Hacker News new | past | comments | ask | show | jobs | submit login
Ask HN: Serious mathematics books that can replace a good teacher?
288 points by newsoul on May 24, 2022 | hide | past | favorite | 168 comments
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?




Been mulling viability of hiring professors and TAs to give 10 day intensive math and physics lecture seminars backed by tutorials in beach settings. Sort of like taking the lectures of theoreticalminimum.com but on the road, where you can join a session with a small group of ~10 students in places like Barbados, BVI, Costa Rica, mostly outdoors. If the economics can work for for yoga, we can probably make them work for an amateur/programmer interest in math. No certifications, maybe just walk through some coursera and khan material as prep.

Want to teach geeky tourists calculus? If we made math tourism a thing, teachers in host countries could skill up pretty fast as well.


I would much rather do this sort of thing in a conference room in a generic hotel. No distractions.

If I’m going to travel all the way to a stunning vacation destination, I’m probably going to want to drink a beer on the beach instead of trying to focus on a class.


Languages and yoga vacations are a huge thing and they typically do them outside conference rooms. (5-7 day probably better) I'd thought about using them, but being totally indoors is a waste of being somewhere. So long as you have shade and a blackboard, you have everything you need. There's no exam, maybe just a final problem set you get on the way in and work your way through.

Idea is ~2-3h interactive lecture session in morning by interesting prof, ideally with no electronics, an optional recommended and personalized TA tutorial in the afternoon to catch anyone up, some fun activities like watching sunset on a sailboat and then using celestial navigation to find your way back, some partnerships with locals on regular stuff like surf lessons, musical performances, etc. Priced at a small premium to whatever all inclusives are.

It's not for everyone, but the people it's for, it's really for. It's good for solo travellers, can qualify as business job training, or employee rewards/incentive/teambuilding. A way for poorly paid TAs to make some money and get a paid trip. Any product manager working in the ML space would need to do it. Developing local teaching talent could become a significant industry as well...

There goes my day, anyway.


Perhaps, tie some of the mathematics to the region/location where the mathcation is. Mathematical thinking does require space (so free time after "classes including learning mathematically about the location) to just think freely in addition to the rigorous focus needed when actively working on a problem or learning a concept.


Different strokes.

I will say that I have experience teaching academic subjects in exotic locations including some of those mentioned (Semester at Sea/Abroad type programs) and it is HARD to effectively teach somewhere that isn't a classroom. A boring/utilitarian classroom (the room itself) is a feature, and classrooms the world over look the same for that reason..

If my company is paying for it though...


Speaking as a physics professor, you can listen to me talk about awesome physics all you want, and maybe feel like you're getting something out of it. But if you actually want to learn math or physics at any level beyond "I've heard those words before and they sounded cool", you absolutely must be doing it: solving problems as practice, struggling with how to apply subtle concepts to new situations, and just absorbing the details for a while.

Are you genuinely envisioning this math/physics beach vacation as "mostly homework"? Because if it's not going to be a completely surface-level experience, it's going to be "mostly homework". :)


Was thinking 15h of lectures, ~20h optional TA tutorials over 5 day period. Undergrad level stuff, as it's not for academics, but interested amateurs and we go home after a week with something to work on, maybe come back for another session once a year.

I'd ask, would you pay $5k to hang out in a yurt on Anegada for a week and learn an undergrad intro to category theory?

This: https://anegadabeachclub.com/

Plus this:

https://ocw.mit.edu/courses/18-s097-applied-category-theory-...

Equals...awesome?

Other appealing ones to me would be from a selection of:

https://ocw.mit.edu/search/?l=Undergraduate&q=geometry

https://ocw.mit.edu/search/?l=Undergraduate&q=proofs

https://ocw.mit.edu/courses/18-781-theory-of-numbers-spring-...


>Are you genuinely envisioning this math/physics beach vacation as "mostly homework"?

That's what I was envisioning. Homework with an onsite PhD for when I get stuck or find an interesting tangent to go on.

At 32, this is literally how I prefer to spend my vacations now anyway. It's just hard to find time when I can focus and don't have to watch the kids.


There's definitely precedent - small math conferences are often held in interesting locations, and everyone gets a travel grant to attend. Just need a couple meeting rooms for the sessions during the day. The more niche the field the smaller the conference - not talking about big resorts here.

Machine Learning Summer School (MLSS) used to have some nice locations, don't know what it's like these days.

You could go with a name brand, Brian Greene type. Like a "MasterClass Live" type thing.


Yeah this idea is pretty killer and I'd love to work on something like that. I have the past decade of experience mostly being in such zones of my own volition and it's just been such an amazing time.

I could completely picture even hiring locals who speak good English in these places to do the job being a thing to get around having to wrestle with stuff like visa requirements. Costa Rica for example has a law where if the job can be done by someone in Costa Rica, you need to hire them - But the country also has an enormously well educated talent pool (They abolished their military in the 40's to put it in on education), so there's no shortage of that at all.


I’m a programmer, almost 10 years out of college now. I’ve been thinking I should re familiarize myself with stats, discrete math, that kind of thing. So I love your idea.


I've never heard of teaching other things with yoga. I really love your idea. I could see it extending to teaching things like programming.


You might work yourself through the exercises in "baby Rudin" to learn analysis, and ask questions on Stackexchange, but it'll probably often be very frustrating.

I think there are so many great books on linear algebra, that it is difficult to single one out. I think LA is also picked up much easier by most people, especially with a programming background.

It isn't really about the book imho, it's about being disciplined and do some practice exercises every day, like you're training a muscle.

Edit: and I recommend you never consult solutions, except to verify yours. "Illusion of competence" is a big thing you might accidentally step into in mathematics. My experience is that looking up solutions gives you a short strong eureka feeling, but you learn almost nothing from it. You have to go and arrive there yourself.


Came here to say Rudin. His presentation is “odd” in that he introduces powerful concepts and whacks large numbers of problems on the head with them, whereas the usual pedagogical approach is to teach techniques that correspond to these concepts but never explain the concept.

I also really liked Herstein on algebra but I believe it’s out of print now.


"Topics in Algebra" is fantastic! It also is in print, if you don't mind paying roughly $240 for a print-on-demand paperback. It's an obscene price, but it is available!


Grief, I picked it up in a bargain bin. Lovely book, goes so deep so clearly.


