Ask HN: Serious mathematics books that can replace a good teacher?
Mathematics is best learned under the guidance of a mentor. But not everyone has access to mentors all the time. That's where books come in. Good books. Books that can be substitute for a mentor or sometimes even better.
Which books (preferably not pop-sci) fall into this category?
173 comments
[ 3.6 ms ] story [ 226 ms ] threadI have tried reading textbooks a few times to teach myself but found it hard to stay motivated, so I found a tutor on Upwork to assign and grade homework problems and answer my questions. Along with math textbooks and YouTube videos, this has been super helpful for fleshing out my knowledge of college-level math that I never learned properly. It's also great because they can go at the pace you want and focus on topics you find challenging, interesting, or useful.
[1]: https://m.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVF...
[2]: https://m.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53Dw...
Mathologer: https://www.youtube.com/c/Mathologer
PBS Infinite Series is not so infinite (they've stopped making content), but it's very good. https://www.youtube.com/c/pbsinfiniteseries
https://www.youtube.com/c/TheMathSorcerer
https://www.youtube.com/c/mitocw
https://www.youtube.com/c/khanacademy
IIRC there might have been other books in a "Made Easy" series about math, but I'm not sure.
An orthodox approach to Analysis would be Terence Tao's Analysis books.
Further recommendations depend on your specific goals.
I found that she was an SRE engineer at uber, later she change corse to full time writing at some newspaper agency, what a change.
From physic to software engineering then fulltime writing, not anyone can do that
You might want to look into what she had to go through while she worked there. It's well known.
Also, Ms. Rigetti has similar guides for physics[1] and philosophy[2].
I intend to slowly go through her math guide over the years. I already started working through How to Prove It by Vellman.
[1] https://www.susanrigetti.com/physics
[2] https://www.susanrigetti.com/philosophy
I see this question as 'Why does math students not take Thermal Mechanics' or 'Why does math students not take Nuclear Methods'. The answer is just that, they do not teach math, they teach an application of math. Your math knowledge has not really grown.
Of course, then there's the fact that statistics builds upon probability theory, and probability theory is, in a sense, a subfield of measure theory which in turn is about as mathematical as it gets (in the discrete setting, it also includes a fair amount of combinatorics).
Well... yeah. I mean, I might be wrong, but my default assumption is that when people on HN ask about learning math, unless they explicitly say otherwise, they are mainly interested in maths from an applied viewpoint. That is to say, I think most such inquiries are rooted in a basis of "I want to learn the math require to DO 'x'" where x might be "machine learning" or "circuit analysis" or whatever, as opposed to "I want to become a mathematician and advance the overall state of mathematics as a field."
I say that at least in part because of an assumption that people who want to become mathematicians per-se are probably asking their questions on Mathoverflow or whatever, and not HN.
EDIT: to be fair the specific sub-thread we're in here does contain this, which I guess justifies taking a "pure mathematics" position in this part of the overall discussion.
Are probability and statistics not part of a regular mathematics curriculum?
Still though, this seems to be a general issue with any maths related discussion on HN. It seems like a lot of people are commenting from a position of assuming that the initial question was based on an interest in pure / theoretical maths and the "I want to become a mathematician" idea. And I am somewhat skeptical that that is normally what's intended by the person asking the initial question.
Other universities may have even stronger probability theory requirements, or none at all. But it's certainly not an uncommon specialisation.
Note that probability theory is not necessarily the same thing as statistics. While the latter builds upon the former, statistics is more about "given the data, what's the most likely distribution ", as opposed to "given the distribution, what's the data gonna look like" (probability theory). For mathematicians, the latter seems to be more relevant, as it's a more deductive form of reasoning.
I got a copy of this book from the 1920s which is really cool because it teaches you math lessons you have to actually go out and physically do stuff with like pegs and strings in a field, from the perspective of the history of mathematics where people were limited to such devices in order to do stuff like trigonometry. Very very different approach, probably not for everyone, but for me I just think it's pretty cool. It definitely was written in the 1920s though so you better get used to that particular writing style if you plan on digesting it like a course. It's designed that way, though, and it's got great reviews. Just keep in mind maybe some of the history is subject to have changed over the years. [2]
Ultimately I've self-taught myself a lot more than I ever learned in school for sure but a wide variety of sources is probably more what you're after in terms of getting a grip on what's interesting enough to pursue further for your own means and ends. I think exploring what fascinates you the most and then just going and finding things from that point is a pretty good start as long as you've got elementary understandings up to a point where the fascination actually happens.
[1] https://www.goodreads.com/book/show/83585.Secrets_of_Mental_...
[2] https://www.goodreads.com/book/show/66355.Mathematics_for_th...
Beyond just learning mathematics, seeing the art and beauty in it is also best taught by someone who knows the subject and the student. Without the beauty, it's just what could be in a textbook, assuming you found the right book.
If you're asking if you can find a good book to teach someone, that depends on your style and theirs . . .
We will almost certainly have good student-focused AI teachers during our lifetimes for things like math and languages. Will AI be able to show us the art? I can't say for sure, but I bet so . . .
