I don't use a lot of advanced mathematics these days -- most of my time seems to be spent shuffling bytes from point A to point B.

I do have two kids in school, though, and my daughter just finished some higher level calculus courses. In my attempts to help her, I've found that we have endless and fantastic resources available online. Every type of problem has a YouTube video explaining, step by step, how to solve them.

In my opinion, there's never been a better time in history to learn advanced (or even basic!) mathematics than right now.

Here are some threads I bookmarked on Hacker News that have a lot of good suggestions:

Ask HN: How to Study Mathematics? [1] Ask HN: How to self-study mathematics from the undergrad through graduate level? [2] Ask HN: Are there books for mathematics like Feynman's lectures on physics? [3] Ask HN: Best resources to gain math intuition? [4] Mathematics for the Adventurous Self-Learner [5]

[1]https://news.ycombinator.com/item?id=23074249 [2]https://news.ycombinator.com/item?id=18939913 [3]https://news.ycombinator.com/item?id=21346272 [4]https://news.ycombinator.com/item?id=20804582 [5]https://news.ycombinator.com/item?id=22400375

Since you don't have a goal, it's hard to recommend anything specific. Some good pointers

- Khan Academy is great because it covers a lot of material at school and some at college level so when you find that you have a gap you can always drop back a little and learn about stuff from school that you've forgotten. The explanations are very accessible and it's a great resource for re-learning stuff.

- MIT Opencourseware has courses for university-level mathematics. This is if you want to learn things more rigorously and the material is a bit more advanced obviously. Watch the lectures, read the textbooks, do the exercises. It's free world-class education. And once you have a foundation, you can always grab a textbook on a topic you want to go even deeper in.

- Youtube math channels like 3blue1brown are wonderful. I don't think they're for learning really but for exposing you to new ideas, giving you a different perspective on things or just allowing you to enjoy a nice explanation. It's infotainment.

In my opinion, this process should have two components.

The core should be working through textbooks, understanding the material as best as you can, and doing exercises. Start with something on the easier side to get things going. Search through online sources or alternative books whenever you get stuck, for more explanations and assistance.

In parallel, try to start forming a "bird's eye" view of the landscape, since there are a lot of areas you might want to focus more on later, by exposing yourself to more advanced, diverse topics and terms without any rigor. Things like watching a video, reading a non-technical book about math, checking out some random Wikipedia article you stumbled on that is way above your head etc.

I think this traditional, safe way is how you should start, and after a while you will know for yourself what to do next. There are a lot of excellent online sources, videos, blog posts etc. but there are also a lot of crappy, superficial ones, and you will need some experience to be able to judge that.

- "A Course of Higher Mathematics", VI Smirnov.

- "Problems in Mathematical Analysis", Demidovich.

- "Differential and Integral Calculus", Piskunov.

That's what our courses in university followed (first and second year), but also high-school teachers were inspired by as they studied these in university themselves. Smirnov takes you from the "concept of number" into the deep. Piskunov is cool and full of examples, including operational calculus and probability theory. Demidovich is a problems book to exercise your hand.

They are concise and to the point. Smirnov's has five volumes across seven books, though, but they're concise (density of information is high).

These books have examples from physics as well. For instance, I didn't understand enthalpy from my physics course until I read about continuity in Smirnov's book as he masterfully gave an example of discontinuity using enthalpy (phases of water).

I've been on a similar mission over the last few years. Youtube channels mentioned like 3Blue1Brown are great for deeper understanding of a topic, channels like Numberphile and Mathologer are great for getting excited about the fun and beauty of math. I bought a bunch of texts looking for the right place to start but ultimately what's worked for me is enrolling in virtual/online classes at my local community college. It's cheap, it's fun, and it's giving me the external motivation and community I've needed to make progress.

Book Of Proof by Richard Hammack[0]. A really nice and FREE introduction to higher math (or "actual" math). No particular prereqs other than comfort with doing arithmetic in fractions and decimals.

[0] https://www.people.vcu.edu/~rhammack/BookOfProof/

This is a good starting point: https://www.cambridge.org/us/academic/subjects/mathematics/l....

Take Poh-Shen Loh's Putnam seminar and his discrete math courses on his youtube channel https://learnaifromscratch.github.io/math.html#CMU%20Putnam%...

I wrote "A Programmer's Introduction to Mathematics" which might interest you. https://pimbook.org

Don't set out to "learn math", do things that require math to get them done. Run your own stats and summaries of Census data or something. Program your own gravity simulator, use any graphics or game engine you like, but do all the physics sims in your own code. Things like that.

I appreciate this methodology for getting stuff done, It's how I do my programming work, but I really do want to just "learn math" for the fun of it, like a puzzle, not make work. I was reading Descartes and he was writing about how the practice of math structured his thinking and I'd like to try to recreate that in myself. Maybe creating a project would be the best way to do it, but that seems like work rather than play to me.

This is a good technique for the engineering/physics side of the coin. For the more academic and "pure" math it might not be such a great goal. That said there's enough nice libraries and IDEs that one could use videos games to explore other mathematical contexts. But I don't get the impression that OP is a game programming enthusiast. Also that's a lot of overhead.

I have tried to do this. I stopped learning math when I was 16 - huge mistake!

What I would really like is something like a study group. It’s kind of difficult when you don’t understand something and there’s no one to turn to