I’ve been struggling with this lately as I’ve been studying ML/Stat. Thanks ColinWright!
How to Read Mathematics
91–100 of 144 posts
Re: How to Read Mathematics
#92Feynman’s method to understand complex problems is so simple and elegant! Surely you’re joking, mr Feynman: ”I can’t understand anything in general unless I’m carrying along in my mind a specific example and watching it go. Some people think in the beginning that I’m kind of slow and I don’t understand the problem, because I ask a lot of these “dumb” questions: “Is a cathode plus or minus? Is an an-ion this way, or t…
So. much. Using simple examples, low dimensions and physical analogies is really a key to advanced math.
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Re: How to Read Mathematics
#93One 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…
But humans can't read it, nor can computers. Maybe it's time to ask if we can do better than traditional formal mathematical notation.
Plenty of humans can read formal mathematical notation just fine.
Re: How to Read Mathematics
#94How 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.
You can replace all that by a trip to sci-hub ;)
Re: How to Read Mathematics
#95One 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…
But humans can't read it, nor can computers. Maybe it's time to ask if we can do better than traditional formal mathematical notation.
What do you mean by "humans can't read it".
Re: How to Read Mathematics
#96Mathematical notation is often too vague, informal and too dependent on context.Frankly there is too much ambiguous overloading of symbols.It seems the language has evolved to be succinct and short to write, rather than being explicit and unambiguous.I know maths people say that it is meant to be read by people and not by computers. More people are acustomed to reading code than ever before.I feel that math notation needs reform that makes it formal, unambiguous and capable of describing imperative manipulations rather than be merely declarative, with clear annotations of type information of entities.
Re: How to Read Mathematics
#97Earlier quoted context omitted.
> I hate that. When I program my computer, I specify all the steps to accomplish something. If it's too verbose, then I abstract it in a function. > I even add comments to make sure that a human reader easily and completely understands what I do. Yeah, honestly, any single time I read a maths-related paper I feel like I would be better off with an example, well-written, implementation in any programming language. The…
So you'd never say to someone "use a bubble sort" or "try a bitwise trie" you'd lay it out in full, commented code each time? When you say "I abstract it in a function" that's surely what a mathematician is doing when they say "then we have a BVP and it naturally follows ...", they're applying the function "treatAsBVP". If you don't understand that step then you go and learn that, just as someone would have to find a…
> When you say "I abstract it in a function" that's surely what a mathematician is doing
In principle, yes.
In practice, I can right-click on any such function and get to see its source (edit: instantly, automatically) whenever I need it. That doesn't work at all with math papers.
EDIT: Furthermore, when writing about something where bubble sort is used, for the general audience, I would simply link to an explanation of what it is, visualization of how it works and an example code, so that anyone who doesn't know it can quickly get up to speed with it, then I'd explain how exactly that sorting method fits with the rest of topic.
In maths, papers are devoid of any links, the teaching materials are up to 50 years old and never updated to take advantage of new presentation techniques, authors frequently name-drop some algorithm and then forget to say what parameters they used or how exactly it connects with preceding and following paragraphs. It's really bad from UX point of view. That's why sites like BetterExplained are so important.
EDIT2: so, in short, I'd do everything I could - without spending huge amounts of time on it - to make the article as accessible and easy to follow as possible and to make it useful even if someone is not able to understand it whole. I'd really like to see mathematicians to at least try doing the same.
Re: How to Read Mathematics
#98Feynman’s method to understand complex problems is so simple and elegant! Surely you’re joking, mr Feynman: ”I can’t understand anything in general unless I’m carrying along in my mind a specific example and watching it go. Some people think in the beginning that I’m kind of slow and I don’t understand the problem, because I ask a lot of these “dumb” questions: “Is a cathode plus or minus? Is an an-ion this way, or t…
There's a lesson somewhere in there that is applicable in software engineering.
Re: How to Read Mathematics
#99Mathematical notation is often too vague, informal and too dependent on context.Frankly there is too much ambiguous overloading of symbols.It seems the language has evolved to be succinct and short to write, rather than being explicit and unambiguous.I know maths people say that it is meant to be read by people and not by computers. More people are acustomed to reading code than ever before.I feel that math notation…
That's what type theory is for. There have been some very interesting developments in the last 40 years that you might want to check out.
Re: How to Read Mathematics
#100A rather obvious but often overlooked piece of advice is to "read the masters".
IMHO it is very easy to waste tonnes of time reading papers / books that are OK but still vastly inferior to an account of a master.
Want to learn index theory: read Atiyah and Singer's original papers.
Interested in the classification of quadratic forms: read Serre's course in arithmetic.
Need to learn about the recent breakthroughs on prime tuples: read Tao!
Find time to read Weil, insist on reading Poincare, come up with an excuse to read Scholze etc.
Top mathematics literature is often spectacularly well written.
See also: https://mathoverflow.net/questions/28268/do-you-read-the-mas...