3 ms·

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

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?

rramadass 3mo ago
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/HoddsAlcockInglisSelfExplanation.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-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_Mathematician https://en.wikipedia.org/wiki/How_to_Think_Like_a_Mathematic...
- hamburgererror 3mo ago
  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_Analysis https://en.wikipedia.org/wiki/Principles_of_Mathematical_Ana... [2]: https://terrytao.wordpress.com/books/analysis-i/ https://terrytao.wordpress.com/books/analysis-i/
  - rramadass 3mo ago
    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 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.
    - nathan_compton 3mo ago
      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.
      - rramadass 3mo ago
        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.
  - BeetleB 3mo ago
    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.
- dtj1123 3mo ago
  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.
dtj1123 3mo ago
I recently started Introduction to Probability theory by Blitzstein and it feels a lot like what you're referring to. 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.
- hamburgererror 3mo ago
  Thanks a lot for the reference, sounds a exactly what I'm looking for!

rramadass 3mo ago

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/HoddsAlcockInglisSelfExplanation.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-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_Mathematician https://en.wikipedia.org/wiki/How_to_Think_Like_a_Mathematic...

hamburgererror 3mo ago
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_Analysis https://en.wikipedia.org/wiki/Principles_of_Mathematical_Ana... [2]: https://terrytao.wordpress.com/books/analysis-i/ https://terrytao.wordpress.com/books/analysis-i/
- rramadass 3mo ago
  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 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.
  - nathan_compton 3mo ago
    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.
    - rramadass 3mo ago
      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.
- BeetleB 3mo ago
  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.
dtj1123 3mo ago
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.

dtj1123 3mo ago

I recently started Introduction to Probability theory by Blitzstein and it feels a lot like what you're referring to. 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.

hamburgererror 3mo ago
Thanks a lot for the reference, sounds a exactly what I'm looking for!