Geach on the Cat on the Mat Problem

One of the key passages for modern mereology is Geach's creation of the cat on the mat problem in Reference and Generality:

How is "---- is the same A as ----" related to a proper name for an A? To attack this problem, I shall first set forth a paradox that I developed from a sophisma of William of Sherwood.

The fact cat sat on the mat. There was just one cat on the mat. The cat's name was "Tibbles": "Tibbles" is moreover a name for a cat.--This simple story leads us into difficulties if we assume that Tibbles is a normal cat. For a normal cat has at least 1,000 hairs. Like many empirical concepts, the concept (single) hair is fuzzy at the edges ; but it is reasonable to assume that we can identify in Tibbles at least 1,000 of his parts each of which definitely is a single hair. I shall refer to thee hairs as h₁, h₂, h₃...up to h_1,000.

Now let c be the largest continuous mass of feline tissue on the mat. Then for any of our 1,000 cat-hairs, say h_n, there is a proper part c_n of c which contains precisely all of c except the hair h_n; and every such part c_n differs in a describable way both from any other such part, say, c_m, and from c as a whole. Moreover, fuzzy as the concept cat may be, it is clear that not only is c a cat, but also any part c_n is a cat: c_n would clearly be a cat were the hair h_n plucked out, and we cannot reasonably suppose that plucking out a hair generates a cat, so c_n must already have been a cat. So, contrary to our story, there was not just one cat called "Tibbles" sitting on the mat; there were at least 1,001 sitting there! Of course this would involve a great deal of overlap and sharing of organs among these 1,001 cats, but logic has nothing to say against that; after all, it happens on a small scale between Siamese twins.

All the same, this result is absurd....

Everything falls into place if we realize that the number of cats on the mat is the number of different cats on the mat; and c₁₃, c₂₇₉, and c are not three different cats, they are one and the same cat. Though none of these 1,001 lumps of feline tissue is the same lump of feline tissue as another, each is the same cat as any other: each of them, then, is a cat, but there is only one cat on the mat, and our original story stands.

(Peter Geach, Reference and Generality, Cornell UP [Ithaca: 1980] 215-216)

Geach then points out that this appears to commit us to the claim that "---- is the same cat as ----" is an equivalence relation, not an absolute identity relation; which, of course, he has no problem with, since he doesn't think there is any such thing as absolute identity. He also argues that it shows "that logic can and must avoid assuming a syntactical category of proper names." A proper name for an A may be the shared name of several Bs, if Bs are in some sense the same A as any of the others. An interesting argument, although I'm not sure this is a wise argument for this conclusion. I'll have to look more closely at what Geach says about proper names.

In any case, I wonder which of William's sophismata Geach has in mind, and whether he mentions it less vaguely elsewhere.

Links and Notes

* Mark Chu-Carroll at "Good Math, Bad Math" has been summarizing sequent calculus and linear logic. Good stuff.

* Lee discusses proportionality in just war theory at "verbum ipsum". Lee's right; there are a lot of things that are said about this in contemporary discussions that don't make much sense.

* "Cliopatria" has a symposium on comparison of British and American imperialism.

* Thomas Williams has an interesting paper (PDF) arguing that the Scotist doctrine of univocity is true and salutary. (h/t: The Prosblogion) Univocity is summarized as:

Univocity: Notwithstanding the irreducible ontological diversity between God and creatures, there are concepts under whose extension both God and creatures fall, so that the corresponding predicate expressions are used with exactly the same sense (in scholastic jargon: have exactly the same significata) in predications about God as in predications about creatures.

Where a hardcore Thomist (or a Thomist-sympathizer) will really stumble is not, as far as I can see, addressed in the paper, though, and that's the assumption that if the concept is the same, "the corresponding predicate expressions are used with exactly the same sense." If the same concept can be predicated in different ways, it isn't clear that they are "used with exactly the same sense." And, in fact, I take it that everyone agrees that the concepts do not apply in the same way to God and creatures -- in God what is signified is transcendental, in creatures categorical; in God unrestricted, in creatures restricted; etc. -- so we seem right back where we originally were in the dispute: are the 'corresponding predicate expressions' really used with the same sense even if the same concept is used in both? Predicates, after all, are not merely concepts, but concepts predicated of subjects in certain ways. But I do think Williams's argument is salutary. Let's have no more bashing of Scotus for such-and-such consequence univocity without taking the trouble to show that it really is a consequence.

