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How to Read Mathematics

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Re: How to Read Mathematics

#71

Post by obastani »

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…

I agree that having possible examples in mind is a great way to learn mathematics. There are whole books on useful counterexamples, e.g.

https://www.amazon.com/Counterexamples-Analysis-Dover-Books-...

These counterexamples are sometimes a bit involved, but I find they are often useful for understanding the purpose of the technical assumptions that accompany many theorems.

Re: How to Read Mathematics

#72

Post by klibertp »

ColinWright wrote: post #59
klibertp wrote: post #53

Earlier 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…

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.

> working at the very boundaries of human knowledge

That's a pretty minor part of all the papers produced every day.

> to be completely up-to-date

Did I say anything about being "completely" up-to-date? Most of the good practices for readability and maintainability of code were known in the 80s, with almost all of the bad practices disappearing in the late 90s. That's twenty years.

> the latest research in software engineering development

That's not even a research, actually. Things like using meaningful function and variable names (I can never, for the life of me, understand why someone would prefer `X` over `training_set` or `x` over `element`), using comments, using uniform, standard formatting, abstracting - but not hiding or omitting - mechanical details and so on don't come from standard research (I think?), but from the practice of programming.

It is completely reasonable to expect people to improve their craft and to change their ways to better fit a changing environment. The arrival of general-purpose computers and later the Internet had a huge impact on almost every other occupation, from accountants to physicists, to biologists and medical professionals, to writers and journalists... just not on mathematicians. I may be wrong on this, of course, it could have changed in the meantime, but when I was a university student I learned algebra - as I've been told to - from a book printed in 1978. And that was ten years ago.

Re: How to Read Mathematics

#73

Post by kqr »

sundarurfriend wrote: post #57
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 bal…

Not from the book but personal experience:

This is also useful to understand preconditions by reduction. I.e. if you want to understand a theorem, it can sometimes be useful to start by figuring out the reason behind the preconditions. "Why does this apply only to balls that have hair?" Simply go, "What would the theorem imply if I start with a smooth ball instead?"

This practise can also lead to generalisations. Oftentimes starting with a smooth ball will make you go "What? That't can't be possible."

But sometimes, starting with a smooth ball leads you to, "Huh, that's really, really weird. But it's not a contradiction in and of itself. I could use that result in another context!"

Re: How to Read Mathematics

#74

Post by klibertp »

ryanmonroe wrote: post #60
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…

You forgot this:

> If it's too verbose, then I abstract it in a function.

Yeah, not everything has to be included in your source and you do rely on abstractions when writing code. On the other hand, these abstractions are always there, and it "easily follows" that you can take a peek at them at any time, up to and including disassembling a binary executable. You can't do this with a maths paper - you can't just go to Github to check the source of a library, you have to "reimplement", again and again, pieces of the paper that an author decided to omit.

Re: How to Read Mathematics

#75

Post by emacsgifs »

erasemus wrote: post #47
Smaug123 wrote: post #36

Earlier quoted context omitted.

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.

It's as if we truly believe that 0 is positive charged and 1 is negative charged.

Or to take us a step further, that a certain arrangements of electrons is "negative" and the inverse arrangement is "positive"

I often tell CS people that everything in computing is metaphor and it honestly disturbs me that 80% of the time I get blank looks.

Re: How to Read Mathematics

#76

Post by lostcolony »

"Mathematical ideas are by nature precise and well defined, so that a precise description is possible in a very short space"

"A well-written math text will be careful to use a word in one sense only"

So many papers are not that careful, and thus, not so precise. There are only so many greek letters, so they end up overloaded. I can't count the number of times I've struggled over an equation, only to find out that the reason it made no sense and/or the value for a specific example came out differently was because the author assumed one variable to mean something differently than I did (and my assumption was perfectly valid in another context. Sometimes another context in -the same paper-). The whole experience is incredibly frustrating.

Echoing another point here, if you just show a sufficiently non-trivial example and walk me through it, I grok it quickly. -Then- give me the equation if you must.

Re: How to Read Mathematics

#77

Post by emacsgifs »

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 endlessly frustrating that the current trend in functional programming fetishizes mathematical notation when describing the same concepts with more verbose versions of the same thing in other programming languages would be clearer and more widely understood.

Re: How to Read Mathematics

#78

Post by joppy »

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 should all mathematics be written using exactly the same language and conventions? Should all computer programs be written in the same language? People modify the language (natural, mathematical, programming) they are using to suit their problem. It is unreasonable to expect that the kind of language you use at the foundations of mathematics for example (set axioms and so on) would be efficient and useful for stating properties of much more complex objects: this exactly parallels computer programming languages.

Admittedly, there are no big tables containing "definitions" of these languages. But I think it is common opinion that if you don't understand the language a paper is using, then there is a very large chance you don't know the underlying theory, and the paper will make no sense anyway.

Re: How to Read Mathematics

#79

Post by lostcolony »

internetman55 wrote: post #68
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…

This is true. The CS folks are my university tended to be quite strict about writing out quantifiers for everything, while a very accomplished math researcher in our department told our class to "please write in English" instead of using those symbols, because he wouldn't be able to read them. I asked a friend why he thought this was, and he suspected that the CS department's focus on formal languages and automatic t…

More computers understand only what they're told. You -have- to be exact.

Humans can infer. If it doesn't make sense, they realize something is wrong and attempt to make sense of it.

As to why the mathematician preferred 'English', I think it's because quantifiers aren't universal. Mathematical notation itself isn't universal. That is, while the student may be internally consistent and exact in their qualifiers, they are not universally consistent between students, nor with the professor's expectation. The professor wanted to immediately grok what the student was trying to do, not have to approach every single turned in assignment as though it was an unfamiliar mathematical paper, using its own notation, that he had to interpret (where some symbols may share a standard meaning across papers, others won't)

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