How to Read Mathematics
<p><pre><code>  > The way to really understand the idea is to re-create what the author left out.
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If reading mathematics requires re-creating what the author left out, why not leave it in?<p>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.<p><i>EDIT</i> 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.<p>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.