I'm looking for resources on this too. I recently started working through this book [1], which might be a good place to start. In the introduction to that, the author also mentions this site [2] and this book [3].
Becoming an expert in one thing takes a lot of time and effort, and this must be maintained even after reaching the goal. But getting a reasonable understanding of many things is not too bad. E.g. working through a freshman level physics text will give you a pretty good (or above average, at least) understanding of the field of physics, and this could be done in a span of months. Maybe try something like this for one subject after another. It could be that this amount of knowledge satisfies you for a given subject and you don't need to go further. But if it doesn't satisfy you, move on to more material. If there really are things you want to spend the time and energy on to reach expert level, you will naturally find them this way. And I suspect that list of things will be a lot smaller than when you started.
Many people recommend learning to write proofs in the context of a class or text that's focussed on another topic, e.g. geometry or linear algebra or real analysis or abstract algebra. But I preferred learning from a book that was more proof-focussed and added context along the way. Here's the text I learned proof-writing from, and which I highly recommend:
https://www.whitman.edu/mathematics/higher_math_online/
The curriculum guides Susan Rigetti provides are an amazing resource for self-study. And the fact that she worked through all of this is truly inspiring.
Not to be greedy, but do any of you know of other thorough curriculum guides like this? I know about https://teachyourselfcs.com already -- another amazing guide. Are there others? I would love to find one for statistics especially, but really any subject would be interesting.
The (unpublished) book "Introduction to Higher Math"[1], along with the professor who taught the course, was probably what got me going on the road to a PhD in math. It's very accessible (no prerequisites to speak of), teaches the reader how to write mathematical proofs, and includes some really interesting material, e.g. cardinality. It's also freely available now.
If you enjoyed logic, you might enjoy reading/working through this text[0]. I learned from that book when it was just written, and it remains one of my favorite books/classes. As you say, it was so very different from previous math classes I'd taken, and it opened up a whole new way of thinking for me. Although it was used for a college course, the prerequisites are minimal -- maybe at most elementary algebra?
"Aronofsky was inspired by such real-life pi-chasers as the Russian mathematician brothers Chudnovski who built a supercomputer in their own cramped Manhattan apartment and calculated pi to two billion digits. But most people don't look for patterns in pi, he says: 'You basically use it to test the speed of a supercomputer. The brothers called me a quack - they told me there's no pattern.'"
Part 3: https://arstechnica.com/gadgets/2023/01/a-history-of-arm-par...