One mistake that would be worthwhile to get out of the way. I don't think Alejandro makes it, although he says things that suggest it, but some people do, and dealing with it will be useful background for the rest of what I have to say. We must not make the mistake of thinking that because the dispute makes use of scenarios like that of Tibbles and Tibbs that this is just what the dispute is about. That would be the same type of mistake as that of the rube who thought that general relativity was about elevators.
In fact, the dispute is not about Tibbles at all. It's a dispute about the partial ordering relation (designated P) and the proper-partial ordering relation (designated PP); about the conditions of countability and classification; and about identity relations. These relations and conditions are not confined to Tibbles, or even to any common-sense object. Even the most basic and simple mereology applies to things that are not ever considered objects in ordinary, everyday life (fusions, indivisibles, temporal parts, mathematical sets, etc.). The relations in question are abstract and perfectly general. There is some question about what conditions must be met to satisfy P, but there is no a priori reason for assuming the scope of the dispute has any limits. The introduction of Tibbles and other such scenarios is simply a matter of convenience and communication, providing a reference point to which people in the dispute can appeal in order to convey their more abstract and general positions in a clear and easily-grasped way.
Now, Alejandro argues that Tibbles has no practical relevance whatsoever. As he says,
By contrast, the question of whether Tibbs is a different cat than Tibbles is one that has no practical significance whatsoever. And I don’t mean "practical" in a vulgar and utilitarian way, but in a broad way that refers to any consequences for our thoughts on other areas. The Problem of the Many, and similar ontological puzzles, seem to me to stand in "isolation" from scientific, legal, moral or other concerns. That it what makes them seem unreal.
So the idea is that some connection between philosophical problems and reality can be found where the problems have consequences for practical purposes, construed broadly; and Tibbles has no such consequences. But it's easy enough to show that this is false as stated. Consider the following simple argument:
Brandon on Wednesday is a human person.
Brandon on Thursday is a human person.
Brandon on Wednesday is not (identical to) Brandon on Thursday.
Therefore, Brandon on Wednesday and Brandon on Thursday are two distinct human persons.
This, in fact, is a problem that arises from the same principles that are in question in the Tibbles problem: identity, parthood, countability and classification. If we accept a many-cats solution to the cat on the mat problem, we are committed to the conclusion that I am a different human person from moment to moment. Now, it's fairly obvious that our standard moral reasoning is just not equipped for discussing every temporal part as a different human person. Since the practical relevance Alejandro has in mind is determining the best way to talk about things for overall purposes of rationality, this is not a small point. If the many-cats solution to the Tibbles problem were right, our moral reasoning as it currently stands would need to be scrapped. We couldn't pass it off as a good enough approximation for most practical purposes, because (1) we would, in effect, be saying it's OK for our moral reasoning to be rationally inconsistent; and (2) if you pass off any moral reasoning as a good enough approximation for most practical purpose, you get Adolf Eichmann. It's simply not a feasible option. So how the Tibbles problem is solved has serious relevance to the best way to talk about moral issues.
There are more general ramifications. For instance, if the many-cats solution turned out to be the only workable solution, we would have to say that "There is one cat on the mat" is true only if it is understood to mean something like "There are cats on the mats, and they do not diverge from unity at this level of precision for such-and-such purposes." In other words, it requires that we always leave open the possibility that at a different level of precision or focus the cats on the mat would have to be counted as two (and this would not be double-counting of one cat). But because the reasoning behind the many-cats solution is general reasoning about general relations, this is perfectly general. Substitute for 'cats' anything you please; the only case in which the reasoning wouldn't apply is if you substituted something that you could prove had no proper parts, i.e., that was simply indivisible even in principle. Cats, tables, people, atoms, universes: it's all the same. To say, "There is one atom in this general vicinity" would have to be always understood to mean, "The many atoms in this general vicinity do not diverge from unity in a significant way for this particular set of purposes at this particular level of precision," allowing for the possibility that for a different particular set of purposes or at a different level of precision there may also be two atoms, or three atoms, or indefinitely many atoms. There is a world of difference in a language that allows you to say, "There is one X," and one that doesn't.
Or consider how the dispute over relative identity affects our understanding of mathematics. It can be shown fairly easily that mathematical equality is not an identity relation on the classical understanding of identity. This is because equality, as a one-to-one matching, is consistent not only with identity (matching a thing against itself) but with non-identical isomorphism (matching two different things against each other. For instance, I can take three apples and six apples and match it against (the same) nine apples; or a I can take three apples and six apples and match them against two oranges and seven oranges. So equality never entails identity on the classical identity view; having in hand an equation we can't ask any identity-based questions unless we prove that the two sides of the equality are related to each other in the way a thing is related one-to-one with itself. However, this is not true on the relative identity view. If relative identity is a coherent and viable solution to the Tibbles problem, it is a coherent and viable form of identity everywhere. And on the relative identity view, equality is an identity relation, because it's an instance of 'X is the same F as Y'. When we have an equation in hand, the question is never whether it is an identity (this is satisfied by its having any sort of equality), but only what identity is being indicated. And since which you choose will affect how your inquiry proceeds, it's clear that the dispute has relevance to the best way to talk about an application of mathematics whenever identity is in question.
So in Alejandro's sense, the dispute over Tibbles has clear practical relevance: that is, it affects what would be the best way to talk about any given cases (whatever they may be) of parthood, identity, or countability and classification, regardless of their context, for the overall purposes of consistent rationality.
