Can a Mereological Sum Change its Parts?

This post is an attempt to understand and evaluate Peter van Inwagen's "Can Mereological Sums Change Their Parts," J. Phil. (December 2006), 614-630.  A preprint is available online here.

The Wise Pig and the Brick House: My Take

On Tuesday the Wise Pig  takes delivery of 10,000 bricks.  On the following Friday he completes construction of a house made of exactly these bricks and nothing else.  Call the bricks in question the 'Tuesday bricks.'  I would 'assay' the situation as follows.  On Tuesday there are some unassembled bricks laying about the building site.  By Unrestricted Composition, these bricks compose a classical mereological sum.  Call this sum 'Brick Sum.'  (To save keystrokes I will write 'sum' for 'classical mereological sum.' ) By Uniqueness of Composition, there is exactly one sum that the Tuesday bricks compose.  On Friday, both the Tuesday bricks and their (unique) sum exist.  But as I see it, the Brick House is identical neither to the Tuesday bricks nor to their sum.  Thus I deny that the Brick House is identical to the sum of the things that compose it. I give two arguments for this non-identity.

Nonmodal 'Historical' Argument:  Brick Sum has a property that Brick House does not have, namely the property of existing on Tuesday.  Therefore, by the Indiscernibility of Identicals, Brick Sum is not identical to Brick House.

Modal Argument:  Suppose that the actual world is such that Brick Sum and Brick House always existed, exist now, and always will exist:  every time t is such that both exist at t.  This does not alter the plain fact that the house depends for its existence on the bricks, while the bricks do not depend for their existence on the house.  Thus there are possible worlds in which Brick Sum exists but Brick House does not.  (Note that Brick Sum exists 'automatically' given the existence of the bricks.) These worlds are simply the worlds in which the bricks exist but in an unassembled state.  So Brick Sum has a property that Brick House does not have, namely, the modal property of being possibly such as to exist without composing a house.  Therefore, by the Indiscernibility of Identicals, Brick Sum is not identical to Brick House.

In sum (pardon the pun!), The Brick House is not a mereological sum.  (If it were, it would have existed on Tuesday as a load of bricks, which is absurd.)  This is not to say that there is no sum 'corresponding' to the Brick House: there is.  It is just that this sum -- Brick Sum -- is not identical to Brick House.  So what I am saying implies no rejection of Unrestricted Composition.  The point is rather that a material artifact such as a house cannot be identified with the mereological sum of the things it is made of.  This is because sums abstract or prescind from the mutual relations of parts in virtue of which parts form what we might call  'integral wholes' as opposed to a mere mereological sums.  Unassembled bricks do not a brick house make: you have to assemble them properly.  And the assembly, however you want to assay it, is an added ontological ingredient that escapes consideration by a general purely formal part-whole theory such as classical mereology.

I assume with van Inwagen that Brick House can lose a brick (or gain a brick)  without prejudice to its identity.  But, contra van Inwagen, I do not take this to imply that mereological sums can gain or lose parts.  And this for the simple reason that Brick House and things like it are not identical to sums of the things that compose them.  I would say, pace van Inwagen, that mereological sums can no more gain or lose parts than (mathematical) sets can gain or lose elements.

The Wise Pig and the Brick House: Van Inwagen's Take

I agree with van Inwagen that "The Tuesday bricks are all parts of the Brick House and every part of the Brick House overlaps at least one of the Tuesday bricks." (616-617)  But he takes this obvious truth to imply that " . . . 'a merelogical sum' is the obvious thing to call something of which the Tuesday Bricks are all parts and each of whose parts overlaps at least one of the Tuesday Bricks." (617)  Well, he can call it that but only if he uses 'mereological sum' in a way different that the way it is used in classical mereology.

Now if we acquiesce in van Inwagen's usage, and we grant that things like houses can change their parts, then it follows that mereological sums can change their parts.  But why should we acquiesce in van Inwagen's usage of 'mereological sum'?

Is Everything a Mereological Sum?

