The Cat on the Mat Problem
Suppose you have a cat on the mat. Let's call this cat Tibbles. Now, let's take that part of Tibbles that includes everything about Tibbles except its tail. We'll call it Tibbs. Here's a question, which, while very odd, nonetheless turns out to be of considerable interest: Are Tibbles and Tibbs two cats?
The reason that it is odd is that we don't normally think of Tibbles and Tibbs as two cats, but as one and the same cat. However, strictly speaking, Tibbles and Tibbs are not identical. Tibbles has a bit (the tail) that Tibbs does not. So it seems we can't strictly say that Tibbles and Tibbs are the same.
We can intensify the paradox with what is sometimes called the 'paradox of the thousand and one cats'. Suppose that instead of Tibbles and Tibbs differing by a tail, they differ only by a hair. There are a lot of hairs on Tibbles. So is every part of Tibbles that differs by a hair from other parts of Tibbles, another cat?
Trying to give the best answer to this question gets into very interesting territory very quickly, because it raises important problems for the ways we think of part-whole relations, part-part relations, identity, and the like. It might be useful, however, to introduce a slight bit of contemporary mereological terminology, which I will use. The term part is reflexive: a thing can be a part of itself. So Tibbles and Tibbs are both parts of Tibbles. We often don't use the term in this way, but we sometimes do; and so it's useful to be a bit more precise. The term proper part is slightly different. If x is a part of y, but y is not a part of x, x is a proper part of y. One thing this means is that Tibbs is a proper part of Tibbles; but Tibbles, being the whole cat, is not a proper part of Tibbles.
So the question raised by Tibbles is this: Some proper parts of a cat only differ from the whole cat by the tiniest little bits; are these proper parts cats, or are they not cats?
Solutions Supposing Many Cats
One way to go is to say that every proper part of a cat that meets certain cat-defining characteristics (whatever they may be) is a cat; and, since each of these cat-candidate proper parts is a cat, there are very many cats on the mat.
This is not a popular solution. It seems to be preserving our normal thinking about identity without preserving our normal thinking about cats. However, there is one proposed solution, which I will call almost identity, or practical sameness, that takes the many cats option but does so with some sophistication. The practical sameness advocate might think along these lines: Technically, Tibbles and Tibbs are each cats, because they meet all the criteria for being cats; and since they are not identical, they are different cats. However, Tibbles and Tibbs overlap to a very high degree. Closely overlapping cats, however, are almost identical, almost the same, and for practical purposes this is usually enough to treat them as one cat. The difficulties with this position are fairly obvious: it seems to be toying with words by saying that there are many cats on the mat, or even that cats are the sort of thing that can closely overlap.
Solutions Supposing One Cat
So this brings us to one-cat solutions. The most popular solution these days is the maximality solution. And the maximality solution is fairly simple. The maximality proponent might be summarized as thinking in this way: The paradox assumes that the proper parts of an X can be X's themselves. But surely it makes sense to deny this? Tibbs is not a cat. Tibbles is a cat; Tibbs is a proper part of a cat; and no proper part of a cat can be a cat. There is one, and only one, part of a cat that is a cat: and that is the maximal part, the one that is not a proper part of the whole cat. There is a lot to be said for this. A problem with this solution, however, is that some of the proper parts of cats are indistinguishable from cats. Think of Tibbs. Tibbs is just Tibbles, not counting the little bit that is a tail. It has a feline nervous system, a feline cardiovascular system, a feline form, etc. So the maximality proponent has to say that many things that look like cats, act like cats, sound like cats, feel like cats, have the biological characteristics of cats, are nevertheless not cats. What is more, we know that we can reduce Tibbles to Tibbs (just take off the tail) and we have a cat. So while the maximality solution makes there to be only one cat on the mat at a given time, it requires us to say that cats are constantly changing into other cats. Further, since Tibbs as a proper part of Tibbles is not a cat, but becomes a cat when it is no longer a proper part of Tibbles, things that are not cats are continually becoming cats without themselves changing. Likewise, since cats can grow hairs and the like, things that are cats are continually becoming things that are not cats, again, without themselves changing. To make matters worse, the maximality solution requires that we be very, very precise about parts -- of all the vast numbers of proper parts of the cat, one and only one is the actual cat. None of the means whereby we ordinarily deal with the world -- sensation, thought, language -- are this precise. If I point to this animal, and say "This is a cat," to which part am I pointing? I cannot say! I cannot distinguish Tibbles from Tibbles-except-one-skin-cell. So the maximality solution means that, while we know that one part of this animal is a cat, we don't ever know which part of the animal is the cat. I know there's a Tibbles on the mat, but I can never actually identify it. And this, surely, is a bit peculiar. The only option seems to be to combine it with practical sameness. All these close candidates are almost identical, so they can be treated for practical purposes as one cat. But if we go this route, maximality starts looking like a many-cats solution; it's just the maximalist says that one of the many cats is really the cat, and the others are just cat-like enough to be treated as cats. That makes the maximality approach an awful amount of work in order to make only a slight improvement over the many-cats solution.
