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I saw this post, and I am pretty sure that it comes from a setting of a first year student.

I wanted to answer this question by using Galois theory (though an overkill I think its more elegant since there are little calculations), but I guess the OP doesn't know it.

Can I still answer the question using the more advanced Galois theory ?

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    $\begingroup$ Yes! It may be very helpful to the future reader. It might be nice to add a short explanation to the OP, though. $\endgroup$ – Lord_Farin Oct 25 '13 at 14:55
  • $\begingroup$ The great thing about stackexchange is that the community will decide if the answer is a good one or not using the voting system (or if it's inappropriate using the flagging system). $\endgroup$ – Dan Rust Oct 25 '13 at 15:26
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    $\begingroup$ Dear @DanielRust: I am rather skeptical on this assertion... It depends on what you call a good answer and on much random reasons. $\endgroup$ – Cantlog Oct 25 '13 at 16:55
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    $\begingroup$ Always possible that someone else will provide an answer at a lower level before or after you post. $\endgroup$ – Will Jagy Oct 25 '13 at 19:41
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    $\begingroup$ How about two answers. One gets accepted, one gets up-votes, and you get lots of reputation. $\endgroup$ – PyRulez Oct 27 '13 at 20:04
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    $\begingroup$ An answer is an answer. $\endgroup$ – copper.hat Oct 27 '13 at 23:19
  • $\begingroup$ There are 4 answers, all offering some degree of support, so I'm surprised you haven't posted your Galois theory answer yet. $\endgroup$ – Peter Taylor Nov 2 '13 at 8:54
  • $\begingroup$ @PeterTaylor - I started working full time and I come home only during the weekends so I had little time to write a full answer $\endgroup$ – Belgi Nov 2 '13 at 9:21
  • $\begingroup$ See also math.meta.stackexchange.com/questions/3913/…. $\endgroup$ – lhf Apr 25 '17 at 12:15
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Unless there's something explicitly in the question prohibiting higher math, I don't see anything wrong with answering a question using more advanced mathematics.

That being said, if you do, understand that your answer is probably not going to be the most useful for the question asker. It however, could be useful for someone further down the road who stumbles across the page.


Caveat: I could see some people downvoting such an answer if it comes across too much like "look at all this math I know" and not enough like "look at all this math that I know will help you." Thus, if you do post something that might be a little too high-level, bear that in mind.

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    $\begingroup$ People are going to arbitrarily vote anyways, so I wouldn't worry too much about the reception. One of my favorite high-profile users used to post the most opaque answers, often with little-to-no English explanation, which I have only begun to appreciate as my knowledge has increased. $\endgroup$ – The Chaz 2.0 Oct 25 '13 at 20:04
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    $\begingroup$ @TheChaz2.0: If we’re thinking of the same ones (with liberal use of \rm), those answers used to drive me nuts, because he almost never gave any indication that they were on the sophisticated side. $\endgroup$ – Brian M. Scott Oct 26 '13 at 8:36
  • $\begingroup$ Can I ask who we're talking about? $\endgroup$ – Bennett Gardiner Oct 26 '13 at 14:05
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    $\begingroup$ The user known as "Gone" (@Bennett). If you've seen color-coded number theory answers, or anything about "telescopy" in an induction answer, that's Gone! $\endgroup$ – The Chaz 2.0 Oct 26 '13 at 14:19
  • $\begingroup$ Gone used to go by the handle of Bill something. After a moderator spat he was hit with a long ban and changed his name to Gone since he says he has no intention of returning. $\endgroup$ – R R Nov 3 '13 at 9:21
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A Q&A website has essentially two purposes: A) Help the OP, by answering the OP's question in a way that will be useful to the OP and B) Offer a service to the current and future community, by answering the question in a way that may be of value to the community in general.

Purpose B permits any level of mathematics while purpose A requires a level of mathematics that can be handled by the OP.

If purpose A is not satisfied, then the OP and its question is turned into just an excuse for posting Knowledge on the web, (and I don't believe that .SE communities view OPs in this way).

If purpose B is not satisfied, then the answer has not reached the "ideal".

From the above it seems clear that the answer should always include a part with a mathematics level that seems understandable by the OP (except of course for cases where it appears evident that the OP really knows nothing of the concepts included in the question, in which case the most helpful answer would perhaps be to point out the concepts that the OP should study first): Purpose A is satisfied.

