Ask HN: How can I discover math?

I recommend seeking out translated Soviet math textbooks. They're dense and unforgiving, but are conceptually correct. You could do far worse.

Get "The Princeton Companion to Mathematics" - http://press.princeton.edu/titles/8350.html

Find a problem that piques your interest, then learn the math that's relevant to the problem. You can choose a problem from most any subject, like math, physics, or economics.

Having a problem that piques your interest--something to work toward--is a good motivator.

Soviet textbooks and the Princeton Companion (recommended by others) are likely to be pretty brutal for you. I suggest problem-solving as a better way in.

Take a look, e.g., at Project Euler (http://www.projecteuler.net/); find some books aimed at K-12 students doing mathematical contests (there are lots; what's best depends on how much you already know) and online resources aimed at the same audience (e.g., http://amc.maa.org/, http://www.mathcomp.leeds.ac.uk/); there are other sources of not-too-routine mathematical problems online, such as http://nrich.maths.org/.

This pretty much guarantees that you'll be doing as well as passively absorbing (one of many problems with which is that it's easy to think you're absorbing when actually you're not). There's a very wide range of levels of difficulty. And it's likely to be fun, if mathematics really suits you at all.

Might be a nice time for GEB. There's even a K-12-ish course for it at MIT OCW:

http://ocw.mit.edu/high-school/courses/godel-escher-bach/ind...

Check out this previous post: http://news.ycombinator.com/item?id=1449799

And also, I'd highly recommend art of problem solving. http://www.artofproblemsolving.com/

Honestly, the best way to discover math is the Hacker's way i.e. find problems to work on and then start working on them even in you don't know how initially. The very digging in process will lead to further discovery and further learning. Now, be sure to check out art of problem solving forums, people ask questions there and there are very good detailed explanations available.

Be also sure to check out Street Fighter math (http://ocw.mit.edu/courses/mathematics/18-098-street-fightin...) and fermi questions (e.g: http://mathforum.org/workshops/sum96/interdisc/classicfermi....)

And remember to print out these questions and carry them with you along with tons of scrap paper. Sometimes, it helps just to read a problem and let it sink in over days...

Get the book "Thinking Mathematically" (http://www.amazon.com/Thinking-Mathematically-J-Mason/dp/020... non affiliate link) which exposes the process of how mathematicians think, using simple problems. You can then adapt the process to your preferred level of problems. Also learn proof technique (Velleman's How to Prove it" is a good book for this) and doesn't make any assumptions of mathematical knowledge.

You asked about beauty and history, with an idea on how to discover... personally, I started with "The Math Explorer: A Journey Through the Beauty of Mathematics": (http://www.amazon.com/gp/product/1591021375). That book got me hooked and from there I went on to pre-cal studies at university, eventually completing sequences in discrete structures, probability, up to and including calculus III, a level I never imagined I'd get to. That book was literally my starting point since I never took high school math seriously nor was I any good at that level. I think it'll serve you well considering the level at which you're at, as it did me. I've reread that book four times to date and still enjoy it.

The one that made me truely see the beauty of maths: Hallucinogenic drugs - due to the hallucinations basis in geometric shapes and patterns.

Not really useful for teaching yourself K-12 math however for the beauty of it nothing better

Hijacking the thread here, but could someone quickly recommend a college-level calculus book?

let's start with "how to solve it" (g. polya), if you love to learn the thinking of math.

One book I'd recommend is "1089 and all that" (http://www.amazon.com/1089-All-That-Journey-Mathematics/dp/0...).

I've bought copies for friends and family whose maths backgrounds range from school to degree level and they've all enjoyed it.

It's not a textbook, more a conversational account of a few topics the author has found interesting through his life from a young boy to an Oxford don. It's very readable and the author's delight in maths and problem solving shine through

It will only teach you a small bit of math (i.e. Trigonometry) but try programming a little 2D rocket ship game with some physics and movement etc. It can be fun and you will get a better feel for the maths involved. don't use libraries. Use javascript with canvas - easy as pie.

"Discovering math", depending on what you mean, could be setting the bar pretty high. Memorizing formula is not what mathematics is about, but you're unlikely to get very far if you just "play with numbers". To learn mathematics and to think mathematically, you still need a structured environment: Lockhart's complaint is that modern (K-12) math teachers neither have that knowledge nor the ability to provide that structure.

If I were you, I'd go to where that knowledge does get taught: in undergraduate mathematics classes. That usually starts with some rigorous calculus (I second the recommendation for Spivak), real analysis (which might seem hopelessly unmotivated if you don't know/remember much calculus), linear algebra or abstract algebra, or discrete math. Depends on what you're interested.

Reading mathematics is hard. Google around for advice. Do lots of problems, which generally means finding proofs. This would very well fit the definition of "discovering mathematics".