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Not sure what you are looking for. Mathematics is just a shorthand, precise language to represent wordy natural language concepts succinctly. So when you see the symbols you have to expand it in your mind (and/or use paper/pencil) into equivalent natural language concepts which the symbols model. There are no shortcuts but only time, effort and patience.

There are a bunch of books on how to do/understand Proofs/Theorems etc. but without knowing what you are specifically looking for i can only make some general recommendations;

1) How to Read Proofs: The ‘Self-Explanation’ Strategy (pdf) - http://www.ma.rhul.ac.uk/~uvah099/Maths/HoddsAlcockInglisSel...

2) How to Read, Understand and Study Proofs - https://mikepawliuk.ca/2014/03/31/how-to-read-understand-and...

3) How to Think Like a Mathematician by Kevin Houston - https://en.wikipedia.org/wiki/How_to_Think_Like_a_Mathematic...

2 comments

Thanks for the references, I'll have a look. I can tell you what I don't want: books that throw theorems one after the other without any context, like Baby Rudin [1] for instance.

I tried once to read Terence Tao's Analysis I [2], it's really good but my main problem was that I wasn't able to know if my proofs are correct when I do an exercise. So maybe the solution is to get a teacher.

[1]: https://en.wikipedia.org/wiki/Principles_of_Mathematical_Ana...

[2]: https://terrytao.wordpress.com/books/analysis-i/

AI is a godsend here. One can use the Socratic (aka Maieutic) method with AI to both explain and verify one's understanding. For some references see https://news.ycombinator.com/item?id=48342887

I have used Google's "AI Overview" and "AI Mode" profitably to have it explain new concepts to me in simple terms and also have had it verify my own understanding of something. The reason i prefer it is because it is a search-driven reasoning engine and thus gives references (in tooltips) directly to the latest sources/articles on the web from which it synthesized its answers.

I've got a PhD in Physics (undergraduate in Math, fair amount of self study) and I have to caution against taking what the AI's say seriously.

One has to know what one is doing and think before doing it.

As an example, there can be "AI Librarian" and "AI Reasoner" personas which set boundaries for what each should do. The former is for organization of data with citations and synthesizing answers objectively with simple explanations while the latter is for more detailed deep dives with logical analysis and thinking.

Note however, that persona's must be distinct in terms of data usage and not role-playing which is largely ineffective.

I'll second the suggestion to use a good LLM.

I haven't tried it with analysis, but I did for linear algebra. It would quickly spot flaws in my proof.

Kevin Houston introduced me to the concept of a number system. Cool guy.

Hard disagree that mathematics is just a shorthand notation though. It's a body of thought, independent of the symbols you choose to represent it.