Note that this does not mean one cannot fully learn a subject from books or that the average teacher is better than books or that a good teacher dealing with 30 other kids will be better than a good book.
I just read the beginning so I don't know how far it goes but anyway here the link https://calculusmadeeasy.org/
I taught myself all of A-level maths and further maths here in the UK from the standard text books at the time (Bostock and Chandler) before I started in the sixth form and then maybe about half of the first year university material before I went. Still better when you have somebody to teach you but not impossible. I did have access to somebody whom I could ask questions but didn't really use that.
Forums, chat groups/channels, pen pal, whatever it takes.
Yes, I know why people think it's a great comment, what I want to know is given how many of us learned about computers as children before we had any idea about functional programming or Turing machines, how do we get recommendation threads to move away from this?
Saying you want a "serious maths book" really is just too large a range. I could suggest an excellent book on computational group theory (my prefered area of maths), but honestly for most people that wouldn't be something they'd want to read.
- Calculus and basics of Analysis: Spivak
- Group Theory: Visual Group Theory
- Complex Analysis: Visual Complex Analysis
- Writing proofs: Proofs: A Long Form Textbook by Jay Cummings
- Abstract Algebra: Dummit-Foote book. First few chapters if you like.
___
Pop-Sci:
1. The Joy of X by Steven Strogatz
2. Fermat's Enigma by Simon Singh
(All this books except AA are very highly recommended by me)
It is the only book (I know of) that brings you from absolute basics to an integrated development of the subjects from its title. And that integrated developments actually leads to a good didactic method.
The search function will show you many comments on HN recommending it.
I disagree. The most important thing is curiosity.
For math, you need to think, question, and play with the concepts. It is much more active than just reading. Develop an intuition by seeing what happens when you maximize or minimize a parameter. Think about the consequences if something were different. Try to derive relations for fun.
This also applies to music and code.
If you figure out how things work on your own, you will retain it far better than going through the motions mechanically.
And having someone to answer questions and validate hypotheses makes the learning process a lot smoother and helps correct mistakes before they solidify.
I mean yeah, but a good teacher can point out your mistakes that are not obvious to you at the entry level. When it comes to i.e. music, such mistakes can make a bad habit that impacts the following development (wrong hand position, wrong fingering, etc). I don't know if that extends to math though.
More important than a good teacher, imo, is collaborators you're learning with who are around your same level. Sometimes they should figure out exercises faster than you, sometimes you should figure things out faster than them, often together. If that's not happening, it's a lot harder (and more frustrating) to learn. This becomes more important the more advanced you get. I never found lectures useful really but if I had no one to collaborate with I was almost certainly going to drop the class.
https://longformmath.com/analysis-home
>This book is the first of a series of textbooks which I am calling “long-form textbooks.” Rather than the typical definition-theorem-proof-repeat style, this text includes much more commentary, motivation and explanation. The proofs are not terse, and aim for understanding over economy. Furthermore, dozens of proofs are preceded by "scratch work" or a proof sketch to give students a big-picture view and an explanation of how they would come up with it on their own. Examples often drive the narrative and challenge the intuition of the reader. The text also aims to make the ideas visible, and contains over 200 illustrations.
>The writing is relaxed and includes interesting historical notes, periodic attempts at humor, and occasional diversions into other interesting areas of mathematics.
Have read this one, a book on advanced high-school math with a ‘problem-solving’ bent from a passionate teacher.
https://www.amazon.in/Educative-Jee-mathematics-PB-Joshi/dp/...
What makes a good book and why there are tons of them on the same subjects?
The best book for you is the one that speaks to your technical preparation and perspective. A few hits the sweet spot for a broad audience - perhaps because they are good at drawing analogies with common experiences - but even some obscure books can be good if it aligns with what your background.
How can a mentor help and can you do without one?
A mentor can help lay out the roadmap to build from simple topics to more difficult ones. Maybe more critically, provide rapid feedback on your understanding. They can also explain things in more than one way. Some textbooks do lay out the roadmap reasonably well, provided that it starts from concepts that you are already familiar with (again, you need to find the right book for you). Problem sets in textbooks are meant to provide feedback on your understanding, but it often fails to provide smaller hints if you can't solve the problem outright. You could get a set of solutions for the problems and that could partially help. Grabbing multiple textbook on the same subject can also help understand the most commonly covered (and by implication most essential) elements on the subject, and also give you multiple explanations of the same concept (though not always).
Takeaway message?
You could potentially try to pick up multiple books on the same subject and try to learn this way. Follow the one that speaks to your background most closely, but the others are also likely to help.
FWIW, one of my favorite maths Youtubers, the "Math Sorcerer"[1], highly encourages this approach. Don't fixate on one particular book, but buy many books on a given topic, and allow yourself to experience different presentations of the material. And as he often points out, if you're willing to accept used books, and older editions (which is often OK if you're not buying a book for a specific class), then you can quite often get copies really cheap from Alibris, bookfinder, etc.
[1]: https://www.youtube.com/c/TheMathSorcerer
I suppose this is really a parallelization of the "third textbook" model (i.e., if the third textbook you try when learning something new seems "much better" than the first two, it might because you actually did learn some things from the first two).