I understand math, but I can't read mathematical symbols. Does anyone know a good website/book for applying and understanding mathematical symbols?
How to Read Mathematics
21–30 of 65 posts
Re: How to Read Mathematics
#22I just had a reviewer criticize a report I wrote for containing "sentence fragments". The paper says things like, "Then x = y/2", which my reviewer interpreted as a one-word sentence ("Then"), followed by an equation. It made me realize how unnatural it is for some people to actively read mathematics-- for example, turning "=" into the verb "equals".
I agree “Then x = y/2” is fine. However, your reasoning why this is OK (“=” is “equal”) is incomplete. Consider “Then the answer = y/2.”, which is grammatically correct but improper.
Re: How to Read Mathematics
#23" Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis? " -Paul Halmos, inventor of "iff" and the ∎ symbol ( http://en.wikipedia.org/wiki/Paul_Halmos )
Interesting:
> Halmos argued that mathematics is a creative art, and that mathematicians should be seen as artists, not number crunchers. He discussed the division of the field into mathology and mathophysics, further arguing that mathematicians and painters think and work in related ways.
Re: How to Read Mathematics
#24Line 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.
You can skip on first reading to get an overview of where the author wants to take you, but don't expect it will clear up later on its own. Read and re-read. Fill in all the little gaps, especially when a sentence begins with "Clearly" or "As we can easily see".
"As we can easily see" == doesn't seem implausible after you read the paper countless times, re-work all the derivations from scratch at least 7 times, read the cited docs, get out your old analysis texts to look up some theorem you'd forgotten existed, and sacrifice a chicken.
I must say, there's nothing like reading a math paper to remind me that there's no shortage of people in the world that are way smarter than me.
Re: How to Read Mathematics
#25Earlier quoted context omitted.
Look at this proof of Fermat's Little Theorem [0], can you spot the indirect bit of the proof? By the way, when you construct a proof you almost always have some leeway in how big your indirect section are. E.g. variant A: Assume X, do bits Y, Z, then contradiction. Variant B: Do bits Y', Z', assume X, then contradiction. In the second variant Y' and Z' could be useful on their own, and might be easier to understand…
I expect you mean this bit: Suppose that ra and sa are the same modulo p, then we have r = s (mod p), so the p-1 multiples of a above are distinct and nonzero ... More completely, I expect you want them to say: Consider the (p-1) multiples of a given by: a, 2a, 3a, ... (p-1)a. (mod p) These are all distinct. To see this, consider otherwise, and suppose ra=sa (mod p) ... and so on. Is that what you meant? The point is…
Yes, I just went for something really basic to give an example of a semi-implicit indirect proof.
I spent years of my life reading mathematics, so I do not trust myself to judge how hard a piece of mathematics is for outsiders. I find the proof cited is easy to read.
About your addendum: You could have a look at Alexander Schrijver's "Combinatorial Optimization: Polyhedra and Efficiency". The interesting thing about its style is, that the author manages to make all lines require constant thought, while in most books there are really hard and really easy parts.
Re: How to Read Mathematics
#26The 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 easily follows that..." is the mathematician's secret handshake that the steps in between are boring, but pointing out an interesting conclusion and then saying the reader deserves an exercise to actually make sense of it, I mean... isn't the point of academic writing to communicate research and results already performed? Wouldn't it be better to show the work for your assertion so that others could critique it or offer even better solutions?
I know somebody is about to point out to me that I'm being "lazy" and should enjoy doing more work to learn so and so, but being asked regularly in the literature to reinvent somebody else's wheel seems to run counter to everything we do in CS (and academia in general, I would think). This article aside, I do sometimes feel that academic writing in the math/CS realm occasionally reeks a little bit of snobbery, where communication is held secondary to keeping up appearances.
Re: How to Read Mathematics
#27http://news.ycombinator.com/item?id=1576969 http://news.ycombinator.com/item?id=85743 http://news.ycombinator.com/item?id=583977 http://news.ycombinator.com/item?id=1509758
Of those, only the most recent one has any substantial comments, and even that is now 10 months old and can no longer be commented on: http://news.ycombinator.com/item?id=1576969
I enjoyed the Calculus for fools link :)
Re: How to Read Mathematics
#28The 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 unicycle - if you don't put in the hours, you won't be able to do it.
You can read all you like about how to program - if you don't put in the hours, you won't be able to do it.
Skills require practice. These instances of "left as an exercise for the interested reader" are to help you really learn and properly understand the material. And if you don't care, don't bother. If you are unwilling to put in the time then the chances are that you would end up thinking you understand an explanation, whereas in fact you don't.
The above is only for good writing, of course. There are plenty of instances of bad writing, but that's not then a complaint about the subject, it's a complaint about the writer.
Someone once said of Feynman that his lectures were beautifully clear, and that those who listened gained real insight and understanding. Until they had to use it. Then they realised that they didn't really understand it at all.
If you want understanding, do the exercises.
So there you are, I've done as you predicted and pointed out that you're just being lazy, and that if you really want to learn then you have to put in the work. Just because you preempted it, doesn't mean it's wrong. It's right, you already know it. Complaining won't help you.
Put in the time.
Re: How to Read Mathematics
#29I 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 for everything and a set of similar macros meaning entirely different things in different files would be considered unreadable by pretty much any programmer.
IMO, the world would immensely benefit from another version of mathematical notation designed specifically for explaining things, rather than doing calculations on paper.
Re: How to Read Mathematics
#30[deleted]