I'm a big fan of Herstein's exposition. The text is plenty rigorous, without too getting bogged down in formalism, and he also manages to come across as personable and funny in a way that doesn't feel condescending.

The people who update/glue stuff onto the 23rd edition of those 1200-page doorstop undergrad calculus texts would do well to revisit books of that era and learn the difference between "engaging" and "patronizing." That's not a dig at the books not being Spivak or Apostol - there's absolutely a place for that kind of undergrad textbook (though maybe not 1200 pages of it, updated every three years) - but in terms of attempts at being 'relevant' that transparently read as 'Hi, fellow kids!'

I also dig Herstein's problem sets, there's a good breadth of difficulty, and they really elucidate key ideas, or point to other topics not covered in the text. Some of the problems are absolutely wicked, but in a good way. (IIRC, doesn't the introduction say that some are meant more to be attempted than actually solved?)


I am searching for a good calculus book. Not those 1200 pages of boring stuff. Something reasonable that teaches in good depth and breadth.


There are lots of cool options, depending on what you want to do with it.

If you're interested in rigorously proving how calculus works, starting from the structure of the real numbers, I hear nothing but fantastic things about Spivak. I've not worked much of it myself, but the exercises seem very well chosen from the sections I've read. It exists in an interesting space between a calculus textbook and an introduction to real analysis, and you'll see stuff like the intermediate value theorem proven rigorously. If you want to practice computational applications, you may have to supplement it with some other exercises and material, from what I've heard, though. Apostol also exists in the same space as Spivak, and was (possibly still is) the calculus textbook of choice at Caltech.

For a more applied approach in line with the general content of the enormous undergrad omnitext, this seems cool: https://ocw.mit.edu/courses/res-18-001-calculus-online-textb...

Dover and Springer have some pretty neat slim volumes that occupy the 'less rigor/intended for applications' side, though the names and authors escape me right now.

(Edit: Morris Kline was the Dover book I was thinking of, and I've heard good things about Serge Lang's introductory calculus book.)

(Second Edit: Kline doesn't seem particularly slim at 900ish pages, but it is well-liked. Let's go with 'comparatively slim.' In my defense, most Dover books ARE comparatively concise.)


Thanks! I think I am either going to go by Apostol or Spivak. I have had calculus years ago. I remember the mechanics faintly. I need a solid revision with the associated theory behind it. The only problem with Spivak is that it covers single variable while Apostol's two volumes cover much much more.


Glad to help!

Posting about this inspired me to review all the calculus that I've barely touched in a decade, so I checked out a library copy of Apostol for myself :)


His "Topics in Algebra", "Abstract Algebra", or both?


I’m thinking Topics. Never read the other.


I couldn't disagree more. I studied Analysis in the Uni, and even in that environment Rudin is pretty bad. For a total newcomer that book will leave you completely helpless. Also, solutions are a must have, without them you are almsot totally lost. In their absence, it is OK to ask on StackExchange or #math on EFnet.

First let's start with a few books to prep you for college-level maths:

* https://www.amazon.com/How-Study-as-Mathematics-Major-ebook/...

* https://www.amazon.com/How-Read-Proofs-Introduction-Mathemat... ; or

* https://www.amazon.com/Numbers-Proofs-Modular-Mathematics-Al... ; or

* https://www.amazon.com/How-Prove-Structured-Daniel-Velleman-... (I believe you can find solutions to the 2nd edition online)

For Single-Variable Analysis

* https://www.amazon.com/Think-About-Analysis-Lara-Alcock/dp/0...

* https://www.amazon.com/How-Why-One-Variable-Calculus/dp/1119...

* https://www.amazon.com/Mathematical-Analysis-Straightforward... (contains solutions to exercises)

* https://www.amazon.com/Understanding-Analysis-Undergraduate-... (there are solutions online for the 2nd edition)

* https://www.amazon.com/Numbers-Functions-Steps-into-Analysis... (this book is a brilliant exercise-guided approach that helps you build up your knowledge step by step + solutions are provided).


Agree, Rudin is excellent as a second course on real analysis, but as a first one it is absolutely terrible.


+1, but nitpicking: There are situations where baby Rudin ("Principles of Mathematical Analysis") can be very good for a first course...but those are corner cases, that you'd want a really good, experienced instructor to sign off on.


Absolutely agree.

Exercises are the key for learning. Like with riding a bicycle -- you learn by trying not reading or watching. Composing an exercise section is hard, so thoughtful books usually have one.

Also, I'd pick just one book for a subject and stick to it, pick it carefully to fit your style and language.


Besides practicing the given exercises (which is a must of course) I find it helpful to try to prove the Lemmas and theorems myself, or at least struggle with them for a while, so that (a) you understand the proof better and (b) the struggle plus reading helps concepts stick far better. In other words try to “own” the concepts as if you had developed them.


I disagree with all of these recommendations about studying analysis.

tl;dr: Study a less careful subject where you can more easily develop a useful intuition. Abstract algebra and linear algebra both fit in that category.

(What follows is a caricature.) I tell people that analysis was born in the 19th century when Weierstrauss killed intutition as a tool for understanding calculus. Analysis might teach you what is "really true" but unless you are going to be some kind of professional I would HIGHLY recommend developing an intuition and understanding of things based on what should be true. It's a terrible thing to learn that what "should be" true is not, and for a long time that doesn't help you make progress.


This is because analysis is built upon the fantasy axiom of "real" numbers (of which, according to the measure theory of real analysis, 100% are impossible to even have names, even in principle, let alone compute with), instead of the solid foundations of algebra.


Can you recommend some books?


Related discussions:

* Susan Rigetti’s “So You Want to Study Mathematics…”: https://news.ycombinator.com/item?id=30591177

* Terry Tao’s “Masterclass on mathematical thinking”: https://news.ycombinator.com/item?id=30107687

* Alan U. Kennington’s “How to learn mathematics: The asterisk method”: https://news.ycombinator.com/item?id=28953781


Terry Tao's course isn't very good. It's a 10,000 foot view of mathematical problem solving presented as entertainment.


The best book for me was "Vector Calculus, Linear Algebra, and Differential Forms" by Hubbard and Hubbard.

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 hear they have an interesting take on the implicit function theorem.


Buy from matrixeditions.com for $98.


