Leo Tolstoy spent over a dozen years of his life writing a book by that very title, and here I plan to tackle it in a single blog entry. :)
A lot of definitions of art are very specific. I like to take a very liberal view of the word. Here's my shot at it: art, n.: Any activity that requires the use of intuition.
What is Intuition?
So what's "intuition"? I see intuition as the human ability to make decisions in complex situations with no rationally clear "right solution".
We'll start with the obvious. There are all kinds of guidelines for story-writing. "Introduction, rising action, complication, climax, denouement," is one. "Know your ending before you begin," is another. These are easy to follow (or ignore). But some guidelines are more vague: "Develop your characters to the point where they write their own dialog." "The writing should flow, so that the reader can be engrossed in the story." There have been some serious attempts at programmed story generation, but Stephen King won't be looking for a new job any time soon.
I actually disagree with the aforelinked researchers' view that computers will never become "truly intelligent" (whatever that means), but what's important here is that human beings can make decisions that we do not deeply understand. A person may know plenty of music theory, but that doesn't mean that they'll be able to produce a new work of Mozart. In fact, Mozart himself actually experimented with algorithmic composition, (Try Mozart's Dice Game!) but it took more than simple algorithms to compose Piano Concerto No. 20.
Human intuition excels (relatively speaking) at making sense of complex human emotions and relationships. There is a great deal that psychological research can tell us, sure, and there are even drugs that can adjust our emotions, but therapy still requires a human being for a reason. There is simple flow chart to mental health. A computer-written story may have the broad outlines right, but it won't capture the nuances of human relationships.
Not just fine art
Any activity? Sure, I think art can apply to any activity. It doesn't have to be restricted to literature or painting or film. You know all those times when someone does something with no clear correct path and says, "It's an art"? Well, I think it is. :)
I've already discussed that therapy is part science and part art. Small talk is a similar art. A person needs a very complex understanding of human behavior and local popular culture in order to be successful at small talk. Knowledge is not sufficient, either; you need to know how to apply that knowledge. Consider that computer programs that simulate conversations can only do so in limited domains and in limited styles. Small talk requires cultural sensitivity, detection of your conversation partner's domain of knowledge given very little information, and the ability to make inventive and convenient segues.
Even software engineering is an art. Good design often relies on intuition as much as formal methods. We tend to call certain coding styles "ugly" without necessarily having an air-tight argument against them. There is much about software engineering that is not well-understood, where there isn't clearly a "right" way to do things, but a human programmer can often get a general sense of "good" versus "bad" design.
The death of intuition?
Of course, the obvious questions are: Isn't intuition just a matter of what we can't yet do algorithmically? When I say "a computer will never do that", what about computers that are as smart as humans? (Whether artificial intelligence will ever be "strong" (truly intelligent and even conscious) or remain "weak" (a mere simulation of intelligence) is still just a matter of philosophy, of course.)
I think that, in a sense, some things do stop being art after a while. Medicine, for instance, is often less of an art than it used to be. As we understand more formal ways of solving certain problems, we need less intuition. In a more traditional realm, many people used to consider realistic paintings to be amazing art. Then came photographs, and the creation of a realistic image no longer required human intuition. That's not to say that photographs cannot be artistic, though, and it's not to say that all realistic paintings are, either.
So, yeah, I think some things that require intuition now will no longer require it in the future, and those activity will cease to be something I consider "art", but there will probably be other realms for human (or strong AI) invention. Perhaps the real question will then be: Will we ever fully understand our own minds? (That's another wide open philosophical topic, of course.) My definition of art hinges on our lack of a complete understand of our own thought processes, so, yeah, maybe if we can one day understand everything there is to know about what we're thinking, there will no longer be any true art. I for one hope that that day never comes.
"So are you saying that the latest Dolph Lundgren action flick is as much a work of art as Dr. Strangelove? For that matter, what if I were to poke you in the eye? It'd require human intuition to do it successfully! Are you saying that Dr. Strangelove is no better than a poke in the eye?" Well, note that I have yet to make the distinction between "good" and "bad" art. I consider bad action movies to contain a great deal of formula and a little bit of art. Since this entry has gotten ridiculously long, I'll talk about what I consider "good" art another time. Stay tuned! And please do comment below!