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How would people view a single user asking or stating minor results or exercises and providing an answer for the sole sake of making the result available on stack exchange? Is this in the spirit of the math stack exchange,since the user does not actually have a question.

Moreover, how would people view a single user asking possibly hundreds of these over time to make these results accessible online?

The asker would provide a prepared solution or proof of the result. Other users can contribute alternative proofs and comments.

There are several aspects I think should be worked out about posts of these sorts. Out of courtesy, should the user never accept their own answer? Should they never accept any answer?

Please let me know what you think about doing this, and how could math stack exchange be used as a mean to make results accessible online.

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Having good online resources is a worthy goal. To me, it is not fully in keeping with what I think of as the purpose of a question and answer site. But I am not too fond of the list of generalizations of common questions either.

When a question is asked, I think the person asking needs to be treated as an individual. This is the case even if objectively the question asked is very much like questions that have already been asked and answered.

Fairly often, an appropriate answer includes directing the person to other resources. However, if a directory of questions-and-answers becomes a large component of the web site, then the people answering questions are turned to a significant degree into librarians. That is a worthy thing to be. However, it is something not likely to appeal to most mathematicians, since it is not what we do professionally. But many of us do teach, and are interested in helping people to understand mathematics.

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There have been at least a couple of times where I Googled something for the purpose of finding an answer to a question I had, and one of the links on the first page of results was either here or MO, which contained an answer to my question. I am surely not the only person to which this has happened. Thus, I would say it is certainly useful to post answers here, even if for the sole purpose of making them available on SE.

That being said, my feeling is that there are other online resources better suited for this purpose. For example, Proof Wiki. I personally haven't used this site myself, but it seems like it would make sense to contribute there if you are looking to "Write down a statement of a problem and its solution.". On the other hand, certainly not every question that appears on here would be appropriate for Proof Wiki, but certainly Proof Wiki is not the only alternative.

Although I've never made use of the feature, I had just assumed that the ability to answer a question yourself was to be used in the case where you had posted a question, and then later, before another user posted a full solution to a problem, found a correct solution, and posted it to answer your question so that others on the site could make use of your solution as well. For that matter, it provides an excellent way of closing the question, as there is no reason to leave it open if you've found an answer yourself, and this is certainly more constructive than just deleting it from the site.

Despite this assumption, I don't see anything wrong with posting the question and answering it at the same time, although, as mentioned above, it might make more sense to do such a thing on community sites whose purpose is providing reference material itself, instead of on a question and answer website.

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    $\begingroup$ I don't think the question should be closed if the OP posts the (first, correct) solution. Rather, the OP should accept their solution if they are satified with it. It's not uncommon that the OP's solution is incorrect, or highly nonoptimal, or nonconceptual, etc, so leaving the question open allows other readers to post better answers. Regarding your final paragraph, please keep in mind that some folks view the site not strictly as a static Q&A site but, rather, more broadly, as a dynamically evolving web of mathematical knowledge (that just so happens to be generated by questions). $\endgroup$ – Bill Dubuque May 29 '12 at 19:11
  • $\begingroup$ @BillDubuque Sorry. I used the word 'closed' incorrectly. I just meant 'accept an answer'. Of course, answers can still be added after one has been accepted, but personally, if I see that a question has an answer that has been accepted, I don't even view the question (unless I am particularly curious) because I assume that the asker is already satisfied with the responses they have received. My guess is that others feel similarly, and so by accepting an answer, you highly decrease the probability of new answers being added (in some sense, a de facto closing of the question). $\endgroup$ – Jonathan Gleason May 29 '12 at 19:14
  • $\begingroup$ @Jonathon Frequently much better answers are posted after an answer is accepted. In many cases the OP does not have the knowledge to evaluate properties of answers (correctness, optimality, conceptualness, etc). In such cases, the OP cannot possibly predict whether or not better answers are possible. Too quick acceptance of answers is one of the major newbie problems. $\endgroup$ – Bill Dubuque May 29 '12 at 19:26

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