7 ms·

How to Read Mathematics

161 points by espeed 14y ago

gajomi 14y ago
A very nice article. >A particular notorious example is the use of “It follows easily that” and equivalent constructs. It means something like this: >One can now check that the next statement is true with a certain amount of essentially mechanical, though perhaps laborious, checking. I, the author, could do it, but it would use up a large amount of space and perhaps not accomplish much, since it'd be best for you to go ahead and do the computation to clarify for yourself what's going on here. I promise that no new ideas are involved, though of course you might need to think a little in order to find just the right combination of good ideas to apply. Even knowing this ahead of time, this kind of thing can be maddening. Namely, when the meaning of the assertion is be sensitive to little sign changes, index shifts and the like these are quite likely to end up in the computation.
- Avshalom 14y ago
  My Basic Concepts of Mathematics teacher was notorious for getting about halfway through a problem before announcing "and from here it's trivial to show...". One day he walks into class one day and addressed us: "I've been told by the head of the department that I can no longer say 'it's trivial'. You have to understand when I say 'it's trivial', I just mean that I can do it"
espeed 14y ago
The article is by Shai Simonson, one of the instructors at ArsDigita University (http://aduni.org/ http://aduni.org/). This was part of the reading material for ADU Course 0: "Mathematics for Computer Science" (http://aduni.org/courses/math/index.php?view=cw http://aduni.org/courses/math/index.php?view=cw), but unfortunately the video lectures aren't available, whereas they are available for all the other courses.
- pm90 14y ago
  Shai is an incredible teacher; his video lectures on theory of computation were a major part of what made me understand the beauty of CS theory (automata, CFG's etc). It was sad to see that they had to shut down ADU; but I guess it was an unsustainable model; if I recall correctly, there was absolutely no tuition charged to the students.
faucet 14y ago
What a boring text! M.Sc.in applied Mathematics
rcthompson 14y ago
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 to read mathematics, remember that equations can be as information-dense as entire paragraphs, or even more so, and just because it is presented in a spatially compact format doesn't necessarily mean that it shouldn't take you at least as long as a text paragraph to read and comprehend.
- JumpCrisscross 14y ago
  Happen to remember the text you used?
  - Dn_Ab 14y ago
    I've read a couple of texts on the history of math. Most of them were dry, a few were very entertaining like Crowe's on Vector Analysis. But I haven't found anything that beats mac tutor. I used to read it nearly everyday many years back. The section on Al-Khwarizmi, a key innovator in algebra shows how laborious the task was (site down linking cache): http://webcache.googleusercontent.com/search?q=cache:nOT6h0c7J2oJ:www-history.mcs.st-andrews.ac.uk/Biographies/Al-Khwarizmi.html+&cd=1&hl=en&ct=clnk http://webcache.googleusercontent.com/search?q=cache:nOT6h0c... a square and 10 roots are equal to 39 units. The question therefore in this type of equation is about as follows: what is the square which combined with ten of its roots will give a sum total of 39? The manner of solving this type of equation is to take one-half of the roots just mentioned. Now the roots in the problem before us are 10. Therefore take 5, which multiplied by itself gives 25, an amount which you add to 39 giving 64. Having taken then the square root of this which is 8, subtract from it half the roots, 5 leaving 3. The number three therefore represents one root of this square, which itself, of course is 9. Nine therefore gives the square.
  - rcthompson 14y ago
    I'm sure I have the text lying around somewhere. I can't remember the title and author, but in any case the text was just a compilation of relevant excerpts from primary sources to be used as reference. It didn't provide any analysis. All the analysis was provided by the professor, Karen Parshall, whose courses I can enthusiastically recommend for anyone at U.Va. interested in the history of math or science (bonus: they satisfy your 2nd writing requirement). Her website is here: http://www.math.virginia.edu/~khp3k/ http://www.math.virginia.edu/~khp3k/ I credit her for pointing out the fact that I note in my original comment, that an equation can be equivalent to a large tract of text. She made the point in the context of why more "advanced" mathematics couldn't have developed before symbolic notation had matured sufficiently to support it. Here, I'm generalizing her point to explain why reading a mathematical text dense in equations might take longer than the page count alone might suggest.
lmarinho 14y ago
During college, in order to plan my time, I used to measure my studying time on more mathematically intense texts. I took 30-60mins per page on average. It is a hard read, at least for me, but intensely gratifying too.
mck- 14y ago
Great article. I think it can very well extend to reading code as well, which may seem daunting at first.
- slowpoke 14y ago
