A plausible way to cash out "X is being arrogant" is "X is claiming more status than I think they have." That means it's a function of my beliefs about how much status I think X has (implicitly, relative to some community), not just a function of X's behavior in isolation.
Isn't arrogance sometimes more about having confidence than anything about social status? E.g. "He is arrogant enough to think he has discovered a grand unified theory of everything." This type of arrogance is virtuous I think.
Believing that you can bluntly tell things you are confident about implies believing that your social status is not too low. (Or a lack of social awareness.) People with very low status know they would get kicked even for saying "2+2=4" too confidently.
Regardless of what the allegedly-arrogant person believes, I think what's happening when someone is accused of arrogance for being "too" confident about a belief is generally that the accuser sees what they say about that belief as claiming higher status than they deserve.
In a (laughably unreal) ideal world, in these cases "status" would mean "reputation for accurate beliefs" and "deserve" would mean "deserve on the basis of known ability to form accurate beliefs in this area". In practice, it often comes down to the same sort of ape-hierarchy status that applies everywhere else.
As you will have guessed, I share Qiaochu's opinion that "arrogant" usually means approximately "trying to claim higher status than they have/deserve". This probably means that those of us who don't think highly of status as a guide to accuracy, trustworthiness, etc. should often prefer terms other than "arrogant" when describing others. casebash's example suggests that "overconfident" and "rudely dismissive" would be good candidates.
I claim that when you describe this situation as arrogance as opposed to something else, you are still parsing it in terms of status. There's a status ladder for physicists, and if someone insufficiently high up on the ladder claims to have a grand unified theory of everything, that gets parsed as them claiming to have more physics status than you think they have. You might also separately object that in fact there's a good argument for assuming on priors that they don't have such a theory, and that they ought to also assume that they don't have such a theory, but describing this state of affairs as arrogance as opposed to, say, a lack of understanding of priors is tapping into mental machinery that I claim is about evaluating status.
People are often confident in a way that other people think they have a right to - confident in a way that doesn't claim more status than they're perceived as having - and then people just call that confidence, not arrogance.
No, I'm making a different point that has nothing to do with status. The prior probability that you actually did come up with a grand unified theory that describes reality is pretty low, even for an actual trained physicist that is the top in her field. Given any one person has limited time and resources, the highest expected payoff thing to do is to NOT look for needles in haystacks but rather assume your new theory is wrong and go do something else. However if everyone takes this approach we will never have progress. So progress is dependant on some people at least acting arrogantly and with audacity to assume, contrary to reasonable priors, that they might actually be on to something and proceed as if they have a fighting chance of being right (despite decades of stagnation and thousands of failed attempts before them).
That is a fine point to make, but I think arrogance is not the right word for the virtue you're describing. Arrogance is how other people might describe a person trying to uphold this virtue, and it's also how a part of you might describe another part of you trying to uphold this virtue, and that's the thing I'm claiming is about status.
should we be dividing arrogance into two components - over-confidence and dismissive behaviour?
Of course we should! Using the same label for two different things is a frequent source of confusion.
I would even add a third component, a "my time is precious" aspect. If there is only one professor of quantum mechanics and thousand youtube-educated quantum magic worshipers, then even if the professor is very humble and if the professor believes these people are nice and good at heart, having a long and detailed discussion with each one of them would be technically impossible, so the professor must optimize for making the debate short, or even avoiding the whole debate.
Okay, in some sense this is also "dismissive", but there is a difference between dismissing people like "go away, you are low status", and dismissing them like "sorry, but there is only one me, and thousands of yous, I can't really give each of you as much time as you would like to get".
There's a lot of nuance to this situation that makes a black-and-white answer difficult, but let's start with the word arrogance. I think the term carries with it a connotation of too much pride; something like when one oversteps the limits of one's domain. For example, the professor saying "You are probably wrong about this" is an entirely different statement (in terms of arrogance) than the enthusiast saying "You are probably wrong about this," because this is a judgement that the professor is well qualified to make. While I can see a person not liking this, I don't think this kind of straight-forwardness is wrong, or arrogant.
The Professor
When I think of an arrogant professor, I don't simply think of a person whose knowledge of a field far superceeds my own. I think of something more than that. I think of a person who seems to have an inflated sense of self-worth because of their knowledge. That is, I think of a person who is overstepping the bounds of their domain; a professor who not only says "You're wrong, and don't have an even basic understanding of quantum mechanics," but does it with an air of regal superiority that implicitly says "Not only are you wrong, but you are not, as a person, worth taking seriously or teaching because you are wrong." That kind of professor is taking the expertise he has in quantum physics and applying it to something well outside quantum physics (in this case, a person's worth). If this is the kind of professor you were describing, then I'd certainly say there's arrogance here. Otherwise, I don't think the professor is being arrogant by saying something like "You don't possess even a basic understanding of the core concepts of quantum physics." Admittedly, a more constructive way to go about this would be to at least show why this is true. As someone who does have an understanding of quantum physics, there's a lot of material that one could easily show an enthusiast that'd be well beyond their current knowledge.