Supposing you have at least high school mathematics under the belt:

1. Book of Proof by Hammack [1]

2. Tom Apostol Calculus Vol 1 (stop at Linear Algebra)

3. Linear Algebra by Insel, Spence & Friedberg

4. Understanding Analysis by Stephen Abbott

5. Tom Apostol Calculus Vol 2

These are books with excellent exposition, and solution manuals are available when needed. You are absolutely required to do all the exercises. To learn mathematics is to do mathematics.

Two other excellent books are Spivak - Calculus, the problems in this book are very good, but much more difficult than Apostol. For more advanced analysis there's Walter Rudin's Principles of Mathematical Analysis (baby Rudin), which on its own is too difficult as a first exposure to analysis, but there's a cool professor that recorded an entire semester of baby Rudin lecture videos [2]. This is as close as it gets without being enrolled in a university.

[1] https://www.people.vcu.edu/~rhammack/BookOfProof/

[2] https://www.youtube.com/watch?v=ab41LEw9oiI&list=PLun8-Z_lTk... (the first video is a bit blurry, it gets better)


Let me ask and answer some questions here that doesn't answer your question directly.

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.


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 have a stack of statistics books that I will flip through if I need explanation or illustration of a new concept. Usually one will help me out better than the others, but the combination of the different explanations usually improves my understanding.

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).


I haven't had many good math teachers, the one that I did was some poor frazzled adjunct who had to run over to CMU after he was done teaching my class at Pitt. (I'm an alumni.)

The textbook isn't the barrier, it's the end of chapter activities.

If I decided to learn, say, geometry, something that wasn't really taught much in my middle school, I'd probably do Khan academy or try to find a set of finished problem sets.

(When you remove the academic dishonesty angle, the real issue with math courses is it's difficult to craft a math question, so then you need to keep the answer and steps leading to it secret lest folks memorize them for an A.

That issue disappears completely if you adopt a more realistic model like "can you figure out which algorithm to use on Wikipedia, search out if it's already used in a common language like Python, then import your data in a simple format like a .csv and use the trustworthy code you found.

- Greg.


I think this is a huge part of it. Realistic hacky workaround is to get one of those "1000 calculus problems solved" books, and work through the examples -- there's a zillion books for math topic X online, you can read through the relevant parts of a few. But your progress / insight will be in proportion to the problems you actively engage with.

I've often thought that a really pedestrian, but possibly world-shaking, application of AI would be generating math problems (and solutions) appropriate to someone's state of knowledge. Such a thing would make problem-solving practically as fun as a game, at least for many people. Imagine the consequences of a generation of kids obsessive solving of interesting multi-area math / engineering problems the way recent kids played Minecraft or whatever.

(Q: Is there anything vaguely like this in the world now?)


If you know a good resource on learning calculus on your own, I'd add it to my todo list. Every algorithm I reverse engineer or develop has been very... I think linear is the phrasing? No curves?

>I've often thought that a really pedestrian, but possibly world-shaking, application of AI would be generating math problems (and solutions)

Maybe read up on plagarism detectors? I can't recall the title but there was one some people used in grad school to spot code re-use that was engineered much differently from Turnitin.

(Play around with the latter, and notice how the "plagarism" score jumps when you tell it to ignore what you cited. I had way too many stories told to me of professors who'd start the academic misconduct proces because they forgot to do that, paired with witnessing some egregious academic misconduct that I'll simply never be able to get away with.)

- Greg.


I really like Arthur Benjamin's work on mental mathematics. I'm not savant-level, doing division in the thousands or huge floating points in my head yet but I sure am a lot sharper than I was coming out of high school from studying his work, and I guarantee you will just have fun with expanding your capability to think about numbers. [1]

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...


The question reminds me of Susan Rigetti's recommendations for math self-study. If your goal is to self-study the equivalent of a university undergraduate mathematics degree, this is one approach: https://www.susanrigetti.com/math


I expected to see something about probability and statistics but they're hidden with dozens of other topics in the electives > Any and every topic imaginable. Are probability and statistics not part of a regular mathematics curriculum?


It will depend on the university. Where I studied, an introductory class to probability and statistics was mandatory, as was measure theory where probability theory was at least hinted at.

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 ended up with a math degree without taking any of these classes.


In this matter, my opinion of statistics is that it does not really teach 'math', it teaches how to use 'math' to derive information. Pretty much applied math.

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.


That's quite close to saying that applied maths is not maths, which many people would disagree with. If you have definitions, theorems, proofs, I'd say that's maths, and you definitely have that in a sufficiently rigorous statistics course.

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).


Pretty much applied math.

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.


No, they're fairly niche topics as far as the rest of mathematics is concerned. You'll see much more emphasis on them, and especially on statistics, in an applied math curriculum.


his recommendation is quite useful, I've read most of it and found most of them are approachable


Susan Rigetti is a woman.


oops

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


> she was an SRE engineer at uber

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


Dummit & Foote is fantastic for algebra. I also really liked Carrothers for analysis. None of the other books I used really stood out, other than Lang's algebra book, but that one was for the opposite reason - once, he said "the following is obvious" and I thought, "oh yes, it is!" and it was one of my proudest moments as a math student. Would not recommend for self-teaching.

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.


For abstract algebra, I'd go with Pinter's "A Book of Abstract Algebra" (ABoAA) [1].

The other recommendations given so far for abstract algebra are fine, but Pinter's organization makes it I think work better for self-study, and it is much more friendly on the wallet because it is a Dover edition. It's currently $14.89 on Amazon ($6.49 eBook).

ABoAA tends to divide the material into short chapters with lots of exercises. A typical chapter is around 5 pages of text and 5 pages of exercises. The ratio of text to exercises varies a bit but mostly will be in the 40-60% range for the text. Chapters are mostly around 10 pages +/- 3.

The exercises for each chapter are split into several sections each section covering a different aspect of the chapter's material. Sometimes there is a section of exercises applying the material to some interesting area. For example, the chapter on groups of permutations has 6 pages of text, then 5 pages of exercises divided into N sections.

The exercise sections for that chapter are computing elements in S6 (5 problems), examples of groups of permutations (4 problems), groups of permutations in R (4 problems), a cyclic group of permutations (4 problems), a subgroup of SR (4 problems), symmetries of geometric figures (4 problems), symmetries of polynomials (4 problems), properties of permutations of a set A (4 problems), and algebra of kinship structures which consists of 9 problems covering how anthropologists have applied groups of permutations to describe kinship systems in primitive societies.