  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.
  - hackinthebochs 14y ago
    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.
    - slowpoke 14y ago
      >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.
johnbender 14y ago
"The same half hour in a math article buys you 0-10 lines depending on the article and how experienced you are at reading mathematics" I'm still reading though the article, but this might be the most important bit of information for anyone getting started reading papers that rely on maths. I wish someone had told me this when I started in with the more complex comp-sci papers because it's hard not to feel dense when you have to go over the same 3 or 4 pages of text time and time again to get the concepts.
- evincarofautumn 14y ago
  Yeah, especially if you’re already a fast reader, it’s frustrating at first to have to drop to what feels like a snail’s pace to properly understand everything. But the information density is a real life-saver once you’re more experienced, because it lets you readily experiment with things at that high level, unencumbered.
lotharbot 14y ago
It can also be helpful to find others who are interested in reading the same bit of math, and talk through it with them. They don't have to be particularly better at it than you, they just have to have a similar level of interest and curiosity. In grad school we read recently published papers in a small group setting we called "journal club". This process helps in a number of ways. It keeps you from reading too fast or too passively because you're constantly asking and answering questions. It makes you less likely to get stuck in a dead end for very long, because others will see alternatives. It gives you an opportunity to ask about notation or background concepts you aren't familiar with. It helps you keep track of the big picture, because while some people are bogged down in a particular detail (like "how do they get from equation A to equation B?") others will be trying to tie it back to the big picture ("how does equation B fit into our overall goal?") And it allows you to see the even bigger picture as others bring in relevant knowledge or experience; it was pretty common to be working through a paper and have someone mention how it tied in to their current research project.
6ren 14y ago
> 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'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. EDIT the reader can skip explanations he can work out himself (or use them as a check); papers are already structured with details in deeper, skip-able sections. One can have a summary that excludes details altogether (like an abstract, or equivalent to a present maths paper). In the article, the parts left out are not "known", but steps that the author could work out themselves, perhaps after many dead-ends to find the right combination. To avoid repetition of known specific concepts (like vocab), one could explicitly reference them, or assume them for a given audience. Perhaps the essential problem is that the omitted steps are not a single concept (like a well-known term), but many concepts, combined according to other concepts (like a complex expression), so they can't be easily be referenced, nor assumed. 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.
- lotharbot 14y ago
  Imagine how poorly your idea would be communicated if you included definitions or explanations for every word longer than six letters. It would be hard to cut through all the noise to get to the interesting part of your message. By assuming a particular level of background information (in this case, vocabulary), you're able to focus on the interesting and important insights. The same is true in math. You don't leave out random steps; you leave out steps that you expect your audience to have already mastered. You leave out steps where the working-out process doesn't contain anything particularly new or useful. Anything that comes down to "apply a bunch of lower-level math in a tedious way" doesn't belong in your paper. Re-creating it shouldn't be necessary for getting the basic idea. When this is done properly, the result is clear and concise communication.
- more_original 14y ago
bearwithclaws 14y ago
Here's the PDF (from Hacker Monthly) if anyone wants to print out in nice magazine format: https://dl.dropbox.com/u/48613464/How%20To%20Read%20Math.pdf https://dl.dropbox.com/u/48613464/How%20To%20Read%20Math.pdf
- espeed 14y ago
  Oh cool, it was in issue #5 (http://hackermonthly.com/issue-5.html http://hackermonthly.com/issue-5.html).
RollAHardSix 14y ago
Slightly off-topic but I will forever remember a math professor of mine, Bryan Snare, declaring that 'You read Mathematics with a pencil!'. True then, true today, true for eternity.
mavelikara 14y ago
Previous discussion: http://news.ycombinator.com/item?id=1576969 http://news.ycombinator.com/item?id=1576969
muyuu 14y ago
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.
- Aeons 14y ago
  The site was made using Word.
X4 14y ago
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 http://pages.cs.wisc.edu/~jyc/papers/fibonacci.pdf Here http://groups.csail.mit.edu/netmit/sFFT/ 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.
tylerneylon 14y ago
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.