* Michael Gilleland has posted yet another translation of the famous ninth-century Irish poem, Pangur Bán. (He links to another post in which he presents three others. The Flower translation of the poem hung in my dorm room for most of my undergraduate years.

* Ocham muses on Tibbles at "Beyond Necessity."

* An interesting discussion is going on at FQI about Kierkegaard and arguments for the existence of God.

Tibbles Again

The original Tibbles argument bled into the comments of another post on another weblog, namely, this discussion of motion and time at "Maverick Philosopher." Instead of picking up the argument as it is continuing in the threads on the Tibbles posts, I'll comment on some comments made there, because I think it offers occasion for some clarifications that seem less easy to make here, given the length of the threads and the number of issues that have to be managed. Ocham summarizes what he sees as the argument in this comment:

The issue is, on the assumption that Tibbles had a tail yesterday, and that he lost it overnight, and now has no tail, what is it that became the cat without a tail? Was it

(a) The part of Tibbles yesterday which had no tail, i.e. Tibbles yesterday minus tail.
(b) The whole of Tibbles, i.e. Tibbles yesterday with tail.

Brandon seems to be saying (rather absurdly in my view) that it is the proper part of Tibbles, i.e. Tibbles-minus-tail that becomes Tibbles today. I say 'rather absurdly' because the proper part, whatever it is, is something that always lacks a tail. So how can it change into precisely what it already is. I maintain, by contrast, that the subject of change is Tibbles himself, i.e. a thing which yesterday had a tail, but became something without a tail.

It should be said that this is not what I intend to convey. In fact, my argument against maximality turns on exactly the point Ocham notes here: Tibbs does not change. On the maximality view, however, Tibbs was originally not a maximal part; therefore Tibbs was not a cat, and now Tibbs, without changing, has become a cat (because Tibbles lost its tail). So the maximality view commits us to saying that things that are not cats can become cats without any intrinsic change. This is due to the fact that maximality requires that all and only maximal parts of cats are cats. Maximality requires us to make the absurd claim that the proper part of Tibbles becomes the cat today.

But it does appear that I've misunderstood Ocham's position, since I took him to be holding the maximality view. The above summary, however, is not maximality at all, but something different. On a maximality view, we have a tension once we recognize that, despite their differences from spatial parts, there are temporal parts. Is the cat the maximal spatial part or the maximal spatiotemporal part? If we accept the former, we have thus committed ourselves to saying that there is a different cat on the mat from moment to moment (now Tibbles, now Tibbs, now something else). If we accept the latter, we have to hold that none of these is a cat at all -- the only cat is the whole of which all of these are merely proper parts. (In something roughly like the sense that the movie, Goonies, is not identifiable with any one scene but is the whole of which all the scenes of the movie are parts.) Either way we get into weird territory.

But Ocham's view is not actually either of these, because, in effect, he posits some subject (let's call it S) which was Tibbles (when it had a tail) and is now Tibbs (when it does not have a tail). So is this a different view from the ones I noted before, or does it fall into one of them? To clarify this, it might be helpful to look at another puzzle Ocham proposes in a later comment. Suppose we take a sentence like:

(A) S was Tibbles and is now Tibbs.

Of this sort of sentence Ocham asks:

In what way does this assert or imply any kind of 'transtemporal identity'? Clearly the pronoun 'he', if it is to make any sense, must refer back to the proper name Tibbles. So this implies an identity, between the object referred to by 'Tibbles', and what is referred to by 'he'. This further implies an identity, if the statement is true, between what satisfies the predicate '- is now lost his tail' and what satisfies '- had a tail'. But how is this identity 'trans temporal'? It's not at all. It's an identity between an object which exists now, an object which once had a tail, and an object which exists now, and has lost a tail.

But while it's true that the identity is between an object that exists now and an object that exists now, this isn't all that (A) implies. For (A) to be true, the object that exists now has to have been an object then (otherwise the object now couldn't be said to have lost a tail). Then it had different properties than it does now -- (A), in fact, gives us one such difference of properties. So for (A) to be true, or even to make any sense, S has to have endured through time, undergoing changes (it is different now than it was), but such that it is the same S. For (A) to be true, S has to have had two mutually exclusive properties. Of course, there is no contradiction here, because it has them at different times. Still there is no (classical) identity between S as it was and S as it now is, even though they are the same S. A relative identity theorist will say that S was identical to Tibbles at that time, and now S is identical to Tibbs; Tibbles and Tibbs are identical qua S, and thus there is a (relative) identity between S as it was and S as it now is. But since Ocham denies relative identity, we see clearly where his position fits: it's a sameness without identity view, at least for temporal parts. Tibbles and Tibbs are the same S, but treating them as the same S is not to treat them as identical (although S is identical to S, of course). And this, I suspect, is why I was completely puzzled by his position; I thought he was putting forward a maximality view, but he kept saying things that suggested sameness without identity.

