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How to Read Mathematics

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Re: How to Read Mathematics

#51

Post by malmsteen »

wimagguc wrote: post #28

Feynman’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.

Re: How to Read Mathematics

#52

Post by nathell »

> The author usually spends months discovering things, and going down blind alleys. At the end, he organizes it all into a story that covers up all the mistakes (and related motivation), and presents the completed idea in clean neat flow. The way to really understand the idea is to re-create what the author left out. Read between the lines.

Pity there's little mathematical writing that actually _does_ contain these blind alleys and mistakes. Part of why I enjoy "Concrete Mathematics" by Graham/Knuth/Patashnik is that it's one of the rare counterexamples.

Re: How to Read Mathematics

#53

Post by klibertp »

wiz21c wrote: post #31

FTA : >>> One can now check that the next statement is true with a certain amount of essentially mechanical, though perhaps laborious, checking. I, the author, could do it, but it would use up a large amount of space and perhaps not accomplish much, since it'd be best for you to go ahead and do the computation to clarify for yourself what's going on here. I promise that no new ideas are involved, though of course you…

> 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.

Then I recall what the code, written by mathematicians, tends to look like, and I'm not so sure anymore...

Seriously, the fact that math people choose to ignore the last 50 years of software engineering development, which was in a large part focused on readability and maintainability, instead of incorporating the techniques for themselves is really baffling. As it is, aside from a couple of really well-written articles, reading any paper has a high chance of frustrating me to tears...

Re: How to Read Mathematics

#54

Post by klibertp »

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.

> The analogy approach works for programming, but it's not applicable to seriously studying mathematics.

Why?

Re: How to Read Mathematics

#56

Post by klibertp »

kalid wrote: post #17
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.

Shameless plug, I write about math on BetterExplained.com For each symbol, I try to make an analogy. For "e", for example, I have the notion of "continuous growth". The formal definition is this: https://betterexplained.com/ColorizedMath/content/img/E_(mat... "The base for continuous growth is the unit quantity earning unit interest for unit time." Once you see the role of each part of the definition, the idea snaps…

Just wanted to thank you for a great job you do on BetterExplained. I was able to understand and put to use some concepts only thanks to your articles, most recently it was Fourier transform. I'd be much better at math if people writing educational materials realized that there are more effective ways to teach (and to learn) than what they had experienced. Thank you!

Re: How to Read Mathematics

#57

Post by sundarurfriend »

wimagguc wrote: post #28

Feynman’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…

Also from the same book:

"I had a scheme, which I still use today when somebody is explaining something that I’m trying to understand: I keep making up examples. For instance, the mathematicians would come in with a terrific theorem, and they’re all excited. As they’re telling me the conditions of the theorem, I construct something which fits all the conditions. You know, you have a set (one ball) – disjoint (two balls). Then the balls turn colors, grow hairs, or whatever, in my head as they put more conditions on. Finally they state the theorem, which is some dumb thing about the ball which isn’t true for my hairy green ball thing, so I say, ‘False!’"

Re: How to Read Mathematics

#58

Post by chestervonwinch »

wiz21c wrote: post #31

FTA : >>> One can now check that the next statement is true with a certain amount of essentially mechanical, though perhaps laborious, checking. I, the author, could do it, but it would use up a large amount of space and perhaps not accomplish much, since it'd be best for you to go ahead and do the computation to clarify for yourself what's going on here. I promise that no new ideas are involved, though of course you…

Of course, ambiguity can still arise even when the author puts the details into the appendices because "details of the details" could still be missing. This can happen when there is a mismatch between the assumed background by the author and the actual background of the reader. There are many hypothetical answers to why the article was not written for a more general audience (and not all require assumed malintent by the author).

Re: How to Read Mathematics

#59

Post by ColinWright »

klibertp wrote: post #53
wiz21c wrote: post #31

FTA : >>> One can now check that the next statement is true with a certain amount of essentially mechanical, though perhaps laborious, checking. I, the author, could do it, but it would use up a large amount of space and perhaps not accomplish much, since it'd be best for you to go ahead and do the computation to clarify for yourself what's going on here. I promise that no new ideas are involved, though of course you…

> 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…

It is unreasonable to expect that people who are experts in their own field, working at the very boundaries of human knowledge, should also be expected to be completely up-to-date with the latest research in software engineering development.

Re: How to Read Mathematics

#60

Post by ryanmonroe »

wiz21c wrote: post #31

FTA : >>> One can now check that the next statement is true with a certain amount of essentially mechanical, though perhaps laborious, checking. I, the author, could do it, but it would use up a large amount of space and perhaps not accomplish much, since it'd be best for you to go ahead and do the computation to clarify for yourself what's going on here. I promise that no new ideas are involved, though of course you…

> When I program my computer, I specify all the steps to accomplish something.

You certainly do not specify all the steps your computer must take to run the program because that would make even the simplest program a nightmare to read. In the same way, specifying all the steps a reader must take would make math papers completely unreadable. There's a line which should be chosen wisely, but it's not as simple as "just explain everything".

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