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How to Read Mathematics

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Re: How to Read Mathematics

#21

Post by alister »

rcthompson wrote: post #5

I took a history of mathematics course at the University of Virginia. One of the I interesting things I learned was that people used to write out equations as sentences before symbolic notation was invented. And so you would realize that a single moderately large equation (such as the quadratic equation) might be equivalent to a full paragraph or more of text. So if you ever get discouraged at how long it takes you t…

> equations can be as information-dense as entire paragraphs

And generally the equation is easier to understand than the same thing written in prose.

But did you notice this from the original article:

“When you add consecutive numbers starting with 1, and the number of numbers you add is odd, the result is equal to the product of the middle number among them times the last number.” (Levi’s theorem)

It's one of those rare cases where the English is much easier to understand than the equation.

Re: How to Read Mathematics

#22

Post by more_original »

6ren wrote: post #14

> The way to really understand the idea is to re-create what the author left out. If reading mathematics requires re-creating what the author left out, why not leave it in? Sure, it will be longer, but if the purpose is communication, wouldn't that be better? The reasons I can think of are not beneficial for communicating knowledge: that's how the game is played (tradition); it excludes the uninitiated/untalented; it…

> > The way to really understand the idea is to re-create what the author left out.

> If reading mathematics requires re-creating what the author left out, why not leave it in?

To really get some feeling for the content of the text, it seems to be really essential that the reader explores the content a bit herself. This is illustrated quite well in Don Knuth's "Surreal Numbers". Knowing this is probably the key to learning to read mathematics.

One can try to write all this out, but that will make the text harder to read for the mathematician, who now has to find the important bits of the text (think of the various 1000 page Visual Basic books). Also, it takes a lot more effort to write down all these "trivialities".

Perhaps the problem here is really with the school system. We study "maths" for many years in school, without ever learning how to read it.

Re: How to Read Mathematics

#23

Post by ColinWright »

6ren wrote: post #14

> The way to really understand the idea is to re-create what the author left out. If reading mathematics requires re-creating what the author left out, why not leave it in? Sure, it will be longer, but if the purpose is communication, wouldn't that be better? The reasons I can think of are not beneficial for communicating knowledge: that's how the game is played (tradition); it excludes the uninitiated/untalented; it…

One of my university lecturers gave what we all thought were dreadful lectures. Muddled, unclear, chaotic, with no discernible thread. It took ages to reconstruct and rework the material to a point where we could attack the problems and old exam questions.

I got nearly full marks on that exam.

Other lecturers were brilliant. Clear, lucid, entertaining. I didn't get full marks on their exams, because I found it hard to do the problems, even though I thought I understood the material from the lectures.

Math is not a spectator sport. You need to get involved, otherwise you're in the situation of someone who has watched a lot of tennis, but never played.

I used to mock the "it is clear that" phrase when it would take two or three pages to show the result, but having done the work to show it, I was then equipped to handle the next stage of the work. Having the explanation given to me as to why it was "clear" would not have done that, my understanding would be meagre, and unsatisfactory, and I would gradually fall behind and not understand what was missing.

So no, it's not because:

* that's how the game is played (tradition);

* it excludes the uninitiated/untalented;

* it's neater to leave out the truth of discovery;

* it makes the author seem superhuman;

* there's satisfaction for the reader in understanding the puzzle.

When done properly it's genuinely for more effective communication. I'm not saying it's always done well - not every writes equally well - and I'm not saying that everyone always has the best motives, but working on what you see as gaps in the presentation really is the best way to understand the material.

Added in edit:

You said:

  > Someone, somewhere will have to work out the combination -
  > I'm suggesting it is more efficient for it to be the one writer
  > than the many readers.
If your purpose is to have it written down, then yes. If your purpose is to communicate effectively to the readers, then no. The "doing" is an essential part of the eventual "understanding".

Re: How to Read Mathematics

#24

Post by jacobolus »

JumpCrisscross wrote: post #7
rcthompson wrote: post #5

I took a history of mathematics course at the University of Virginia. One of the I interesting things I learned was that people used to write out equations as sentences before symbolic notation was invented. And so you would realize that a single moderately large equation (such as the quadratic equation) might be equivalent to a full paragraph or more of text. So if you ever get discouraged at how long it takes you t…

Happen to remember the text you used?

This book is pretty good: http://www.amazon.com/The-Nature-Growth-Modern-Mathematics/d...