If we're going to go this route for temporal parts, however, why wouldn't we do the same for spatial parts? Tibbles and Tibbs are the same S, even though what has a tail and what does not have a tail are not identical. There is no need to appeal to maximality and no need to say that only Tibbles is a cat or that Tibbs is not a cat -- in fact, Tibbles is just S considered as a totality of parts, and Tibbs is just S considered without its tail. In all these cases S is the cat, so Tibbles is the cat considered in all its parts, and Tibbs is the cat considered in all its parts except the tail. Tibbles and Tibbs are the same cat, even though they are not identical, just as the cat before its amputation is the same cat as the cat after its amputation, even though they are not identical.

More on Mereology

Alejandro provided a good set of comments to my previous post on mereology. They raise some interesting questions about life, the universe, and everything, so I thought I'd use them to clarify a few things.

I realize of course that the discussion is not about Tibbles, but I am not so sure it is not primarily about the common notion of object –which I perhaps conflated in my post with "common objects" like cats, overlooking things like fusions, indivisibles, etc. This "common notion" involves an object as a "thing", a subject of properties, that can be in relations with other objects, and that (at least if it physical) can be located in space, or in space and time. Isn’t this the idea of object with which mereology works?

This is a good question, since it allows me to make more clear something that should have been made more clear in the previous post. Mereology as such doesn't deal with objects of this sort, except in the sense that objects of this sort always exhibit mereological relations. When we talk about objects in mereology -- it's often avoided, but sometimes not -- we mean something capable of being the value of a variable and able to be the term of a mereological relation. An example of a mereological object that does not have location in space or time is a mathematical set; membership in a set is definitely a mereological relation, although the mereology of mathematical sets involves a lot of complicated issues I don't know much about (e.g., how the relation element-of is connected to the relation part-of). Another example would be parts of a theory (understood as a collection of equations). Another related sort of mereological object is the fusion, which can be completely arbitrary, since you can have a fusion of any a and b. (x is a fusion of elements when x has all the elements in question as parts, and no parts that are not those elements.) For instance, we can have fusion of my nose, your hand, and the equation E=mc². It isn't clear in what context there would be any point in considering this object in particular; but it is a legitimate mereological object. (There are complicated issues about fusions that I'm glossing over here. I'm assuming what's called 'unrestricted composition', which is actually a controversial thesis about what can be admitted as part of a fusion. But note that this is a real debate; whether the unrestricted composition thesis is right or not, mereological principles don't automatically rule out such odd fusions. And this is really a dispute about objects, not about the mereological relation itself.) Further, even things that have no proper parts -- like geometrical points -- seem to be able to exhibit some mereological relations. (A geometrical point, it seems, is capable of the part-of relation, because it is reflexive -- even though the point has no proper parts, it is part of itself.) So the upshot is this: anything capable of being in a mereological relation to anything is fair game for mereology.

Of course, it is true that people doing work in mereology have tended to focus on a lot of common objects. In part this can be explained by the fact that common objects are easily accessible mereological objects; if you put forward a strong mereological thesis, it's easier to see if it's refuted by a common object than to see if it's refuted by Quantum Field Theory. Since mereological relations are general, any mereological object should (in principle) be able to falsify a strong mereological thesis. So the easiest way (again, in principle) to refute such claims is to show that they break down even in the easiest and most accessible cases. Of course, I have to add the 'in principle' qualifications because in practice we run into the same problem with mereology that we run into elsewhere: things don't falsify cleanly, because the theses can't be applied in an isolated way -- you have to make additional assumptions to apply them. Falsification just shows that you've gone wrong somewhere in the whole system of theses and assumptions; on its own, it doesn't tell you where. Sometimes it's easy to figure out what's throwing things off, and sometimes it's not. But in principle, it makes sense for mereologists to pay attention to things like cats.

