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How to Read Mathematics

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Re: How to Read Mathematics

#51

Post by Someone »

nothis wrote: post #31

That's a very elegant way of saying: "Communication in math is just as fucked up as you'd think." As an outsider who only ever needs to use maths as a "tool" for very specific things, maths books and general attitude by professors and teachers is just nightmarish. Somewhere behind that curtain of inaccessibility, it has to be their fault. Especially, since once you actually do understand certain mathematical concepts…

I have my doubts about that claim. In most cases, the 'you' that thinks something 'is easily explainable with a few words of plain text' often is a different 'you' from the one that struggled grasping the concept. That struggle is what made you think the concept is 'easy'.

Next time you encounter such a 'aha erlebnis', try saying the 'few words of plain text' to someone who still has to master the subject, and _immediately_ test him or her on the subject. I predict that, oftentimes, (s)he understood nothing of what you said.

Case in point: at university, I had a prof who said that it was perfectly OK if you could not do any practice integral. Trust me, he said, if, a month from now, you do these exercises, you will not understand what is hard about them.

Re: How to Read Mathematics

#52

Post by jamii »

ColinWright wrote: post #46
nocipher wrote: post #33

Earlier quoted context omitted.

Most math concepts -are- fairly easy to explain and, from the sounds of it, you don't actually want a math book. You want a book that gives you the executive summary or a run-down of specific tools for specific situations. The flaw is not in the books or the teaching, it is what you are expecting from mathematicians. Math is, in a very general sense, about understanding and reasoning about highly structured objects.…

Do you really think that continuity is an obvious concept? Are you aware, for example, that there are functions that are continuous at all irrational points and discontinuous at all rational points?

Which comes back to the problems with the OPs argument. Sure, you can explain continuity in a few simple sentences but sooner or later you will start to hit corner cases and you find that those few simple sentences are really quite ambiguous. It's much like using pseudocode to describe an algorithm. I've found that the best mathematical texts provide both - a 'morally correct' definition in plain english to describe the intention and a mathematical description which defines the precise meaning.

Re: How to Read Mathematics

#53

Post by gambler »

nocipher wrote: post #36
romaniv wrote: post #29

Earlier quoted context omitted.

That's natural, since mathematical notation is heavily slanted towards writing on paper and symbol manipulation. Use of one-letter variable names and highly context-sensitive notation reduces the number of pen strokes, but doesn't help the reader. Modern programming languages and conventions are quite a bit better in that respect, since people design them to be maintainable. A program using one-letter variable names…

The problem is that the mathematical notation, while not perfect, works really well for those adept with its use. It's an efficient shorthand. For explaining things in depth we have pictures and and natural language. I fail to see how creating a new set of symbols would do anything but hinder communication within the math community. If you have any deeper argument to support your position, I'd be interested to hear i…

You speak about "math community" as if is some exclusive, elite club. In reality, anyone who needs to read a formula or some proof is part of that group.

Also, a formula can be written once and read hundreds of thousands of times. Because of this, I consider ease of reading math notation far more important than efficiency of writing in it.

Re: How to Read Mathematics

#54

Post by pyninja »

pak wrote: post #26

The most irritating trope of math and CS literature has to be whenever something interesting or useful is "left as an exercise to the reader." 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 appendices. Calling something "trivial" is swallowable, and perhaps this article's example of "it easil…

> 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 appendices.

The constraint is not page length, of course, but time. It is a much better use of a mathematician's time to leave easily reproducible proofs to the reader, and focus on the interesting ideas.

Anyway, doing these kinds of exercises is actually very useful in understanding new ideas.

Re: How to Read Mathematics

#56

Post by nocipher »

gambler wrote: post #53
nocipher wrote: post #36

Earlier quoted context omitted.

