Is there something in the order of a Goedel Escher Bach type book? If you've read it you know what I mean. Something compelling that you have to read a couple of times in order to start to get it, but it's so interesting you can't put it down!
7 Answers
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I'm going to answer your title question instead of your body question (which to my mind is completely different): what you're looking for in the title is the Princeton Companion to Mathematics.
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1$\begingroup$ A very nicely put-together book. Seems like we have some thread duplication going on? mathoverflow.net/questions/8609/favorite-popular-math-book $\endgroup$ Commented Jan 11, 2010 at 5:05
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4$\begingroup$ On the other hand, I've put the Princeton Companion down many times. It's not a read-in-one-night kind of book. :) $\endgroup$ Commented Jan 11, 2010 at 5:13
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1$\begingroup$ Yes, but I can't think of a better "good general overview of math and its various branches." Like I said, I think the title and the body ask very different questions; in particular I am not convinced GEB is a good general overview of anything (as much as I like it). $\endgroup$ Commented Jan 11, 2010 at 5:43
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The classic answer to this is Courant and Robbins, What is Mathematics? A bit dated, but certainly worth looking at if you haven't yet.
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Mathematics and its History by John Stillwell
This book aims to give a unified picture of Mathematics through it's history. The good things about this book are the extremely beautiful figures, interesting exercises and emphasis on the interplay of Algebra and Geometry.
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Penrose's The Road to Reality covers large portions of mathematical physics. This isn't a textbook, and omits many details, but it is as meaty as GEB.
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Mathematics: Its Content, Methods and Meaning is an excellent overview of the full body of mathematics. It is large (3 volumes), but comes in a paperback edition that includes all three.
The draw is that it is edited by three well-known Russian mathematicians (Aleksandrov, Kolmogorov, Lavrentev) who wrote some of the articles and solicited the rest from many other Russian luminaries. It was developed as a compendium able to communicate both the vibrancy as well as the importance of each of the areas of the mathematics so that science ministers in Russia could better understand mathematics as mathematicians do.
The translation into English is excellent.
The first article, a General View of Mathematics, is highly recommended from a philosophical, historical, and phenomenological point of view.
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Saunders MacLane, Mathematics: Form and Function. Very good overview of undergraduate mathematics, showing interconnections between different areas. As might be expected from one of the inventors of category theory, MacLane defends categories as a foundation for mathematics.
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Modern Mathematics in the Light of the Fields Medal, which is pretty darn good for all its flaws
A Panorama of Pure Mathematics, which looks good but I haven't read
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2$\begingroup$ Maybe also Tim Gowers' Mathematics: A Very Short Introduction (amazon.com/Mathematics-Short-Introduction-Timothy-Gowers/dp/…) depending on the level desired. $\endgroup$ Commented Jan 11, 2010 at 6:26
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$\begingroup$ PS. I've put a wiki up (and deleted the question on MO) for a proposal along the lines of an updated version of the first book in my answer above. See modmath.wikia.com/wiki/Modern_Mathematics_Wiki $\endgroup$ Commented Jan 11, 2010 at 6:31
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$\begingroup$ Dieudonné's is quite idiosyncratic! $\endgroup$ Commented Jan 11, 2010 at 6:32
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$\begingroup$ That's a big reason why I think something like this oughta be crowdsourced. $\endgroup$ Commented Jan 11, 2010 at 6:35