While there are no specific instances I can think of offhand, this issue really bothers me. Sure, there are some posts that are complete nonsense (homework, etc.), but in other cases, leaving a downvote without an explanation is really unhelpful. If someone has posted something that deserves a downvote, he should at least be given some feedback in the comments explaining it, or else he'll be unable to fix what he's done wrong in the downvoted post or in the future. I don't know what can be done, but it would be nice if there were some way we could encourage people to explain downvotes in cases that aren't flagrant violations of the "interesting to mathematicians" rule.

My impression was that the founders of the site see the downvote as having a broader purpose. Essentially, if you should upvote an answer that you think is completely correct and/or adds something crucial that has not already been said, it seems that you should downvote an answer which is (i) fundamentally incorrect, (ii) needlessly unclear, or (iii) is lacking something crucial that you want to see in the correct answer.

I don't think a downvote necessarily expresses personal disapproval in any way. I also don't think it is necessary to justify a downvote: in some situations that might be appropriate and useful, but in other situations it can be unnecessarily confrontational.

It is interesting to look at experienced users and examine their ratio of up/down votes. My ratio used to be about 20:1 but it is coming down closer to 10:1 as I become more familiar with the site. There are others with somewhat higher and significantly lower ratios.

No, I do not think that. As I said, the reason for a downvote is not necessarily bad behavior, and a downvote need not be punitive. It may just be an appraisal of the mathematical value of the answer. That seems to be a big part of how the site works: you want the best answers to float to the top and the worst to sink to the bottom.

I am, with some effort, refraining from commenting on fpqc's behavior on MO. I think it is up to the moderators to set the tone.

Hanche's case study is interesting. Ben's reason for the down vote was:

I'm not familiar enough with the subject to write a good answer like that, but I'm familiar enough with the subject to say I don't think an answer which leaves it out should be at the top.

Given that another answer has been accepted, and so is "sticky" and will remain at the top, I presume that Ben will now remove his down vote.

My serious point is that I don't think that the third item on the list that plclark has, namely "is lacking something crucial that you want to see in the correct answer", is a valid reason for a down vote. Unlike in the programming world, I think that we will tend to be much more likely to iterate to a good answer, and partial answers should be actively encouraged. Harald's first sentence makes it abundantly clear that he is not attempting a full answer:

One way to get started ...

I often do something similar, saying "This isn't a complete answer but ...", to flag that I'm recording some initial thoughts that might be of help. As yet, none of my answers have received a vote against (indeed, my only negative behaviour on the site so far has been to ask about running in the rain and the metric space question). Maybe I should change my behaviour and only answer questions when I'm absolutely sure that I know the full answer.

As I've said many times, I'd rather praise good behaviour than punish bad behaviour, especially when that is borderline. And since there are only a limited number of tools for actually punishing bad behaviour, it's all the more important to explain the reasoning.

To clarify my third point, the rationale behind the downvote in that case is to alert readers that (in the voter's opinion) something crucial is missing. If the person who answers the question says this openly, this is not necessary.

Let me give an example of a recent downvote of mine that had nothing to do with disapproval of the user. Late last night Andrew Critch asked a nice question about "abstract linear disjointness" of fields. The first to answer was Greg Kuperberg. I thought (and still think) that GK's answer is not correct. I gave an answer myself, which was somewhat tentative (and I indicated as much) but I think pointed out two points that GK was missing. As I was writing my answer, Jim Milne also gave an answer which was brief but helpful. After I finished I noticed that there was little immediate response (not surprising, considering the time of day) and that the three answers were basically just sitting there undifferentiated. In my professional opinion, a reader looking for the fastest route to the answer should read GK's response last if at all, so I downvoted it.

Was this an attempt to censure Greg Kuperberg? Of course not. He is one of the very best contributors we have on MO. He answers many questions rapidly, and his answers are, with very high probability, fantastic.

It would be interesting to hear GK's opinion...

The voting mechanism must be anonymous because if it weren't, people would obsess about who's voting (or not voting) for whom, but I agree that a downvote should almost always be followed by a comment explaining the downvote or by upvoting an existing comment which contains anything you'd put in your explanation. It's extremely frustrating to get a downvote with no explanation, and I'm continually surprised by how easily people change their behavior if you give them constructive feedback.

I'd like to stress the words constructive feedback. Whenever you downvote, I think you should leave a comment actually explaining what the poster can do in the future to avoid similar downvotes. Make it as easy as possible for the person to improve. If there is something wrong with a solution, say what's wrong in a comment. It should be clear to the other person what they would have to correct in their post to get you to change your vote. If they've violated some etiquette, your comment should explain what rule they've violated and how they can change their post or future posts. If they've asked a question that doesn't belong, do your best to explain why it doesn't belong and point them to another forum if possible (where they can direct future questions of the same sort).

Comments like "I wish I could downvote this twice" or "This is awful, -1 to an odd power!" are at least as bad as leaving no comment at all. If you make it clear that you're downvoting the material and why, then people will change their behavior. If you downvote the person (which is what people will assume by default!), then they will get defensive and find any reason they can not to change their behavior.

I feel like non-constructive comments are fine for obviously bad questions. Is that wrong?

"You will also come to learn that there is no such thing as an obviously bad question."

I disagree, and I think that most people here do not agree with such a radical viewpoint, but they're free to join this discussion as they wish.