This combination of small chapters with lots of exercises organized in small groups of related exercises makes it a lot easier to fit this book into a self-study plan if you are like the typical self-study student who has other things (like work) taking up much of their time and so can't get in many long study sessions.

[1] https://www.amazon.com/Book-Abstract-Algebra-Second-Mathemat...


- "A Course of Higher Mathematics", V. I. Smirnov.

- "Differential and Integral Calculus", N. Piskunov.

- "Problems in Mathematical Analysis", B. Demidovich.

These are great books. Demidovich's book is a collection of more than 3000 exercises. Smirnov's book is a course that takes you from what a value means, to advanced topics (5 volumes, in 7 books). Piskunov's book is in-between (course and exercises).

These are the books we've used and went to during the first two years of Engineering (all engineers civil/mechanical/electrical/etc. and phys/maths/chem students go through the maths/phys/chem heavy common core except CS/SE students).


These are MIR books afaik, where would you get them?


Two appear to be freely available for download from the Internet Archive.

https://archive.org/details/DemidovichEtAlProblemsInMathemat...

https://archive.org/details/n.-piskunov-differential-and-int...

Another has at least some of the volumes available to borrow from IA.

https://archive.org/details/courseofhigherma0005smir

And of course there's always Ebay, Alibris, Bookfinder, etc. Or the various pirate e-book sites if one is interested in that sort of thing.


I don't know where you live. There's a website https://mirtitles.org

We have the Office of University Publications here in Algeria that sells them. I know the Piskunov one is available in their physical stores because I bought a copy a few years back (though I couldn't find the beautiful small red hard-cover with fabric that smelled so nice) and I have inherited the Smirnov's and others from my siblings. They're not trivial to find, but they're worth it.


I used the ODE book by Krasnov/Kiselyov/Makarenko (which doesn’t seem to be easily findable online, nor even physical copies) to teach a class at a US R1 university. My only complaint was the many show-stopping typos.


You can find many of the MIR (and other former Soviet era) books published by low-cost Indian Publishers on Amazon India. The paper quality might be iffy but the contents are always great.


You don't say what you're looking for, so I will assume this is a general inquiry.

The Art of Problem Solving books introduce subjects with a problem-solving, "inquiry-based" approach. As part of the text, you can read the question, attempt it on your own, then read how to do it if you need that guidance. In my experience, this method is great for building up knowledge what's true and how to approach a new unknown problem... which to me is proof of true understanding.

A drawback is that they are organized along more or less traditional US high school lines. Don't be fooled though, "Intermediate Algebra" is lots deeper than expanding (x+y)^2. Read the exercises if you're checking out these books, not just the "content". Intermediate Counting and Probability might have content outside of your academic experience, if you're looking for one to try.

If you post more about your mathematical background and goals, you can get better advice.


The other benefit is that all of the AoPS books have extremely complete solutions manuals so you'll get immediate feedback on the quality of your solution, or you can just look if you've been banging your head against something for a while.

Solutions manuals are the autodidact's best friend.

Other high quality books with complete solutions included or in separate solutions manuals include Spivak's Calculus, Hubbard and Hubbard's Vector Calculus, Tenenbaum and Pollard's Ordinary Differential Equations (a really cheap Dover text from the 60s!), and Knuth et. al's Concrete Mathematics.

Going through the AoPS books and the above will give you an extremely solid foundation to learn any further math you might need as a software engineer as needed.

If you have a career spanning 3-4 decades, spending a few years mastering this material can be extremely useful.


Great introductory books along with videos:

  - Mathematics and Its History (John Stillwell)
  - Journey through Genius: The Great Theorems of Mathematics (William Dunham)
  - Proofs Without Words: Exercises in Visual Thinking (Roger B. Nelsen)
  - The Mathematical Universe: An Alphabetical Journey Through the Great Proofs, Problems, and Personalities (William Dunham)
  - Odds & Ends: Introducing Probability & Decision with a Visual Emphasis (Jonathan Weisberg)
  - Introduction to Probability (Joseph K. Blitzstein, Jessica Hwang)
  - The Secret Life of Equations: The 50 Greatest Equations and How They Work (Richard Cochrane)
  - Euler's Gem: The Polyhedron Formula and the Birth of Topology (David S. Richeson)
  - 3Blue1Brown Essence of Linear Algebra: https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab 
  - 3Blue1Brown Differential Equations: https://www.youtube.com/playlist?list=PLZHQObOWTQDNPOjrT6KVlfJuKtYTftqH6 
  - Dynamics: The Geometry of Behavior (Ralph Abraham, Christopher Shaw)
  - Geometry, Relativity and the Fourth Dimension (Rudolf Rucker)
  - 3Blue1Brown Essence of Calculus: https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr 
  - Calculus for the Practical Man (J.E. Thompson)
  - Div, Grad, Curl, and All That: An Informal Text on Vector Calculus (Harry M. Schey)
  - Who Is Fourier? a Mathematical Adventure
  - Physics, Topology, Logic and Computation: A Rosetta Stone (John C. Baez, Mike Stay)
  - An Illustrated Guide to Relativity (Tatsu Takeuchi)
  - The Mathematics Of Quantum Mechanics (Martin Laforest)
  - A Student's Guide to Maxwell's Equations (Daniel Fleisch) or if you're also interested in a visual guide, I wrote one as well: https://github.com/photonlines/Intuitive-Guide-to-Maxwells-Equations 
  - Introduction to Elementary Particles (David J. Griffiths)
  - An Invitation to Applied Category Theory: Seven Sketches in Compositionality (Brendan Fong, David I Spivak)
  - Solving Mathematical Problems: A Personal Perspective (Terence Tao)


I'd highly recommend checking out https://www.mathacademy.us

It's current targeting students, but works well for adults too. There is no better online math training around, and it goes up to graduate level math.

The founder talks about it regularly on his podcast and it's pretty amazing to hear the progress and how comprehensive it is: https://techzinglive.com


Whichever books you choose, one thing I'd recommend is trying to find some good college that used that book and has their syllabus and schedule online. You can use that to get an idea of how to pace yourself through the material.

I've found that pacing is one of the hardest parts of self-studying college or graduate level STEM subjects. In many STEM books each chapter expects that you have a certain understanding of and are comfortable applying the previous chapters.

