Varzi, Sums, and Wholes

Achille C. Varzi, "The Extensionality of Parthood and Composition," The Philosophical Quarterly 58 (2008), p. 109:

Suppose we have a house made of Tinkertoy pieces.  Then the house qualifies as a sum of those pieces: each piece is part of the house and each part of the house overlaps at least one of the pieces . . . . Are there other things that qualify as the sums of those pieces?  UC says there aren't; the house is the only candidate: it is the sum of those pieces.

UC is Uniqueness of Composition: 

UC  If x and y are sums of the same things, then x = y,

where

(1) x is a sum of the zs =_df The zs are all parts of x and every part of x has a part in common with at least one ofthe zs. 

Perhaps commenter John, who knows some mereology and the relevant literature on material composition, can help me understand this.  What I don't understand is what entitles Varzi to assume that the Tinkertoy house -- 'TTH' to give it a name -- is identical to a classical mereological sum.  I do not deny that there is a sum of the parts of TTH.  And I do not doubt that this sum is unique.  Let us name this sum 'TTS.'  (I assume that names are Kripkean rigid designators.)  What I do not understand is the justification of the assumption, made near the beginning of his paper, of the identity of TTH and TTS.  TTH is of course a whole of parts.  But it doesn't straightaway follow that TTH is a sum of parts.

Please note that 'sum' is a technical term, one whose meaning is exactly the meaning it derives from the definitions and axioms of classical mereology.  'Whole' is a term of ordinary language whose meaning depends on context.  It seems to me that one cannot just assume that a given whole of parts is identical to a mereological sum of those same parts.

I am not denying that it might be useful for some purposes  to think of material objects like TTH as sums, but by the same token it might be useful to think of material objects as (mathematical)  sets of their parts.  But surely it would be a mistake to identify TTH with a set of its parts.  For one thing, sets are abstract while material objects are concrete.  For another, proper parthood is transitive while set-theoretic elementhood is not transitive. 

Of course, sums are not sets.  A sum of concreta is itself concrete whereas a set of concreta is itself abstract.  My point is that, just as we cannot assume that that TTH is identical to a set, we cannot assume that TTH is identical to a sum.

What is the 'dialectical situation' when it comes to the dispute between those who maintain that TTH = TTS and those who deny this identity?

It seems to me that the burden of proof rests on those who, like Varzi, identify material objects like TTH with sums especially given the arguments against the identity.  Here is one argument. (a) Taking TTH apart would destroy it, (b) but would not destroy TTS.  Therefore, (c) TTH is not identical to TTS.  This argument relies on the wholly unproblematic Indiscernibility of Identicals as a tacit premise:  If x = y, then whatever is true of x is true of y, and vice versa.  Because something is true of TTH -- namely, that taking it apart would destroy it -- that is not true of TTS, TTH cannot be identical to TTS.

The simplicity and clarity of modal discernibility arguments like this one cast grave doubt on the opening assumption that TTH is a sum.  I am not saying that Varzi and Co. have no response to the argument; they do.  My point is that their response comes too late dialectically speaking.  If you know what a sum is, you know that the identity is dubious from the outset: the discernibility arguments merely make the dubiousness explicit. Responding to these arguments strikes me as too little too late; what the identity theorist needs to do is justify his intitial assumption as soon as he makes it.

My main question, then, is this.  What justifies the initial assumption that material particulars such as Tinkertoy houses are mereological sums?  It cannot be that they are wholes of parts, for a whole needn't be a sum.  TTH is a whole but it is not a sum.  It is not a sum because a sum is a collection that is neutral with respect to the arrangement or interrelation of its parts, whereas it is essential to TTH that its parts be arranged house-wise.

 

Comments

[John]

Perhaps this passage from Kathrin Koslicki's book "The Structure of Objects" will help make clear why Varzi et. al. take themselves to be entitled to that assumption:

"But suppose, for the moment, that Lewis is right in thinking that standard composition is the only genuinely mereological form of composition there is. We began by taking it as an intuitive datum that ordinary material objects are wholes composed of parts; and everyone except the Nihilist (who believes that nothing has proper parts) will concur. If we combine this intuitive datum with Lewis' thesis that standard composition is the only genuinely mereological notion of composition, then we of course get the result that ordinary material objects must be wholes in the standard sense of composition, i.e., that ordinary material objects must be mereological sums" (p. 22).

Briefly, Lewis is suspicious from the start of your distinction between sums and wholes. Our only understanding of wholes, he thinks, is of mereological wholes. And our only understanding of mereological wholes is that provided by classical extensional mereology. So our only understanding of mereological wholes comes from classical extensional mereology. Therefore, all wholes are sums.

In response to your point about the manner of arrangement of the parts, he would presumably claim that the notion of arrangement is implicit in mereology. Certain supplementation principles that secure extensionality, for instance, rule out the idea that arrangement matters for mereological sums. So "arrangement" is, in a certain sense, a mereological term. As such, I think Lewis would object to the idea that we can distinguish between sums and wholes by pointing out that how the parts are arranged matters for the latter but not the former. And so, again, he would say our only understanding of wholes is provided by classical extensional mereology. So: all wholes are sums.

Lots of people disagree with this of course. Or, at least, there is a growing chorus of dissenters, most of whom are inspired in some way or another by Aristotle (like Koslicki herself, or Jonathan Schaffer). But the foregoing is simply an effort to make clear where Varzi et. al. are coming from.