The problem is that the mathematical notation, while not perfect, works really well for those adept with its use. It's an efficient shorthand. For explaining things in depth we have pictures and and natural language. I fail to see how creating a new set of symbols would do anything but hinder communication within the math community. If you have any deeper argument to support your position, I'd be interested to hear i…

You speak about "math community" as if is some exclusive, elite club. In reality, anyone who needs to read a formula or some proof is part of that group. Also, a formula can be written once and read hundreds of thousands of times. Because of this, I consider ease of reading math notation far more important than efficiency of writing in it.

I disagree. By "math community" I refer specifically to those people who create new mathematics, i.e. mathematicians. That is not at all as inclusive as you argue. Math notation comes from mathematicians and is designed to facilitate communication between people of that profession. The notation is very functional and efficient for its designed task.

To create a new set of symbols and notation designed specifically for those who are not mathematicians is a waste of resources. Professors, whom would be responsible for teaching such a thing, would have to translate all of their concepts from "professional notation" to "novice notation" when lecturing. This involves them learning a totally new set of notation designed for no reason other than teaching. Furthermore, anyone who wanted to pursue mathematics professionally would have to then learn "professional notation".

Aside from these logistical difficulties, I fail to see how a different set of notation would be any clearer or easier to understand than the current notation, which is already widely accepted and considered useful. Essentially, you are arguing that math notation is too difficult and so it should be simplified. That is like saying Faulkner or Joyce is too difficult to read so should be rewritten for those without the ability to comprehend the source document: it sort of misses the point.

Re: How to Read Mathematics

#57

Post by nocipher »

kenjackson wrote: post #49
ColinWright wrote: post #45

Earlier quoted context omitted.

Yes, but there's more that you can say. In particular, there is more than one infinite family. How many are there? And do all solutions belong to an infinite family? Or are there isolates/sporadics? Can you characterise the solutions? How many pieces do they contain? Some solutions have two pieces. Some have four. Are there other possibilities?

I don't know what an infinite family, isolate, or sporadic is. A quick Google search didn't make it obvious. I think 8 pieces also works, but I think that may be it.

An infinite family seems to refer to an infinite collection of solutions that are all related in some sense. An isolate or sporadic solution is a solution that is unique and unrelated to other solutions. They are both terms used to categorize solutions.

Re: How to Read Mathematics

#58

Post by pak »

pyninja wrote: post #54
pak wrote: post #26

The most irritating trope of math and CS literature has to be whenever something interesting or useful is "left as an exercise to the reader." 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 appendices. Calling something "trivial" is swallowable, and perhaps this article's example of "it easil…

> 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.

Re: How to Read Mathematics

#59

Post by kenjackson »

nocipher wrote: post #57
kenjackson wrote: post #49

Earlier quoted context omitted.

I don't know what an infinite family, isolate, or sporadic is. A quick Google search didn't make it obvious. I think 8 pieces also works, but I think that may be it.

An infinite family seems to refer to an infinite collection of solutions that are all related in some sense. An isolate or sporadic solution is a solution that is unique and unrelated to other solutions. They are both terms used to categorize solutions.

Thanks nocipher for the definitions. From this I see no sporadics. And two infinite families, at least how I'd classify a family.

Re: How to Read Mathematics

#60

Post by ColinWright »

kenjackson wrote: post #49
ColinWright wrote: post #45

Earlier quoted context omitted.

Yes, but there's more that you can say. In particular, there is more than one infinite family. How many are there? And do all solutions belong to an infinite family? Or are there isolates/sporadics? Can you characterise the solutions? How many pieces do they contain? Some solutions have two pieces. Some have four. Are there other possibilities?

I don't know what an infinite family, isolate, or sporadic is. A quick Google search didn't make it obvious. I think 8 pieces also works, but I think that may be it.

nocipher's definition are correct.

And I have, so far, 5 infinite families with, respectively, 2, 4, 8, 16 and 32 pieces.

But details matter, and you might start to question what it really means when I say "piece." Does the definition I've given really capture your intuition?

But I need to go back to writing the article.

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