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.

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.

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.

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.

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

a great (even if expensive) math book

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.

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...

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

http://www.amazon.com/gp/product/0691118809?ie=UTF8&tag=...

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

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

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