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How to Read Mathematics

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Re: How to Read Mathematics

#41

Post by heavenlyblue »

catnaroek wrote: post #25
criloz2 wrote: post #13

Earlier quoted context omitted.

haha true, I envy people that just sit down read a book/chapter/article/blog linearly and then comes enlightened, I can't, I need to first Skim almost 90% of the material, and then start again from the bottom, reading the last part and trying to figure out how the first part is connected or are important for understand what is written. I think that people that read it linearly don't miss important info, but I just ca…

Nobody is born knowing how to read effectively. It is a matter of practice and discipline.

Have you actually read any of the papers? While I tend to agree with you in an ideal situation where you may assume the writer is good - that is usually not the case with real-life papers.

It's not even the writing per se - it's the structure, or omissions instead. I would rather skim first to find which parts I may not be able to grasp right away and to find out if there's any point spending time on this, rather than go on and read the pieces from the citation list.

Re: How to Read Mathematics

#42

Post by jonnybgood »

tw1010 wrote: post #37
catnaroek wrote: post #24

Earlier quoted context omitted.

When you are a student, rather than a researcher, formal definitions must come first, and intuition can be developed later. Otherwise, you are just allowing yourself to say nonsense.

I worry about the fact that you're being downvoted, because what you're saying is utterly crucial and important. I fear those who ignore it will waste a lot of time with really misguided ways of doing mathematics. The analogy approach works for programming, but it's not applicable to seriously studying mathematics.

I wouldn’t dismiss analogy so easily. George Polya opened my eyes to the power of analogy in math. I can’t recommend his “Mathematics and Plausible Reasoning” enough.

Re: How to Read Mathematics

#43

Post by brango »

hcoyote wrote: post #21
sulam wrote: post #16

I’m curious if anyone has advice on how to understand all the symbols used in math papers. Frequently I find this is the first barrier for me in reading them.

Probably the first thing to do is find a copy of Lawrence Chang's Handbook of Spoken Mathematics (also referenced in a previous HN post: https://news.ycombinator.com/item?id=10284052 ) This has been great for helping me figure out what to even call various symbols so I can then go look up what they actually intend you to do.

Thanks! I've been looking for something like this for so long. It's always been a pain to study something like graph theory, graphics, crypto, etc. without even being able to read the formulae.

Re: How to Read Mathematics

#44

Post by remus »

retox wrote: post #32
jordigh wrote: post #2

One thing that I really would like to impress upon computer people is that mathematics is not (usually) a formal computer language. Symbols and technical terms change meaning depending on the author, people write with different "accents". There's a little ambiguity and informality at times, but a whole lot less than other kinds of writing. Mathematics is (usually) written for humans, not computers. Don't attempt to r…

Why is that? Why is there not a universal way of writing mathematics that is not ambiguous and can be read by anyone that understands the 'language'? I have a deep dread and fear of numbers and mathematics in general because I don't understand them and I have never learned. Now I learn that there isn't one thing to learn but a vast array? No thanks. I had this apparently romantic view that an equation is an equation…

> Now I learn that there isn't one thing to learn but a vast array? No thanks. I had this apparently romantic view that an equation is an equation (the same equation) anywhere in the world.

It's not as bad as that! You're very unlikely to hit this sort of issue until you're reading research papers (unless you decide you're very interested in mathematical history). Pretty much all the material youll come across up to ~undergrad level will use very, very similar notation.

Re: How to Read Mathematics

#45

Post by TuringTest »

retox wrote: post #32
jordigh wrote: post #2

One thing that I really would like to impress upon computer people is that mathematics is not (usually) a formal computer language. Symbols and technical terms change meaning depending on the author, people write with different "accents". There's a little ambiguity and informality at times, but a whole lot less than other kinds of writing. Mathematics is (usually) written for humans, not computers. Don't attempt to r…

Why is that? Why is there not a universal way of writing mathematics that is not ambiguous and can be read by anyone that understands the 'language'? I have a deep dread and fear of numbers and mathematics in general because I don't understand them and I have never learned. Now I learn that there isn't one thing to learn but a vast array? No thanks. I had this apparently romantic view that an equation is an equation…

> Why is that? Why is there not a universal way of writing mathematics that is not ambiguous and can be read by anyone that understands the 'language'?

