How to Read Mathematics • Hacker News Board

Re: How to Read Mathematics

#61

Post by pyninja » Wed, Jun 29, 2011, 8:45 AM UTC

pak wrote: post #58

pyninja wrote: post #54

Earlier quoted context omitted.

> I mean... isn't the point of academic writing to communicate research and results already performed? Obviously mathematicians don't leave "exercises for the reader" in research papers, but it is common in textbooks or expository writing. > I always felt that phrase has no place in the internet age, where the concept of a "page limit" is laughable and a simple hyperlink can point me to chapters upon chapters of appe…

>Obviously mathematicians don't leave "exercises for the reader" in research papers ORLY? I'll leave this right here for you: http://scholar.google.com/scholar?hl=en&q=left+as+an+exe... That's >100k hits from what Google Scholar can index alone--a few are from books, but the vast majority are journal articles.

I looked at the first few pages and saw only a few math papers, and in those the exercises were uninteresting details (e.g. just computation, or the second of two analogous cases where only the first is proved in the paper).

Do you really think anyone would benefit from such proofs in a paper? I would be interested if you could give an actual example of a paper where a reader might be inconvenienced by something being left as an exercise.

Re: How to Read Mathematics

#62

Post by dlib » Wed, Jun 29, 2011, 10:17 AM UTC

dfan wrote: post #16

My secret for reading difficult math and physics texts: write out, by hand, every equation as you encounter it. Firstly, it slows me down so I'm not tempted to skim it like a novel. More importantly, my standards for what I write are much higher than my standards for what I read; if I see something I'm not completely convinced of, I may shrug and move on, but I'm not willing to write something down unless I really un…

This is what I also do, any step in a derivation I would glance over when read must be fully understood and proved whenever I write it down. Every non-trivial result in my math books is written down so I can fully comprehend what's going on.

Re: How to Read Mathematics

#63

Post by tintin » Wed, Jun 29, 2011, 1:15 PM UTC

roundsquare wrote: post #55

tintin wrote: post #20

I understand math, but I can't read mathematical symbols. Does anyone know a good website/book for applying and understanding mathematical symbols?

Sorry, what exactly does that mean? Can you give some examples?

Well I learned to read math the 'computer' way. So here two examples of 'Math' vs 'Computer'

  ≠ -> !=
  √ -> sqrt

Re: How to Read Mathematics

#64

Post by Someone » Thu, Jun 30, 2011, 12:31 PM UTC

eru wrote: post #25

ColinWright wrote: post #14

Earlier quoted context omitted.

I expect you mean this bit: Suppose that ra and sa are the same modulo p, then we have r = s (mod p), so the p-1 multiples of a above are distinct and nonzero ... More completely, I expect you want them to say: Consider the (p-1) multiples of a given by: a, 2a, 3a, ... (p-1)a. (mod p) These are all distinct. To see this, consider otherwise, and suppose ra=sa (mod p) ... and so on. Is that what you meant? The point is…

Yes, I just went for something really basic to give an example of a semi-implicit indirect proof. I spent years of my life reading mathematics, so I do not trust myself to judge how hard a piece of mathematics is for outsiders. I find the proof cited is easy to read. About your addendum: You could have a look at Alexander Schrijver's "Combinatorial Optimization: Polyhedra and Efficiency". The interesting thing about…

Maybe I am too well-trained in this, but I think using 'Suppose' (or "assume") is a dead giveaway for a proof by contradiction.

The only semi-implicit ways to start a proof by contradiction I can think of are the phrases "if x is..." or (less implicit) "if x were...".

Re: How to Read Mathematics

#65

Post by roundsquare » Fri, Jul 01, 2011, 2:26 AM UTC

tintin wrote: post #63

roundsquare wrote: post #55

Earlier quoted context omitted.

Sorry, what exactly does that mean? Can you give some examples?

Well I learned to read math the 'computer' way. So here two examples of 'Math' vs 'Computer' ≠ -> != √ -> sqrt

Ah. Well, its not like your missing or unable to follow a fundamental concept, you just use different symbols. A bit of practice with reading things the "math way" and you'll be fine.