161 points by espeed 14y ago

A very nice article. >A particular notorious example is the use of “It follows easily that” and equivalent constructs. It means something like this: >One can now check that the next statement is true with a certain amount of essentially mechanical, though perhaps laborious, checking. I, the author, could do it, but it would use up a large amount of space and perhaps not accomplish much, since it'd be best for you to go ahead and do the computation to clarify for yourself what's going on here. I promise that no new ideas are involved, though of course you might need to think a little in order to find just the right combination of good ideas to apply. Even knowing this ahead of time, this kind of thing can be maddening. Namely, when the meaning of the assertion is be sensitive to little sign changes, index shifts and the like these are quite likely to end up in the computation.

Avshalom 14y ago
My Basic Concepts of Mathematics teacher was notorious for getting about halfway through a problem before announcing "and from here it's trivial to show...". One day he walks into class one day and addressed us: "I've been told by the head of the department that I can no longer say 'it's trivial'. You have to understand when I say 'it's trivial', I just mean that I can do it"

espeed 14y ago

The article is by Shai Simonson, one of the instructors at ArsDigita University (http://aduni.org/ http://aduni.org/). This was part of the reading material for ADU Course 0: "Mathematics for Computer Science" (http://aduni.org/courses/math/index.php?view=cw http://aduni.org/courses/math/index.php?view=cw), but unfortunately the video lectures aren't available, whereas they are available for all the other courses.

pm90 14y ago
Shai is an incredible teacher; his video lectures on theory of computation were a major part of what made me understand the beauty of CS theory (automata, CFG's etc). It was sad to see that they had to shut down ADU; but I guess it was an unsustainable model; if I recall correctly, there was absolutely no tuition charged to the students.

faucet 14y ago

What a boring text! M.Sc.in applied Mathematics

rcthompson 14y ago

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 to read mathematics, remember that equations can be as information-dense as entire paragraphs, or even more so, and just because it is presented in a spatially compact format doesn't necessarily mean that it shouldn't take you at least as long as a text paragraph to read and comprehend.

JumpCrisscross 14y ago
Happen to remember the text you used?
- Dn_Ab 14y ago
  I've read a couple of texts on the history of math. Most of them were dry, a few were very entertaining like Crowe's on Vector Analysis. But I haven't found anything that beats mac tutor. I used to read it nearly everyday many years back. The section on Al-Khwarizmi, a key innovator in algebra shows how laborious the task was (site down linking cache): http://webcache.googleusercontent.com/search?q=cache:nOT6h0c7J2oJ:www-history.mcs.st-andrews.ac.uk/Biographies/Al-Khwarizmi.html+&cd=1&hl=en&ct=clnk http://webcache.googleusercontent.com/search?q=cache:nOT6h0c... a square and 10 roots are equal to 39 units. The question therefore in this type of equation is about as follows: what is the square which combined with ten of its roots will give a sum total of 39? The manner of solving this type of equation is to take one-half of the roots just mentioned. Now the roots in the problem before us are 10. Therefore take 5, which multiplied by itself gives 25, an amount which you add to 39 giving 64. Having taken then the square root of this which is 8, subtract from it half the roots, 5 leaving 3. The number three therefore represents one root of this square, which itself, of course is 9. Nine therefore gives the square.
- rcthompson 14y ago
  I'm sure I have the text lying around somewhere. I can't remember the title and author, but in any case the text was just a compilation of relevant excerpts from primary sources to be used as reference. It didn't provide any analysis. All the analysis was provided by the professor, Karen Parshall, whose courses I can enthusiastically recommend for anyone at U.Va. interested in the history of math or science (bonus: they satisfy your 2nd writing requirement). Her website is here: http://www.math.virginia.edu/~khp3k/ http://www.math.virginia.edu/~khp3k/ I credit her for pointing out the fact that I note in my original comment, that an equation can be equivalent to a large tract of text. She made the point in the context of why more "advanced" mathematics couldn't have developed before symbolic notation had matured sufficiently to support it. Here, I'm generalizing her point to explain why reading a mathematical text dense in equations might take longer than the page count alone might suggest.

lmarinho 14y ago

During college, in order to plan my time, I used to measure my studying time on more mathematically intense texts. I took 30-60mins per page on average. It is a hard read, at least for me, but intensely gratifying too.

mck- 14y ago

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

slowpoke 14y ago
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.
- hackinthebochs 14y ago
  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.
  - slowpoke 14y ago
    >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.

johnbender 14y ago

"The same half hour in a math article buys you 0-10 lines depending on the article and how experienced you are at reading mathematics" I'm still reading though the article, but this might be the most important bit of information for anyone getting started reading papers that rely on maths. I wish someone had told me this when I started in with the more complex comp-sci papers because it's hard not to feel dense when you have to go over the same 3 or 4 pages of text time and time again to get the concepts.

evincarofautumn 14y ago
Yeah, especially if you’re already a fast reader, it’s frustrating at first to have to drop to what feels like a snail’s pace to properly understand everything. But the information density is a real life-saver once you’re more experienced, because it lets you readily experiment with things at that high level, unencumbered.

lotharbot 14y ago

It can also be helpful to find others who are interested in reading the same bit of math, and talk through it with them. They don't have to be particularly better at it than you, they just have to have a similar level of interest and curiosity. In grad school we read recently published papers in a small group setting we called "journal club". This process helps in a number of ways. It keeps you from reading too fast or too passively because you're constantly asking and answering questions. It makes you less likely to get stuck in a dead end for very long, because others will see alternatives. It gives you an opportunity to ask about notation or background concepts you aren't familiar with. It helps you keep track of the big picture, because while some people are bogged down in a particular detail (like "how do they get from equation A to equation B?") others will be trying to tie it back to the big picture ("how does equation B fit into our overall goal?") And it allows you to see the even bigger picture as others bring in relevant knowledge or experience; it was pretty common to be working through a paper and have someone mention how it tied in to their current research project.

6ren 14y ago

> 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'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. EDIT the reader can skip explanations he can work out himself (or use them as a check); papers are already structured with details in deeper, skip-able sections. One can have a summary that excludes details altogether (like an abstract, or equivalent to a present maths paper). In the article, the parts left out are not "known", but steps that the author could work out themselves, perhaps after many dead-ends to find the right combination. To avoid repetition of known specific concepts (like vocab), one could explicitly reference them, or assume them for a given audience. Perhaps the essential problem is that the omitted steps are not a single concept (like a well-known term), but many concepts, combined according to other concepts (like a complex expression), so they can't be easily be referenced, nor assumed. 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.

lotharbot 14y ago
Imagine how poorly your idea would be communicated if you included definitions or explanations for every word longer than six letters. It would be hard to cut through all the noise to get to the interesting part of your message. By assuming a particular level of background information (in this case, vocabulary), you're able to focus on the interesting and important insights. The same is true in math. You don't leave out random steps; you leave out steps that you expect your audience to have already mastered. You leave out steps where the working-out process doesn't contain anything particularly new or useful. Anything that comes down to "apply a bunch of lower-level math in a tedious way" doesn't belong in your paper. Re-creating it shouldn't be necessary for getting the basic idea. When this is done properly, the result is clear and concise communication.
more_original 14y ago

bearwithclaws 14y ago

Here's the PDF (from Hacker Monthly) if anyone wants to print out in nice magazine format: https://dl.dropbox.com/u/48613464/How%20To%20Read%20Math.pdf https://dl.dropbox.com/u/48613464/How%20To%20Read%20Math.pdf

espeed 14y ago
Oh cool, it was in issue #5 (http://hackermonthly.com/issue-5.html http://hackermonthly.com/issue-5.html).

RollAHardSix 14y ago

Slightly off-topic but I will forever remember a math professor of mine, Bryan Snare, declaring that 'You read Mathematics with a pencil!'. True then, true today, true for eternity.

mavelikara 14y ago

Previous discussion: http://news.ycombinator.com/item?id=1576969 http://news.ycombinator.com/item?id=1576969

muyuu 14y ago

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.

Aeons 14y ago
The site was made using Word.

X4 14y ago

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 http://pages.cs.wisc.edu/~jyc/papers/fibonacci.pdf Here http://groups.csail.mit.edu/netmit/sFFT/ 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.

tylerneylon 14y ago

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.