A second way to have a one-cat solution is to accept relative identity. The relative identity theorist thinks roughly along these lines. The problem here is not how we understand cats, but how we understand identity and difference. The paradox arises because we assume that two things that are not identical in every way are not identical in any way. But perhaps this is false. Tibbles and Tibbs are not identical in every way -- Tibbles is Tibbs plus an additional bit of cat. But, setting aside this additional bit, they are identical in every other way. So we can say that Tibbles and Tibbs are identically the same cat, even though (because Tibbs has no tail) they are not identically the same cat-with-a-tail. While this is not a particularly popular solution to the problem, it has a lot to recommend it. It clearly gives us one cat, it doesn't have any of the bizarre consequences of maximality, it satisfies those of us who think that Tibbles and Tibbs need to be considered the same cat, and we do (at least at first glance) seem to talk about sameness in this way. In fact, whenever anyone asks whether X is the same as Y, it's very natural to ask, "The same what?" And if we take all sameness to be identity, this is the same as to allow that two things may be identical in one way but not identical in another. It can be made rigorous and, apparently, logically consistent. The problem, however, is that this seems to stretch our notion of identity beyond all recognition, because it requires us to say that things may be identical without having all the same properties. But a lot of our reasoning seems to require the denial of such a claim.
One could perhaps allow for some sort of sameness without identity, and say that Tibbles and Tibbs are not identical at all, but are still, in a weaker-but-still-genuine sense of 'same', the same cat. As David Wiggins, I think, has said somewhere, merely because we think it important to qualify sameness sentences (e.g., by asking "The same what?" in response to the statement "These are the same") it doesn't follow that we are committed to saying identity is relative. But then we need some account of what this non-identity sameness is. (One possibility that seems promising to me, but which I haven't seen developed anywhere, is to deny that parts and wholes exhaust our mereology. Suppose we distinguish parts, wholes, and subjects, where subjects are the things of which we predicate parts and wholes. Then it might be possible to argue that while Tibbles and Tibbs are not the same whole cat, because Tibbs is only part of the whole cat, they are the same cat subject. On this view two things could be the same subject without having to be considered identical at all. But, as I said, this would require development, and likely would have problems of its own that would need to be resolved.)
If a lot of this sounds like word-chopping, you're almost right. It certainly does seem like a lot of the mereological literature does get into word-chopping, and weird word-chopping at that. But it's not purely verbal, because it matters a great deal to the way we reason about identity, parts, and wholes. If relative identity is true, for instance, we have to admit that two subjects not identical in every way can be identical in some ways. This would affect a lot of our reasoning. Likewise, maximality and practical sameness have a lot of implications for our reasoning about classification. Which position you take can change the sort of objections you can make to other positions (for a discussion in which it matters crucially to whether certain kinds of objections can be put forward to a claim, see here); it can change the sort of things you think are particularly important for saying that two things are the same; it can change the way you think proper parts are related to wholes.
Which position would you choose?