For original source material (more essays than papers, but a pretty big mix of stuff), this set is fabulous: http://www.amazon.com/The-World-Mathematics-Four-Volume-Set/...

Both can be found pretty cheap used online.

Re: How to Read Mathematics

#25

Post by mmcnickle »

6ren wrote: post #14

> The way to really understand the idea is to re-create what the author left out. If reading mathematics requires re-creating what the author left out, why not leave it in? Sure, it will be longer, but if the purpose is communication, wouldn't that be better? The reasons I can think of are not beneficial for communicating knowledge: that's how the game is played (tradition); it excludes the uninitiated/untalented; it…

Imagine someone explaining a new feature in their web framework while all the time taking time out to explain the concept of arrays, dictionaries, list comprehensions, URL routing, MVC etc. This may be a great idea for a book aimed at beginners though it would be frustrating for experienced programmers to have to constantly revisit basic concepts. Assuming the reader to have a solid background allows the author to concentrate on the novel thing they are trying to show.

Also historically some journals have restrictions on paper length, which increases the incentive to cut the idea down to the essential, novel material.

Re: How to Read Mathematics

#26

Post by slowpoke »

mck- wrote: post #10

Great article. I think it can very well extend to reading code as well, which may seem daunting at first.

Except that (good) code is written to be read, and to be understood - and not be terse and cryptic, just so that people writing it have less to do. That's essentially what most math texts do. I don't want math to be prose, but often, a little more verbosity or communication of intent would be nice. Just like good comments and documentation. In CS, this is universally accepted as good style, and for very good reasons - and for the same reasons, it should be in math, as well.

Re: How to Read Mathematics

#28

Post by lusr »

alister wrote: post #21
rcthompson wrote: post #5

I took a history of mathematics course at the University of Virginia. One of the I interesting things I learned was that people used to write out equations as sentences before symbolic notation was invented. And so you would realize that a single moderately large equation (such as the quadratic equation) might be equivalent to a full paragraph or more of text. So if you ever get discouraged at how long it takes you t…

> equations can be as information-dense as entire paragraphs And generally the equation is easier to understand than the same thing written in prose. But did you notice this from the original article: “When you add consecutive numbers starting with 1, and the number of numbers you add is odd, the result is equal to the product of the middle number among them times the last number.” (Levi’s theorem) It's one of those…

The problem I have with the way equations are often presented e.g. on Wikipedia is that the authors are so focused on conciseness that they don't stop to explain the idea in straightforward terms. For example, a while ago I was looking into the Radon Transform.

The wikipedia page is full of concise technical explanations and equations: http://en.wikipedia.org/wiki/Radon_transform

After reading a number of articles and eventually stumbling across one on MathWorks (http://www.mathworks.com/help/toolbox/images/f21-25938.html) with some great worked out examples I immediately got the idea: if you put your x-ray glasses on and look at an object from many different angles, and make note of what you're seeing at each angle, you can use your notes to rebuild the image or figure out how similar this object is to another object. Pretty intuitive.

With that understanding of the idea, the Wikipedia article made much more sense. Most mathematics is only difficult for me because I spend all my time trying to understand why the author is " the integral transform consisting of the integral of a function over straight lines". Once I get that, the rest is straightforward.

Re: How to Read Mathematics

#29

Post by muyuu »

Off-topic, but who uses an IE document icon as a favicon? got freaked out here in the office thinking IE managed to crawl back anywhere I can mistakenly open it somehow.

Re: How to Read Mathematics

#30

Post by X4 »

Hi,

that's exactly the kind of problem that was going on through my mind for about 2 weeks now. We've never learned howto analyze whitepapers or dissertations. Let alone howto convert an unknown mathematical formula into usable code. I see this as one of the most essential skills we never learned. Could someone please teach that skill? =))

I'd be very grateful if someone could help me and others understand how to dissect and codify the main point of a whitepaper.

This is a very interesting paper for example, that I've tried to understand, but I still only have a vague idea of it: http://pages.cs.wisc.edu/~jyc/papers/fibonacci.pdf

Here http://groups.csail.mit.edu/netmit/sFFT/ is another very interesting algorithm I knew for a long time, but even though I understand the principle. And tried to put it into code, I wasn't able to identify what the important part of their whitepaper is. They even provided some pseudo-code which I've seen in many other papers, but I've never seen where and how they standardized the pseudo-code notation. The pseudo-code looks ambiguous to me. (NOTE: They provide the code now, but that wasn't the case when the paper was published first).

This kind of thing makes me feel stupid.

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