Found in 14 comments on Hacker News
vmilner · 2023-01-19 · Original thread
Something else I’ve found extremely useful in getting into maths topics is the Princeton Companion to Mathematics - it doesn’t have exercises but gives excellent overview essays of a wide range of maths topics - expensive to buy (mine was a present) but should be available in academic libraries, say.

throwlaplace · 2020-05-22 · Original thread
this is his article in the princeton companion to mathematics

a great (even if expensive) math book

rramadass · 2019-07-16 · Original thread
I suggest the following approach;

Start with some school textbooks for grades 8-12 i.e. Secondary Education. This is more for a refresher course in the absolute basics.

The above can be supplemented with the following books to develop intuition;

1) Who is Fourier -

2) Functions and Graphs -

After this is when you enter undergraduate studies and you have to fight the dragon of "Modern Maths" which is more abstract and conceptual. In addition to standard textbooks; i suggest the following;

1) Concepts of Modern Mathematics -

2) Mathematics: Its Content, Methods and Meaning -

3) Mathematical Techniques (i am linking this so you can see the reviews but get the latest edition) -

Finally, if you would like to learn about all the new-fangled mathematics your best bets are;

a) The Princeton Companion to Mathematics -

b) The Princeton Companion to Applied Mathematics -

One important piece of advice that i have is to become comfortable with the Symbols, Notation and Formalism used in Mathematics. Most students are intimidated by the Formalism (which is nothing more than a precise form of shorthand to express abstract concepts) and give up on studying Mathematics altogether. This is a shame since it is merely the Form and not the Function of Mathematics.

synthmeat · 2018-03-11 · Original thread
I’m going to go with a few assumptions here:

a) You don’t do this full time.

b) By “bottoms up” you just mean “with firm grasp on fundamentals”, not logic/set/category/type theory approach.

c) You are skilled with programming/software in general.

In a way, you’re ahead of math peers in that you don’t need to do a lot of problems by hand, and can develop intuition much faster through many software tools available. Even charting simple tables goes a long way.

Another thing you have going for yourself is - you can basically skip high school math and jump right in for the good stuff.

I’d recommend getting great and cheap russian recap of mathematics up to 60s [1] and a modern coverage of the field in relatively light essay form [2].

Just skimming these will broaden your mathematical horizons to the point where you’re going to start recognizing more and more real-life math problems in your daily life which will, in return, incite you to dig further into aspects and resources of what is absolutely huge and beautiful landscape of mathematics.



pavelrub · 2015-09-20 · Original thread
Nice! This book is modeled after the The Princeton Companion to Mathematics [1] which is simply awesome in every sense of the word, and extremely recommended for any person who is interested in mathematics.

I'm very glad to see more books being modeled after it, and I hope this trend will continue with things beyond math. My only hope is that those new books will match the quality of the original.

[1] -

webnrrd2k · 2015-04-07 · Original thread
I have a hard copy of the Princeton Companion to Mathematics and find it incredibly useful -- highly recommended.

pm90 · 2014-11-24 · Original thread
Please please please have a copy of these books in your house:

Its certainly too advanced for a 6 year old (or even a 16 year old, TBH) but just having it around is really great, I think. I remember when I was younger, I would look up stuff in more advanced books even if I couldn't understand them right away. The feeling I had was always: "Someday, I will be able to understand this..." which made me learn more physics and math.

"How to Solve it" is especially great if you do/will teach her in the future.

cschmidt · 2012-11-28 · Original thread
The link was direct when I tried it. However, to answer your question, it is a chapter from The Princeton Companion to Mathematics. An amazing work covering much of mathematics.

zyfo · 2012-01-21 · Original thread
Tim Gowers is also the editor of the extraordinary mathematics companion Princeton Companion to Mathematics (
ffmmjj · 2011-06-02 · Original thread
The Princeton Companion to Mathematics (
Maro · 2011-02-10 · Original thread
kqr2 · 2009-07-19 · Original thread
Book recommendation: Princeton Companion to Mathematics

It's a good way to skim a lot of different mathematical topics for further exploration.

dschobel · 2009-06-19 · Original thread
I'll third this recommendation. Fantastic text.

harpastum · 2009-01-31 · Original thread
If you're really interested in learning a lot about mathematics, I would definitely recomment the Princeton Companion to Mathematics (see below for URL). It came out very recently, and while i'm only about 100 pages in (of well over 1000), it's down to earth writing and low prerequisites are very appealing to me.

or if you'd rather not use my affiliate link:

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