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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?

#181

Post by gnulinux »

aoki wrote: post #130
School-Cotton wrote: post #109

Earlier quoted context omitted.

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 voc…

I think there is absolutely nothing in math one should memorize. "In abstract algebra failing memorize might kill you" is nonsense. Memorize what? Axioms of group? Tactics that can be applied to problems?

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

#182

Post by hyperpallium »

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…

I have a lot of sympathy for this approach, and of course you learn more by having fun and continuing, then being meticulous and stopping.

However if at each fork, you choose the fun road, you might get a long way until you run out of options (which may be fine). But the terrible difficulty is necessary knowledge that is not explicitly stated. These gaps ("mathematical maturity") are difficult to even identify, let alone fill. Story time:

---

In high school, I missed a week or two, and later got stuck on a problem in calculus. Discussed it with the teacher, and he eventually seemed to see my difficulty was, but wouldn't tell me, instead saying "you can work it out". I couldn't.

Years later, I started asking online, and people responded helpfully, but didn't actually help. I read wikipedia and math websites about it. Watched videos about it, that were interesting and gave a new perspective, but didn't help my particular issue. I looked it up in famous textbooks (Spivak, Hardy), still nothing.

Finally, I did all the problems in the derivatives section of Khan Academy, but when I came to this problem, it still didn't address my specific issue.

Thinking I had gaps in my knowledge, some known to me, and suspecting others unknown, I went back to even earlier material.

Some time later, I reviewed the problem again, and realized that the issue was a completely trivial part of limits - an earlier section. Which was what I missed in those 2 weeks of high school, all those years ago.

A nice maths teacher can save you years.

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

#183

Post by gnulinux »

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.

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

#184

Post by dunham »

wbl wrote: post #43
mlevental wrote: post #24

Earlier quoted context omitted.

goofy suggestion. those are both either grad or senior undergrad books depending on where you are. much better suggestion is A Book of Abstract Algebra by Pinter (really gentle) or A First Course in Abstract Algebra by Fraleigh

I used on in high school and the other my second year of undergrad. They have lots of material but they also have good exercises and explanations.

Herstein was my undergrad book back in the early 90's. I remember really liking it. It's been a very long time, but my recollection was that the groups to rings progression felt a lot more natural to me than the rings to groups progression of some of the other books at the time.

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

#185

Post by InclinedPlane »

oarabbus_ wrote: post #97
madhadron wrote: post #20

There's a set of basics that you will want no matter which direction you go: calculus/real analysis, linear algebra, differential equations/dynamical systems, and sets, groups, rings, and lattices. Calculus: learn to extract qualitative information about a function (it goes up here, has a maximum there, goes down there, oscillates with an increasing period, goes to this value at infinity...) and to numerically comput…

Most engineers I know have not learned set theory or groups/rings/lattices. They still seem to be doing pretty well.

Most people I know have not bothered to learn any math beyond algebra, and try to forget what algebra they did learn, they still seem to be doing pretty well. You can always limit yourself and pretend that's "enough" but it can very well be self-limiting. You simply don't know what you don't know, and that includes not knowing all of the things you might miss out on by not studying. Set theory and abstract algebra has a huge range of applications, simply being able to earn a living without studying those things is an insufficient reason to avoid studying them.

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

#186

Post by aoki »

oefrha wrote: post #166
aoki wrote: post #130

Earlier quoted context omitted.

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 voc…

I have to disagree, especially about abstract algebra. Most concepts and theorems feel like abstract nonsense (not specifically talking about category theory here) when you don’t understand them, but should become pretty natural once you do. For true mastery you need to work with the concepts and results on a day to day basis for a while, by applying them; continually reading the text of definitions and theorems hard…

i agree that memorization by internalization (concepts) is different from (and superior to) memorization by rote (text); i was agreeing with the other fellow that rote memorization is not the end goal. however, to me they're both "memorization" because they both represent work to achieve fluency in application.

however, i made my comment because i think i disagree that spaced repetition has no place in learning. i think that if you dig around in the Polar guy's earlier comments, you'll find threads where folks like michael nielsen are talking about using spaced repetition tools for much more than purely textual memorization of theorems - basically, cycles of repetition and (re)synthesis. so i don't feel it's right to completely shut him down about card decks. you may disagree, of course.

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

#187

Post by aoki »

gnulinux wrote: post #181
aoki wrote: post #130

Earlier quoted context omitted.

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 voc…

I think there is absolutely nothing in math one should memorize. "In abstract algebra failing memorize might kill you" is nonsense. Memorize what? Axioms of group? Tactics that can be applied to problems?

please consider the statement from the position of a student just learning the material (like the OP). if you are working through a book with 1000+ problems, like pinter (as some people are suggesting), and you have to keep looking up stuff like how to verify a subgroup, how long is that book going to take you?

to me, memorization means to recall the important parts of something accurately and precisely so that they can be used fluently. that is not to say textually (rote memorization). i do not think it is enough to say "well, that's what learning is" because i doubt most people learn most things to that level.

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

#188

Post by vladislav »

I don't at all recommend limiting yourself to self-study when you start this endeavor, it's too daunting given all the unknown unknowns. To start with it helps a lot to take courses, even if it means sitting in/enrolling in night courses at the local community college. It provides a reference point to what others understand, allows real-time querying with an instructor, and is more motivating than self-study. Once you have completed and fully grasped a difficult proof-based course (real-analysis, algebra, etc), then you can possibly embark on a successful self-study. I'm not saying it can't be done, but you will get to mathematical maturity faster this way and there will be less risk of early abandonment.

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

#189

Post by momofarm »

School-Cotton wrote: post #163
makeset wrote: post #147

Earlier quoted context omitted.

I am a scientist/engineer/mathematician who studied the hell out of abstract mathematics at an extremely rigorous undergrad program. In the 20 years since, not once has that knowledge been useful in my academic or industrial work, not even remotely. I am all for studying theory for its own sake, for the career theoretician and the interested hobbyist, but as an investment I regret those four years of my life as a col…

Math is not just a tool; it is an area of intellectual exploration. By the same logic studying history or philosophy is a colossal waste of time, too.

so, in real world application, what's the most useful/valuable topic in mathematics?

Although I wonder for 90% people of this world that math is just a tool to pass the exam at school, not any real application (or they just can't sense it).

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

#190

Post by hyperpallium »

Khan Academy has multivariable calculus, differential equations and linear algebra --- with many exercises, solutions, and worked solutions (under the "hints" link after a problem), and gamification as a motivation aid.

The videos are above average, but seem about 5x too slow on the steps, then rushing through the interesting stuff almost too quick to catch: boredom punctuated by rewinding. Probably because it targets highschool kids.

BTW You may find you need to cover even earlier material anyway (I did), to refresh and fill gaps. You can cope with some gaps and hazy recall, until there are too many... and some gaps cannot be filled piecemeal (because you don't know you have a gap).

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