The Princeton Companion to Mathematics

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1. infinetNov 2025
   First impression from this list is that Russia has a lot of mathematicians!

   From the list, I have only read What is Mathematics? by Courant, which is a well-known and excellent book. I’d add Princeton Companion to Mathematics as a more advanced version, and Mathematical Methods in the Physical Sciences by Mary Boas as a more detailed alternative to What is Mathematics, to the list. Other titles on the list, such as those on probability or linear algebra, are perhaps less well-known.

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2. rramadassOct 2025
   There is no one book which can give you the overall sweep of Mathematics. However, you might find the following (though not textbooks per se) useful.

   1) The Princeton Companion to Mathematics by Timothy Gowers et al. and The Princeton Companion to Applied Mathematics by Nicholas Higham et al. - The closest you have to a Modern Encyclopedia of Mathematics. You get unmatched breadth after which you can move on to dedicated books as needed. Well worth the money.

   2) Mathematics: Its Content, Methods and Meaning by Aleksandrov, Kolmogorov et al. - Absolutely brilliant overview of Basic Mathematics. Published by Dover and hence very affordable.

   3) Elements of Mathematics: From Euclid to Gödel by John Stillwell - Written as sort of an update to the great Felix Klein's Elementary Mathematics from an Advanced Standpoint books. The Topics are particularly well chosen given modern advances; they include Arithmetic, Computation, Algebra, Geometry, Calculus, Combinatorics, Probability, Logic.

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3. reikonomushaFeb 2025
   This document reminds me of the kinds of things I wrote at a smaller scale when I'd self-teach math. The usual pattern was:

   Step 1. I don't know how X works.

   Step 2. I collect several sources about X and try to understand it.

   Step 3. I put in a lot of effort to understand X by reading all these sources repeatedly. I try to do exercises, do calculations, etc. I'm desperately seeking the moment it "clicks".

   Step 4. I finally kinda sorta "get" X.

   Step 5. I feel, "why didn't anybody simply explain X in this way?" / "why was everybody so overly formal?" / "why was everybody so overly informal?"

   Step 6. I'm motivated to write a short note about X that makes it (allegedly) easier to understand X.

   Step 7. I write it, and I realize it's actually hard to weave together a narrative that doesn't over- or under-assume prerequisites, that captures nuance, that has good examples, etc.

   Step 8. "There are 15 competing standards."

   Step 9. Find the next topic X and go back to Step 1.

   The Infinitely Large Napkin is a really cool consolidation of a ton of undergrad/early grad pure math topics. It's so incredibly expansive in its scope and, if it were in book form, I'd have been ecstatic to have it as a 16 year old.

   But paging through it, I find that they're very much in the style of quasi-formal lecture notes. A lot of topics are mentioned by their formal definition, and it's followed by a very anemic (if any) discussion, sometimes preceded by a very informal (sometimes humorous) introduction. Often such definitions are immediately followed up by a relatively technical exercise that presumes a fully synthesized understanding of material preceding. This can make it very difficult to learn from as a primary/sole source. It does make it fun to flip through, though, when you already have familiarity with the topics.

   In my view, this isn't the kind of book you work through. It's not "math distilled". Instead, it can serve as a great diving-off point for a new subject, or an inspiration to know where to look further on a given subject, or even a useful document to find a topic that piques your interest. Other books like this are those of yore that were encyclopedic in nature, such as:

   - VNR Concise Encyclopedia of Mathematics (1975–1989) edited by Gellert et al. The math here doesn't get terribly advanced (complex and numerical analysis), but it's a good, expansive treatment to dive into.

   - Mathematics From the Birth of Numbers (1997) by Jan Gullberg: This is another grand tour of math, albeit "only" to differential equations. It's refreshingly written by somebody who was a surgeon/anesthesiologist and amateur mathematician.

   - The Princeton Companion to Mathematics (2008) edited by Timothy Gowers. This is a massive book that covers just about everything, up to and including some of the latest problems in mathematics. It's 1000 dense pages. (There's also the Princeton Companion to Applied Mathematics edited by Nicholas Higham.)

   - The CRC Encyclopedia of Mathematics (1999) by Eric Weisstein. This is an anti-digitization of the Wolfram MathWorld into book form. Expansive, and also famous for some of the drama around its copyright. :)

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4. nextosJul 2024
   Garrity's All the Math You Missed book, mentioned elsewhere in the comments, draws a nice map of subjects, along with little introductions and book recommendations. The map is good for continuous mathematics, but IMHO fails to consider logic and type theory, which is a bit odd given the separate chapter for category theory. It also does not do proper justice to computation and clumps everything together under the label "algorithms".

   Good alternatives are The Princeton Companion to Mathematics by Gowers and Mathematics: Its content, Methods and Meaning by Alekxandrov, Kolmogorov, et al. Those present much more detailed maps so YMMV.

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5. rramadassApr 2024
   "Approachability" might vary somewhat but you might find the following useful;

   * For an introduction (and a reference) to various areas of Modern Mathematics that one didn't even know existed, The Princeton Companion to Mathematics and The Princeton Companion to Applied Mathematics are a must.

   * All the Math You Missed: (But Need to Know for Graduate School) by Thomas Garrity - A survey and a good adjunct to a textbook.

   * Mathematics: Its Content, Methods and Meaning by Kolmogorov et al. - Classic text from the great Russian Mathematicians.

   * Methods of Mathematics Applied to Calculus, Probability, and Statistics by Richard Hamming - Unique text from the great Richard Hamming (also checkout his other books).

   There is plenty more of course, specifically; checkout "Dover Publications" texts, many of which are classics and affordable.

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6. gautamcgoelApr 2024
   For those of you who don't know: this is a companion to the Princeton Companion to Mathematics, which is excellent but focuses mainly on pure math. Also, the editor of this new volume was a highly respected applied mathematician named Nick Higham, who sadly died a few months ago.
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