As I use 'mereological sum,' not everything is such a sum.  The Brick House is not a sum.  It is no more a sum than it is a set.  There are sums and there are sets, but not everything is a sum just as not everything is a set.  There is a set consisting of the Tuesday Bricks, and there is a singleton set of the Brick House.  But neither of these sets is identical to the Brick House.  Neither of them has anything to fear from the pulmonary exertions of the Big Bad Wolf -- not because they are so strong, but because they are abstract objects removed from the flux and shove of the causal order.  Sums of concreta, unlike sets of concreta,  are themselves concrete -- but the Brick House is not a sum.  Van Inwagen disagrees.  For him, "Everything is a mereological sum." (618)

His argument for this surprising claim is roughly as follows. PvI's presentation is tedious and technical but I think I will not be misrepresenting him if I sum up the gist of it as follows:

1. Everything, whether simple or composite, has parts.  (This is a consequence of the following definition: x is a part of y =_df x is a proper part of y or x = y.  Because everything is self-identical, everything has itself as a part, an improper part to be sure, but a part nonetheless. Therefore:

2. Everything is a mereological sum of its parts.  Therefore:

3. Everything is a mereological sum. Therefore:

4. ". . . mereological sums are not a special sort of object." (622)  In this respect they are unlike sets."'Mereological sum' is not a useful stand-alone general term." (622) 'Set' is.

What's At Issue Here?

I confess to not being clear about what exactly is at issue here.  One could of course use 'mereological sum' in the way that van Inwagen proposes, a way that implies that everything is a mereological sum, and that implies that there is no conceptual confusion in the notion of a mereological sum changing its parts.   But why adopt this usage?  How does it help us in the understanding of material composition?

What am I missing?

 

Comments

[John]

Hi Bill,

First, in the interests of full disclosure, let me say that I haven't read this particular PVI article. That said, here are some thoughts.

Premise 2 of PVI's argument is false. Simples are objects lacking proper parts but, as you say, they are parts of themselves (since parthood is reflexive). But they are not mereological sums of their sole part. If I'm not mistaken, the axiom of Unrestricted Composition says that if the xxs are two or more, then there exists a y such that y is the mereological sum of the xxs. In the case of simples, the xxs are not two or more, so there is no mereological sum to speak of.

Moving to your assessment of the Brick House/Brick Sum situation, I'm not sure I agree. Part of the purpose of Uniqueness of Composition is to rule out the idea that objects having the same parts, but having them in different arrangements, can be distinct objects. Imagine this variant on the Ship of Theseus: suppose that the Ship of Theseus is made out of the xxs, where the xxs are some planks. Suppose further that the Ship is sold to a landlubber who decides to dismantle the ship and construct a house out of its planks. Thus, the house in question is made out of (all and only) the xxs. Intuitively, we want to say that the Ship and the House are distinct. But Uniqueness of Composition denies this. Because the Ship and the House have the same parts, they must be the same object.

A similar analysis would surely be applied to your distinction between Brick House and Brick Sum. If the xxs compose Brick Sum, and Brick House is made out of (all and only) the xxs, then by Uniqueness, Brick House = Brick Sum. This is why I think your Non-Modal Historical Argument is problematic: it already assumes that Brick Sum is not identical to Brick House; that is, it already assumes that Uniqueness of Composition is false.

In general, if you want the arrangement of the xxs to constitute an additional ontological ingredient, it seems to me that you're going to have to reject Uniqueness of Composition. I myself am inclined to think that Uniqueness is false. My point is simply that your view isn't going to hold in classical extensional mereology, given that (I think) your view must reject one of the axioms of that system (viz., Uniqueness).

Maybe I'm not understanding your view. I am not certain I follow your comparison of sums with sets, for instance. I also am not sure what you mean when you say that PVI is using 'mereological sum' in a different way than it is used in classical extensional mereology. What he said sounded fine to me. But again, maybe I'm not understanding.

[Bill Vallicella]

Thanks John, but I think you need to read the paper to see what PvI is up to. I linked to an online version

(2) is not a premise but a conclusion from (1). Intuitively, if everything is a part of itself, then a simple is a part of itself in which case it is a whole of itself as part.

Do you really want to say the Brick House exists on Tuesday? That makes no sense to me at all.

