| As I a software developer with no degree (working since 2009 in 3 countries), I can share my experience of attempts to learn advanced math. In 2010, I was very interested in foundations of mathematics, an extremely abstract math branches: https://en.wikipedia.org/wiki/Foundations_of_mathematics In particular I spent huge amount of time on: https://en.wikipedia.org/wiki/Nicolas_Bourbaki (Set theory) https://en.wikipedia.org/wiki/Principia_Mathematica https://en.wikipedia.org/wiki/The_Foundations_of_Arithmetic http://www.jhtm.nl/tudelft/tw3520/Introduction_to_Mathematic... What attracted me is that these books doesn't require any specific knowledge of classical math. I.e. they are self-contained. It was fun and ... the experience to delve into highly abstract view on entire math. The big problem is that while I read that for more than a year, I had no experience in problem solving and just ignored exercises (thinking that concept is everything). As a result of that, my entire knowledge is completely evaporated and I literally can't solve any of exercises. After that year, I dropped math till recently. Now, I have completely different approach. I learning elementary olympiad style math and most importantly solving problems all the time. Currently, I'm into series of books: https://www.artofproblemsolving.com/store These books made for math olympiad preparation. While I solving exercises, I feel how solid my knowledge is. So if you want to learn advanced mathematics, learn elementary olympiad-style math first. It will give you solid background to start learning advanced math (not just knowledge background but most importantly problem solving skills). |
I recommend (surprise surprise) programming. Implement fast fourier transform in C and then Common Lisp. Write a finite difference PDE solver. Try solving actual problems to motivate you. Signals analysis can be a fun way to exercise your knowledge. Try analyzing your favorite songs and figuring out what makes them sound the way they do. Maybe implement some audio filters. If that's not your cup of tea, write physics or chemistry simulations instead. Then use OpenGL to visualize them. Then make them interactive.
I can go on and on, but I'll just leave two book recommendations for those who might enjoy programming advanced mathematics.
Structure and Interpretation of Classical Mechanics : https://mitpress.mit.edu/classical_mech
Functional Differential Geometry : https://mitpress.mit.edu/books/functional-differential-geome...