadter’s “sequel” to GEB. It’s not quite as playful as GEB, but it’s definitely whimsical and thought-provoking. Anyone who liked GEB would probably like “I am a Strange Loop”. And if you like both of them, look for another book by Hofstadter, “Le Ton Beau de Marot”.