Recommended survey of mathematics

Question

What do you see as the most explanatory and beautiful survey of mathematics book...For short is there a book like Feynman lectures but for math?I've looked at Elementary Mathematics from an Advanced Standpoint by Klein and Mathematics: Its Content, Methods and Meaning by A. D.Aleksandrov ,Kolmogorov and Introductory Mathematics: Algebra and Analysis by Geoffrey C. Smith:,which one did you read and recommend, any other suggestion,please?

Answer 1

There are some Soviet style mathematical survey books available. But I do not know what their American translations are. I admit in general it is difficult to find survey level books enjoyable to read while not bogging down the reader with technical details. There are some options availbe:

1. Try to read the Notice of AMS. It is freely available online, but mostly for a mathematical educated audience. You can also try to read American Mathematical Monthly.

2. One can try to find expository papers online explaining the idea behind a proof. In a lot of cases nowadays it happened to be in forums like mathoverflow and this site. Sometimes googling does not really help because no one has thought about referring this paper before; and you need to check the publication list or note list of the experts in the field. But this is very time consuming. I remember when I was an undergraduate I used to download whole Harvard math department homework database for personal review, for example.

3. One can try to read some introductory level books at a cheap price, especially the Dover series. Since you mentioned Kolmogorov, his other book with Fomin on real analysis is a classic that assumes almost no formal background. And there are numerous other ones available. For Klein, a good book I know is "Lectures on Mathematics". But it is not that easy to read.

4. I think nowadays blogging is surprisingly common among established mathematicans. So you can probably find contents from the sum of geometric series to real mathematical research online. I guess instead of reading "classics", knowing what is going on nowadays might help, especially to satiate your personal curiosity.

Answer 2

I think you find a mixed bag of things sprinkled here and there. For example:

History

• The Math Book From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics by Pickover, Clifford A.
• A History of Mathematics, Second Edition , Carl B. Boyer , Uta C. Merzbach , Isaac Asimov (Foreword)
• A Concise History of Mathematics: Fourth Revised Edition, Dirk J. Struik

Topical Areas

• Mathematical Thought from Ancient to Modern Times by Morris Kline
• Hilbert by Constance Bowman Reid
• Six Books of Euclid by Euclid
• Hilbert-Courant by Constance Bowman Reid
• Great Feuds in Mathematics: Ten of the Liveliest Disputes Ever by Hal Hellman
• The Calculus Wars: Newton, Leibniz, and the Greatest Mathematical Clash of All Time by Jason Socrates Bardi
• Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics by John Derbyshire
• The History of Statistics: The Measurement of Uncertainty Before 1900 by Stephen M. Stigler
• Episodes in the Mathematics of Medieval Islam by Lennart Berggren
• Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics by Jóse Ferreirós

There are also specific biographies of mathematicians that are very good and show some of the correspondence between them and others.

Answer 3

A really great survey of much of modern mathematics is Mathematics Form and Function by Saunders Mac Lane. The book gives the big picture on many topics and explains the interconnections between different mathematical fields. I think everyone with a serious interest in mathematics should eventually read it.

The reader should have some experience with abstract mathematics on the advanced undergraduate level though.

A more elementary, yet still demanding, text is The Nature and Origins of Modern Mathematics: an Elementary Introduction by Andrew McLennan, which is (for now) available for free.

Answer 4

I've just borrowed Mathematics and Its History by John Stillwell from the library. The first couple of chapters are quite nice, and the table of contents suggests a broad coverage, if at undergraduate level.

Allow me to also second Mac Lane's Mathematics: Form and Function (not able to vote yet).