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Beauty
Mathematics

Two different sets cannot have the same reason for membership, so if beauty is the reason why a painting is in the set of beautiful paintings, then beauty cannot be the reason why the painting is in any other set, such as the set of good paintings. Is that fair?

No. There is a set of even numbers. There is also a set of numbers that are even or prime. (Note, by the way: something can be even and prime: the number 2.) The number 8 is in the first set because it's even. It's also in the second set because it's even, hence even or prime. Not all good paintings are beautiful, but for present purposes, we can still assume that all beautiful paintings are good paintings. A beautiful painting clearly fits the membership condition for the set of beautiful paintings. But it also fits the membership condition for being in the set of paintings that are beautiful or good and it fits it by virtue of being beautiful. There's nothing peculiar here at all. If X and Y are both sets, their union is also a set. That's elementary set theory, and it's so whether or not X and Y are mutually exclusive.
Beauty
Biology

Are 'dangerous' and 'aesthetically ugly' one and the same thing? I read somewhere once, that arachnophobia evolved as a defence mechanism against dangerous spiders. Even though most spider species are harmless, this evolved response is still there, as it is better to avoid all spiders, even the harmless ones to avoid being bitten by the really deadly ones. Seeing as this aesthetic disgust and fear arose for the purpose of keeping one safe, and very few spiders are actually dangerous, would it be incorrect to view the harmless ones as ugly? Similarly, there are some dangerous animals I consider quite beautiful: tigers, for example. Would it be incorrect to view them as beautiful because they are dangerous? Basically, what I'm trying to ask is, because perception of ugliness evolved to keep us from danger, is danger synonymous with ugliness and is any visual beauty we ascribe to a dangerous animal simply an illusion? Conversely, are non-dangerous animals that we find ugly actually visually beautiful even...

I think the answer is pretty clear and is implicit in things you've said. Yes: something dangerous can be beautiful. Tigers would be a widely-accepted example. "Dangerous Beauty" isn't just the name of a movie that got a 70% fresh rating on Rotten Tomatoes. It's an idea that's something of a cultural touchstone. Maybe the perception of ugliness evolved to keep us from danger, or maybe the story is more complicated than that. (I'd strongly suspect the latter.) But however things evolved , the concepts have long since come apart. If someone commented on the beauty of a tiger, and someone else tried to correct her on the grounds that tigers are dangerous, a blank stare would be an appropriate response. We find the appearance of tigers beautiful. They'd look the same way if, somehow, they magically became the protectors of humans. We also find their movements graceful; same comment. The second question you ask is whether non-dangerous animals that we find ugly might actually be beautiful. The...
Beauty

Can being constantly surrounded by ugly things and people ruin own's sense of the aesthetic? Conversely, can constantly being surrounded by beautiful things and people ruin own's sense of the aesthetic?

Let's start with the phrase "ruin one's own sense of the aesthetic." There might be different ways to interpret that, but the reading that first occurred to me was something like "undermine one's ability to appreciate things aesthetically" or perhaps "undermine one's ability to make sound aesthetic judgements." In either case, the question isn't just philosophical. It's partly a matter of what the actual psychological effects of being surrounded by beautiful—or ugly—things actually is. And although we might have guesses about the answers, our guesses might not be good ones. With due regard for the fact that philosophers can't really answer the question, however, we can still ask some rather more conceptual questions. What about aesthetic value might suggest that being surrounded by ugly things could ruin our aesthetic sensibilities? One possible reason is that if we're surrounded by ugliness, we may have trouble noticing things that are beautiful or in some other way aesthetically rewarding....
Beauty

Are all beautiful paintings good paintings?

It's an interesting question. An obvious preliminary: whether or not all beautiful paintings are good, not all good paintings are beautiful. "Beauty" is sometimes quite beside the point in judging that a painting is good. But your question was about the converse implication. Here's one sort of problem case. Suppose I see a painting that strikes me both as beautiful and as a genuinely good painting. I then find out that it's completely and totally unoriginal. It's slavishly derived from another painting. The extreme case might be a forgery, but highly derivative works raise the same sort of problem. The original might be both beautiful and a good painting; the derivative work might be every bit as visually beautiful as the original, and it might have taken enormous skill to create. It might be a good copy ; it might even be a good forgery . Whether we want to say that it's a good painting is not as clear. We might think that to count as a good painting, a work has to have at least some measure...
Beauty
Value

In his response to an earlier question about physical beauty, Nicholas D. Smith responded: "Unfortunately, a lot of good-looking people are not very beautiful in any way other than the way they look." Though there might be some rare exceptions in the world, for the most part I agree with his statement. And I'm wondering about the relationship between physical beauty and virtue... If, hypothetically speaking, Mr. Smith's claim were a natural law (Good-looking people are not very beautiful in any way other than the way they look) what then would be the most likely cause for its validity? In other words, do external factors such as our society/culture make it difficult for good-looking people to develop in more internal ways, such as through character, morality, kindness etc. Or does physical beauty itself inherently impede the good-looking ones from ever becoming beautiful in more virtuous ways?

