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Mathematics: Its Content, Methods and Meaning

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Re: Mathematics: Its Content, Methods and Meaning

#31

Post by throwaway81523 »

helterskelter wrote: post #6

I've been relearning trigonometry lately by myself for navigation and astronomy; not for work, just curiosity I guess. One book I've really enjoyed is Heavenly Mathematics by Van Bremmelen. It's a spherical trig textbook, but it's written by a math historian who describes how trigonometry was gradually developed over human history and he discusses its early proofs, methods and applications. I have to confess that the…

Is spherical trig still a thing? Calculation is so cheap now. If you want to find the spherical distance between cities from first principles, it's easiest to convert to 3D rectangular coordinates and find the angle ABC from the vector dot product, where A and C are the cities and B is the center of the earth. Same for other types of navigational calculations like headings. But, I've never looked into it really.

Re: Mathematics: Its Content, Methods and Meaning

#32

Post by throwaway81523 »

alok-g wrote: post #27
clmul wrote: post #23

Earlier quoted context omitted.

Look for books from "Mir publishers", there are some nice soviet books. https://mirtitles.org/tag/mathematics/

Very very cool! Somehow most names are unfamiliar. I have very much enjoyed the OP (Mathematics: Its Content, Methods and Meaning). Any particular books that you would recommend?

Look also at https://www.sumizdat.org/ which is Alexandre Givental's small press.

Re: Mathematics: Its Content, Methods and Meaning

#33

Post by rramadass »

hamburgererror wrote: post #28

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?

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...

Re: Mathematics: Its Content, Methods and Meaning

#34

Post by hamburgererror »

rramadass wrote: post #33
hamburgererror wrote: post #28

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?

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/Theor…

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/

Re: Mathematics: Its Content, Methods and Meaning

#35

Post by rramadass »

hamburgererror wrote: post #34
rramadass wrote: post #33

Earlier quoted context omitted.

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/Theor…

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/…

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.

Re: Mathematics: Its Content, Methods and Meaning

#36

Post by soulofmischief »

helterskelter wrote: post #6

I've been relearning trigonometry lately by myself for navigation and astronomy; not for work, just curiosity I guess. One book I've really enjoyed is Heavenly Mathematics by Van Bremmelen. It's a spherical trig textbook, but it's written by a math historian who describes how trigonometry was gradually developed over human history and he discusses its early proofs, methods and applications. I have to confess that the…

This is the approach I take to learning just about everything: Ask the five W's: Who, what, when, where, why?

It compounds, as you learn more history, you're able to connect more and more dots, and learning new information becomes easier on average.

Re: Mathematics: Its Content, Methods and Meaning

#37

Post by BeetleB »

hamburgererror wrote: post #34
rramadass wrote: post #33

Earlier quoted context omitted.

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/Theor…

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/…

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.

Re: Mathematics: Its Content, Methods and Meaning

#38

Post by dtj1123 »

hamburgererror wrote: post #28

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?

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.

Re: Mathematics: Its Content, Methods and Meaning

#39

Post by dtj1123 »

rramadass wrote: post #33
hamburgererror wrote: post #28

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?

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/Theor…

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.

Re: Mathematics: Its Content, Methods and Meaning

#40

Post by helterskelter »

throwaway81523 wrote: post #31
helterskelter wrote: post #6

I've been relearning trigonometry lately by myself for navigation and astronomy; not for work, just curiosity I guess. One book I've really enjoyed is Heavenly Mathematics by Van Bremmelen. It's a spherical trig textbook, but it's written by a math historian who describes how trigonometry was gradually developed over human history and he discusses its early proofs, methods and applications. I have to confess that the…

Is spherical trig still a thing? Calculation is so cheap now. If you want to find the spherical distance between cities from first principles, it's easiest to convert to 3D rectangular coordinates and find the angle ABC from the vector dot product, where A and C are the cities and B is the center of the earth. Same for other types of navigational calculations like headings. But, I've never looked into it really.

Yep, it's mostly used in for navigation, especially as a backup in case GPS fails (in the case of a lightning strike on a sailboat, for instance, probably all your electronics are toast). The military still uses it with slide rules/log tables, in case of jamming, etc. It kind of "disappeared" sometime after WWII, and there aren't many new books on it, but you'll see it come up in computer graphics and animation quite a bit, and I'm pretty sure it's used extensively in surveying still.
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