[John]

I'll try to take a look at the paper, but I've got quite a bit of reading and writing for class going on this coming week, so it's unlikely that I'll be able to take a look at it. Which is a shame, because I clearly must be failing to see what PVI is doing. As far as I can see, (2) simply doesn't follow from (1). Everything is indeed a part of itself. So a simple is a part of itself. But it does not follow that the simple is a mereological sum of itself. Again, it seems to me that this is because the very notion of a mereological sum is tied up with the idea of there being TWO objects which come together to compose a sum. Thus, in his SEP article on Mereology, Achille Varzi writes: "Conditions on composition are many. Beginning with the weakest, one may consider a principle to the effect that any PAIR of suitably related entities must underlap, i.e., have an upper bound...A somewhat stronger condition would be to require that any PAIR of suitably related entities must have a minimal underlapper—something composed exactly of their parts and nothing else. This requirement is sometimes stated by saying that any suitable PAIR must have a mereological “sum”, or “fusion”, though it is not immediately obvious how this requirement should be formulated in the formal language" (my emphasis). One way to get at what I'm thinking is this: summation is an operation of sorts that is "performed" (I speak metaphorically here) on two (or more) entities. To speak of a simple being a mereological sum of itself is just confusing. In this case, then, it seems to me that PVI is using 'mereological sum' in too broad a way. You also object to his use of the term, but for different reasons. But perhaps we can still call this an agreement between us.

As for the Brick House, I don't want to say that it has to exist on Tuesday. But it seems to me that anyone who accepts classical extensional mereology HAS to say that, given Unrestricted Comp. and Uniqueness. I don't accept classical extensional mereology, and so I don't have to (and do not) say that Brick House exists on Tuesday. My concern is just that, if you accept both of those axioms, then you're committed to saying that Brick House exists on Tuesday. It's just arranged in a different way on Tuesday than it is on Friday.

[Boram Lee]

Bill, John's response to your discernibility arguments (as I shall call them) seems promising. It is the line that Varzi takes and develops in defending uniqueness of composition (or as he calls it, "extensionality", on analogy with a similar axiom in set theory), "Mereological Commitments", section 3.

One could also take "Brick Sum" to be like a substance sortal, and "Brick House" to be like a phase sortal. Then Brick Sum and Brick House have different persistence conditions, but that's ok. So does a person and a child, but the child is nevertheless a person. Likewise, it could be argued, so can a brick house be a brick sum. (In terms of Varzi's strategy, this could be seen as a de dicto reading of your arguments that seem to block their conclusions.)

[Bill Vallicella]

John,

Thanks for the discussion. PvI's paper is tedious as hell, and I appreciate that you don't have time to read it. But you can skim through it quickly. If you do so, you will notice that the first section is entitled "Everything is a mereological sum." Yep, that is what our iconoclast is arguing!

Now if everything is a mereological sum, then simples are mereological sums. Now every simple has a part: itself. So a simple is a mereological sum of itself.

Now it may be that in standard mereology summation is an operation the operands of which must be at least 2 in number. But PvI is doing his own thing which ought to be obvious from the fact that he is arguing the wildly contrarian thesis that sums can change their members.

Maybe I'm dense, but WHY must I say that the Brick House exists on Tuesday if I accept class. ext. mereology with Unr Comp and Uniqueness of Comp? Why can't I say that class. mereology is true of sums but not true of houses? What can't I accept mereology as a formal theory of those wholes called sums while denying that many of the wholes we encounter in the world are sums?

After all, you accept set theory, but you are not inclined to say that your house is a set -- or at least I hope you are not.

[John]

Bill,

I'll refrain from saying any more about PVI's paper since I haven't read it, except to say this: I'm not sure that I understand, or agree with, his claim that everything is a mereological sum, for reasons I've already discussed. But without having read his paper, I shouldn't suppose that what I've said actually counts against what PVI has said, and I won't say any more about it.

The reason I think you have to say that the Brick House exists on Tuesday (given your acceptance of classical extensional mereology) is that it's not clear on what grounds you distinguish Brick House from Brick Sum. You can't say that it's because Brick House has its parts arranged in a certain way, whereas on Tuesday, when there was only Brick Sum, those parts were not arranged in any particular way (or, if you like, were arranged in some other way). Uniqueness of Composition says that arrangement of parts is irrelevant; if two objects have the same parts (as Brick House and Brick Sum do), then those two objects are in fact identical. Until you give me a story about what it is that makes Brick House distinct from Brick Sum - where this story has nothing to do with the arrangement of parts - then I say you are committed to the claim that Brick House exists on Tuesday.

