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How to Read Mathematics

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Re: How to Read Mathematics

#31

Post by hackinthebochs »

slowpoke wrote: post #26
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…

I think you're misunderstanding the purpose of the terseness of mathematics. The point is to convey the important concepts as cleanly as possible, assuming your reader has a certain level of prerequisite knowledge. The same can be said about code: you assume your reader has a certain level of understanding of the domain that you're modelling through code. The terseness is a virtue, in the sense that only the critical bits of new information need be conveyed. All the prerequisite knowledge is hidden behind abstractions that the reader should be familiar with. Math, and code, is cryptic if one doesn't have the prerequisite knowledge, but this is by design.

Re: How to Read Mathematics

#32

Post by slowpoke »

hackinthebochs wrote: post #31
slowpoke wrote: post #26

Earlier quoted context omitted.

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…

I think you're misunderstanding the purpose of the terseness of mathematics. The point is to convey the important concepts as cleanly as possible, assuming your reader has a certain level of prerequisite knowledge. The same can be said about code: you assume your reader has a certain level of understanding of the domain that you're modelling through code. The terseness is a virtue, in the sense that only the critical…

>The same can be said about code: you assume your reader has a certain level of understanding of the domain that you're modelling through code.

No, this is not at all the same. Code can be read without understanding everything if it is well documented. Yes, code should be short and simple, too. But not terse and cryptic - and that's what so many math texts do. So yea, cool, you can compress half a page of written English into half a line of mumbo-jumbo. Great. You have gained nothing.

I'm not saying that the half page of written English is better. Not at all. But there's something somewhere between those two where math should be. And not at the cryptic end of the scale.

Take just simple things such as meaningful variable naming. There are tons of good reasons why we do this in programming. I can't count the number of times I've tried to decipher math texts and had to look up over and over and over again what some x or f or lambda or phi is actually supposed to represent. It gets worse when they start making distinctions based on the bloody way a symbol is typeset (x vs x).

It's perfectly fine to do this in quick calculations on paper (just as it is fine to use one character variables in quick one-off scripts). But if you are writing a prolonged mathematical text, then take the time to give variables meaningful names. It's not that hard, and would go a long way for improving readability.

Re: How to Read Mathematics

#33

Post by hackinthebochs »

slowpoke wrote: post #32
hackinthebochs wrote: post #31

Earlier quoted context omitted.

I think you're misunderstanding the purpose of the terseness of mathematics. The point is to convey the important concepts as cleanly as possible, assuming your reader has a certain level of prerequisite knowledge. The same can be said about code: you assume your reader has a certain level of understanding of the domain that you're modelling through code. The terseness is a virtue, in the sense that only the critical…

>The same can be said about code: you assume your reader has a certain level of understanding of the domain that you're modelling through code. No, this is not at all the same. Code can be read without understanding everything if it is well documented. Yes, code should be short and simple, too. But not terse and cryptic - and that's what so many math texts do. So yea, cool, you can compress half a page of written Eng…

I don't think descriptive variable names would be useful in math. The thing in math is that "x" you see can be repeated 20 times in a set of equations. Having to read "running_total" 20 times instead is a hindrance here rather than a benefit.

The difference between math and code is the number of variables. A piece of code can have an order of magnitude more variables than a set of math equations. But in math, a few variables will usually be repeated many times. You're simply optimizing different usage profiles.

So the benefit in descriptive names in code is being able to distinguish easily the many different variables. The benefit in single letter variable names in math is that you're able to write the information in a more compact, digestible manner.

The reason this latter point is important is for the same reason short, compact programming languages are a boon to comprehension. The faster you can read a set of related items, the more of it is in your working memory at any given time and thus the better you're able to understand it. I fully believe the time it takes to input a chunk of information through your visual system is inversely related to one's understanding of it. To put it another way, working memory's decay function is parameterized by time.

Re: How to Read Mathematics

#34

Post by tylerneylon »

I agree that typical math writing must be absorbed very slowly and carefully to be understood well.

Unfortunately, in the world of mathematicians, this isn't what really happens most of the time. In grad school classes (for math), students don't have time to absorb the material this carefully - they must 'learn' too much too quickly (or at least this was the case at my grad school), so that reading math becomes learning just enough to pass exams. Writing a dissertation involves a lot of talking to people and skimming papers to hopefully grab relevant bits. Reviewing papers, for professors, involves handing papers to their grad students and asking them to read it. Even writing a paper for journal submission feels primarily about satisfying reviewers and getting in to the best possible journal - being readable becomes a lower priority than space constraints and the quibbles of a particular reviewer.

Re: How to Read Mathematics

#35

Post by tylerneylon »

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?

I took a course that used this book (Boyer and Merzbach, A History of Mathematics), which I enjoyed:

http://www.amazon.com/A-History-Mathematics-Second-Edition/d...