If you move on way too soon, you'll quickly run into a wall and probably realize you weren't ready to go forward, and go back. But there is a terrible spot between "way too soon" and "ready to move on" where you might not notice your struggles with the current chapter are because you aren't as proficient at the earlier material as you should be, and you can get mired down.

It is also possible to get stuck the other way, not moving on when you should.

When you do it right, you move on when you understand enough of the current chapter that it won't hold you back in the new chapter. This generally comes before you are fully comfortable in the current material, but are comfortable in the parts necessary to handle the new chapter, and using the current stuff in the new chapter will increase your comfort in the old material. That gets you learning new material and reinforcing the prior material when you use it with the new material.

A good experienced teacher will know about how long it takes students to get to the "I'm not fully comfortable with this but know enough for the next thing" stage, and the course pacing will be designed around that.


I post this a lot when people ask similar questions: get the Open University books (such as MST124, MST125, M208). They're not cheap but they are designed to teach undergraduate level mathematics without a teacher and they work!


Entirety of math is too large. Are you trying to learn to learn math? Or are you trying to learn actual math? Which topic are you interested in? By that, I mean something like say topology or algebra etc. Of course they have overlaps, but you have to start somewhere. Did you have even more specific ideas in mind? Maybe commutative rings or quasi groups are your jam.

Once you have a vague idea of what you want topic wise, the best thing to do is sample as many books as possible. It seems that some people click with one book and other not so much. Take (baby) Rudin for example. Some people love it as their first analysis book. It was too dense for my liking and I needed something that did more spoon feeding then that. But I can see Rudin making for a good reference book if I wanted such a thing.

Tangentially, the premise is that mentors aren't always available. Sure, but might not have access to them all the time but you'll probably want some one to talk to once in a while. Say you pick a highly recommended book X and you don't like it at all. What do you do? If you find someone who has not only read X but other books on the same topic, they can nudge you in the right direction. It can also help with discussing and understanding concepts. HN seems to have a decent number of mathematically aware people. There is also math stack exchange and friends (which I haven't used a lot). There also exists more niche solutions like the ##math channels in some irc servers (which i used extensively and was the best thing to happen personally).

Lastly, a lot of the popular/most recommended style books never worked out that well for me. I ended up looking at lecture notes scattered across the internet by various professors on very narrow topic and also very niche youtube videos by random professors and such.


Less wrong has a collection of "best" books per subject [1]

[1] https://www.lesswrong.com/posts/xg3hXCYQPJkwHyik2/the-best-t...


If you're like me, of average intelligence and not an abstract thinker, I don't believe any maths book can replace a human until you have a pretty solid experience & background established. There's too many ways to misinterpret what's being said, or just not 'get' it. One needs another mind. IME anyway.


If you're like me, of average intelligence and not an abstract thinker, I don't believe any maths book can replace a human

I don't disagree, but what's the alternative if one doesn't have a human instructor available? Reading books and asking questions on the Internet is one of the few remaining options. About the only option I can think of is to seek out recorded video lectures of a human lecturer teaching. The downside is that it's not interactive and you can't ask questions for further clarification, but at least you get to hear somebody explaining the material.

The good news is, vis-a-vis recorded lectures, between Khan Academy, Youtube, etc., there's a TON of material out there, and quite a bit of it is pretty high quality.


> but what's the alternative if one doesn't have a human instructor available?

My point is, there isn't one. Modelling the student's thought processes is essential. Maybe GPT could be trained to emulate such?

But there are some solutions such as book publishers not uncritically accepting an author's braindump as being the last word, but putting it in front of people to see what how they interpret it and where they get lost etc and feeding that back into the writing process, a few rounds of that and it'll be much better. I really wish they did that, the quality of so many maths books make them little better than lecture notes.


My point is, there isn't one.

So a person who's interested in math but doesn't have a teacher handy should just give up?


The universe won't bend to what a person needs and deliver it. Many people truly deserve better than they have but never get it. Maybe you can find an answer, I can't.


It is not the spoon that bends, it is only yourself

Maybe if the universe won't bend and deliver us we want/deserve, we need to bend and accept a form of satisficing[1]. Eg, if I want an instructor to learn math (optimal case) but one is not available, then I go buy a stack of books, find some relevant Youtube videos, and find a reddit forum or something similar to ask questions, and dive in and do my best.

Is that "as good"? No. But is it "good enough"? Well, maybe. That's obviously a personal choice, but my argument is only that for some people the answer is "yes, teaching yourself from books and other resources is a satisficing choice given the absence of other preferred choices."

[1]: https://en.wikipedia.org/wiki/Satisficing


Generally, I think lecture notes + books works much better for this than just books alone. The pacing/style/presentation of both is different for a reason and they work well together. In particular, books are great for a second look, practice and revising, but might be too condensed at first.

So instead of searching for a good book, I would recommend to search for good lecture notes/scripts first and then also get the book(s) these list as references.


- Linear Algebra: Gilbert Strang videos + book

- 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)


After Strang I recommend working through Hoffman and Kunze


Hoffman/Kunze have zero as an eigenvector for reasons I can’t quite fathom.


Agreed, that is weird. I like the rest of the book though.


Most any textbook should do provided that you read it in full, take notes, and do the assignments at least in part. As a mathematics major in college I didn’t actually understand a thing my professors were talking about. It wasn’t until I spent time with the book that I understood. Sometimes it took multiple semesters to get a single concept like Fourier transforms.


I recently saw Sheldon Axler's Linear Algebra Done Right recommended here, and have been working through it. In my nonexpert opinion, it's really good, and has a great series of videos reviewing each section in a clear and understandable fashion.

https://linear.axler.net/


Depends on what mathematics you want to learn and whether it's specific to a particular domain.

Many years ago I used Engineering Mathematics[1] when I was doing electrical engineering and my GCSE maths knowledge wasn't cutting it.

Fabulous book.

[1] https://www.amazon.co.uk/Engineering-Mathematics-K-Stroud/dp...

To quote from a reviewer

> Firstly, it's not just for university engineering/science students. The first half of the book would be great for A level. I wish I'd known about it when I did my mine! I wouldn't have needed a teacher.


The book that opened my eyes to what Math really was is a biography on Paul Erdös, The man who loved only numbers. Another really invaluable resource for me was How to solve it by Polya.


