When I posted different answers to a question, some people seemed to frown upon it.
Doctor, it hurts when I do this
You're getting some push-back for something you're doing on the site. The feedback from other members prompted you to ask this question, but even prior to that the software itself gave you a bit of resistance when you went to add that second (and third) answer:
Are you sure you want to add another answer?
You could use the edit link to refine and improve your existing answer, instead.
Of course, it is allowed - there are times when it's very useful - but they're not common; generally, you do want to just edit your existing answer.
If you want feedback on your actions, explain their purpose
Let's ignore the abstract question here. As robjohn and Bill have pointed out in their answers, there are occasions when multiple answers are useful, and (many more) occasions where you really just want one answer. But that's not what you asked. You linked to your multiple answers, and a cursory examination makes one thing blatantly obvious:
You aren't making any effort to explain why one answer might be preferable to another.
There are often multiple ways to solve a problem. One of the big advantages of the Stack Exchange system is that, when posted as separate answers, the most generally useful solution can be ranked higher by those reading than the rest. However, this falls apart when there is effectively no difference, or when the difference is subtle. So... Which is it? You must've had a reason for bothering to write multiple responses - yet you didn't bother to include that reasoning in the answers themselves. It's almost as though you expect others to do this for you...
Asking - and answering - in good faith
Indeed, this meta question suffers from the same problem. Right before asking it, you posted this comment below one of your answers, in response to specific criticism of your answering strategy:
@CameronBuie Maybe someone would down vote for a false proof. This is only an example. There are other confusing factors. As for your last comment, I didn't get any reps. Anyway, what's wrong with separate answers? – Makoto Kato
Uh, yeah, maybe that would happen. Is that what you hoped would happen? Being unsure of one or more of your proofs, you posted them separately so as to elicit specific critiques of each? Then why not say that? And when raising the discussion of a specific scenario here, one already under discussion elsewhere, why not summarize your already-stated rationale for doing what you're doing, rather than leaving folks to debate the general case? There's a time and a place for everything - but what you're really concerned with here is, was this the right time and place for multiple answers?
Frankly, I think you would garner a lot more sympathy - or at least understanding - if you were a bit more transparent as to your intent when you engage in activities that are uncommon on this site. Don't assume that others will assume good intentions on your part, and you may find they're a bit more likely to do just that in the future.
When answers legitimately take different approaches, separate answers are usually better since they each come with different direct links and comments sections.
One reason for preferring a single answer is given when you try to add another answer, which suggests refining and improving an existing answer.
The answers you gave are not especially distinct from one another. By combining them into a single answer, the relationships between them can be seen implicitly through juxtaposition -- or even better, it could be explained explicitly. The combined answer could easily be much more illuminating than three individual answers.
I imagine that the answers aren't especially distinct aggravated the negative reaction to multi-answering.
There are various honorable reasons that one may have for posting more than one answer to a question. For example, often I am interested in judging how well-known various approaches are, so that I can use that feedback to revise my teaching methodology. This is more difficult to do if one places multiple answers in one post since then it is not clear which particular answer the votes and comments target (which, by the way, is a huge problem with votes on multiple-answers on meta).
For a concrete example, I explicitly chose not to place the well-known degree-based approach into my answer on why polynomial rings are not fields, because I sought to determine what sort of response the given universal approach would receive. The surprisingly large number of votes it received (along with the tone of the comments) helped provide further evidence to confirm my suspicion that many algebra textbooks do not effectively teach such universal ideas.
It is important to keep in mind that - in this way - the site provides useful feedback to teachers. Having sharp modular boundaries between logically independent answers serves to optimize the value of such feedback. This allows teachers to learn as well as teach, which provides much further motivation for teachers to participate. And, of course, the more teachers we have participating, the more the site will prosper.
Another reason for posting multiple answers is to draw modular boundaries between knowledge levels. Sometimes I post answers that include both elementary and advanced perspectives on topics. These combined posts often receive fewer votes than if I separate the answer into an elementary post and an advanced post. Probably this is because the advanced parts scare away those with no knowledge of the advanced topics, so they stop reading - mistakenly believing that the entire post is too advanced.
Similarly, students may have honorable reasons for posting multiple answers in separate posts. For example, a student with two answers may be confident of the first, but may have doubts about the second answer. Thus it makes sense to post the second answer separately, so that feedback on it can be appropriately isolated. Otherwise, if both answers were combined into one post and the second answer had problems, then the combined answer might receive a mix of upvotes and downvotes, which might relay false impressions about the first answer, e.g. misleading readers into thinking that the first answer also has problems. Further, the first answer might not get elevated to the proper level of exposure, possibly leaving an inferior answer to be the highest-voted.
Generally, readers may desire to upvote some answers and downvote others, which is not possible if they are all combined into one answer.
Finally, one tangential point is worth emphasis. Before one leaps towards accusations such as "reputation farming" etc, it is important to keep in mind that not everyone pays attention to reputation, so it may not even occur to some folks that posting multiple answers might be viewed that way. Indeed, some of us are here simply to share our knowledge of mathematics, and have been doing so for years in other forums (e.g. usenet newsgroups) where reputation plays no role. Please try to keep in mind the diversity of our community before jumping to possibly unfounded conclusions about such matters.
There is nothing wrong with giving multiple answers, as long as they are sufficiently different. However, it is customary to give them all in the same post, headed by a bold title and separated by a line
If one of the answers is very long, it might be necessary to make an additional post, but that is a very exceptional case.
One reason we sometimes see multiple answers to the same question is some doubt as to the mathematical knowledge or experience of the person who has asked the question.
In such cases I think it is of general benefit to the person who has asked the question - and to future explorers of the site - to have the links between levels indicated.
My oldest daughter (aged 15) said to me yesterday - when I ask you about homework you often go off onto things I haven't learnt yet, and which I don't quite understand - but then when my teacher deals with them in class, I do get them, and my classmates don't. She was pondering why that was, and I was thinking - it's how I learned mathematics - from my dad.
This is a great site for sensing the direction of travel and the challenges ahead.
The issue here is with reputation farming. Therefore, if you wish to post multiple answers and do not wish to suffer any hopstilities from anyone there is a simple solution.
Have one "main" answer. Make the rest community wikis.
If you are not simply after reputation then you should have no issues with this, and noone can criticise you for it. Simples!