Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

How to Read Mathematics

web.stonehill.edu

Post reply on HN
Print view View original

31–40 of 65 posts

Re: How to Read Mathematics

#31

Post by nothis »

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 you suddenly realize they're easily explainable with a few words of plain text or --gasp-- a "childish" drawing to illustrate.

Re: How to Read Mathematics

#32

Post by pak »

ColinWright wrote: post #28
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…

The most irritating thing to me is when someone says they want to understand something, but then they are unwilling to put in any real effort. It is a misapprehension, misunderstanding and misrepresentation to say: ... reinventing the wheel runs counter to everything we do in CS. You can read all you like about how to juggle - if you don't put in the hours, you won't be able to do it. You can read all you like about…

You must have only seen this phrase in places where it was used in the bright, sparkling, professorial manner assumed by your comment. I take no offense when it appears in a textbook, where the intent is to have me practice a skill. I do object to its appearance in a journal article, however; that is a context where you are supposed to communicate findings as clearly and concisely as possible. In particular, I've come across its use more and more often where one of the following likely applies:

1. it is doubtful that the author has actually done the work to prove his own assertion, and uses the phrase to cast his own burden on the reader,

2. the author has possibly done the work, but can't be bothered to condense it to the quality required for publication,

3. showing the work would clearly be useful for the target audience, but the author is more concerned with making the material appear difficult, or

4. the author is being ironic, because the assertion is either superfluous, outright incorrect, or known to be unprovable.

In fact, all these bad use-cases are common sources of humor among math nerds (http://www.jargon.net/jargonfile/e/exerciseleftasan.html), (http://abstrusegoose.com/12), (http://fasterdonuts.tumblr.com/post/4516834904/the-proof-is-...) ... so I think I'm being fair in calling this phrase a trope. As in, its overuse is well-known, tolerated, and occasionally ridiculed.

tldr: I am indeed complaining about bad writing, not the subject--specifically that "left as an exercise" has become a common idiom behind which bad writers in math and CS hide, and its inappropriate use is now all too common.

Re: How to Read Mathematics

#33

Post by nocipher »

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…

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. Mathematicians have developed methods and notation to achieve that end, not to make it easily digestible by the uninitiated. Rigor is an important part of this. It is what allows mathematicians to be so sure their work is correct and it is often what makes math seem so arcane. It is not always necessary for teaching, but professional mathematicians are often the teachers and the ideas are intimately tied to their rigorous formulations in their head.

For example, continuity is a fairly intuitive concept in calculus, but the rigorous epsilon-delta definition is necessary in proofs. The basic concept, while easy to explain and understand is useless, while the precise formulation, though much more opaque, is ubiquitous in analysis simply because it is an incredible tool... for mathematicians.

An engineer likely just needs the machinery built on top of the analysis: the derivative and integral. If that's all you need, then pick a book that is focused on applications and a development of general intuition, not a book designed for mathematicians-in-the-making.

Re: How to Read Mathematics

#34

Post by btilly »

ckuehne wrote: post #7
Rhapso wrote: post #5

The big ones: Watch out for proof by contrapositive and contradiction. Until you really get used to proving things by assuming bits of them are false it throws you off to see it used in textbooks and papers because most papers do not tell you they are using them.

Could you cite an example?

To find the maximum of a function on a closed interval it suffices to look for places where the derivative is 0, undefined, or at the boundary of the interval.

Proof, Suppose that x in the middle of the interval has a derivative > 0. Then from the definition of the derivative there exists h > 0 such that x+h is in the interval and 0 f(x) and therefore x is not where the maximum is achieved. The argument for f'(x) Therefore if f achieves a maximum at x, then x is either a boundary point, or a spot where the derivative is 0 or undefined.

Re: How to Read Mathematics

#35

Post by nocipher »

scotty79 wrote: post #8

Line by line carefully. Never skip anything in hopes it will clear up later. You may allow to yourself not to know with what intention author wrote the line but you should never allow yourself not to know why the author could write this line.