I will always remember my discovery of Paul Erdös at my university. The lecturer was a true admirer and passed the n their passion to the class.

Every mention of him is now a fond memory of these times. So thanks for yet another memory triggered :)


Books that are given nicknames by grad students. :) For a foundational understanding of Calculus, and just becoming a better mathematician, a good start would be ->

Baby Rudin (Principles of Mathematical Analysis, Walter Rudin)

Papa Rudin (Real and Complex Analysis, Walter Rudin)

Grandpa Rudin (Functional Analysis, Walter Rudin)


This reminds me of the Little Liddell and the Big Liddell of classical Greek scholarship fame.

The little Liddell is the small version of the famous Greek lexicon, while the Big Liddell is the big version. To make matters weirder, the small version is huge.


Haven’t read this one, but has been recommended highly to those trying to pick up rigorous math starting from non-quant training:

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/...


> Mathematics is best learned under the guidance of a mentor.

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.


> The most important thing is curiosity.

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.


> For math, you need to think, question, and play with the concepts.

And having someone to answer questions and validate hypotheses makes the learning process a lot smoother and helps correct mistakes before they solidify.


Spivak's Calculus! (It's really a book about real analysis.) It's extremely well written and starts by rebuilding your understanding of your fundamental mathematical building blocks, only using things you can prove. It also teaches you how to prove them.


I'm not sure what it's but all books suggested for all self learning seem to be not like books I am used to having as textbooks. I expect textbook to have lot of diagrams, colors (as dumb it sounds, pictures/colors help me remember things), tips, questions and answers in the back. But all recommended books seem to boring novel style books and everyone swearing by it. How can you learn math by reading novels?


We need to replace teachers with mentors. People you can actually get in touch with on a regular basis that can drive you in a specific direction while letting you study alone most of the time


If someone has a nonstandard analysis text that they endorse as really good, I would love to hear about that. It would be a shame to tell a kid about deltas and epsilons.


One reason for a good tutor is having some direction. I feel this is like asking for a "serious computer book". Do you want a book on C++ or Dummies guide to Windows or regular expressions or Turing machines or chip design or functional programming or hash functions or ...?


Semi-serious question; how come you never hear of a 10 year old math prodigy where someone stopped them at age 4 and said "ah, ah, ah I refuse to tell you anything about math until you've decided what you want to learn specifically and what you are going to use it for", yet it's everywhere on the internet, multiple times in every thread on all kinds of topics?

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?


Honestly, with a child I would give them a book on a topic I think they would enjoy. If an adult said "suggest a book I might like", I'd also give them something on a topic I thought was interesting.

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.


Mathematics is very broad. You might try something like "Mathematics and its history" by John Stillwell to get some perspective on the subject.


To add to this, I would really suggest that you get some perspective before you create a long list of tasks/books. Stillwell is not a popular science book but covers modern branches of mathematics. You could easily take a course in college and still have no idea what the point is at the end if your teacher is bad enough. You might also want to ask/search on math stack exchange rather than HN.


No. Books can't understand how you think and learn and help guide you. There is no substitute for a good teacher. Is there a best book for that student? Probably, but who is that student? We don't know!

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 . . .


> There is no substitute for a good teacher.

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.


Of course you’re not guaranteed a good teacher by studying the conventional way. Studying from books is a great skill to develop regardless.


I find brilliant.org to be pretty good. It may not go as deeply into each topic as a good book, but it's a great start. I recommend going through all the math courses there, starting from the level you're at and then go deeper with some books.


Do you have specific mathematical topics you're interested in?

I 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.


Do you have favorite YT channels that you can recommend?


3Blue1Brown has great series on Linear Algebra [1] and Calculus [2]

[1]: https://m.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVF...

[2]: https://m.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53Dw...


3Blue1Brown's whole catalog is great. https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw

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/ProfessorLeonard

https://www.youtube.com/c/TheMathSorcerer

https://www.youtube.com/c/mitocw

https://www.youtube.com/c/khanacademy


There are a couple of textbooks I haven't seen mentioned elsewhere in this thread which I found very useful as a first year undergraduate - they are concise in style but not to the point of being too difficult to follow (it's personal preference but I found I got on much better with this style of textbook):

Davenport - The Higher Arithmetic

Burkill - A First Course in Mathematical Analysis (and also A Second Course in Mathematical Analysis by Burkill and Burkill)

Hardy and Wright - An introduction to the theory of numbers


Walter Rudin's books are the only ones you need. Start with Principles of Mathematical Analysis and when you're done with it read the more advanced ones.


Back in 1960 I learned Boolean algebra from Suzanne Langer's Symbolic Logic Dover paperback. In '63-'64 I learned calculus from Johnson and Kiokemeister's Calculus (with some tutoring in the summer of '63 from a future MacArthur grantee).

https://archive.org/details/in.ernet.dli.2015.523033


What type of mathematics? What level?

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.


Jeffreys and Jeffreys is an old favourite of mine, and the copy shown at the link is in reasonably good condition.

Jeffreys, Harold, and Bertha Swirles Jeffreys. Methods Of Mathematical Physics. Cambridge At The University Press, 1950. http://archive.org/details/methodsofmathema031187mbp.


Nothing can replace a teacher, much less a good one, but I recently stumbled upon an introduction book on calculus from 1910 (yes last century) that had a really nice way to explain things.

I just read the beginning so I don't know how far it goes but anyway here the link https://calculusmadeeasy.org/


Geometry by Brennan. Even it was writen for self study at the Open University.

Concrete Mathematics by Knuth.

Information Theory, Inference and Learning Algorithms by Mackay.


It's not just Mathematics, the discussion is about teaching vs learning.

Ideally the recipe is: good mentor + good student + good teaching tools

IMO for most disciplines, in the year 2022 CE, the critical point is student, not the mentor, not the tools.

Internet resources can hack the mentor/tools stack, when WE are eager and hungry student WE will find a way.


Since this thread is very general, I'd like to specifically request anything that'd help one dive into Model Theory. Everything I've found labeled introductory is very dense, and I feel like I'm missing some prereqs it assumes. It's not easy to know what those are though.