If I understand your Nonmodal 'Historical' Argument, it says that Brick Sum has a property that Brick House lacks (namely, that of existing on Tuesday), and so the two must be distinct. But this argument already presupposes a rejection of Uniqueness, because given Uniqueness, Brick House DOES have the property of existing on Tuesday.

It might be that your Modal Argument succeeds; I haven't given it a close reading. If it does, then Brick House and Brick Sum are distinct after all.

All of this being said, I don't want to make it seem that I am opposed in general to your proposals. I'm inclined to think favorably of ontological views which leave classical extensional mereology alone, and add some other ontological ingredient. This is why I think some of Schaffer's work on grounding is promising. My only worry about your position is that I'm unclear on what the additional ontological ingredient is. But I AM sympathetic to your project here.

In any case, in the future I'll try to avoid commenting on posts of yours referencing articles I've not read. I only just discovered your blog recently, and as it happens, many of your recent posts are concerned with one of my favorite areas of philosophy, and I simply wanted to join the discussion.

[Bill Vallicella]

John,

This is a good discussion. I'm grateful for your comments. You write,
>> Uniqueness of Composition says that arrangement of parts is irrelevant; if two objects have the same parts (as Brick House and Brick Sum do), then those two objects are in fact identical.<<

You are right that the principle in question implies that arrangement of parts is irrelevant: Brick Sum is the same sum whether the bricks are piled randomly or made into a wall or made into a house fit for a pig and proof against the pulmonary exertions of a certain wolf. So I agree completely with the first independent clause of the above quoted sentence. But in the second clause you slide over to talk of 'objects' whatever they are. You seem to be suggesting that the second clause is a logical consequence of the first. But it isn't. You are assuming that Brick House is a sum. But you can't assume that. If the whole question is whether or not Brick House is a sum, you cannot answer that question by simply assuming that it is.

And don't forget: I gave two arguments why Brick House is not identical to Brick Sum. Are you saying they beg the question? But then you too are begging the question by just assuming that Brick House is identical to a sum.

[Bill Vallicella]

Boram,

You makes some interesting suggestions, e.g. >>One could also take "Brick Sum" to be like a substance sortal, and "Brick House" to be like a phase sortal. Then Brick Sum and Brick House have different persistence conditions, but that's ok. So does a person and a child, but the child is nevertheless a person.<<

Consider Boy Tom and Human Tom. You couldn't validly argue that they are not the same human being on the ground that BT exists only 20 years while HT exists for 70 years. Both are human beings and the same human being despite the fact that boyhood is but a phase in a male human being's life.

But while a boy is uncontroversially a human being as a matter of analytic necessity, Brick House is (arguably) not a mereological sum. So the analogy fails. This failure is also indicated by the implausibility of thinking of a mereological sum as a substance -- it is the exact opposite lacking as it does any integral unity.

[Peter Bruin]

Why should it be the case that if a house is built from the Tuesday Bricks and nothing else, the bricks themselves are the only parts of the house? It appears that building the house does not only make the house come into existence, but also a number of walls, for example. It seems reasonable to say that these are parts of the house. An argument for this is the following: We could first have built the walls separately and then put them together to form the house. Although the end result would be the same, we could then say that the house is constructed from the walls instead of directly from the bricks.

In fact, strictly following this line of thought makes "The whole is the sum of its parts" sound trivially true. More precisely, it follows from two assumptions: (1) the whole is a part of itself; (2) a sum of objects O_i is an object O that has all the O_i as parts and such that any object P having all the O_i as parts also has O as part. (Note that "sum" in this sense is more analogous to the set-theoretic concept of "union" rather than that of "disjoint union".)

This would mean that Brick House is indeed the sum of its parts. Yet it is different from Brick Sum, which is only a proper part of Brick House. This is consistent because the bricks are not the only parts of Brick House.

[John]

Hi Bill,

I agree that this is a good discussion. I'm grateful for the opportunity to discuss mereology with someone.

Here is my assessment of the situation. On Tuesday, we have some xxs lying about. By Unrestricted Composition, they compose a sum. By Uniqueness of Composition, the xxs in question compose a unique sum. You call this sum 'Brick Sum'. I'm with you.

Now, on Friday, those xxs have been arranged into the shape of a house. By Uniqueness of Composition, the sum on Friday is identical to Brick Sum. So there is a classical mereological sum on Friday, Brick Sum, which is arranged in the shape of a house. Okay.