[David Brightly]

Hello Bill,

I've been struggling to understand formulations such as the one you quote from Varzi, so I'd be interested to know if the following interpretation makes sense.

Let's read 'sum of the zs', or 'Sum(z1,...)' as a concept. Sum is a function mapping a plurality of concrete parts onto a concept. Let's read 'x is a sum of the zs' as 'x is an instance of Sum(z1,...)'. Unrestricted Composition assures us that there is at least one instance of Sum(z1,...) and Uniqueness of Composition assures us that there is no more than one. Sum(z1,...) behaves like a definite description such as 'Tallest man in Texas' except that a referent is always guaranteed. So, taking the zs to be parts that comprise TTH, we are entitled to define TTS as the unique x such that x is an instance of Sum(z1,...). But this is just TTH, for the zs are indeed parts of TTH and every part of TTH has a part in common with at least one of the zs, since the zs comprise TTH. Hence TTS=TTH.

But this *abstract* reading of 'sum of zs' is rather different from the *concrete* one we have been using in recent threads. There is scope for vacillation.

[Bill Vallicella]

Thank you for that, John. Koslicki's last sentence, however, strikes me as a non sequitur despite her "of course." Or she is equivocating on 'mereological'? Does she intend this term in the sense of mereology? Or does she mean it in the sense of our ordinary understanding of wholes and parts?

Lewis position has nothing to recommend it that I can see. If we didn't have an intuitive understanding of wholes and parts to begin with, we wouldn't be in a position to work out mereology, i.e., formulate definitions and axioms. The latter are an attempt to make precise our intuitive understanding of part-whole relations. So of course there is distinction between wholes and sums.

[Bill Vallicella]

David,

But a sum is not a concept. A sum is an individual. You and your cat form a sum. This sum is not a concept but an individual. So I've lost you right at the get-go.

[David Brightly]

Bill,

Certainly. What I am suggesting is that the term 'sum of the zs' behaves like the concept term 'dog'. Thus the individual TTH can be a sum of the zs just as the individual Fido can be a dog.

[David Brightly]

Perhaps a diagram will make my meaning clearer.

Sum is a function from power set of Object to Concept
InstanceOf is a relation over Object and Concept
comprise is a function from power set of Object to Object

          -----> Sum(zs)                           Sum:  P(Object) --> Concept
         /           |                      InstanceOf:  Object <--> Concept
        /            |                        comprise:  P(Object) --> Object
       /             |
     Sum         InstanceOf
     /               |                      comprise (zs) InstanceOf Sum (zs)
    /                |
   /                 |            
 zs  -------------> TTH
        comprise

The object the zs comprise is an (the) instance of the concept sum of the zs.

[David Brightly]

Would it also help if I were to suggest that, on the evidence of your quote, Varzi is in the PVI camp with regard to interpretation of mereological talk. He is not saying that a mereological sum is a radically different kind of object that, for example, transforms from cube-shaped to house-shaped. I think he is just introducing a new bit of language or conceptual machinery. He is proposing that whenever some parts zs and an object x are in a certain relation then we should say that the x is a 'mereological sum of the zs'. That relation holds whenever the zs are all parts of x and every part of x has a part in common with at least one of the zs. Or, in my terminology, the zs comprise x. As Halmos says of general set theory, it's 'pretty trivial stuff really'. With this stipulation as to how to use the term 'sum of zs' I think it does follow straightaway that TTH is a sum of parts. Forgive me if I'm wrong, but are you not bringing the other, concrete, sense of sum to the Varzi paper, and inevitably finding a clash?

[Bill Vallicella]

David,

Sorry to misunderstand you. Just as Fido is a dog, TTH is a sum. Whereas what I am saying is that TTH is not a sum. TTS is a sum the parts of which are the parts of TTH; but TTH is not a sum.

For you, TTH is a sum with the further qualification of being such that its parts are arranged house-wise. (Is that right?) For me, TTH is not a sum. For me it makes no sense to say of an item that it is both a sum and a house. For then what one would be saying is that it is both such that the arrangement of its parts is irrelevant to its being what it is, AND that the arrangement of its parts is relevant to its being what it is. And that's a contradiction.

Fido's being a dog does not exclude Fido's being an Irish Setter, a young Irish Setter, etc. But TTS's being a sum does exclude it from being a house or a fort or any other integral whole. As I see it, a sum can no more be a house than a set can be a house.

Is the difference between our positions clear?

[David Brightly]

Hello Bill,

Yes, I agree that the difference between your position and the view I'm taking in this thread is clear. What I'm hoping to persuade you is that this is exactly the difference of interpretation that PVI points up in his 'Can Mereological Sums Change Their Parts' paper that you referred us to earlier. My comment here was aimed at showing that, under the definition of sum you have quoted from Varzi we can define

TTH=the object the zs comprise (going along the bottom of the diagram),
TTS=the unique instance of the concept sum of the zs (going round the other three legs),

and since the two right hand sides are equal it follows that TTH=TTS.

My fear is that we risk Straw Man arguments if we don't recognise this radical difference of interpretation.

[Bill Vallicella]

David,

I think you are right about PvI, though I am not convinced that TTH = TTS. So expect another post!

Thanks for your comments, and that is a nifty diagram!

[David Brightly]

I'll look forward to your further post. Glad you liked the diagram!