Are you familiar with a general treatment of first order logic, including the completeness theorem? If not, I suggest starting there, for which I'd recommend:

- Chiswell & Hodges, "Mathematical Logic" - This is very clear and careful and gives lots of motivations for introduced concepts, but moves somewhat slowly (introducing FOL in three stages, first propositional, then quantifier-free and then full FOL)

- Leary & Kristiansen, "A Friendly Introduction to Mathematical Logic" (available online for free) - this moves somewhat faster and skips "lower" logics such as propositional logic and uses a different proof system. If you read up to Löwenheim-Skolem, you already have a little bit of Model Theory (the rest of the book is more about computability and logic).

If you've already done FOL, I would recommend:

- Kirby, "An Invitation to Model Theory". This doesn't presuppose much more than FOL (and even recapitulates it briefly) and some basic familiarity with undergraduate math concepts (e.g. groups, fields, vector spaces) and explains everything very carefully. I also think it has great exercises.


Introductory model theory has no particular prerequisites beyond elementary set theory and logic, but it does demand a fair amount of mathematical maturity. If this is your first exposure to higher math, you're probably better off starting with another topic. Real analysis or group theory would be the traditional choices.


The Wikipedia article has over a dozen references, including a number of online books.

https://en.wikipedia.org/wiki/Model_theory


I'd honestly spend my energy on finding a platform which gets you closest to a mentor. Books are good, but in no way, shape, or form a substitute for a mentor. Mentoring is two-way communication, a books is almost always a one-way.

Forums, chat groups/channels, pen pal, whatever it takes.


It's going to very much depend on which field you are interested in. The subject is huge. If you want to study statistics or number theory, you're going to be looking at very different skills and knowledge, with only basically high school maths in common.


That's not quite true, as there are overlaps. To name just one big topic: Pseudorandom number generators. Here you have a (number) theory, including things like finite fields, to generate and understand deterministic numeric sequences, but also a lot of statistical methods tonassess whether these "look" random. Knuth has an entire chspter on these.


The Schaum's Outline Series. It has tons of solved problems and exercises, which is exactly what is critical for getting a handle on mathematical concepts and methods. There's also the Demystified series, which is good but has fewer exercises.


Not sure if they can replace a good math teacher but some top picks for math books include "Geometry" by Euclid, "Algebra" by Harold Jacobs, "Trigonometry" by Charles P. McKeague, and "Calculus" by Michael Spivak.


George Polya's "How to solve it" taught me how to think better, in engineering, mathematics & problem solving. After going through the whole thing, and solving a few of the exercises, I feel like I get stuck less compared to before.


A teacher has two fundamental roles: first, to explain the concepts in a way that is accessible to a given person, and second, to check if they actually understood.

As for 1), there are so many resources to choose from and everybody has their favorites. But 2) is crucial, so today I tend to choose resources with exercises AND answers/key/solutions so that I can verify if I understood the material correctly myself.

There are also online books with solutions integrated, such as[0].

[0] http://linear.ups.edu/fcla/section-WILA.html


I repeat my comment from https://news.ycombinator.com/item?id=11274264

„A single book is enough to learn mathematics: Riley, Hobson, Bence: Mathematical Methods for Physics and Engineering: A Comprehensive Guide It has a whopping 1300 pages, but it has everything you need.

And if that is not enough for you get Cahill: Physical Mathematics This will give you advanced topics like differential forms, path integrals, renormalization group, chaos and string theory.“


I believe Richard Feynman has mentioned in interviews that he learned calculus from "Calculus Made Easy".

IIRC there might have been other books in a "Made Easy" series about math, but I'm not sure.


That's one of the books my grandpa had that I wish I had managed to get before that whole collection got dispersed because I remember him reading it over like a ten year time period. He just liked repeating variations on the exercises I think


He also liked "Advanced Calculus" by Woods (possibly not for his first calculus text, but it happened to be six inches from my hand as I read this.)


I liked that book. It uses differentials to give simple explanations of various derivatives, which seems to be frowned upon these days. You can download the book for free if I'm not mistaken. I once recommended it to a calculus class I was TAing for, though I don't think they were very impressed.


See also: https://news.ycombinator.com/item?id=14192894


I had two acceptable maths teachers in my life. Good teachers seem so rare.


For independent study, get workbooks with exercises and solutions. Best way of learning is by doing. Solving problems is an invaluable way of proving to yourself that you understood the material.


“Unknown Quantity: A Real and Imaginary History of Algebra”

This is a great history of algebra. Math comes alive for me when I know more about how things come to be, the problems people we’re working on, etc.


Not an overall answer, but some parts of an answer:

- Notes on Discrete Mathematics (James Aspnes)

- Mathematical Proofs (Chartrand, Polimeni et al)

- TrevTutor videos on YouTube

- Professor Leonard videos on YouTube


A good alternative to a book would be Khan Academy.

https://www.khanacademy.org


I would recommend looking not only at books, but at online video lecture recordings (from reputable sources). There is so much intuition you get from someone actually discussing and problem-solving in front of you, not to mention (especially in higher pure mathematics at least) a lot of intuition which is spoken or drawn on the board, which never makes it into a book.


Burn Math Class by Jason Wilkes is interesting.


Seconded. also consider Lockhart's "Arithmetic". Bear in mind. These are both non-orthodox style books. Burn Math Class least orthodox.

An orthodox approach to Analysis would be Terence Tao's Analysis books.

Further recommendations depend on your specific goals.


Lockhart wrote another book, called measurement, with a different scope. It covers Geometry and Calculus in his whimsical style. It can give you many aha moments.


This is a nice recommendation! The chapters on circles and Trigonometry were amazing. But, man, it takes a lot of patience to slog through the dialogues.


Mathematics is extremely broad. Find translations of books used for mathematics courses at Russian universities.


Curious, why is that?

I know there's been great Russian mathematicians, but is there also a tradition of pedagogy that's also a part of Russian mathematics?


Yes, many Russian maths, physics, chemistry books are excellent. They balance rigorousness, insight, and difficulty like no other.


I am skeptical that a book can substitute for a mentor (or teacher) at the start of someone's dive into mathematics.

Once you know your basics, sure, go study a book. But by this point, you already know whether a given book helps you or not...


I'd really want books that cover topics of pre-requisites of General Relativity with the following attributes:

- Accessible, does not go too deep into the topic

- Exercises with printed answer (could be in separate student solution manual)

- Well written for self study


It is a long time since I learned, so I dont have specific book, but dont underestimate exercises. Whatever math topic you are learning, look for one of those books with only exercises in them and solutions in the end.


