To me as an educator, one of the more interesting, and challenging, questions is: what is intelligence? Further, we can ask how many "intelligences" are there and how do we recognize intelligence? For what it is worth, here are some of my thoughts on these questions:
What Is Intelligence
The word "intelligence" has proven quite resistant to any single definition (or discovery of any necessary or sufficient conditions). I think this has a large amount to do with its function as an abstract noun, a noun that doesn't refer to an existant so much as an abstract generalization. As with a term like "justice," we all might have our own ideas as to what we think intelligence is, but there is no "thing" that we can point to to prove or disprove any competing conception. (If we suggested that a dog must have x condition, we can point to dogs that may not have x condition in order to disprove the assertion, but we cannot do the same with abstract nouns, as there are no concretes to point to.)
on the online dictionaries, a most common definition goes like this: intelligence is the "ability to comprehend, or, to acquire and use information." I have yet to find fault with this definition except to say that if we use it as our guide, there must be as many "intelligences" as there are things to be "intelligent" about. (I will get to that in the proceeding section.)
But ultimately, I do think that the "abstract noun" problem faces a very Wittgensteinian dilemma in that it seems to prove incapable of any obviously correct definition in the same way concrete nouns are. Further, attempts at defining words like "intelligence" seem like exercises in circularity; in order to define "intelligence," we must think about examples of intelligence we know of, and abstract their common traits. But doing this presupposes a working definition (or way to recognize) what counts as intelligence. This, in a circular way, defining a term like "intelligence" requires a pre-existing idea of what intelligence is (as we define by abstracting from concrete examples).
In short, I don't hold out any real hope of any once-and-for-all correct definition of intelligence, nor am I worried about it. Like other ab tract nouns - beauty, justice, evil - we as often as not rely on the "Potter Stewart" definition: we know it when we see it.
How many intelligences are there?
As mentioned above, If we use the above definition - intelligence is the "ability to comprehend, or, to acquire and use information," it seems quite obvious that there are as many intelligences as there are things to be intelligent about.
First, I do stress the contingency of the above sentence: there is no a priori reason to use that definition over others, other than the fact that it is nondescript enough to accord with most intuitions I think we have about intelligence.
There is one big problem with this definition, though. For some, it will be too broad and for others, too narrow. This mostly hinges on whether people want to see things that we might otherwise call "talents" be labeled as "intelligences." To see this, take the example of the songwriter. Some want to narrow the definition of "intelligence" to disallow the "talent" of songwriting from being an "intelligence." Others, though, want to see songwriting included as an intelligence. (As mentioned, we often circularly define words like "intelligence" based on what we want that definition to allow for.)
The problem here seems to be that most people debating this issue seem to talk about different disciplines as "either/or" scenarios; either it an example of intelligence, or talent. Songwriting, they say, is a talent; math is an intelligence. What they often don't see is that both activities involve admixtures of both "talent" ( the "natural" and unteachable predilection) and intelligence (for now, the ability to acquire and use information). Math, generally thought of as an intelligence, requires both of these as much as songwriting does (just in different proportions). Just like songwriting seems to be more a talent than an intelligence because you can't really teach it - either you have it or you don't - we can easily see math in the same way; some have the intuitive number sense and those who don't have immense difficulty learning it.
I do think that those taking a more restrictive view of intelligence, and wish to preclude art, music, sports, etc. - from being called "intelligence" take an unusually strict view because they fail to see that just because one has a talent for something does not negate the fact that they also can develop those talents in ways deserving of the "intelligence" categorization. To take a favorite example of tennis player Roger Federer, one might be tempted to suggest that tennis playing is a talent rather than an intelligence. One could only do this, of course, if they neglect the amount of skill and learning involved in the enterprise. Like any other genius, Roger Federer's ability to acquire and use knowledge about his craft surpasses, in sensitivity in breadth, what most will achieve. Talent? In part. But talent only goes so far, and it is HERE where I think it becomes a crime not to call Roger Federer's tennis ability an intelligence.
I am hard pressed to come up with examples of things that qualify all the way as talents or all the way as intelligence. It seems that just about every constructive human activity requires both some sort of natural predilection (talent) and an ability to acquire and use information (intelligence). That is why I think that under our dictionary definition of intelligence there are as many intelligences as things to be intelligent about.
How do we recognize intelligence?
Much of the dispute over what to call an intelligence and what not to has to do with whether the proposed activity can be measured? It is uncontroversial that ma thematic ability and spatial reasoning should be called intelligences because we can objectively measure them. It is, however, controversial to say that creative writing ability should be called an intelligence and the reason is because it cannot be objectively measured (it can, of course, be subjectively measured, as any lover of writing will tell us.)
This, I think, is a mistake. We should not base whether x intelligence exists by deciding whether it can be measured. We do not, for instance, say that talent for acting does not exist because it cannot be measured. What we do say is that while there is such a thing as talent for acting, such a thing cannot be objectively determined or measured, but is a matter of subjective judgment. We should say the same when it comes to "intelligences" that cannot be objectively measured. We should not say that they simply don't exist because htey can't be measured. Rather, we can recognize that a thing like "social intelligence" is recognizable, even if it cannot be objectively measured.
I am as big a fan of measurement as the next, and I think that IQ is a valid measure of a person's scholastic aptitude. What I don't think is that IQ exhausts the possibility of what can qualify as an intelligence simply because other "intelligences" cannot be objectively measured. (The only persons who would object are psychometricians, and this is no sup rise, because their profession only allows them to target those capacities that can be measured, even if they realize that other capacities exist in the same way a lawyer's profession demands that she recognizes only codified law but can acknowledge that uncodified morality exists.)
To conclude, I see no reason to suppose that "intelligence" will ever yield to any once-and-for-all agreed-upon definition, nor would I expect it to (as an abstract noun). Partly because of this, I think it is folly to suggest any narrow definition of intelligence and support a more pluralistic view: intelligence exists wherever there is a recognized ability to acquire and use information to inform one's actions. In holding this view, I see it as backwards to suggest that intelligences exist only when they can be objectively measured. To hold this is to put the cart before the horse.