The Princeton Companion to Applied Mathematics

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1. 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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2. kobenniApr 2025
   For mathematics, there are the Princeton Companion to Mathematics and the Princeton Companion to Applied Mathematics.
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3. 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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4. rramadassJan 2024
   Definitely!

   I have both the Princeton Companion to Mathematics and Princeton Companion to Applied Mathematics and both are well worth the money. One of the difficulties in studying the breadth of mathematics is that the huge amount of jargon/notation of the different branches/models/tools all seem complex/unrelated and hence a single volume where you can jump back and forth between different views is very much necessary to appreciate the unity of Mathematics. It is also an invaluable aid in understanding new terminologies/techniques that one may come across while studying other subjects. The volumes are truly comprehensive w.r.t. "Modern Mathematics" and should be a part of every educated person's "personal library".

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5. rramadassNov 2020
   Assuming you are asking in general and not specifically Bourbaki authored;

   * Mathematics: Its Content, Methods and Meaning by Aleksandrov, Kolmogorov, Lavrentev

   * The Princeton Companion to Mathematics by Timothy Gowers et al.

   * The Princeton Companion to Applied Mathematics by Higham et al.

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6. photon_linesFeb 2020
   The author isn't right. He's right in some aspects (I'd say in 5% on what he conjectures), but for the most part, programmers and programming will be in quite a high demand for quite a foreseeable time in the future. Yes, C / C++ programming might become less important (see: memory wall) and I'd say highly parallel computation will take precedence, but this won't replace computer science nor programming...it'll just create the need to programmers to transition from one skill set into another.

   Also, as a note: the field of computer science is a branch of mathematics. Whenever you deal with data structures or algorithms, you're dealing with how to procedurally design information structures and procedures to solve problems and model the real world. Mathematics? Well, depending on what branch you're into, it's using symbols and information ...to, well, do the same thing!

   Out of all of the professions, it seems rather silly to me to think that our own profession is in danger of being replaced. I would think that it's the LEAST likely one, since programmers, to me at least, represent workers who are paid to communicate and find ways to solve problems in the real world using a tool which will continue to improve and enable our species to excel and automate jobs filled with repetitive drudgery. Out of all of the professions, I’m not sure why the author would think that ours is likely to go out of demand soon. I would conjecture that us playing a prominent role in automating other jobs ensures that we stay in demand for quite a long time!

   As far as the most useful mathematical branches are concerned – if you’re interested, the branches I find have the highest ability to help in solving real world problems are: calculus, linear algebra, probability and statistics, computer science (as already mentioned), convex and computational optimization, quantum mechanics, complex analysis, ordinary and partial differential equations, data mining and analysis, information theory … well, I could keep going, but I think you’d greatly benefit from checking out a great book which gives a pretty good overview on the key areas in applied mathematics: The Princeton Companion to Applied Mathematics. The above are just my own guesses, and it’s highly dependent on which problem you’re looking to solve.

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