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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?

#111

Post by ivanhoe »

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…

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 day when you run into a problem you'll know where to look for more details. This is IMO a common problem with self-thought programmers as well, they often end up inventing a wheel simply because they just never heard that solution to their problems already exist in some 70s CS textbook.

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

#112

Post by postit »

You first need to define your boundaries and limits. Don't get me wrong, but mathematics is grinding, and if you think you have an idea where you're getting yourself into because you've studied calculus and algebra as an undergrad CS, you will fail miserably. Don't make the same mistake I did :)

Undergrad math is tooling; I like to use the analogy that it's like entering a workshop where you have all the tools available, but you're blindfolded, and you have no idea what each stuff is used for. So you'll have to touch everything. Look down for the tooling advice.

When venturing in Mathematics, for your own sanity, please have an objective in mind. I'm down serious! Understand what you want to research and go in that direction.

One last piece, Find a mentor, share, talk to people. You won't advance from yourself.

As a piece of tooling advice, I have the following I've stolen from Reddit a few years ago (sorry I haven't found the source to it) __ TOOLING __ First of all, most important, GO LEARN ALGEBRA. Seriously, I know you think its bullshit but its the most basic skill in some ways that any mathematician should know. Second learn Calculus: Single and Multivariable. If you are still interested here are some things to go onto next:

Discrete mathematics: This includes equivalence relations (probably one of the most important things for you understand ever), propositional calculus (logic) proof techniques (induction) and some basic combinatorics (Pigeonhole principle). You can literally find any textbook and start reading. The theory is kinda a hodgepodge, but those are the major themes.

Linear Algebra: Again, one of the most important subjects you will ever study. Once you understand this, you are really on your way, and this stuff comes up everywhere. Many mathematicians have said many of the biggest proofs in the world come down to "just some linear algebra". The major point here is to understand that there is only one vector space for each dimension over a field and understand how a linear transformation becomes a matrix only after a choice of basis. Here equivalence relations come up again!

Differential equations: Unless you're focused on engineering math or serious applied stuff, don't worry too much about this. Seriously, it's not that integral (haha get it!).

Complex Analysis: Yes, mathematicians and Engr. Actually, do study "imaginary" numbers, but there is nothing imaginary here. This is serious stuff, do it.

Okay, so now you're about as a sophomore/junior level place in mathematics. How to finish it off? It's not that unclear:

Abstract Algebra -- Grab any book read about groups, rings, fields, vector spaces, and modules. Proofs will be difficult here but work through it. There are so many books here, avoid Lang (good book but not for starting out), Dummit/Foote is okay. As a undergrad I had a good time with Rotman's An introduction to abstract algebra.

Analysis -- Grab Baby Rudin. No seriously, Grab this book, sit in a room for a semester and just fuckn' read it. You will basically be "redoing" calculus. This is a trial by fire, go!

Topology -- Grab Introduction to Topology by J. Munkries. Its so well written it might as well be a coffee table book.

There now, you have done everything a math major would. Yes, there are lots of things that are missing, arguably the most important things depending on what your goals are. Typically one studies Number Theory along with Abstract Algebra, or studies Analysis and Differential Equations together or Analysis and Topology. Seeing the links across different topics is essential, but I'm just giving the overview here.

Not every mathematician studies logic, and there are LOT of fringe topics that I'm omitting (including some of my favs: Projective Geometry, Varieties, Lattice/Order theory, Combinatorics, Elliptic Curves, Coding theory, Harmonic Analysis, etc.). However, none of these are required courses at more than a 1% of programs

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

#114

Post by you_owe_me »

analog31 wrote: post #108
oarabbus_ wrote: post #97

Earlier quoted context omitted.

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

From what I've observed, most engineers are glad to be done with math when they finish college. Most engineering is qualitative: Organizing and arranging things, making things fit together, and troubleshooting. Maybe 10% of engineering is quantitative, and that work often goes to the handful of people in the department who have maintained an interest in it. Some of the engineers who attract quantitative work are peop…

When you refer to engineers, do you mean actual engineers (BSc in Engineering)? or your web designer with jquery skills who calls himself engineer?

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

#115

Post by MattHeard »

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 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.

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

#116

Post by geebee »

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.

