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?
Want to teach geeky tourists calculus? If we made math tourism a thing, teachers in host countries could skill up pretty fast as well.
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.
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.
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". :)
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-...
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.
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.
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.
Dead Comment
Dead Comment
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.
I also really liked Herstein on algebra but I believe it’s out of print now.
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).
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.
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.
* 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
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.
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)
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).
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'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?)
>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 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...