Ask HN: How or where to begin learning mathematics from first principles?

Very similar question: http://www.reddit.com/r/compsci/comments/2notz5/how_do_you_p...

(with a good answers regarding Khan Academy, Polya "How to Prove", lamar.edu, math.stackexchange.com, universityofreddit.com, lots of online curriculums from different universities, curricula aimed at data science (Prob/stats, linear algebra, calculus). These're good listings of resources for precalc and for data science:

http://www.reddit.com/r/math/comments/2mkmk0/a_compilation_o...

http://www.reddit.com/r/MachineLearning/comments/1jeawf/mach...

http://www.zipfianacademy.com/blog/post/46864003608/a-practi...

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The threshold question are,

- can you locate like minded folks to bootstrap a study group, or tutor(s) who are willing to devote time?

- (if you're in US/Canada) how about community colleges by you, in a lot of places they're still well funded and will efficiently pull you up to first year college calculus and linear algebra, and maybe further

- What level of high school / college math did you last attain, because reviewing to that level shouldn't be too stressful. At least, in my very biased view of math education.

I got started on "real" math with Spivak's Calculus. Some people start with Topology by Munkres, which is not a difficult book but is very abstract and rigorous so makes a good introduction. If you feel like you have ok calculus chops, maybe Real Mathematical Analysis by Charles Pugh. Other good books are Linear Algebra Done Right by Axler, or the linear algebra book by Friedberg, Insel, and Spence. Maybe even learn linear algebra first. It's so useful.

Do plenty of exercises in every chapter, and read carefully. Count on about an hour per page (no joke). Plenty of math courses have their problem sets published, so you can google a course which uses your chosen book and just do the exercises they were assigned.

If you don't feel comfortable with basic algebra and other high school math, there's Khan Academy, and some books sold to homeschoolers called Saxon Math.

If you haven't had a course in calculus before, maybe you should skim a more intuitive book before or alongside reading Spivak. I don't know of any firsthand, but I heard Calculus for the Practical Man is good. Scans are freely available online (actually, of all these books) and Feynman famously learned calculus from it when he was 12.

This is a bit of a divergence, but a fun way to practice what you learn is Project Euler, which will link it to your programming. https://projecteuler.net/ It's more of an applied problem set of increasing difficulty than learning from first principles though.

Betterexplained.com has many very good intuitive explanations of mathematical concepts. Elements by the publishers of Dragonbox will give you reasonable intuition for geometry. If you just want to use calculus Silvanus P. Thom(p?)son's Calculus Made Easy is excellent. Linear Algebra Done Right and LAD Wrong are both good books. LADW is free, legally.

The Art of Problem Solving series of books are uniformly excellent.

'The Nature and Origins of Modern Mathematics: an Elementary Introduction' could be a nice complement to the suggestions already here. It's very much first principles, but in a comprehensive and interesting way. You can download it here: http://cupid.economics.uq.edu.au/mclennan/NatureOrigins/natu...