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Ask HN: How to self-study mathematics from the undergrad through graduate level?

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Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?

#211

Post by oefrha »

aoki wrote: post #192
oefrha wrote: post #169

Earlier quoted context omitted.

> The most difficult part for a person who hasn't done a lot of math to become a person who does a lot of math is to read and understand rigorous proofs. I’m afraid you haven’t delved into any advanced topics. That’s actually about the easiest part, and could be mastered by ten year olds (certainly myself when I was ten).

alnar is likely referring to the US system, or something like it: unless you are tutored externally (rich), an autodidact outlier (gifted), or selected for honors courses, you basically take computational courses for 14 years (with one cursory stop for euclidean geometry) and are then thrown into proofs at the age of 19-20, if at all. it's widely recognized as a problem in the math pipeline, which is why many US univ…

This has little to do with education systems, I was talking about objective difficulty. Try to tell any mathematician that the hardest part of math is rigorous proofs; if they don’t laugh in your face, they are just being polite. One may think the transition is “hard”, until one actually gets into more advanced topics. As you said, it’s just unfamiliar to the uninitiated, at best.

Reading and presenting rigorous proofs in elementary number theory, Euclidean geometry, etc. is easy for gifted ten year olds and definitely manageable for a lot of fifteen year olds. You asked about material — it doesn’t actually matter, and I don’t recall specifics; any entry level treatment of elementary number theory should do (really beautiful subject with a very low barrier of entry). For kids who have eyes on IMO, it’s very common to be throwing around perfectly rigorous proofs at young ages.

Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?

#212

Post by kowdermeister »

sweeneyrod wrote: post #80
netvarun wrote: post #68

As an alternative I would suggest a top-down approach. Start with the theorems/results you truly wish to understand and work backwards. There was a great quote from an interview of Peter Scholze (one of last year's Fields' Medallists), which has really changed how I view learning: At 16, Scholze learned that a decade earlier Andrew Wiles had proved the famous 17th-century problem known as Fermat’s Last Theorem, which…

An approach might work well for a Fields medallist but less so for normal people.

I love generative art, so I quite often run into things like Lorenz attractors. I looked up how they are made and that lead me to differential equations which lead me to better understand calculus. So my love of pretty graphics lead me to learning math with a purpose.

I haven't won any awards yet.

Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?

#213

Post by kkmazur »

There are some awesome resources on github on many topics including math. Just search for "github awesome-X" like https://github.com/rossant/awesome-math

And here is a root project for all resources https://github.com/sindresorhus/awesome

Personally I'd like to have something like a learning path. Not only a list of resources/topics but also some guide how to approach the learning process. O'Reilly has something like this but it's not mature yet https://www.safaribooksonline.com/learning-paths/ in my opinion.

MIT OCW has some guide for prequesities https://ocw.mit.edu/courses/mit-curriculum-guide/#map

I hope that helps you

Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?

#214

Post by ivanhoe »

tbabb wrote: post #135
ivanhoe wrote: post #111

Earlier quoted context omitted.

The issue with this approach is that if you learn just "the fun parts" you might be left with some huge gaps in your knowledge, all that in-between stuff - especially if you're not a genius like Scholze. Standard approach is perhaps less motivating and you learn a lot of stuff that frankly you'll never need and you'll probably forget most of it, but it ensures that you've at least heard about all the major ideas. One…

I think fun is an extremely important part of keeping up the pace of learning. I would rank that much higher over "methodically combing through everything", which could cause you to quit from boredom long before you reach your goal. I'd hypothesize that the problem you describe is due to insufficient curiosity about the way things are already done. Especially with Google, Wikipedia, and well-populated internet forums…

You're totally right, it's an approach that requires more motivation, but I don't think it's really that hard. University curriculums are made for young people who're not necessarily super motivated to study (many study just to finish the course, not to learn it) and who also have to study a lot of other things in the same time (and to party, and to fall in love and feel miserable and all other things that you do when you're young that are way more important to you than math). Compared to that your starting point is not that bad at all. Being older and more mature, plus genuinely interested in learning that subject you're probably way more motivated, plus you don't need to pass 6 or 10 courses that year, you can concentrate on just that one, at your own pace. Again, it's really up to ones own personality, there's no one-size-fits-all here, but IMO chances are that you'll learn it way better than someone who had that course on Uni.

And regarding wikipedia and google, they're much more effective when you know what you're looking for. If not exact name of theorem or algorithm then at least "there was that thing that we learned related to that other thing". Having at least a faint idea like that can save you tones of time when researching.

Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?

#215

Post by gmatbarua »

IMHO the books being advised here are too high level. Its like trying to read about Computer networking basics from RFCs. Some advise are equivalent to searching for a reason to learn "C" by appreciating what can be done with "C" by reading books on Linux Kernel programming.