Here's where I get lost. You say that on Friday, there is an object, Brick House, which is distinct from Brick Sum. There seem to me to be two ways one could say this. First, you could say that Brick House is a sum having different parts than Brick Sum, so that by the Indiscernibility of Identicals, Brick House is not identical to Brick Sum. Clearly, this is not what you say. Second, you could say that Brick House and Brick Sum are both sums, and share all the same parts, but reject Uniqueness of Composition. Two objects having the same parts can nevertheless be distinct. Clearly, again, this is not what you say.

Instead, you opt for a third route: you say that Brick House is NOT a sum. It is an object which, you say, "corresponds" to a sum (namely, Brick Sum), but is not identical to it. What does it mean for an object to "correspond" to a mereological sum? One thing you might be saying is that Brick House, unlike Brick Sum, is something over and above its parts. Brick House "corresponds" to Brick Sum, but adds an additional ontological ingredient, such as its arrangement into a house. To further support this position, you adduce both a Historical and a Modal argument to show that Brick House and Brick Sum have different properties, and are thus distinct.

I hope this is a fair assessment of your position. I confess to simply not understanding it. First point: I don't think it is question-begging on my part to say that arrangement in a particular fashion cannot be an "additional ontological ingredient", given your adherence to Uniqueness. You are drawing an ontological distinction between objects and sums, a distinction which you say cannot be found in classical extensional mereology. But classical extensional mereology does say something about the arrangement of parts, namely, that it makes no difference. So, I say, the arrangement of the xxs into the shape of a house cannot ground an ontological distinction between objects and sums. If that's right, then your distinction is unfounded, which means we have no choice but to treat Brick House as a sum having all and only the same parts as Brick House. By classical extensional mereology, Brick House = Brick Sum. And this is why I think both your Historical and Modal arguments don't succeed: they presuppose the ontological distinction between objects and sums for which you have not provided justification.

Now, I anticipate that you will respond in the following way: you will grant, of course, that Uniqueness holds for sums. As evidence, you will cite the fact that you say Brick Sum on Friday is the same sum as Brick Sum on Tuesday, despite the difference in part-arrangement. And so you will accuse me of begging the question against you, on the grounds that I have assumed that Brick House is a sum (and thus must be identical to Brick Sum), as opposed to an object. Thus, you say, the very point at issue is whether or not Brick House is a sum.

I respond: Yes, the point at issue is indeed whether or not Brick House is a sum. But the burden of proof is upon you to show that it isn't. It has all the same parts as Brick Sum. Why isn't it just a sum, and thus identical to Brick Sum? It cannot have anything to do with the arrangement of its parts, for Brick Sum, on Friday, has its parts arranged in the very same way. So you must think it is because of your Historical and Modal arguments. But, to take the Historical argument, you presuppose that Brick House does not exist on Tuesday. That supposition REQUIRES that Brick House not be a sum governed by the axioms of classical extensional mereology, because if it WERE such a sum, it would exist on Tuesday (because it would be identical to Brick Sum, which DOES exist on Tuesday). So your argument has already built in an ontological distinction between objects and sums, a distinction not captured by mereology, but for which you have not provided any justification.

Here's one way to go from here. Take the second route of distinguishing Brick House and Brick Sum that I mentioned earlier, and reject Uniqueness of Composition.

Or here's another way to go: admit that everything, including Brick House, is a mereological sum, but claim that there is an ontological distinction to be drawn between sums. I think this is much closer to the spirit of what you write. For, in your Modal argument, you write that while the house DEPENDS for its existence on the existence of its parts, its parts do not DEPEND for their existence on the house. This suggests drawing an ontological distinction between two sums by appealing to relations of ONTOLOGICAL DEPENDENCE between the whole and the parts. Jonathan Schaffer takes a similar line in his (2009), as I've mentioned before, though he would describe the relations of dependence differently. Here's what he'd say:

Brick Sum is a "mere aggregate" - though, to be sure, a mereological sum - in that its existence is GROUNDED by the existence of its parts. By contrast, Brick House is an "integral whole" - though, again, a mereological sum! - in that its existence GROUNDS the existence of its parts. Here, grounding is to be understood as an irreflexive, transitive, and asymmetric relation.

