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

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.

> "Topics in Algebra" by Herstein. this is a lovely book and beautifully written but some of the notation is a bit dated

Now you make me feel old! I had the second edition just as it came out in 1984. A long time ago but I remember it as one of my favourites.

I heard from somewhere that Herstein was an Emil Artin student at Princeton.

Herstein went to the extra trouble to make his linear algebra also work for finite fields.

In the back he has group representation theory is a small nutshell.

Also in the back he does linear programming, but his treatment is obscure and for no good reason. Since then nearly every treatment, beginner or advanced is not obscure at all.

Math genealogy project says Herstein was a student of Max Zorn (the guy the lemma is named after), but Zorn was a student of Artin! Zorn doctorized under Artin in Hamburg in 1930, before Artin moved to Princeton.

https://www.genealogy.math.ndsu.nodak.edu/id.php?id=6526

Have an apoplectism

https://agorism.dev/book/math/ag/royal-road-algebraic-geomet...

Also iirc, various books titled Royal Road to Geometry, which might make more sense given that IMO geom was the first to get crushed by chatbots

Ymmv, but denying existence of royal road to math seems like a narcissism limited to the most elite layity

Of course there is no single best road to mathematics, mainly because the way humans are able to think isn't uniform.

There are people with more or less Aphantasia, so people who can't or have a hard time forming mental images. Then there are others who can rotate 3 bezier curves in their head and plot the intersections.

For students of the latter category relying on mental images is a great way to teach them, for the former it is catastrophical.

Anybody who has thought anything mathematical should be aware of the fact that different people prefer different ways of learning.

Thanks for the recommendation. That book looks very interesting although I'll need a bit more algebra before I can really tackle it properly.

Not sure I fully understand your point about narcissism and laity. If Euclid and I are laity then I think I'm happy to be called that though. For me the "royal road" refers to the idea that you can take shortcuts and understand things without putting in the work. That just does not ring at all true to my experience with maths anyway.