I'm looking for mathematics books that take the time to explain with words and sentences what is actually going on when they introduce a new theorem, something that focuses on meaning.
Anyone knows if such books exist?
Anyone knows if such books exist?
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...
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...
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.
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 haven't tried it with analysis, but I did for linear algebra. It would quickly spot flaws in my proof.
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.
The author introduces something called 'proof by story', where instead of just running through algebra to prove equivalence of two expressions, you describe a single physical scenario in two ways that each intuitively correspond to the two expressions. I've never seen maths done this way before, but I have to say it works brilliantly.