Schaffer clearly thinks differently than you do about the grounding relations between parts and wholes, but the spirit is there: everything's a mereological sum, but only some sums are integral wholes. And, crucially, his distinction, like yours, is not captured by classical extensional mereology.

Anyhow, I hope all of that makes sense. I suspect that we're at something of an impasse, for you think I am begging the question against you, and I think you are begging the question against me. But maybe the foregoing can at least elucidate our differences.

[Bill Vallicella]

Thanks for that, John. I will try to respond tomorrow. But now, to bed!

[David Brightly]

I'd like to concentrate on PVI's apparent contention that there are (at least) two concepts of mereological sum in currency.

The first, simpler, concept we might call an 'abstract mereological sum' (AMS). 'x is an AMS' means, by definition, 'x has parts'. And 'x is an AMS of parts p1, p2,...' just means 'x has exactly the parts p1, p2,...' . This is PVI's preferred usage. Since everything appears to have parts, everything is an AMS. Hence he says '[A]MS is not a useful stand-alone general term'. It picks out no special kind of object. And since objects can change their parts it seems that an AMS can change its parts. The second, more interesting and more problematic concept we might call a 'concrete mereological sum' (CMS). Starting with parts p1, p2,... Unrestricted Composition says there is a CMS whose parts are precisely p1, p2,... We might denote this as CMS(p1, p2,...). In contrast with an AMS a CMS does appear to be a special kind of object. If those parts are simples then a CMS is a compositionally unchanging object. It has, by definition, always the same parts. Now suppose at time t an ordinary object x has simple parts p1, p2,... We might say that at time t object x 'coincides' with CMS(p1, p2,...). Since x can change its parts x will be coincident with a different CMS at different times. We can associate with x a 'compositional history', a sequence [s1, s2,...] of CM sums such that x coincides with sn at time n.

How does this fit with Bill's interpretation? I think it's clear that Bill takes 'mereological sum' to mean CMS. Denoting the Tuesday bricks b1, b2,... he accepts BrickSum=CMS(b1, b2,...). Although BrickSum coincides with BrickHouse at some time it's clear that BrickHouse!=BrickSum, and indeed BrickHouse is not identical to any CMS. Bill says "This is because sums abstract or prescind from the mutual relations of parts in virtue of which parts form what we might call  'integral wholes' as opposed to a mere mereological sums. " I think it's simpler than this. Most ordinary objects cannot be identified with a CMS because the former undergo change of parts and the latter do not. The compositional history of BrickHouse clearly differs from that of BrickSum.

A CMS is a really strange object. At the start of PVI's story BrickSum is in neat stacks on pallets as delivered to Wise Pig's site by his builder's merchant. A bit is detached and comes to rest on WP's newly laid footings. Then another bit. Gradually the neatly stacked part of BriskSum shrinks and a wall-like, then house-like part grows. Finally, the stacked part of BrickSum vanishes. BrickSum now looks exactly like BrickHouse and remains so for a while. Then one of the parts falls out and lies in the garden and its vacated place is taken by a wholly new part. BrickSum once more looks unlike BrickHouse. It's interesting to draw spacetime diagrams showing the worldlines of the parts. Then one quite literally sees that BrickSum and BrickHouse are not identical. But perhaps this is not so strange after all. Consider a living cell isolated from the rest of the world undergoing cell division. With our view of ordinary objects we are forced to describe this as one object becoming two objects. But we could view it as a CMS rearranging its disposition in space.

[Bill Vallicella]

Hi David,

>>And 'x is an AMS of parts p1, p2,...' just means 'x has exactly the parts p1, p2,...' . << As you know, parthood, unlike elementhood in set theory, is transitive, so I wonder if you want the word "exactly." If p1 is one cat and p2 is a second, then a sum of the two cats will also have as parts the parts of the cats, and so on.

>>I think it's simpler than this. Most ordinary objects cannot be identified with a CMS because the former undergo change of parts and the latter do not. The compositional history of BrickHouse clearly differs from that of BrickSum.<<

I agree.

I like your final paragraph, though I don't understand the bit about the cell. The main thing is that we agree that Brick House isnot identical to Brick House. I now take the following step: Given that Brick House is a whole of parts, which it obviously is, it follows that that there are wholes that are not sums.