Mathematics for the Adventurous Self-Learner

https://www.youtube.com/watch?v=WUvTyaaNkzM&list=PLZHQObOWTQ...

Just reading doesn't get much at all. Not even a superficial overview. I tried it. It's essentially a meaningless combination of words after a certain point.

Reading extremely thoroughly is actually marginally useful. Stopping to think, do all these assumptions matter, why, what if one of them changes, etc, pencil in hand, making notes, testing things out. I've managed to "understand" the topics when doing this, and so far it's been the highest ROI method. But it does still leave one feeling like something is missing. Just because you can sight read music doesn't mean you're an expert on the piano.

Doing exercises is a huge jump on investment, and the return on that investment is a bit questionable from my experience. A couple reasons: first you don't know if you did them right. If you did them wrong then that's negative ROI. Second you don't know what a "reasonable" workload is. It varies by author. Is it three problems per chapter, is it all of them, are some orders of magnitude more difficult than others? Without some guidance it's hard to know if your difficulties are due to not understanding basic material, or due to that problem being a challenge geared toward Putnam medalists. So they may cause you to question your understanding and thus mentally roadblock you unnecessarily. And finally with proofs (and this may be a me thing), it's pretty easy to say "I guess this is okay(?)" and move on, even if you're not sure. Since nobody is ever going to review it, and it's just a homework problem, it's very very hard to will oneself to make sure every assumption is correct and you're not missing anything, even if you feel like there's a good chance you are. Or perhaps I just don't have the constitution to do so.

So while I think doing exercises is necessary for a deeper understanding, I don't know whether the ROI is worth it outside of a classroom perspective. You need feedback for exercises to be beneficial. At least, I feel like I do.

Finally, is even taking a class that useful if the end state is that two years from then you'll have forgotten most of it and so what was the point. Can you claim knowledge of a subject that you've never actually used beyond some homework problems and exam questions, or is this still a superficial understanding? Having an ends where that knowledge gets used seems critical.

I feel like I have some knowledge but I don't feel like I'm there yet. But I don't know if I know where there is. Maybe that's the biggest challenge. Does completing a Ph.D. even get you to there? No idea. But, I guess it's up to the individual to decide what they want out of it. Nobody can determine that for you.

So in my first year of studying maths, I had 8 hours of maths lectures (and 4 hours of a minor which was of negligible effort). The exercises that came with the maths lectures made this a full-time program (and I estimate that while I didn't usually study all weekend, I typically only had 1-2 weekends per year in which I didn't look at anything at all). So one thing that can easily go wrong is underestimating that for every minute spent reading / listening to a class, one would want to spend 4 minutes working the problems.

The other comment I would have is that, yes, university level mathematics is (at least it was for me) incredibly hard. The reward is also astonishing: All these hard exercises I struggled with one year, are easy to do on a napkin during breakfast in the next year.