Mathematics: Its Content, Methods and Meaning | Hacker News Reader

Mathematics: Its Content, Methods and Meaning

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85 pointsteleforce43 comments

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.

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.

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).

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

Sorry, author's name was Van Brummelen, not Van Bremmelen.

You know you can edit your comment right. You are still in that edit window.

Edit window is 10m now.

Oh! My bad.

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.

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.

Yep, build up your interior knowledge graph, each connection reinforces the nodes it joins.

Full 3 volumes of Free pdfs on IA:

https://archive.org/details/MathematicsItsContentsMethodsAnd...

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.

In this context, people should also read;

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/

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

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.

FWIW the Preface says it's written for people with secondary school mathematics education - whatever that meant in the Soviet Union in the 1950s.

Yeah, but they meant Soviet secondary school...

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.

> 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.

Right. The book is accessible to a smart student in grades 9-10 and most everybody in grades 11-12.

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

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...

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/

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.

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.

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.

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.

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.

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.

Thanks a lot for the reference, sounds a exactly what I'm looking for!

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.

Where can I find mathematical book titles like this one?

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?

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

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

Thanks a lot! Very helpful.

Wonderful!

I grew up fascinated by some of the books from MIR publisher sI could lay my hands on.

> Many of the titles are also published in hard copy form by low-cost Indian publishers on Amazon India.

Any way to quickly filter for these? (Other than of course searching for each book by title)

Thanks!

EDIT:

The one or two titles I found were not really priced economically --

https://www.amazon.in/Satellite-typhoon-Eye-Science-everyone...

https://www.amazon.in/Our-Planet-Earth-Alexei-Byalko/dp/0828...

https://www.amazon.in/Yakov-Perelman%C2%92s-Physics-Entertai...

> Any way to quickly filter for these? (Other than of course searching for each book by title)

Search for author name and/or title name. Generally the Indian publishers only publish well-known and famous textbooks/exambooks since they are the ones which sell the most. Eg. Piskunov (Calculus in 2 vols), Irodov (Physics problems) etc. The publishers are CBS Publishers, Arihant Publishers, Aargon Press etc.

The links you have listed are for the original Mir publications from 1970s/1980s which are collector's items today and hence some enterprising individuals are selling them at a markup on Amazon.

Finally, the best way to build up your personal library and enjoy books today, is to get the largest screen Android tablet that you can afford and install a ereader app like Moon+ReaderPro (https://www.moondownload.com/) and start downloading ;-)

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

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