The best way I've found to read math texts is to take it concept by concept. Line by line is too mechanical. Approach each concept and get a "gist" for what is trying to be accomplished. Once you understand where they are starting from for a given concept, where they are going with it, and have a rough idea of how they'll get there, then you read the details. The details are essentially meaningless if you don't know the bigger picture.

Re: How to Read Mathematics

#36

Post by nocipher »

romaniv wrote: post #29
kruhft wrote: post #3

I remember back in school that doing the math was not really a problem, but actually reading the textbooks was. I was the same way with programming when I started, where I could write but not read the language, but I'm suprised that I got through 4 years of university math without really being able to read the proofs that explained it.

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 it.

Re: How to Read Mathematics

#37

Post by contravert »

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 completely agree. This phrase appears in even some of the most hallowed tomes of computer science, such as Introduction to Algorithms. The textbook puts important theorems and proofs in the exercises section, and does not even provide solutions online (at least for the later sections). Keep in mind that this book is used as a reference manual for many implementations of algorithms and data structures.

Re: How to Read Mathematics

#38

Post by ColinWright »

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'm working on something at the moment to try to explain exactly why what you've said here is, to a large extent, simply wrong. It's too long to include here, and it's not yet ready to "publish". You're about two weeks too early.

But let me ask you this. It's easy to cut a square into identical pieces so that all the pieces touch the center point.

In slightly more detail, the pieces are disjoint sets such that their union is the whole square. The pieces are identical except perhaps for details as to the boundaries. To say that they all "touch" the center point means that every non-zero radius disk centered at the center contains some points from each piece.

So now, how many ways can this be done? No, it's not five. And no, it's not six either.

When you start trying to work it out you find that the details matter, and they can't just be covered by a "childish" drawing to illustrate.

Details matter, and some of them are hard.

Yes, most math teaching is atrocious. We all know that. But it's not always just the teacher's fault. Sometimes it's at least partly the fault of the readers expecting everything to be made simple and immediately accessible with neither work nor effort.

Re: How to Read Mathematics

#39

Post by ColinWright »

pak wrote: post #32
ColinWright wrote: post #28

Earlier quoted context omitted.

The most irritating thing to me is when someone says they want to understand something, but then they are unwilling to put in any real effort. It is a misapprehension, misunderstanding and misrepresentation to say: ... reinventing the wheel runs counter to everything we do in CS. You can read all you like about how to juggle - if you don't put in the hours, you won't be able to do it. You can read all you like about…

You must have only seen this phrase in places where it was used in the bright, sparkling, professorial manner assumed by your comment. I take no offense when it appears in a textbook, where the intent is to have me practice a skill. I do object to its appearance in a journal article, however; that is a context where you are supposed to communicate findings as clearly and concisely as possible. In particular, I've com…

Yes, it's used too often. Yes, it's used sometimes when it's inappropriate. Yes, sometimes I suspect that the author doesn't really know the answer. I'm less sure that the author uses it it all seriousness when the know the result is unproven or unprovable.

But it is a phrase that has its place, there are times when it's exactly right to use it. Don't reject it outright. I've used it in publication where the editor actually asked me to remove details and put the phrase in its place.

It's not always the author.

Re: How to Read Mathematics

#40

Post by slowpoke »

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…

>I fail to see how creating a new set of symbols would do anything but hinder communication within the math community.

I don't think this is just about the math community. The problem - at least as I see it, as a student myself - is that Math is a required part of many fields of studies. Fields which not necessarily have much to do with Math. And that's a problem for many of the students. I won't say that Math is entirely unneeded and should be completely abolished, but what I am saying is that many of these students (including myself) don't want to be part of said "math community".

We don't want to have to learn the arcane notations and "math language" just to be able to use some of the tools we might indeed need. Yes, I might sound like a whiny student who's butthurt over having to actually study for something. I may be. However, seeing as /so many/ other students (who, for the most part, are competent in their own chosen field) are in danger of failing their studies or struggle heavily with their math lectures simply because they are /not/ competent at getting into the "math community" is ridiculous to say the least.

For the record, I study Informatics at a university in Germany, so the situation might be different elsewhere, but I've read about similar issues at American colleges and universities, and every now and then, related articles pop up here at HN.

Post reply on HN