I will mention a few books which by no means are undergraduate level, but are per-requisite for undergraduate level study. Are you thorough with books given below? These books are available through http://gen.lib.rus.ec.

1. Higher algebra - Hall and Knight 2. Trigonometry I - Loney 3. Coordinate geometry I - Loney 4. Calculus of one variable - Maron

Download the books and check if you already know all those. Once done, then go for the higher level books most the people advising here.

Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?

#216

Post by abnry »

proginthebox wrote: post #207
abnry wrote: post #38

Here'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 terrible advice. Apart from the last sentence. A better advice would be to specify which subject to learn. For example, (since I don't really have much time) 1. Topology (book by Munkres) 2. Real Analysis and Measure Theory (book series by Stein Shakarchi) 3. Algebra (book by Aluffi) 4. Linear Algebra (book by Friedberg Insel) 5. Measure Theoretic Probability (book by Cinlar) 6. Differential Geometry (book Sm…

The advice I gave is not exclusive to working through the typical undergraduate books.

I was questioning why the OP wants to self-study an undergrad math curriculum to begin with.

It's probably not to become a pure mathematician. So I suggested, instead of creating a massive goal of getting through a collection of books just for the sake of being a completionist, to have a concrete personal goal. Otherwise, people can throw books "you have to read" at you until the cows come home. Especially since this person is talking about applied math.

Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?

#217

Post by _recommendation »

oefrha wrote: post #171
enraged_camel wrote: post #149

Earlier quoted context omitted.

>>Depending on your background, you probably don't have enough information to pick a long-term goal anyway. Nah, it's totally possible for newbies to pick high-level long-term goals. This can be something like "I want to teach my computer to tell apart dogs and cats", or "I want to create a website where people can buy and sell yarn." From there, Google searches can direct someone towards concepts and various methods…

Not in math. Unless you are a mathematician, I challenge you to pick a math equivalent of "I want to create a website where people can buy and sell yarn." I’ll wait. People just assume learning math is the same as learning everything else. That is not even remotely true.

Genuinely curious - why do you think that? I have been self studying math for about 4 years now and find it to be the same as everything else that's worth learning: hard! But I haven't found that it's some entirely different realm divorced from all other intellectual pursuits.

Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?

#218

Post by jim_bailie »

gnulinux wrote: post #183
jim_bailie wrote: post #173

My $0.02: Make a goal to fully understand General Relativity as Einstein published it. Then work through the evolution of some of its mathematics.

Why set an arbitrary goal like that? Why not understanding Curry-Howard correspondence or Godel's Incompleteness or Hilbert's 17 instead.

That was my $0.02 as someone who majored in Physics. You just gave $0.04. Good suggestions.

Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?

#219

Post by enraged_camel »

oefrha wrote: post #171
enraged_camel wrote: post #149

Earlier quoted context omitted.

>>Depending on your background, you probably don't have enough information to pick a long-term goal anyway. Nah, it's totally possible for newbies to pick high-level long-term goals. This can be something like "I want to teach my computer to tell apart dogs and cats", or "I want to create a website where people can buy and sell yarn." From there, Google searches can direct someone towards concepts and various methods…

Not in math. Unless you are a mathematician, I challenge you to pick a math equivalent of "I want to create a website where people can buy and sell yarn." I’ll wait. People just assume learning math is the same as learning everything else. That is not even remotely true.

What makes math special, exactly?

Re: Ask HN: How to self-study mathematics from the undergrad through graduate level?

#220

Post by oefrha »

_recommendation wrote: post #217
oefrha wrote: post #171

Earlier quoted context omitted.

Not in math. Unless you are a mathematician, I challenge you to pick a math equivalent of "I want to create a website where people can buy and sell yarn." I’ll wait. People just assume learning math is the same as learning everything else. That is not even remotely true.

Genuinely curious - why do you think that? I have been self studying math for about 4 years now and find it to be the same as everything else that's worth learning: hard! But I haven't found that it's some entirely different realm divorced from all other intellectual pursuits.

I don’t really have time to give a thoughtful answer (it would be quite long), but the exact post you responded to gave an obvious difference. To roughly summarize that difference, producing anything of value in mathematics requires learning a tremendous amount of prior art, and without a tremendous amount of work you won’t even know what’s of value. It’s no wonder that many crackpots choose to work on high profile number theory problems, like Goldbach’s conjecture and previously Fermat’s Last Theorem, since the formulations are simple enough for laypeople to understand, yet the theories behind developed over hundreds of years are incredibly deep.

> everything else that’s worth learning: hard!

I disagree. I’ve learned many things worth learning that are not hard at all, but to each their own.

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