Sunday, July 16, 2006

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.