Of course, the danger from this is that mereologists will only build mereologies for common objects, without regard to other mereological objects. To the extent that this happens (and I think it does happen), something has gone wrong, and I am in agreement with Alejandro. I don't think this always happens, and I don't think it's endemic to the discussion. An area in which the question always needs to be raised, though, is maximality. Maximality tries to resolve mereological paradoxes by appealing to a principle of classification: no proper part of an F is an F. But it isn't clear that this applies to all mereological objects. For instance, if the elements are proper parts of the mathematical set, maximality clearly fails for mathematical sets. (Are elements proper parts? I can think of arguments both for and against; but, as I said, this is an area of mereology with which I am only slightly acquainted, so I don't know what, if any, work has been done on the question in recent years.) If they are not, however, then we can have parts of an object that are not proper parts and are not the object itself. And that's an unusual and interesting thesis that would have to be explored. Does maximality hold up for all fusions? It appears not, because you can have a fusion of fusions. So perhaps maximality is merely a principle for doing mereology in a restricted domain -- e.g., perhaps it only applies to common objects. If that's so, however, we need to be very careful (as maximalists rarely are) to distinguish between what follows strictly from our mereological principles and what follows when our maximality principle is added. And we also need to justify the use of maximality for every field in which we use it. Otherwise, we are beginning to confuse the general issues of mereology with the particular issues of maximal parts, and mereological objects with common objects (which are only one kind of mereological object).

I don’t see how your "Brandon on Wednesday" argument proves relevance of mereology for ethics. If we decided on general, abstract grounds that temporal parts of objects are different objects, so B on Wed is different than B on Thurs, that would make us rewrite most of our ethics and our talk about persons, but it would be just a change of language –it would not affect substantive moral issues. If now we say a man can only be responsible for what he did himself, then we would say that he can be responsible only for what his "temporal predecessors" did, or something like that. There seem to be no problems internal to ethics that require "help" from a more precise mereology, as in my example the problem of abortion needs a more precise concept of personhood.

I'm not entirely sure what 'internal' means here. It's true that ethics is not a fragment of mereology, and that mereology is not a fragment of ethics; so they aren't internal to each other in that way. But the two may overlap by sharing some common concern. And the common concern here is identity or sameness; which is an issue in mereology (for mereological reasons) and an issue in ethics (for ethical reasons). Thus our conclusions in the one case have ramifications for our conclusions in the other case. It's not necessary, of course, for those ramifications to be straightforward; but they are clearly there.

Consider the matter this way. How would we go about making our concept of personhood more precise? One of the questions we would have to ask is, "What is required for two things to be the same person?" We've thereby added the question of sameness. Now, we can means sameness without identity or sameness as identity. If the latter, we can mean either classical identity or relative identity. One thing that will be relevant is whether 'person' is better used for the whole four-dimensional entity through time, or if it is better used for the thing at a time; which gets us into questions like maximality. This raises further question. For instance, do I ever see a whole person, or do I only see the part of the person that exists at a given time? And should we associate responsibilities to, for, and of persons with the whole person or the person-part. (To put it in a crude analogy, for ethical purposes are persons mereologically like lawns, where we can see the whole lawn without having to rely on memory, or like movies, where we never actually see more than a proper part of the movie. It seems clear that we will evaluate differently depending on which we say, because one links the evaluation to what's happening at a given time, and another links it to what has happened over the whole history of the person.) I'm not saying that these would be the driving, make-or-break questions of the ethical inquiry into personhood; but they, or questions like them, do come up.

And the fact that they come up seems sufficient to say that they have practical relevance in the way Alejandro's post suggested. That is, it affects our evaluation of the best way to talk about persons for ethical purposes. For the best way to talk about anything for ethical purposes can't be completely divorced from the best way to talk about them for other purposes; because the best way to talk about them for other purposes will be the best way to talk about them in light of certain facts, and ethics has to take some account of the facts, or it becomes blind.

I also can’t see your justification for the claim that "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'." Why can’t relative identity be posed as a sui generis solution to the particular problem of parts and wholes, without exporting itself to other areas? Again, I don’t see any issues internal to mathematics that could be affected by mereology in its philosophical sense.

Again, it depends a bit on what's meant by 'internal'. It certainly wouldn't change any equations; formal systems would remain as they were. But it would have some effect on how we inquire in interpreting and applying them, because accepting or rejecting relative identity shifts how mathematical equality relates to logical identity. And how one formal system is connected to or not connected to another does seem reasonably 'internal' to mathematics as a discipline, even if it made no practical difference what position we took. It's one of the things mathematicians might be expected to do.

