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

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95 points atomicnature | 1 comments | 31 Aug 25 04:31 UTC | HN request time: 0s | source
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seanhunter ◴[31 Aug 25 13:32 UTC] No.45083039[source]▶
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.

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1. atoav ◴[01 Sep 25 07:19 UTC] No.45090310[source]▶
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.