I could give you a list of books that are used at the graduate and undergraduate level that are considered good. However, almost none of them really discuss the motivations behind topics. My suggestion is to start with this book by Arnold on Abel’s Theorem. If it interests you then delve deeper into the topics once you are done with it.
Ask HN: How to self-study mathematics from the undergrad through graduate level?
121–130 of 231 posts
Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?
#122A nice foundational textbook to start with is Michael Spivak's text "Calculus." This text is fairly conversational, it motivates its concepts well with many examples, and it will help you build a strong foundation in writing proofs and reasoning mathematically.
As others have said, it might be helpful to have a friend or two read with you so you can answer each other's questions. Also, make sure you do a ton of exercises!
Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?
#123Here's my personal opinion about how you should approach this. It's all well and good to want to cover undergraduate math courses. When you are actually enrolled in a university, you will have enough inertia and motivation to complete the courses. However, when you are self-studying you are doing it all on your own. It's hard to be as thorough and cover everything. And so I ask, what really is your goal here? You don…
This is exactly the kind of advice I give to people who ask me about teaching themselves to "code". 1. Find a thing you want to make. 2. Find out how to make it. 3. Try to make it. 4. Learn the skills that previously prevented you from making it.
I don't think this is a good comparison. People who are skilled in both areas might not realize that, but learning how to program is a hundred times easier than working through any advanced topic in higher mathematics and coming up with your own proofs.
Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?
#124Be a social human and find a local math prof or someone in industry to mentor you.
That aside: Haskell Road to Math and Programming. Concrete Mathematics. Pick your favorite computer graphics book to learn working with vectors and physics equations. From there chase your passions.
Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?
#125I'm actually building an app specifically for this use case: https://getpolarized.io/ Polar is basically a personal knowledge repository. The idea is that you use Polar for all of your education. Either official education or continued learning. Every textbook or technical paper should go into Polar. You can then annotate the textbooks directly including text highlights, comments, etc. It supports spaced repetition sy…
Learning math isn't really about memorizing things.
It's funny that the article (http://augmentingcognition.com/ltm.html) displayed on polars feature page has this:
"I now believe memory of the basics is often the single largest barrier to understanding. If you have a system such as Anki for overcoming that barrier, then you will find it much, much easier to read into new fields."
Also I just made an account to say thank you - polar looks awesome. I just wanted to start using anki and stumbled upon it.
Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?
#126Earlier quoted context omitted.
group sessions keep you honest and help you work through high level stumbling blocks. You're expected/obligated to do a fair amount of solo pick-and-shovel work in between.
Well in my experience they are a waste of time, but that's just me.
you have to have the right group. It's nontrivial to manage, unfortunately, and if you do not have the right group, it is indeed a waste of time.
Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?
#127Earlier quoted context omitted.
This is exactly the kind of advice I give to people who ask me about teaching themselves to "code". 1. Find a thing you want to make. 2. Find out how to make it. 3. Try to make it. 4. Learn the skills that previously prevented you from making it.
I don't think this is a good comparison. People who are skilled in both areas might not realize that, but learning how to program is a hundred times easier than working through any advanced topic in higher mathematics and coming up with your own proofs.
This thread seems bizarre to me. There's one guy claiming he "worked himself up to graduate level math" in the last two years of high-school. So either he's a literal IMO-gold-medal-level genius, or people here don't quite understand what undergraduate math studies actually involve. Even if he is that intelligent there's no way he actually did the amount of work required on such a broad number of topics.
Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?
#128Here's my personal opinion about how you should approach this. It's all well and good to want to cover undergraduate math courses. When you are actually enrolled in a university, you will have enough inertia and motivation to complete the courses. However, when you are self-studying you are doing it all on your own. It's hard to be as thorough and cover everything. And so I ask, what really is your goal here? You don…
Personally, I think this is bad advice, because without an undergrad+ background, the projects above will either be impossibly frustrating or you will make up some crackpot bullshit. Plus, the undergraduate curriculum is its own reward.
Every example project suggested by the GP can be easily externally verified, so it has built-in protection against crackpotism. Secondly, it may be very difficult, but not impossible.
I think maybe the most likely outcome, if they were really motivated and somewhat capable, is that they learn lots of mathematics well—but maybe not quite to, "... through graduate level" (that phrasing is a bit ambiguous though; self-teaching 'up to' graduate level is definitely doable).
Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?
#129I disagree with the common advice that before you study math you should try to develop love for it or find an application you care about. If you don’t know any math, how can you know what you like or want?
What I think one should do, initially, is try to be better than other people in some way. That’s a motivation everyone has accessible. One way is what you’re doing already — trying to learn math on your own. That’s pretty cool.
From there I’d use Reddit and Quora to build yourself a curriculum. Most people start with calculus, linear algebra, differential equations, and real analysis. Look for textbooks that are known to be pedagogically well-written for self-study (rather than intended to be used as a manual in conjunction with lectures). Skim the table of contents and the first chapter of several books to get a feel for which ones you’ll be able to stand going through. I like Spivak, Arnol’d, Axler, and Pugh.
In each subject, find a book full of problems, and do the hardest ones in each section.
Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?
#130I'm actually building an app specifically for this use case: https://getpolarized.io/ Polar is basically a personal knowledge repository. The idea is that you use Polar for all of your education. Either official education or continued learning. Every textbook or technical paper should go into Polar. You can then annotate the textbooks directly including text highlights, comments, etc. It supports spaced repetition sy…
Learning math isn't really about memorizing things.
very true, in the sense that rote memorization is not the point.
but false in the sense that doing math requires fluency - in applying a small amount of technique up to lower division math, and in applying a large number of definitions/results after that. in areas like abstract algebra, failing to memorize will kill you. it is as disfluent as writing text in a foreign language without having memorized the working vocabulary and grammar; even if you can eventually recall, derive or look up what you need, it's just not practical to work like that. (and i say this as someone who prefers to mentally index information rather than memorizing...)