How to Learn Advanced Mathematics Without Heading to University • Hacker News Board

Re: How to Learn Advanced Mathematics Without Heading to University

#31

Post by graycat » Fri, Mar 11, 2016, 6:06 PM UTC

ska wrote: post #26

graycat wrote: post #24

Earlier quoted context omitted.

Ah, a favorite is a function that is differentiable but its derivative is not Riemann integrable! Recall, a function is Riemann integrable if and only if it is continuous everywhere except on a set of measure 0. So, the derivative has to be discontinuous on a set of positive measure. Now, to construct one of those! Here's another favorite: For positive integer n and the set R of real numbers, suppose C is a closed su…

Fun book indeed! Your first example you won't typically run into until an introductory measure theory course.

No, it's doable at the level of Rudin's Principles: He shows that a function is Riemann integrable if and only if it is continuous everywhere except on a set of measure 0.

For the definition of measure 0, he gives that quickly, and don't really need a course in measure theory. Besides a course in measure theory likes just to f'get about Riemann integration, and thankfully so.

Re: How to Learn Advanced Mathematics Without Heading to University

#32

Post by kenjackson » Fri, Mar 11, 2016, 6:08 PM UTC

goc wrote: post #29

Any one know any books or resources that concentrate on invariant programming? Translating recursive code into properly tail recursive or iterative code can be pretty difficult.

I recall that Concrete Mathematics has a pretty section on solving recurrence relations. That's probably as close as I can get you to this.

Re: How to Learn Advanced Mathematics Without Heading to University

#33

Post by jasode » Fri, Mar 11, 2016, 6:08 PM UTC

ska wrote: post #21

The main thing to remember is that nobody learns mathematics to any significant depth by reading. The only way to learn it is by doing it. Doing it carefully, and in full detail, not falling into the trap of "and I understand it from there". So the biggest barriers to doing it on your own aren't source material (there is lots of that) it's a good source of correction. There is also the usual problem of self study, in…

>There is also the usual problem of self study, in that you don't have a roadmap and can waste time easily.

Can you expand on what you mean by "roadmap"?

To me, a roadmap is very easy to get and can be arrived at from different angles.

method 1) Pick up an "Advanced Mathematics" book and start at page 1. The sequential chapters of that book would start a roadmap. If page 1 looks incomprehensible, look at the preface/introduction to see what the author lists as prerequisites. Seek out the book(s) on the prerequisites and start on page 1 of that book. If that prerequisite looks like gibberish, then look at that book's prequisite. And so on.

method 2) Google "advanced mathematics study roadmap" and look at various answers from math.stackexchange.com, reddit.com, blogs, etc.

method 3) Look at the undergrad curriculum of math courses for degree requirements (e.g. Bachelor of Mathematics, Bsc Electrical Engineering, etc) published by universities. (e.g. go to http://mit.edu).

It seems like a "roadmap" for self-study is readily accessible for anyone curious.

Re: How to Learn Advanced Mathematics Without Heading to University

#34

Post by mathattack » Fri, Mar 11, 2016, 6:15 PM UTC

Amazing how thorough the Math curriculum is in the UK! I got a minor at my undergrad just by doing stats, linear algebra and calc through multivariate. I think I was just 2-3 courses shy of a BA in the subject.

I'm curious how many people have met self-trained mathematicians in real life. I know there are storied examples, but is it feasible? (Compared to say - self-trained programmers or writers)

Re: How to Learn Advanced Mathematics Without Heading to University

#35

Post by danharaj » Fri, Mar 11, 2016, 6:18 PM UTC

goc wrote: post #29

Any one know any books or resources that concentrate on invariant programming? Translating recursive code into properly tail recursive or iterative code can be pretty difficult.

`Pearls of Algorithmic Design` by Bird - Beautiful little book. It's a series of problems that are solved by first writing the naive program and then transforming it rigorously to make it more efficient.

`Algebra of Programming` by Bird and De Moor - This a treatment of the theory that is implicit in the methods of the previous book.

Re: How to Learn Advanced Mathematics Without Heading to University

#36

Post by danharaj » Fri, Mar 11, 2016, 6:19 PM UTC

Learn it in a group. Mathematics is a social activity. Meeting even once a week to discuss the material you're trying to learn will vastly improve the learning rate for everyone involved.

Re: How to Learn Advanced Mathematics Without Heading to University

#37

Post by ska » Fri, Mar 11, 2016, 6:20 PM UTC

graycat wrote: post #31

ska wrote: post #26

Earlier quoted context omitted.

Fun book indeed! Your first example you won't typically run into until an introductory measure theory course.

No, it's doable at the level of Rudin's Principles : He shows that a function is Riemann integrable if and only if it is continuous everywhere except on a set of measure 0. For the definition of measure 0, he gives that quickly, and don't really need a course in measure theory. Besides a course in measure theory likes just to f'get about Riemann integration, and thankfully so.

Agree, it's do-able with baby Rudin. But as I recall it's a typical example used (early) in measure theory, not so much intro analysis. Hence "typical".

Ymmv, of course.

Re: How to Learn Advanced Mathematics Without Heading to University

#38

Post by Patient0 » Fri, Mar 11, 2016, 6:22 PM UTC

danharaj wrote: post #36

Learn it in a group. Mathematics is a social activity. Meeting even once a week to discuss the material you're trying to learn will vastly improve the learning rate for everyone involved.

Brilliant suggestion!

Re: How to Learn Advanced Mathematics Without Heading to University

#39

Post by h1srf » Fri, Mar 11, 2016, 6:25 PM UTC

mathattack wrote: post #34

Amazing how thorough the Math curriculum is in the UK! I got a minor at my undergrad just by doing stats, linear algebra and calc through multivariate. I think I was just 2-3 courses shy of a BA in the subject. I'm curious how many people have met self-trained mathematicians in real life. I know there are storied examples, but is it feasible? (Compared to say - self-trained programmers or writers)

Where did you go to school? Those were just the prerequisites for my engineering degree.

Re: How to Learn Advanced Mathematics Without Heading to University

#40

Post by forgetsusername » Fri, Mar 11, 2016, 6:32 PM UTC

kafkaesq wrote: post #28

bmer wrote: post #10

Earlier quoted context omitted.

It is very easy to get most mathematics textbooks online, through slightly unsavoury means. It is far more difficult to figure out which textbooks are worth reading---this is a process that requires trial and error, and browsing through recommendations (math.stackexchange and mathoverflow.net have many good textbook recommendation questions, with many excellent answers). Also, it is very easy to audit courses at univ…

"Unsavoury", as in?

Piracy.

Though, between price fixing and booksellers going under and not being able to guarantee that you will retain access to the books you bought, I'd say that this form of piracy is morally ambiguous.