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 historical approach has really helped me develop a more complete mental picture and appreciation of the math itself. Understanding the "how" and "why" of its development, and seeing the early practical need and implementation for some of this stuff has made the topic a lot more engaging.
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.
It seems like you'd get a lot deeper understanding by doing it that way, and be much more able to adapt the knowledge to the real world, vs only knowing how to solve problems in the exact form they were presented to you. I had so many semesters of undergrad math, did fine, but feel like I took basically nothing from it.
This is a very entertaining hobby to have. Wishing you a lot of fun.
Next stop, making sundials and reading astrolabe.
I was so surprised to know that Chaucer had such interest in the workings of an Astrolabe. It's not much of a surprise if you think that Astrolabe were the pocket GPS, pocket watch, pocket star chart of those times.
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.
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;
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.
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.
This is a great series. I was awed by it in high school. Some parts were too advanced for me (and probably still are), but I got a lot out of other parts.
This is a great book if you already know good amount of Math. It helps you fit things into a bigger picture. Really appreciate the fact that something like this exists.
Soviet primary and secondary education on math was one of the few good things of the regime.
Culturally, mastery of mathematics, engineering, chess and technical inteligence were a source of social status and prestige.
Yes, an engineer could have been badly paid, he was not free in a liberal sense, be subject to the vagaries of political paranoia waves, but he commanded a certain level of social respect that even good paid engineers in the sillicon valley can hardly imagine. A soviet physicist could be underpaid and constrained by bureaucracy, but being introduced by your parents as a professor of the Keldysh Institute of Applied Mathematics or the Steklov Mathematical Institute in your home town give you almost an aura. Kids would look at you and dream of being admitted to secondary institutions like Kolmogorov's boarding school at the Moscow State University.
Given that, it was as likely for people in political positions of power to have a good mathematical background as it is to find a lawyer in the US Capitol.
This book is really fascinating because it contains a surprising amount of Soviet ideology. The authors repeatedly state that mathematics is posterior to the material world, not prior to it. That is, mathematics is just the observation of regularity in the world, particularly those discovered by people working to create things. Contrast this with the still heavily idealistic world of western mathematics, where mathematicians are more likely to sympathize with the notion that numbers are real things somewhere out there whose structures the real world supervenes upon in some way.
Interesting stuff!
Even though I favor the Soviet view of mathematics personally (I do not think numbers "exist" out there independent of the material world), I think this approach hampers the didactic goals of the text and probably hurt Soviet mathematics as well. The examples in the text are all highly concrete (literally things like rubber mats when discussing curvature). This very down to earth style makes the abstract notions of curvature in other contexts (for example, general relativity) more difficult to grasp, in my opinion.
On the other hand, some people prefer strong, material, examples of mathematical ideas. This book definitely provides that. The section on affine maps in terms of fixing the plane of a surveillance airplane photograph is beautifully concrete.
I thought everybody knew of this site especially Indians since it was started and is lovingly maintained by an Indian :-) A lot of hard work has gone into it with not just scanning of the books but also typesetting some of them in latex for better rendering and reading. The website author deserves all the credit/acclaim he can get.
2) Books by Lev Tarasov. Some of these books are structured as a dialogue and thus walks you through the conceptual process. Titles are on Calculus, Probability, Quantum Mechanics, Physics Q/A etc. All worth reading; some of them are latex remastered versions - https://mirtitles.org/?s=Tarasov
3) Higher Math for Beginners(with Yaglom) and Elements of Applied Mathematics(with Myskis) by Zeldovich - https://mirtitles.org/?s=zeldovich
4) Mathematical Handbook (two vols); Elementary Mathematics and Higher Mathematics by M.Vygodsky. - https://mirtitles.org/?s=vygodsky
Next stop, making sundials and reading astrolabe.
I was so surprised to know that Chaucer had such interest in the workings of an Astrolabe. It's not much of a surprise if you think that Astrolabe were the pocket GPS, pocket watch, pocket star chart of those times.
I don't that's against code (yet).
