I'm slowly reading through Feynman's Lectures on Physics and I find myself wondering, is there an analogous book (or books) for math?
The Princeton Companion seems to me to be an attempt to achieve a similar mixture of depth, accuracy, content, motivation, and context. However, because math is a different kind of subject, this is a very different kind of book.
I tend to agree with Adam-the sheer scale and difficulty level of most mathematics beyond the level of basic calculus would make a book like this almost impossible to write. I think the closest anyone's ever come to writing the kind of book you're suggesting is Kolomogrov, Alexandrov and Laverentev's Mathematics:Its Content, Methods And Meaning. This 3 volume overview-originally in Russian-attempts to give an overview of all mathematical fields for students without much background-only some high school algebra,geometry and calculus is needed. Admittedly,though-in the Soviet Union in the 1960's, most of these students had stronger backgrounds then most of today's undergraduates in America! It's currently available in Dover paperback-I think you'll find it worth a serious look.
As for me, Vladimir Arnold's writing style is sometimes similar to Feynman's style. For instance, Arnold's Ordinary Differential Equations may be appealing to those, who appreciate Feynman's lectures. I can also recommend the following books:
- V. I. Arnold, Mathematical Methods of Classical Mechanics (Graduate Texts in Mathematics)
- V. I. Arnold, Lectures on Partial Differential Equations
- V. I. Arnold, A. Avez, Ergodic Problems of Classical Mechanics (Advanced Book Classics)
Tristan Needham's Visual Complex Analysis has sometimes been compared to Feynman's Lectures.
"...it is comparable with Feynman's lectures in Physics. At every point it asks 'why' and finds a beautiful visual answer. ...I believe that this book can make every student understand and enjoy complex analysis. If its methods could be applied in teaching more generally, mathematics would become a flourishing subject" -- NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY
It's much more specific in scope then Feynman. But it remains the best written math text book I've read.
Joe Harris's textbooks often remind me of Feynman's style, in that they frequently omit details, and may cause the casual reader to think he knows more than he does, but do a wonderful job conveying the most important points of the theory from an expert's perspective. I am thinking here of Representation Theory, Moduli of Curves and Algebraic Geometry: A First Course. (Since this is rather mixed praise, let me add that Harris and Feynman are among my favorite authors; just that the reader needs to be vigilant about filling in the gaps.)
However, none of these books attempts anything like an overview of all of math.
I am surprised that no one has mentioned the book "What is Mathematics?" by Richard Courant.
Although it is quite expensive to buy (since it is out of print), perhaps you could borrow MacLane's Mathematics: Form and Function from a library. I found it to be a beautiful overview of mathematics and interconnections between topics you may have seen.
First 16 chapters from Penrose's "Road to reality" could be quite close. He starts from fractions and goes to calculus on manifolds, group theory, complex analysis, Rieman geometry, Lie algebras, etc. All on ~350 pages! Would be great if he could spend the rest 700 pages on math alone. That could be something comparable to Feynman's lectures...
Another one (similar in style, not popularity) would be a small book by Lars Garding "Encounter with Mathematics" - quite advanced topics described in a pleasant manner - a nice relief after so popular dry "definition, theorem, proof" approach.
This should really be a comment, but I don't have the reputation. I don't think any such book exists. In fact, I don't think that such a book is possible. There are two reasons for this.
Math, even at the undergraduate level, is much bigger than physics. It's not that it is impossible for anyone to understand everything that is taught to undergrads -- I certainly feel comfortable teaching any undergraduate-level course in my university. Rather, there are an enormous number of topics (calculus, geometry, linear algebra, abstract algebra, topology, partial differential equations, combinatorics, probability, etc) each of which has its own pattern of thought. At some point in your mathematical life, you will start to view them as one subject, but I don't think there is a way to teach undergraduates the foundational materials without having the topics fragment. A book that tried to describe all of them would be just too disjointed and incoherent.
You can learn a lot of physics without getting your hands dirty too much (via informal thought experiments, easy calculations, etc). This is basically the pattern in Feynmann's book -- it's all intuition and (almost) no detail. Math, however, doesn't work that way. You can't learn math without getting down to the details in a serious way. I guess you could tell a fun story, but the students would learn nothing from it.
I don't know if these are in english, but in Spain (and Russia) there are a collection of books of the URRS editorial called "Lecciones de Matemática" (Math lessons) of a russian mathematician called V. Boss which covers a lot of advance and modern mathematics in a fresh way not going into the particular details, but more centering on the intuition about each mathematical topic with its structure -something like giving perspective about the mathematical topic-.
If you are looking for a treatment of a mathematical subject which is unconventional, then I'd recommend "Concrete Mathematics" by Graham, Knuth & Patashnik [ISBN-13: 978-0201558029] which is reasonably far-reaching, and perhaps "On Numbers and Games" by Conway [ISBN-13: 978-1568811277] which is specifically concerned with the theory of numbers and number-like entities. Both of these are very 'elementary', in the sense of operating from first principles.
If, on the other hand, you're looking for breadth-of-coverage, then consider the enormous dictionary "CRC Concise Encyclopedia of Mathematics" by Weisstein [ISBN-13: 978-1584883470] (also on-line http://mathworld.wolfram.com/; new shorter edition forthcoming), which is very good to "dip into", but would not really be suitable for cover-to-cover reading.
Another option, available both in paper and on-line is "NIST Handbook of Mathematical Functions" [ISBN-13: 978-0521192255] or "NIST Digital Library of Mathematical Functions" http://dlmf.nist.gov/. Obviously, this work is primarily concerned with various special functions, including those of trigonometry and combinatorics.
There are also various handbooks of mathematics, which arrange the material by topic, but with very little discussion. I do not have a strong view on which of these is best, as various options have their own merits, but perhaps that of Bronshtein & Semendyayev [ISBN-13: 978-3540621300] has relatively large breadth and contains much more explanatory text than is typical for a handbook.
This is a small start, but in my (somewhat unqualified estimation) fabulous. It's lecture notes of a real analysis course given by Fields Medal winner, Vaughan Jones. They are elegant, self-contained, and beautifully typed up by an anonymous student. Here is a link to download them:
I would recommend Stephen Hewson's "A Mathematical Bridge"; it is similar in tone to Feynman's lectures. While it's perhaps not as comprehensive (500 pages vs 1500), Hewson manages to cover a very impressive range of topics (all the "highlights" of an undergrad math course).
His explanations of the major concepts are the best I've read anywhere, and it does a good job of giving the reader a sense of what the major fields of mathematics are and how they relate to one another.
I.M Gelfand's books on trigonometry,algebra,functions and graphs and calculus of variations(and much more) are comparable to Feynman Lectures. He has even stated his effort to write a book like feynman's in the book's preface. I strongly recommend the books. You can search the books in amazon for user reviews.
I want to mention two resources:
Differential Geometry Reconstructed by Alan U. Kennington which is freely available and a video lecture series by Fredric Schuller called "Geometric Anatomy of Theoretical Physics".
'John Baez's lectures on Mathematics'. You can find something similar in the notes of John Baez on the blog by the same author. It's a super nice and laid back approach. But it is still arduous in some points. And it requires the reader a certain mathematical maturity. Analogous to the Feynman's Lectures on Physics ask.
Can't believe no one has (yet) mentioned George Polya's incomparable How to Solve It. It's very dissimilar to Feynman in that it covers very few (if any) specific subfields of mathematics - but it's very similar in that it attempts to give the student an understanding of how to approach the discipline, and to build their intuition so they can grapple with problems in the field.