Sheafification – The optimal path to mathematical mastery: The fast track (2022)

And vibe-coding is barely coding, nevermind software engineering. (;

Linear Algebra and Geometry by Igor Shafarevich, coupled with Linear Algebra by Friedberg, Insel, and Spence; the latter has great problems to work with, whereas the former is for a lucid exposition.

The Feynman Lectures are in a very different category from Landau & Lifshitz. You shouldn’t use L&L as your first university-level physics textbook (too difficult and divorced from the real world), but you probably shouldn’t use the FLP as your second, either, provided your first was good (too easy to be worth the time spent, though it’s still useful as pleasure reading that also fills in things you might have missed).

I have to admit I like the FLP less than the typical reader—it’s immensely fun in the moment, but I’ve always found the material too disjointed to build a coherent picture (ah, and now we have the tools to understand this random fun thing that I’ve never mentioned before and never going to mention after). As far as classic introductory books, the Berkeley course covers less, but the things it does cover fit together much better.

As for L&L, it’s just very uneven in quality. General relativity is great; electromagnetism is kind of bad. (And the two are in the same book!) Theoretical mechanics is adequate but way too much of a slog despite how short it is. Elasticity is surprisingly good. QM is OK but there’s a dozen approaches to QM depending on your background and it’s a toss-up whether this one will work for you. QED is good at the things it covers but like half of those things are relegated to the status of obscure specialist topics these days, while the point of view is something you should be aware of eventually but definitely not at your first go through the subject. And so on.

There is no royal road to mathematics[1], and it's incredibly arrogant to think that any person can provide a single optimal path. For me for example the next steps are Axler, Abbott and Herstein[2]. That's where I am at the moment, and it's way earlier than the books listed here. It would be far from optimal for me to try to bang my head stubbornly on this list. Mathematics demands you put in the work to build a foundation - you cant just skip steps. For some people those books I listed are very rudimentary. For others they are definitely too advanced for where they are and they'll need something else.

Even more so is the idea that you can actually cover the material listed in that page in 3 years. If you were to blast through it in that time you would only be skimming the very surface of the topics. There's simply no way you could possibly do all of those subjects justice in that time.

[1] As Euclid is supposed to have said about geometry to the Pharoah Ptolemy when Ptolemy said he wanted to learn geometry but because of all the concerns of his kingdom he didn't have time to read the Elements.

[2] "Linear Algebra done Right" by Sheldon Axler

"Understanding Analysis" by Stephen Abbott

"Topics in Algebra" by Herstein. this is a lovely book and beautifully written but some of the notation is a bit dated. I have two more recent algebra books but they are a bit advanced for me until I work through Herstein. They are Aluffi "Algebra Chapter 0" which is a good modern algebra book which introduces category theory at the start and Hien I forget the title but it's a springer one that he claims is good for an introduction but it's definitely not. It assumes you know a lot. It's very good though.