Come to think of it, most of the math majors know (who didn't get PhDs and now work as programmers or data scientists) have forgotten the greater part of what they learned about groups/rings/lattices, and they seem to be doing pretty well too! ;)

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

#117

Post by susam »

wrycoder wrote: post #48
hprotagonist wrote: post #2

The hard but maximally useful thing to do, in my opinion, is to regularly meet with at least two other people and a blackboard and beat your heads against it together for about 3 hours at a stretch. Do this at least weekly--and preferably more often. Self study in between meetings is obligatory. One of those people should be at about your level. The other should be farther along. All three of you should trust each ot…

Like Einstein's Olympia Academy! https://en.wikipedia.org/wiki/Olympia_Academy

Can we start something like this online for Hacker News community who are interested in Mathematics?

I have created a group here if someone is interested to join: https://groups.google.com/d/forum/projectfermat (If you think that there is a better place to have a forum like this that anyone can easily view or participate in, please let us know. I mean we could also create an IRC channel, Slack workspace, etc. but there should be one main starting point and a mailing list/web forum like this seems like a good place for that.)

I am thinking we could also host a web meeting to present, discuss, or share interesting topics and problems regularly. We can form our own mathematics discussion community here.

I have been doing this kind of thing at my workplace as well as outside work and it has been an incredible source of learning. I believe something like this for the Hacker News community would be very helpful and we can learn a lot of mathematics from each other if we can interact with each other on a more topic-focused forum.

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

#118

Post by joe_the_user »

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…

The OP did specify what he wanted - the basic undergraduate and starting graduate curriculum. That's a pretty well defined area: Algebra, Real Analysis, Geometry and Topology with maybe some complex analysis, number theory, statistics, CS or etc thrown in.

I personally did work myself up to the graduate in math during the last two years of High School & first year of college. I was motivated by exploring ideas and gaining knowledge. I would guess that each person has a somewhat unique motivation strategy. Maybe solving problems gets some people doing stuff. I'm sure simply learning stuff can motivate others. Probably each person has to experiment to discover what works for them - I would pick up a calculus book and read it - well, I'd skim repeatedly and then read in depth, solving a few problems. Math is difficult, of course, so having a bit of patience with your until it gets the ideas on it's own is probably necessary.

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

#119

Post by aoki »

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.

in the US, most engineers take the standard two-year lower division sequence (calculus, linear algebra, a bit of diffeqs). for the most part, you learn technique rather than proving things. upper division engineering math courses teach more technique (e.g., more diffeqs).

but as madhadron says, you can't read/write proofs of upper division or graduate level math without the "foundations" material, which includes naive set theory.

do you need any of that to do engineering math? well, there are a couple of standard quotes, relating to the fact that the technique taught is brittle, in weird and subtle ways. the claim is that understanding the proofs tells you what the limits of applicability are.

"[F]or more than 40 years I have claimed that if whether an airplane would fly or not depended on whether some function that arose in its design was Lebesgue but not Riemann integrable, then I would not fly in it." - richard hamming, "mathematics on a distant planet"

"It is customary to begin courses in mathematical engineering by explaining that the lecturer would never trust his life to an aeroplane whose behaviour depended on properties of the Lebesgue integral. It might, perhaps, be just as foolhardy to fly in an aeroplane designed by an engineer who believed that cookbook application of the Laplace transform revealed all that was to be known about its stability." - tom korner, fourier analysis

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

#120

Post by jasonmorton »

Here is a proven approach for at least the first part, building foundations and being ready for graduate work. Many Berkeley Ph.D. students passed through this route. Get the book "Berkeley Problems in Mathematics." It contains historical problems from the Berkeley math prelim exam, and solutions. Now don't look at any solutions yet.

This is the exam all Berkeley math Ph.D. students must pass within three semesters of arriving to stay in the program, and the fail rate is about 50%.

You will also need reference books, advanced undergraduate and beginning graduate textbooks. Buy, download, or borrow as appropriate.

Pick a problem (start with the older ones, they are easier). Set aside 30-60 mins and try to solve it. No devices, no references at all, go to a library or a coffee shop without your devices. Dont' give up till time is over. If you cannot (usually the case), still don't look at the answer. Hit the reference books (don't look up the problem online either, it will go right to the answer and you won't learn much). Read and try to understand enough so that you can solve the problem. It is ok if you solve it this way (in the course of reading about it).

For bonus points, students studying for the exam will typically take entire old exams (available from the Berkeley website), take that to the library and just sit down for three to six hours and try to solve all the problems correctly. Then self-grade harshly. When you can do that for a recent exam (and get a good score), you will have more or less mastered undergrad math to the point that you could teach it.

Most important: you have to struggle to solve problems. Reading a solution is about as useful as watching someone else lift weights: you get minor tips on form but not any stronger.

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