6 ms·

Mathematics: Its Content, Methods and Meaning

85 points by teleforce 4mo ago

ykonstant 4mo ago
This is one of the best generalist books on mathematics ever published. I highly recommend it.
o4c 3mo ago
Full 3 volumes of Free pdfs on IA: https://archive.org/details/MathematicsItsContentsMethodsAndMeaningVol3 https://archive.org/details/MathematicsItsContentsMethodsAnd...
rohityin 3mo ago
The only mathematics books I ever read was textbooks in school but now as adult I want to start from scratch.
- deleted 3mo ago
  [deleted]
deleted 3mo ago
[deleted]
helterskelter 3mo ago
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.
- shanusmagnus 3mo ago
  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.
- srean 3mo ago
  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.
  - helterskelter 3mo ago
    Honestly I've been thinking about putting some standing stones in my yard to act as solar clock and calendar, maybe doing a lunar calendar as well... I don't that's against code (yet).
    - srean 3mo ago
      All the best for your Stonehenge. 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://en.wikipedia.org/wiki/Solarigraphy
- helterskelter 3mo ago
  Sorry, author's name was Van Brummelen, not Van Bremmelen.
  - srean 3mo ago
    You know you can edit your comment right. You are still in that edit window.
justincredible 3mo ago
[dead]
__rito__ 3mo ago
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.
- mmooss 3mo ago
  FWIW the Preface says it's written for people with secondary school mathematics education - whatever that meant in the Soviet Union in the 1950s.
  - kmt-lnh 3mo ago
    Yeah, but they meant Soviet secondary school...
  - elzbardico 3mo ago
    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.
    - rramadass 3mo ago
      > Culturally, mastery of mathematics, engineering, chess and technical intelligence were a source of social status and prestige. 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.
mohamez 3mo ago
Where can I find mathematical book titles like this one?
- clmul 3mo ago
  Look for books from "Mir publishers", there are some nice soviet books. https://mirtitles.org/tag/mathematics/ https://mirtitles.org/tag/mathematics/
  - alok-g 3mo ago
    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?
    - rramadass 3mo ago
      Not the person you asked the question of; 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 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-complete/ 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 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 https://mirtitles.org/?s=zeldovich 4) Mathematical Handbook (two vols); Elementary Mathematics and Higher Mathematics by M.Vygodsky. - https://mirtitles.org/?s=vygodsky https://mirtitles.org/?s=vygodsky 5) Handbook of Physics by Yavorsky and Detlaf - https://mirtitles.org/?s=Handbook+of+Physics https://mirtitles.org/?s=Handbook+of+Physics 6) Fundamentals of Physics by Ivanov - https://mirtitles.org/2018/04/21/fundamentals-of-physics-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 https://mirtitles.org/?s=Yakov+Perelman
nathan_compton 3mo ago
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.
- rramadass 3mo ago
  In this context, people should also read; On Teaching Mathematics by V.I.Arnold - https://archive-dsweb.siam.org/The-Magazine/All-Issues/vi-arnold-on-teaching-mathematics.html 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://ufn.ru/en/articles/1999/12/c/
hamburgererror 3mo ago
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
throwaway81523 3mo ago
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.

85 points by teleforce 4mo ago

This is one of the best generalist books on mathematics ever published. I highly recommend it.

o4c 3mo ago

Full 3 volumes of Free pdfs on IA: https://archive.org/details/MathematicsItsContentsMethodsAndMeaningVol3 https://archive.org/details/MathematicsItsContentsMethodsAnd...

rohityin 3mo ago

The only mathematics books I ever read was textbooks in school but now as adult I want to start from scratch.

deleted 3mo ago
[deleted]

deleted 3mo ago

[deleted]

helterskelter 3mo ago

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.

shanusmagnus 3mo ago
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.
srean 3mo ago
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.
- helterskelter 3mo ago
  Honestly I've been thinking about putting some standing stones in my yard to act as solar clock and calendar, maybe doing a lunar calendar as well... I don't that's against code (yet).
  - srean 3mo ago
    All the best for your Stonehenge. 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://en.wikipedia.org/wiki/Solarigraphy
helterskelter 3mo ago
Sorry, author's name was Van Brummelen, not Van Bremmelen.
- srean 3mo ago
  You know you can edit your comment right. You are still in that edit window.

justincredible 3mo ago

[dead]

__rito__ 3mo ago

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.

mmooss 3mo ago
FWIW the Preface says it's written for people with secondary school mathematics education - whatever that meant in the Soviet Union in the 1950s.
- kmt-lnh 3mo ago
  Yeah, but they meant Soviet secondary school...
- elzbardico 3mo ago
  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.
  - rramadass 3mo ago
    > Culturally, mastery of mathematics, engineering, chess and technical intelligence were a source of social status and prestige. 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.

mohamez 3mo ago

Where can I find mathematical book titles like this one?

clmul 3mo ago
Look for books from "Mir publishers", there are some nice soviet books. https://mirtitles.org/tag/mathematics/ https://mirtitles.org/tag/mathematics/
- alok-g 3mo ago
  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?
  - rramadass 3mo ago
    Not the person you asked the question of; 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 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-complete/ 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 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 https://mirtitles.org/?s=zeldovich 4) Mathematical Handbook (two vols); Elementary Mathematics and Higher Mathematics by M.Vygodsky. - https://mirtitles.org/?s=vygodsky https://mirtitles.org/?s=vygodsky 5) Handbook of Physics by Yavorsky and Detlaf - https://mirtitles.org/?s=Handbook+of+Physics https://mirtitles.org/?s=Handbook+of+Physics 6) Fundamentals of Physics by Ivanov - https://mirtitles.org/2018/04/21/fundamentals-of-physics-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 https://mirtitles.org/?s=Yakov+Perelman

nathan_compton 3mo ago

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.

rramadass 3mo ago
In this context, people should also read; On Teaching Mathematics by V.I.Arnold - https://archive-dsweb.siam.org/The-Magazine/All-Issues/vi-arnold-on-teaching-mathematics.html 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://ufn.ru/en/articles/1999/12/c/

hamburgererror 3mo ago

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

throwaway81523 3mo ago

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.