A bit of nuance: Using this same standard for arrogance, the real answer is something like "it depends on what they were talking about." For example, some philosophical theories that emerge from quantum mechanics can be understood and thought about by the laymen (key word: some; many arise from a misunderstanding of quantum mechanics, and thus can easily be debunked with knowledge about quantum mechanics). If the professor and enthusiast were talking about one of these theories, then the professor dismissing this legitimate theory would, in fact, be arrogant (because it would be a step outside of the professor's bounds).
In the case that the professor doesn't admit to being a professor, indeed it would seem arrogant—but only because it seems as if the incognito professor is overstepping his bounds. That is, since no one knows the professor is a professor, there's no reason to assume quantum physics is within his area of expertise. (This may also be somewhat pedantic, but in something like quantum physics, because of this gap in knowledge, it'd be very obvious who the professor was to an audience that doesn't know quantum physics, even if it wasn't made explicitely clear beforehand.)
The Enthusiast
We can apply the same standards of arrogance to the enthusiast. I've seen many people who are simultaneously engaging, calm, polite and arrogant. If a professor, someone whose studied the field very rigorously, with mathematics that would take the enthusiast years to learn, claims that the enthusiast doesn't have a working knowledge of quantum physics—then it's probably true. If the professor says "If you knew enough about quantum physics, you'd see why this idea is strictly wrong," and the enthusiast is still convinced he's correct, then I'd say the enthusiast is stepping outside of his bounds (i.e. - is being arrogant). Again, admittedly, if the professor doesn't explain why the enthusiast is wrong, it's difficult for the enthusiast to gauge whether or not whatever is being spoken of is a legitimate theory or one that dissolves with sufficient knowledge of quantum mechanics.
A bit of nuance: It's easy with quantum mechanics, since so much of it is based in physics and mathematics in ways where the metric for correct or incorrect are fairly straight-forward and clear. If the professor were an economist, the lines defining what the bounds are could be much blurrier.
In the case that the professor doesn't admit to being a professor, the enthusiast might not have a way of knowing whether or not bounds are being overstepped, and the bystanders would probably see the professor as being arrogant. Though, I reiterate: Especially in something like quantum physics, the party who has the expertise would be apparent.
To come back to your distinction between over-confidence and dismissive behavior, I think these things are both addressed by considering arrogance to be synonymous with an overstepping of one's bounds. Dismissive behavior is relevant only insofar as it implies this overstepping. For example, does it indicate that the professor thinks less of the enthusiast as a person? Maybe the professor is being dismissive because the enthusiast's vehemence is blinding? Similarly, over-confidence is relevant only insofar as it implies an overstepping of one's bounds (this is a bit easier, because "over-confidence" is almost synonymous with "stepping over the bounds of your knowledge-limitations.")
Kind-of Disclaimer:
1) I might be underestimating the amount of knowledge you intended for your enthusiast. In my experience (I am not a professor), I've never met a physics-enthusiast who has a working knowledge of the actual physics with which they are enthused—and the types of physics that people find most interesting are usually the ones that require the most knowledge (i.e. - string theory, quantum mechanics, etc.; few non-physicists are that enthused with Newtonian Mechanics!).
2) "Overstepping one's bounds" might be a bad term. I'm willing to use a better one, it's just the one that came to mind. I hope it's clear what I mean from the context.
"This may also be somewhat pedantic, but in something like quantum physics, because of this gap in knowledge, it'd be very obvious who the professor was to an audience that doesn't know quantum physics, even if it wasn't made explicitely clear beforehand." - I met one guy who was pretty convincing about confabulating quantum physics to some people, even though it was obvious to me he was just stringing random words together. Not that I know even the basics of quantum physics. He could actually speak really fluently and confidently - just everything was a bunch of non-sequitors/new age mysticism. I can imagine a professor not very good at public speaking who would seem less convincing.
Positive reinforcement for exactly the sort of post I'd like to see. (specifically, it's starting a useful conversation, and it makes it's point quickly and enjoyably)
I think dividing arrogance into it's component parts is pretty key. I'll hopefully write more on this when I have more time. I also think it's helpful to dive into why (and when) dismissiveness is a problem.
Arrogance is an interesting topic.
Let's imagine we have two people who are having a conversation. One of them is an professor in quantum mechanics and the other person is an enthusiast who has read a few popular science articles online.
The professor always gives his honest opinion, but in an extremely blunt manner, not holding anything back and not making any attempts to phrase it politely. That is, the professor does not merely tell the enthusiast that they are wrong, but also provides his honest assessment that the enthusiast does possess even a basic understanding of the core concepts of quantum mechanics.
The enthusiast is polite throughout, even when subject to this criticism. They respond to the professors objections about their viewpoints, to the best of their ability throughout, trying their best to engage directly with the professors arguments. At the same time, the enthusiast is convinced that he is correct - equally convinced as the professor in fact - but he does not vocalise this in the same way as the professor.
Who is the most arrogant in these circumstances? Is this even a useful question to ask - or should we be dividing arrogance into two components - over-confidence and dismissive behaviour?
Let's imagine the same conversation, but imagine that the enthusiast does not know that the professor is a professor and neither do the bystanders. The bystanders don't have a knowledge of quantum physics - they can't tell who is the professor and who is the enthusiast since both appear to be able to talk fluently about the topics. All they can see is that one person is incredibly blunt and dismissive, while the other person is perfectly polite and engages with all the arguments raised. Who would the bystanders see as most arrogant?