Re: How to Read Mathematics

#36

Post by GlennS »

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…

Maybe mathematics papers would be improved by hypertext links to background topics or explanations of sub-problems at the places where they are needed?

Re: How to Read Mathematics

#37

Post by slowpoke »

hackinthebochs wrote: post #33
slowpoke wrote: post #32

Earlier quoted context omitted.

>The same can be said about code: you assume your reader has a certain level of understanding of the domain that you're modelling through code. No, this is not at all the same. Code can be read without understanding everything if it is well documented. Yes, code should be short and simple, too. But not terse and cryptic - and that's what so many math texts do. So yea, cool, you can compress half a page of written Eng…

I don't think descriptive variable names would be useful in math. The thing in math is that "x" you see can be repeated 20 times in a set of equations. Having to read "running_total" 20 times instead is a hindrance here rather than a benefit. The difference between math and code is the number of variables. A piece of code can have an order of magnitude more variables than a set of math equations. But in math, a few v…

>The benefit in single letter variable names in math is that you're able to write the information in a more compact, digestible manner.

You missed my point entirely. I said that you should not optimize for writing math, you should optimize for reading math. And that's where the same argument as for code holds true as well: It doesn't matter if it takes longer to write, it will be read magnitudes more often, and that is what matters most.

Again, if you're just doing quick calculations on paper or hacking something into your shell/REPL, I really don't care if you use cryptic variable names. I do, too. But I don't want to ever see this in a good program, and I think that it doesn't belong into a good math text, either.

>The reason this latter point is important is for the same reason short, compact programming languages are a boon to comprehension.

And again you missed what I explicitly stated: it's good to write in a concise way, but only as long as it doesn't become cryptic. And I disagree with your belief about information processing. It's better to build semantic relations by having meaningful names than it is to process a lot of information in a short time. It really doesn't matter how fast you are able read something - if it doesn't make sense, you won't understand it.

Re: How to Read Mathematics

#38

Post by hackinthebochs »

slowpoke wrote: post #37
hackinthebochs wrote: post #33

Earlier quoted context omitted.

I don't think descriptive variable names would be useful in math. The thing in math is that "x" you see can be repeated 20 times in a set of equations. Having to read "running_total" 20 times instead is a hindrance here rather than a benefit. The difference between math and code is the number of variables. A piece of code can have an order of magnitude more variables than a set of math equations. But in math, a few v…

>The benefit in single letter variable names in math is that you're able to write the information in a more compact, digestible manner. You missed my point entirely. I said that you should not optimize for writing math, you should optimize for reading math. And that's where the same argument as for code holds true as well: It doesn't matter if it takes longer to write, it will be read magnitudes more often, and that…

>You missed my point entirely. I said that you should not optimize for writing math, you should optimize for reading math.

My point is that these goals are nearly one in the same when you get to the high level. Many mathematical relationships are very complex, usually not the step-by-step procedures that is common in code. Thus being able to hold the entire relationship in your head at once is crucial. Single letter names for variables and functions are critical here (for the previously mentioned reason working-memory-decay).

I basically have a bachelors in math, and I could not imagine reading complex equations with full variable names. The hard part is understanding the whole, not remembering what x or i means. If you find an equation cryptic, that just means you don't have the requisite knowledge to really understand it.

Re: How to Read Mathematics

#39

Post by AlexDanger »

lusr wrote: post #28
alister wrote: post #21

Earlier quoted context omitted.

> 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 an…

+1 absolutely agree. Wikipedia is terrible for learning mathematical concepts, not surprising given that its a reference site. Its appropriate to be concise in this context.

People are getting excited about education being an area ripe for disruption. What about a site that rips all the Wikipedia maths topics and allows people to add some worked examples and arrange the content to be more pedagogically pleasing? Providing a nice little HTML5 widget toolbox to create simple interactive visualisations would be cool...

Re: How to Read Mathematics

#40

Post by slowpoke »

hackinthebochs wrote: post #38
slowpoke wrote: post #37

Earlier quoted context omitted.

>The benefit in single letter variable names in math is that you're able to write the information in a more compact, digestible manner. You missed my point entirely. I said that you should not optimize for writing math, you should optimize for reading math. And that's where the same argument as for code holds true as well: It doesn't matter if it takes longer to write, it will be read magnitudes more often, and that…

>You missed my point entirely. I said that you should not optimize for writing math, you should optimize for reading math. My point is that these goals are nearly one in the same when you get to the high level. Many mathematical relationships are very complex, usually not the step-by-step procedures that is common in code. Thus being able to hold the entire relationship in your head at once is crucial. Single letter…

Sorry, if all you have left is the (very typical) argument of "if you think that way you just don't understand it", then I'll consider this discussion over. I have no interest in hearing endless appeals to tradition. I'm sure people made similar arguments for GOTO back in the day. Good thing we moved on.
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