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In my experience, Math.SE has a large number of very good mathematics teachers at a variety of levels: some of the regular, reputed users are really skilled at explaining topics within their [sometimes broad] range.

A lot of the answers focus on general methodologies, explanations, generalizations, etc. as opposed to just solving a given problem. This is very valuable knowledge.

Because of these two points, I thought that Math.SE would be a great place to create the definitive mathematics textbook whose primary characteristics would be:

  • be accessible at all levels: Equipped with just my brain and the English language, I should be able to start right from the beginning (okay, so 'beginning' might actually be algebra, but that's a separate issue). Equipped further with whatever prerequisites for a topic, I should be able to learn that topic.
  • be replete with examples: No matter what topic it is, Math.SE has had a large number of questions and probably a few questions with illustrative answers which would be helpful case-studies for teaching the topic. Further, being a dynamic textbook, you can add/substitute examples as needed (e.g. when someone posts a brilliant question/answer).
  • be a useful reference: With appropriate moderation and peer-review, this textbook could be used as an easy-to-cite and accurate reference which would, of course, be backed by references of its own. There would be links between pages for the often-found yet not-so-well-known connections between distant areas of mathematics, for optional reading.
  • be free to use (vs. print textbooks/journals), complete (vs. Wikipedia and to some extent, ProofWiki), dynamically community-driven (vs. print textbooks/journals and MathWorld) and rigorous (vs. Wikipedia) but not terse or hard to understand (vs. MathWorld): Of course the parentheses represent my personal opinion, but the adjectives should hold for the proposed textbook nonetheless. Most importantly, the mathematical knowledge of humanity should be freely accessible to humanity (with an Internet connection) and should keep getting better through constant content generation, peer-review, and editing.

In my view, this would be the perfect ultimate textbook; in fact, I didn't realize that my dream for a central math repository and Math.SE could actually be so compatible until a few minutes ago. What do you think?

(inspired by this question and a comment on an answer by @TBongers: "... I'm inclined to think that the mathematics version of the program is just a well-written textbook.")

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  • $\begingroup$ These older discussion seem to be related to some extent (although the proposal there is somewhat different): Some older related posts: Could we make a book with MSE answers?, Compiling an introductory resource for (say) Abstract Algebra, from MSE answers (and maybe also other posts linked there). $\endgroup$ – Martin Sleziak Aug 26 '16 at 6:01
  • $\begingroup$ I must ask: What would be advantage of compiling such a book here to using some site which was created specifically for such purpose, such as Wikibooks. If you want to ask help from math.SE users while contributing to one of the mathematical Wikibooks, you can use chat for that. $\endgroup$ – Martin Sleziak Aug 26 '16 at 6:06
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    $\begingroup$ @MartinSleziak Would it be okay for a Wikibook to have a lot of links to illustrative MSE questions/answers? Would there be enough quality content generation and review, when the book is not part of SE and in the absence of 'reputation'/'badges'? (Just asking for argument's sake.) $\endgroup$ – shardulc Aug 26 '16 at 6:12
  • $\begingroup$ shardulc: I am not sure about things you are asking about, but I think that Wikibook (as well as other typical textbook) should preferably be self-contained. $\endgroup$ – Martin Sleziak Aug 26 '16 at 6:22
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While this might be a nice thought, I cannot think that it would work in practice. Here are a couple reasons off the top of my head.

  1. With (necessarily, I think) dozens of different authors, such a textbook would more than likely fall far short of being the "perfect ultimate textbook". Different authors have different styles, and use different conventions. IMHO the chances of everyone agreeing on specific conventional choices, even within a specific mathematical area, are slim. It would end up a hodge-podge without any coherence.

  2. Our blog is effectively dead at this moment. If as a community we cannot muster enough stamina to have even one post of personal interest come up every month, then the chances of there being enough sustained effort to complete this task is at best negligible.

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  • $\begingroup$ "different conventions": While authors would be encouraged to follow conventions, we could have 'moderator-editors' who would maintain coherency, style, format, etc. This would be a separate task from writing itself. $\endgroup$ – shardulc Aug 26 '16 at 21:37
  • $\begingroup$ Also, the blog has a different intention and takes different skills to write than the proposed textbook. It would be a generalized version of what happens everyday on MSE as opposed to a special "personal interest" post. $\endgroup$ – shardulc Aug 26 '16 at 21:40
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    $\begingroup$ @shardulc "Moderator-editor" sounds awfully like the type of chore that I've heard mathematicians abhor. ​ ;-) ​ While a blog post is different than a text, I feel that the effort required to pull off the latter is magnitudes more than the former. The lack of sustained interest in posting to the blog would seem to imply that something similar would happen with this text. $\endgroup$ – бір-төрт-төрт-үш-жеті-бес Aug 26 '16 at 22:35
  • $\begingroup$ True, it would be a chore. :) However, this might just be me, but I really didn't even know about the blog and I still don't understand its purpose or what would be appropriate there. A textbook would be more focused and well-defined, and we could hope it gets more publicity. :) $\endgroup$ – shardulc Aug 26 '16 at 22:42
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I would be in favour of this.

I wonder to what extent a documentation-like feature (as we have on Stack Overflow) would work for this. But since SO Documentation is still in beta, we will probably have to wait >1 year before getting documentation.

Writing good textbooks is hard. One of the major mistakes (in my opinion) is making it too long, as in, sticking too long on one topic. When you have so many editors, that might happen. The art of writing a good book is having a reasonably short explaination that is understandable for most, and a few short examples that do, however, explain most of what you can do with the method.

On the other hand, I've seen very few comprehensive textbooks, so we might have a chance to cover more than most textbooks do - something I would definitely be in favour of, I am a bit annoyed to have to check seven books or so before finally knowing most of the subject.

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