And an active author/publisher who still posts corrections, or on older book. Maths books tend to have errors in the solutions.


I am not sure if they can replace a teacher, but some good mathematics books are "Euclid's Elements", "Introduction to Algebra" by Richard Rusczyk, and "Calculus" by Michael Spivak.


The Elements is of course historically significant, but it would be a horrible introductory textbook.


I try to find lectures online of professors who wrote books. Good example is Mr. Gilbert Strang. Good books and lectures. For theory of computation Sipser Michael. Signals and Systems by Opprnheimer. And so on.


If you're willing to pay a monthly subscription, check out Jason Roberts (of https://techzinglive.com) online Math Academy.


Visual Complex Analysis.


Loved that book ! Got me inspired to do this visualisation after reading it https://www.youtube.com/watch?v=CMMrEDIFPZY


After reading all the comments I am sharing my choices

1. The foundations of mathematics by Stewart and Tall

2. Problem Solving Strategies by Arthur Engels

3. Journey into Mathematics by Rotman

4. Concrete Mathematics by Knuth et al.


art of problem solving


J.Steward's "Calculus: Early Transcendentals" is pretty on point for calculus. Mathematics is more than calculus though.


Bluntly, with only occasional exceptions, have to learn from the books. Period. No biggie: Once are in grad school or a prof, have to learn from papers, and the good books are typically MUCH easier to read than the papers!

For first calculus, read one of the well respected books. In my experience and opinion, don't need high school advanced placement calculus -- regard it as a waste of time. Also the last time I looked at Khan Academy, I concluded that it was a waste or worse and by people who didn't have a good background in calculus. Generally the Internet sources are too elementary. Solution: Just use some good books. Protter and Morrey is good (I taught from it), but it is a bit elementary. I learned from Johnson and Kiokmeister; I like it. At the time it was also being used at Harvard. The exercises are especially good. It's old, but calculus hasn't changed much since 1950 or so about when it gave up on teaching infinitesimals.

But might check out the Maxima software for symbolic indefinite integration -- as I recall, its algorithms are so good they do any indefinite integration that can be done in closed form.

For linear algebra, sure, Nering (a student of E. Artin at Princeton) and then Halmos (a finite dimensional introduction to Hilbert space and written when Halmos was an assistant to von Neumann, office not far from Einstein), etc. For still more, R. Bellman has a good book -- he went all over active and put in everything including the kitchen sink. Then for numerical work, old but a start, Forsythe and Moler and/or the documentation of LINPAK.

For multivariate calculus through the divergence theorem and as needed, e.g., for Maxwell's equations, start with the most elementary treatments of vector analysis and f'get about the lack of proofs. Actually notice, can see via a Google search, that the divergence theorem on a box is just a dirt simple application of the fundamental theorem of freshman calculus. Then the difficulty is proving the thing on other shapes. Then do multivariate calculus again via, say, Apostol, Mathematical Analysis: A Modern Approach to Advanced Calculus. That he uses the Jordan curve theorem is a bit much but just go with the flow. Fleming's Functions of Several Variables does it with both measure theory and exterior algebra. Or W. Rudin's, as I recall, third edition, Principles of Mathematical Analysis. For calculus done via Lebesgue's measure theory, Royden's Real Analysis and/or the first half of Rudin's Real and Complex Analysis.

My favorite author on ordinary differential equations is Coddington, and his elementary An Introduction to Ordinary Differential Equations is nicely done and likely enough for nearly any applications might encounter. Will want some of this if do A/C circuit theory or deterministic optimal control.

For basic abstract algebra Herstein, Topics in Algebra should usually be enough and is well written. With that will be well prepared for, say, algebraic coding theory which actually has some important applications.

Once have a background in measure theory, for probability, of course, Loeve or either of his students Breiman or Neveu. Breiman is really nice to read. Neveu is a good candidate for the most polished and elegantly done math book or writing of any kind in all of civilization. Be sure to get to the good stuff on the Radon-Nikodym theorem, conditional expectation, martingales, and ergodic processes.

For relatively good results, better than hardly any students get just from courses, study the theorems and proofs until can do them yourself and can also explain them to any common man in the street and also explain where they fit in more generally. Also be able to do the exercises. E.g., Rudin's Principles has the inverse and implicit function theorems as exercises! Fleming applies them to Lagrange multipliers. These two theorems are the main prerequisites of differential geometry -- some parts of physics, now popular, touch on this math!

In my Ph.D. program I led the class in 4 of the five qualifying exams, and for the analysis and linear algebra exams, the books above were the secret. After I did that, the department tried to have the students do better by offering a course from Rudin's Principles; the course didn't work; that book is too hard for a course and needs weeks of quiet time where a student can chew on the text slowly line by line. Rudin has some of the most precise presentations on Fourier theory. So when physics uses Fourier theory or engineers use the fast Fourier transform, you will likely know Fourier theory quite a lot better than they do. Right, the uncertainty principle in quantum mechanics is just a basic result in Fourier theory. With Fourier theory and linearity, Schroedinger's equation starts to make sense. A good exercise in early measure theory is differentiation under the integral sign, Leibniz's rule -- be sure to cover that with a good proof.

With the background from the books above will be well prepared for a wide range of more narrow topics in pure and applied math.

Nearly all these books are available used in nearly new condition for low prices. In a move I lost some of my library and rebuilt with used copies.


Nothing can replace a good teacher, as nothing can replace natural interest and motivation. A good teacher makes topics graspable... Shows why this is an awesome thing to know and how this can propel the child and society further... Most stories of people I know that are great in something was a person who lead them to find the purpose and interest in a thing


Probability Theory: The Logic of Science by E. T. Jaynes.


Only a better teacher can replace a good teacher.


In order:

1. Book of Proof

1a. How to Prove It: A Structured Approach

2. Elements of Set Theory - Enderton

2a. Naive Set Theory - Halmos

3. Foundations of Analysis - Landau

4. Basic Mathematics - Lang

5. Principles of Mathematics - Allendoerfer & Oakley

6. A Transition to Advanced Mathematics

6a. Concept of Modern Mathematics - Stewart

7. Calculus - Apostol or Spivak

Don't skip any




Guidelines | FAQ | Lists | API | Security | Legal | Apply to YC | Contact

Search: