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