Just knowing that the Sun doesn't really rise on the East (barring exceptions) is a fun reward in itself.
Solarigraphy and Analemma tracking are great fun if you have the luxury of an undisturbed access to the skies.
https://en.wikipedia.org/wiki/Solarigraphy
https://archive.org/details/MathematicsItsContentsMethodsAnd...
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...
[2]: https://terrytao.wordpress.com/books/analysis-i/
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.
Culturally, mastery of mathematics, engineering, chess and technical inteligence were a source of social status and prestige.
Yes, an engineer could have been badly paid, he was not free in a liberal sense, be subject to the vagaries of political paranoia waves, but he commanded a certain level of social respect that even good paid engineers in the sillicon valley can hardly imagine. A soviet physicist could be underpaid and constrained by bureaucracy, but being introduced by your parents as a professor of the Keldysh Institute of Applied Mathematics or the Steklov Mathematical Institute in your home town give you almost an aura. Kids would look at you and dream of being admitted to secondary institutions like Kolmogorov's boarding school at the Moscow State University.
Given that, it was as likely for people in political positions of power to have a good mathematical background as it is to find a lawyer in the US Capitol.
This is what has severely degraded in current-day global society/culture and due to globalization exported everywhere.
However, Soviet-era Science/Mathematics books are still highly regarded/prized in many countries by the older generation.
Interesting stuff!
Even though I favor the Soviet view of mathematics personally (I do not think numbers "exist" out there independent of the material world), I think this approach hampers the didactic goals of the text and probably hurt Soviet mathematics as well. The examples in the text are all highly concrete (literally things like rubber mats when discussing curvature). This very down to earth style makes the abstract notions of curvature in other contexts (for example, general relativity) more difficult to grasp, in my opinion.
On the other hand, some people prefer strong, material, examples of mathematical ideas. This book definitely provides that. The section on affine maps in terms of fixing the plane of a surveillance airplane photograph is beautifully concrete.
On Teaching Mathematics by V.I.Arnold - https://archive-dsweb.siam.org/The-Magazine/All-Issues/vi-ar...
Mathematics and Physics: mother and daughter or sisters? by V.I.Arnold - https://ufn.ru/en/articles/1999/12/c/
https://mirtitles.org/tag/mathematics/
I have very much enjoyed the OP (Mathematics: Its Content, Methods and Meaning). Any particular books that you would recommend?
I thought everybody knew of this site especially Indians since it was started and is lovingly maintained by an Indian :-) A lot of hard work has gone into it with not just scanning of the books but also typesetting some of them in latex for better rendering and reading. The website author deserves all the credit/acclaim he can get.
All of the books are hosted on internet archive which you can browse here - https://archive.org/details/mir-titles
Many of the titles are also published in hard copy form by low-cost Indian publishers on Amazon India.
Some recommendations which are not typical textbooks;
1) Little Mathematics Library series - https://mirtitles.org/2024/05/11/little-mathematics-library-...
2) Books by Lev Tarasov. Some of these books are structured as a dialogue and thus walks you through the conceptual process. Titles are on Calculus, Probability, Quantum Mechanics, Physics Q/A etc. All worth reading; some of them are latex remastered versions - https://mirtitles.org/?s=Tarasov
3) Higher Math for Beginners(with Yaglom) and Elements of Applied Mathematics(with Myskis) by Zeldovich - https://mirtitles.org/?s=zeldovich
4) Mathematical Handbook (two vols); Elementary Mathematics and Higher Mathematics by M.Vygodsky. - https://mirtitles.org/?s=vygodsky
5) Handbook of Physics by Yavorsky and Detlaf - https://mirtitles.org/?s=Handbook+of+Physics
6) Fundamentals of Physics by Ivanov - https://mirtitles.org/2018/04/21/fundamentals-of-physics-iva...
7) Ever popular Science/Maths books by Yakov Perelman viz. Physics for Entertainment, Mathematics can be Fun etc. - https://mirtitles.org/?s=Yakov+Perelman