But the more interesting question is why relative identity can't be proposed as a solution for this type of problem and this type alone. And the reason, I think, is this: relative identity theorists try to handle the mereological paradoxes by asserting that they are based on a misunderstanding of identity itself. And the identity relation is perfectly general; it's found just about everywhere. Now, if identity has to be understand in a certain way (as determined by mereological questions raised), this applies to identity in all its uses. There is some room to restrict the significance of this move, in that we can hold that Leibniz's Law (which is the feature of classical identity relative identity theorists most definitely reject) still obtains under some conditions. But this doesn't make the relative identity thesis any less general; it just means that you can add suppositions to relative identity to get classical identity. So the proper understanding even of classical identity will require understanding it as a development of relative identity.

Admittedly, the reason why relative identity is so general is that, while it is relevant to mereology (since it comes up in mereological questions), the feature with which relative identity is concerned is more general than any particularly mereological feature. Identity is not a mereological relation; it holds even in cases where there are no mereological relations. It's just that a complete mereology, like everything else, can't avoid the identity relation, and it just so happens that mereology is a field in which questions about identity can become especially intriguing. But the fact that mereology can provide interesting test-cases for questions about identity is just one of the reasons why mereology is relevant to everything. Of course, it's only indirectly relevant (on this point, at least) -- but this should still be enough for practical relevance in Alejandro's sense.

I agree of course that the existence of cats and their parts is, in a sense, a fact more certain that any physical theory. But it doesn’t follow from this that the language of "objects" and "parts" that philosophers use is the "correct" one, in the sense that their questions are well-framed taken as "deep" questions about reality (instead of "shallow" questions about pedantical refinements of ordinary language). My idea is that the truth-value of ordinary sentences like "there is a cat on the mat" supervenes on fundamental physics (given such-and such distribution of quantum fields or strings or whatever, whether there is or not a cat on the mat follows), so it is a fact that cats exist, but we should not invest too much seriousness into the concepts of "object", etc., which we normally use to express this fact. Our ultimate physical theories may very well use conceptualisations utterly different to these. Just to take an example, I do not think there is anything in Quantum Field Theory corresponding to "objects", with or without "parts". What most people intuitively picture as ultimate physical objects, particles like electrons and quarks, turn out to be at best ambiguously defined and observer-dependent, and at worst completely ficticious entities (in popular formulations the "number of particles present" means only "number of clicks in a particle detector") Sigfpe has an understandable explanation of this. Quantum fields themselves are more likely candidates to be "real" than particles are, but they have purely abstract mathematical definitions. And things are even messier in string theory.

The first part of this quotation is definitely true. The claim would have to be conditional (as I formulated in my last post), not categorical. If the questions are well-framed, they are relevant to deep issues. They might not be well-framed, of course. However, whether or not this is so can only be decided by looking at each of the questions in particular and doing exactly what mereologists are doing -- examining whether they are general or restricted, whether they allow us to make sense of some mereological object, or just lead to confusion; etc. It's possible (and even likely) that mereologists could do better than they are doing, e.g., by paying more attention to uncommon mereological objects, but the point is that there's no other way to determine whether our current mereological questions are well-framed except by mereological inquiry and discussion.

I have (a lot) more to say on this subject, and on this passage in particular; but I'll have to leave it for a later post, given how long this one looks like it will be. (BTW, Alejandro I'm assuming that the post by Sigfpe you had in mind was the one I link to above? The link in the comments wasn't working properly.) Expect more on this.

Slightly More Technical About Tibbles

Alejandro has an excellent post reflecting on my recent post on mereology. It raises the interesting question of how the dispute actually relates to the real world, so I thought I'd post a few musings on it.

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.

Tibbles and Bits

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.)

The Upshot

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?

Note

I'll be gone for a few days. I thought I'd point out a few places where I have been engaging in comments-box discussion, both on issues related to posts I've been preparing:

* Suspicious of Numbers: A Bias Against Probability at "Fides Quarens Intellectum" (I had already intended to say something about Fogelin's A Defense of Hume on Miracles when this discussion began)

* "The Doctrine" - Part 3 at "Trinities" (I want to say something about Tibbles and mereology)

Cheers,
Brandon