There is one, it's called formal logic. The problem with it is that it's not adequate to explain things, only to prove theorems formally; any strict unambiguous proof written in formal logic looks more like a debug trace than a scientific article.

To provide a proper explanation of a mathematical result, at many points in the proof you need to summarize the hundreds of intermediate mechanical steps and write the insight behind them in common language, using potentially ambiguous words.

Re: How to Read Mathematics

#47

Post by erasemus »

Smaug123 wrote: post #36
retox wrote: post #32

Earlier quoted context omitted.

Why is that? Why is there not a universal way of writing mathematics that is not ambiguous and can be read by anyone that understands the 'language'? I have a deep dread and fear of numbers and mathematics in general because I don't understand them and I have never learned. Now I learn that there isn't one thing to learn but a vast array? No thanks. I had this apparently romantic view that an equation is an equation…

Why is there not a universal way of writing anything that is not ambiguous and can be read by anyone that understands the 'language'? Because humans have a tendency to defy attempts at classification and constraint. Constructed real-world languages (Lojban, Esperanto) haven't really taken off, while non-constructed languages mix and match whenever they feel like it ("le parking", "Schadenfreude"). The same is true in…

Because whenever we read something we have to guess the meaning. To the human mind, all communications are inherently contextual and metaphorical. To put it another way each individual has his own internal private language into which everything he reads must be translated.

Re: How to Read Mathematics

#48

Post by tw1010 »

jonnybgood wrote: post #42
tw1010 wrote: post #37

Earlier quoted context omitted.

I worry about the fact that you're being downvoted, because what you're saying is utterly crucial and important. I fear those who ignore it will waste a lot of time with really misguided ways of doing mathematics. The analogy approach works for programming, but it's not applicable to seriously studying mathematics.

I wouldn’t dismiss analogy so easily. George Polya opened my eyes to the power of analogy in math. I can’t recommend his “Mathematics and Plausible Reasoning” enough.

Polya's text was targeted towards researchers and people who already understood the basics pretty well. But I don't think the analogy approach is valid (or at least, not recommended) in the context of this thread, where the parent seemed to assume that the audience wasn't even particularly comfortable with limits. It can be a dangerous and seductively-easy path to go down, is all I'm saying. You're not going to understand mathematics without having a firm firm grasp of the rigorous definitions and by doing mountains of exercises (where it will often be quickly apparent that using an analogy is not sufficient).

Re: How to Read Mathematics

#50

Post by giomasce »

vog wrote: post #40
zitterbewegung wrote: post #3

How I read Mathematics. First try to find the paper. First use http://front.math.ucdavis.edu . If you can't find it then use google scholar. If that fails then go on the authors academic website. If that fails then see if the author is live and try to contact them. Then repeat for their grad students. If all else fails post it to r/math or HN and ask for the pdf. First you want to see if this paper is even worth read…

> First try to find the paper. First use http://front.math.ucdavis.edu . If you can't find it then use google scholar. If that fails then go on the authors academic website. If that fails then see if the author is live and try to contact them. Then repeat for their grad students. If all else fails post it to r/math or HN and ask for the pdf. Or, go directly to arxiv, scihub or a similar service. Fully agree with the…

This is still the easy path. Sometimes you have to hope that the article you need was translated from Russian to English, find which is its translated title, search the title, understand that you could not find anything because you have to search the title the volume it was published into, not its original one; then search again the new title, find nothing on GenLib, SciHub and similar, go the university library, discover that the section this article belongs to is currently packed because they are renovating (or moving, who remembers) and there is not ETA and they do not allow you to fetch it from the packages; speak of the problem with your advisor, who tries to find it in his university's library; in the meantime, speak of the problem with a friend, that encourages to you to ask the library again, this time the library agrees to get the paper for you and you photocopy it; more or less at the same time your advisor manages to give you a photocopy. At last, scan the paper (because you do not want to carry a hundred pages in your backpack all the time, do you?) and put it on GenLib for the sake of future souls that might need it.

All of this just to have the paper. During reading, you still need to check the Russian original and ask a Russian friend (as Google translator is not necessarily to be trusted on mathematical texts) for some points that make no sense and turn out to be mistranslations. Not to mention the references to other Russian papers from the sixties that could be even more difficult to find.

On the other hand I have to mention that the paper actually contained the theorem I wanted and the proof was reasonably clear. The author also replied helpfully to an email I sent him! So this was definitely worth the effort. I was lucky!

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