There's difficulty that stand in the way of answering your question. In the actual world, it's not a law that physically beautiful people aren't virtuous. Some are, and some aren't. So your question is about a world with different laws than this one and you're asking what would be the explanation for a regularity in that world that doesn't hold up in this one. Now such questions aren't necessarily meaningless. One way to understand them: think about a world that's otherwise as much like this one as possible, except that beautiful people aren't virtuous. Do we have any hope of getting a grip on that question? Possibly. Though there certainly are people in the actual world who are both beautiful and virtuous, perhaps there's some statistical correlation between beauty and lack of virtue. (I'm skeptical, but let that pass.) If so, then the way to answer your question would be to investigate whatever it might be in this world that underlies the statistical pattern, and extrapolate from that to a...
Beauty
Biology

Can Darwinian science explain the uncanny fact that a crow both looks and sounds ugly whereas as a prettier bird makes a prettier song? What possible purpose could such an aesthetic unity serve and why would humans be able to recognize it?

The first question is whether there's a fact to be explained. Do "pretty" birds typically have "pretty" songs? And do "ugly" birds typically have "ugly" songs? I'm no expert, but I'm betting not. Peacocks are usually considered attractive; their songs not so much. Swans are (conventionally, at least) beautiful; their honking (at least to my ears) isn't. Many people like the cooing of pigeons. But pigeons aren't usually seen as "pretty." I'm sure a real bird aficianado could multiply examples. Of course, there are also questions about whether "pretty" and "ugly" are objective notions. That's a big question, but you can no doubt see that it's relevant. But leave that aside. If your speculation were correct, it would be an interesting fact. What might explain it is something that it's very hard to say in the abstract. We'd need a lot more detail, but in any case there wouldn't be much reason to expect philosophers to come up with the best answer.
Beauty

Why is aesthetics so concerned with beauty? When I listen to music or appreciate art I respond to it in all sorts of different ways and beauty is only a small but significant part of the experience of art.

The answer is that it isn't. Here are links to recent tables of contents from two major aesthetics journals: http://bjaesthetics.oxfordjournals.org/content/51/2.toc http://www.temple.edu/jaac/archive/69.2.htm As you'll see, beauty doesn't make much of a splash here; only one essay on the topic. If you do some archival digging, you'll see that this is pretty typical and has been for a long time. Aestheticians would agree with you: beauty is only one bit of our experience of art, and often not the most important bit.
Beauty

Why do we enjoy the beautiful? Or, what is the nature of aesthetic appreciation (it seems like a special type of enjoyment)?

On the one hand, it seems safe to say that not all aesthetic appreciation is enjoyment. There are some works of art that are profoundly disturbing, and yet we still value them. An example: I remember vividly the first time I saw one of Ad Reinhardt's large black canvases. I was taken by surprise: I didn't expect to have much of a reaction, and yet I felt something for which the word "despair" is about the best label I can come up with. I found the experience moving, but it feels wrong to call it enjoyable. Still, there are other works of art that we do enjoy and that are beautiful. So let's turn to those. Take an example of some work that you find beautiful -- perhaps the third movement of Beethoven's opus 132 A Minor quartet. If someone asked "Why do you enjoy listening to that?" saying "Because it's so beautiful" would be a perfectly good answer, though there's a great deal more that one could add. If your friend then asked "But why do you enjoy beautiful things?" you might find the question...
Beauty

We are often told that beauty is in the eye of the beholder. But I wonder if this is not an over-simplification. Surely some things are beautiful regardless of our response to that beauty. Is there not a case to be made for completely objective beauty?

This much seems plausible: whether something is beautiful doesn't depend on the actual responses that anyone has. It might be that no one has seen the thing. It might be that everyone who's seen it so far doesn't have the discrimination to appreciate it. It may be that no one who's ever been born or even will be born will have that capacity. All of that could be true, and yet the object might still be beautiful. But does that mean that it's beautiful apart from all possible responses? I don't think so. Does this make sense? Object X is beautiful, but no sentient creature that the universe could possibly produce would find X beautiful. I have a bit of trouble understanding what it would mean here to say that X really is beautiful. And if that's right, it suggests (as many philosophers are inclined to think) that whatever exactly beauty may be, it has something to do with the kinds of responses that the right sort of creature would have upon contemplating it. That's not...