There is an additional part of Alejandro's argument that I don't think is quite right. Alejandro argues:
But also not for theoretical purposes; because if one has the philosophical goal of finding a sort of "ultimate vocabulary", sharp and precise, which reflects in some sense the structure of the world, then one should not use in the discussion ordinary things like cats and tables. One should make first an analysis of our deepest scientific theories, like Quantum Field Theory or perhaps String Theory, and design a conceptual frame that fits well with them. Of course, ordinary notions of "objects", "parts", "properties", "time" and so on would probably not map at all into the mathematical structure of these theories, and so the ontological puzzles I started with could perhaps not even be stated, let alone solved, by taking this route.
But this involves a confusion (or, at least, I am inclined to think so). If, for instance, it were to turn out that our 'deepest scientific theories' were suddently to be shown to make impossible the demonstrable existence of cats, this would indicate a failure of our deepest scientific theories. This is for the straightforward reason the existence of cats is a fact -- a scientific fact, independent of any physical theory you choose to build -- and the inconsistency of the theory with it would be an inconsistency of theory with reality. We know that reality is structured so as to have cats and their parts, however confused or limited our ordinary talk about such things may be. Whether reality is structured as Quantum Field Theory or String Theory say has to be proven by consideration of how the theory relates to this very type of fact on the level for which the theory is supposed to be useful. And what they will do if successful -- what would make them genuinely deep scientific theories -- is conform to the real world as we know it to exist.
Now, in fact, there is reason to think that an adequate physical theory would shed an immense light on the mereological problems. But it would do so not by dissolving the problems themselves, which are based on general considerations of parthood, identity, and countability, or even really by showing us different elements of reality, but by showing us the same elements of reality at a vastly more precise level of analysis than we are usually able to obtain. For instance, if we came up with the ultimate physical theory, and found it to be well-founded it would tell us quite clearly what, for all purposes relevant to the discussion of the physical world, has to be treated as countably distinct. If the ultimate physical theory told us that we have to treat proper parts of atoms as, basically, distinct atoms with regard to a given property (if we had to treat spatiotemporally overlapping atoms as different for some aspect of the theory), to the extent that this could be shown to be well-founded, it would be an interesting datum that would have to be taken into account in any adequate mereology. The ultimate physical theory's being reality-relevant is (as far as real parts and wholes go) nothing other than its ability to make such a factual-analytic contribution to mereology; because it cannot be the reality-relevant ultimate physical theory unless it sheds light on that little bit of reality that is the physical object Tibbles. (This begins to touch on the issues of scientific realism Alejandro discusses toward the end of his post. These get very complicated very quickly, and are sometimes rather murky anyway, so I won't get into them here. But the basic point here is that we start with reality, as it were at the beginning of the theory, and keep our pulse on it the entire time; what we get as a conclusion of theory --electrons or quarks or what have you-- may be relevant to reality, but there's no chance of that at all if we don't keep in mind that we already have a handle on reality, albeit an incomplete one, which includes things like cats and stars and so forth, that we are simply trying to make more complete.)
So I'm inclined to think the rational direction of inquiry is actually the opposite Alejandro suggests. What is necessary is to see what issues are raised by the real-world facts of Tibbles and everything Tibbles-like; if the issues raised are legitimate, and are relevant to things qua physical objects, the adequacy of our physical theory is tested by its ability to clear up these mereological issues by deeper analysis and newer facts. (In other words, it explains what needs explaining about physical objects like Tibbles.) This is because the issues themselves are raised by physical reality itself; and a physical theory is pointless, at least on its own, if it doesn't shed light on physical reality.
But, of course, even if this is not so (and it is admittedly a controversial sort of claim), the issues in question in the Tibbles dispute are only indirectly relevant to our common notions of part, time, etc.; because the issues in question are general issues about parthood itself, identity itself, etc. And this leads in to the practical relevance question again. Alejandro suggests that the dispute isn't relevant to practice because
our ordinary language is good enough for describing things like parts of cats; I am not aware of practical concerns that require a more logically precise conceptualisation (as for example, a discussion about abortion may require a more precise conceptualisation of personhood than our unreflective one).
But, of course, the problem with talking about practical concerns is that it depends on what practical concerns you mean. Alejandro almost recognizes this. He recognizes that we have common notions about part, time, etc.; and he recognizes that our best scientific theory might not mesh well with these notions. Now, for practical concerns like petting a cat, this is not an issue, nor is the mereological problem of the cat on the mat. But for practical concerns like scientific popularization, scientific pedagogy, and so forth, it appears to be much more relevant. One of the key plagues scientific pedagogy continually has to deal with is this very disparity. In this context, our ordinary language is not obviously good enough for describing things like parts. If we understand the relevance of philosophy to practice in the way Alejandro does, as getting us closer to an 'ultimate vocabulary' that is maximally useful for overall rational purposes, then we can see that the sort of questions raised by the disparity are: What are the implications of each? Do they share anything in common (they have to if there is any genuine conflict) and, if so, what? Is there any modification of language that we could make that would reduce the disparity? And so forth. If we take philosophy to have practical relevance in the sense of building a language that allows us to speak over the widest possible range of rational concerns with the least danger of language-induced error and equivocation, we can see at once that the mereology of Tibbles is relevant in this context. Because the mereology of Tibbles is not about how we use the terms 'part' and 'identity', although this is a fact taken into account in the inquiry, but over the best way to think of parts and identities generally. And this means that, since both theoretical and common notions of parts and identities fall within the range of discussion, the discussion about Tibbles is discussing issues relevant to the best way to talk about parts and wholes that allows us to talk about them scientifically and practically with the least chance of error. And that, I would suggest, is not a small thing.