I got this [1] a few years ago deluding myself into believing that I'll learn a bit from it everyday. Alas, the only way it's useful to me on a daily basis now is as a monitor stand. I remain cautiously hopeful that I'll begin pecking at it someday.

[1] https://www.amazon.com/Princeton-Companion-Mathematics-Timot...

this is his article in the princeton companion to mathematics

https://www.amazon.com/Princeton-Companion-Mathematics-Timot...

a great (even if expensive) math book

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 - https://www.amazon.com/Who-Fourier-Mathematical-Adventure-2n...

2) Functions and Graphs - https://www.amazon.com/Functions-Graphs-Dover-Books-Mathemat...

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 - https://www.amazon.com/Concepts-Modern-Mathematics-Dover-Boo...

2) Mathematics: Its Content, Methods and Meaning - https://www.amazon.com/Concepts-Modern-Mathematics-Dover-Boo...

3) Mathematical Techniques (i am linking this so you can see the reviews but get the latest edition) - https://www.amazon.com/Mathematical-Techniques-Dominic-Jorda...

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

a) The Princeton Companion to Mathematics - https://www.amazon.com/Princeton-Companion-Mathematics-Timot...

b) The Princeton Companion to Applied Mathematics - https://www.amazon.com/Princeton-Companion-Applied-Mathemati...

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.

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.

[1] https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...

[2] https://www.amazon.com/Princeton-Companion-Mathematics-Timot...

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] - http://www.amazon.com/Princeton-Companion-Mathematics-Timoth...

I have a hard copy of the Princeton Companion to Mathematics and find it incredibly useful -- highly recommended.

http://www.amazon.com/Princeton-Companion-Mathematics-Timoth...

Please please please have a copy of these books in your house:

http://www.amazon.com/Princeton-Companion-Mathematics-Timoth...

http://www.amazon.com/How-Solve-Mathematical-Princeton-Scien...

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.

I would recommend the Princeton Companion to Mathematics[1]. Its got a broad range of topics in short digestible articles. (though the first section has preliminary material for understanding the later articles)

[1]http://www.amazon.com/gp/aw/d/0691118809/ref=redir_mdp_mobil...

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.

http://www.amazon.com/Princeton-Companion-Mathematics-Timoth...

Tim Gowers is also the editor of the extraordinary mathematics companion Princeton Companion to Mathematics (http://www.amazon.com/Princeton-Companion-Mathematics-Timoth...)