If the same answer can be provided in a more general, elegant, insightful etc way using higher-level mathematics, then the answerer could/should also include this alternative answer, in order to also serve Purpose B above.

If the question posed cannot be answered at a mathematics level that suits the OP, then at least Purpose B should be serviced, and the answerer should go ahead with the only available alternative, i.e. an answer at a higher math level, so at least the community can benefit from it (Purpose B). But here, the situation should be explained to the OP , so that the OP understands that it is not out of a desire to show-off that the OP gets an answer that (s)he probably cannot use.

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I suppose that for the question at hand, an answer using Galois theory is welcome and helpful (to the general audience), but should (and by expereinece quickly will) be accompanied by an alternative, fully elementary answer. The OP will surely pick the elementary answer for their concrete situation, but maybe also get a glimpse into higher levels and maybe even get an idea how the two answers are in fact related, thus encouraging their mathematical interest.

I think it is even more ok to add "theory-laden" answers in many cases (though maybe with not as much of an overkill factor as in Galois theory vs. indirect proof with fraction and prime factor manipulations), such as not using bases, matrices, assumptions about finite dimension, assumptions about ground field characteristic when answering questions about vector spaces and linear maps that (unnecessarily) make use of them.

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I think this is a delicate point.

My view on the topic is that the answers should first address the OP, then address the rest of the world. So one can do one of two things, really.

  1. Write an elementary answer, and add a second answer/second part with an advance approach using better tools.

  2. Wait until one (extensive) or more answers are posted to the satisfaction of the OP, and then write an answer with the disclaimer "Now that an elementary answer has been given ..."

There is some problem with that. The concept of a "dangerous knowledge". When studying mathematics properly, you don't get all the information dumped on you, for you to sort it out. You sit in classes, or read books, and the information is structured so you could understand it better. Some topics require more maturity than others. That much is a fact of life.

When someone who only had one semester in mathematics is introduced to Godel's incompleteness theorems, they are unlikely to fully grasp the theorems properly. The result would be "But I heard there are no complete and consistent theories." sort of reply later on. Similarly introducing cardinals at the wrong time and place would end up with people thinking that they can apply calculus based tools to compute things about infinite cardinals.

These mistakes result from someone reading material which is meant for readers with a stronger base. I know because I'd done a whole lot of these things when I was a freshman, and I was just lucky that this site (and MO) weren't around during my undergrad (I joined MO on the very last day of my undergrad). Otherwise I would have never learned set theory properly.

So while advance answers can be useful, it might be good to try and gauge the OP first and see if the answer won't be harmful to them. With time, one gains the experience for better estimating when an advance answer is in place, and when it's not. (For example if the question is given by a high school student, an advance answer should be carefully constructed and include some exposition to the topic; or otherwise is going to either be completely useless and frustrating to the student, or becoming "dangerous knowledge".)

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  • $\begingroup$ Speaking of such, Asaf, I want your recommended reading list for someone (myself) who feels about ready to jump from elementary-set-theory to set-theory. I tried Smullyan and Fitting, but their approach is unorthodox (making it hard to discuss with others) and their book, even revised, has a somewhat significant number of substantial errors. The approach they choose for the "elementary" topic of ordinals is a bit bizarre. $\endgroup$ – dfeuer Nov 3 '13 at 23:20
  • $\begingroup$ In what sort of context do you mean that? Are you looking for a read for learning modern set theory? The canonical reference book is Jech. If you want a specific topic, like large cardinals then Kanamori should be about right. $\endgroup$ – Asaf Karagila Nov 3 '13 at 23:54
  • $\begingroup$ Which Jech? And I was looking for the book for learning. Is Jech that, or just a reference? $\endgroup$ – dfeuer Nov 4 '13 at 0:47
  • $\begingroup$ Set Theory, the 3rd Millennium edition. It's excellent for both. I am not a huge fan of his approach to forcing, though. I haven't read any other book about forcing, though, so I can't give a better reference. $\endgroup$ – Asaf Karagila Nov 4 '13 at 0:51
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I think it is important to offer both. Give the advanced theory with hopes that they can figure it out (or if anyone else stumbles upon it), but offer the less advanced one too in case they cannot figure it out.

I appreciate getting more advanced information that will perhaps put me ahead of the competition at school, but at the same time I don't want to be left hanging if I cannot figure out a question.

Either way, the voting system will take care of itself.

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