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When do two objects touch?

October 7, 2008

I’ve been reading through Casati and Varzi’s “Parts and Places” recently. It’s a really fun book – I highly recommend it to anyone interested in the metaphysics of parthood and location.

Anyway, I have come to this very curious passage in their discussion of mereotopology. It has to do with the first “substantial” bridge principle relating parthood and connection (i.e. when two objects “touch”, or “kiss”.) They call it C8, which they write in their notation as:

  • Cxy \rightarrow \exists z(SCz \wedge Ozy \wedge Ozx \wedge \forall w(Pwz \rightarrow Owx \vee Owy))

which translates as: if x and y are touching, then there must be a selfconnected object which overlaps both x and y and is completely within their sum (i.e. all of its parts overlap one or the other.) A selfconnected object is one that can’t be divided into two objects which don’t touch (so, like what a bikini isn’t, but a mankini is.) The first thing that threw me was that they said the Cantor bar is an object such that none of its parts is selfconnected (thus by C8 cannot touch anything.) The Cantor bar, if you don’t already know, is constructed by taking a finite line, removing the middle third, then removing the middle thirds of the remaining two bars, et cetera et cetera. The limit of this process is just the intersection of all the stages. Now surely, if this is what they mean by the Cantor bar, this is just a fusion of points, each of which is selfconnected. So maybe I’m not understanding the example. But I’m actually finding it difficult to come up with an example of an object, none of whose parts are selfconnected. The best I can think of are slightly esoteric: (1) I guess a bilocated mereological atom, or (2) an extended simple which isn’t selfconnected. [I’m assuming that x touches y just in case x’s location touches y’s location. Or even stronger: x touches y in virtue of x’s location touching y’s location. Note how (2) is different from (1) in that (2) postulates an object that is singularly located at a disconnected location, whereas (1) is completely located at two selfconnected locations.]

The other curious thing was I can’t think of any reason whatsoever for Casati and Varzi to accept C8. For their preferred interpretation of ‘touching’ in ordinary space, is that x touches y if and only if either x’s closure overlaps y, or y’s closure overlaps x. But if you let x be the selfconnected block below, and let y be the fusion of the bars, then x and y are touching by this definition. However, clearly, there is no selfconnected object running from one to the other.

Maybe we should interpret ‘touches’ as path connected – i.e. that we can get from x to y by drawing a continuous line from some point in x to some point in y, without ever leaving the two. But I’m still not quite sure if that’s right.

This is a sin(1/x) type curve. The fusion of the block and the sin wave appears to be self connected, however there cannot be a continuous line running from the block to the curve. This violates the converse of C8, which I take it is supposed to be true too (I think they took this direction to be obvious.) So path connected doesn’t seem to be the right definition of ‘touching’ either.

A couple of things to note about this example. Firstly, if the amplitude of the sine wave gets smaller as it approaches the block, then they are path connected, because you can draw a continuous line between them. But, intuitively, there shouldn’t be a difference between the two examples – they should both count as touching each other.

Here’s something I find a bit weird. Suppose further that the amplitudes of the peaks, starting from the left to the right, is 1/2, 1/4, 1/8 … 1/(2^n) … . Then the informal explanation of path connectedness applies: you can draw a line between the two without taking your pencil off the page. But if the amplitudes of the peaks go 1/2, 1/3, 1/4, 1/5 … 1/n … etc. then the informal explanation breaks down. The distance you have to travel to get from the curve to the block is infinite! Even if you have a giant novelty pencil that extends off to infinity in one direction, your pencil will still run out before you get to anywhere properly inside the block. Yet they are still path connected because this line is still continuous.

One more cool fact. Another plausible definition of ‘touching’ is that x touches y iff their closures overlap. However, Frank Arntzenius pointed out to me that on this interpretation of touching, the four colour theorem comes out false! (Just draw the sin like curves above with a rainbow paintbrush to see how.)

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29 comments

  1. Wow. Topology. Exciting! I may need to check this book out. Terminology is a little bit different than I’m used to, but I’m a little confused about x’s closure “overlaps” y or y’s closure “overlaps” x. I’m guessing that that just means the intersection is nonempty.

    I’m not sure if this definition is philosophically sound, since as you point out two objects touching does not imply connected in either of the two standard senses. But I think this issue solely stems from the fact that you have embedded them in the standard plane and interpret from there. If you treat in your first example the fusion of bars as an abstract entity in itself, then I think it is selfconnected.

    On the other hand, by your interpretation of C8, I think what they mean is that if you center your first example on the plane so that (0,0) is where your fusion of bars is converging to, then the line segment inside the block and ending at (0,0) might do the trick.

    I really have to run, but I’ll think about this more tonight when I have more time. It is all very interesting.


  2. Sorry – I’m flitting between mereological and set theoretic terminology. What I meant was “x’s closures intersection with y is non empty, or y’s closures intersection with x is non empty”. But I think that’s how you interpreted me, right?

    I don’t think I understood what you were saying here: “two objects touching does not imply connected in either of the two standard senses.'” What are the tow standard senses? My conclusion from all this is that it is hard to say what the standard definition of ‘touching’ should be.

    “If you treat in your first example the fusion of bars as an abstract entity in itself, then I think it is selfconnected.” If I’m understanding this right, then it’s a pretty weird way of thinking about things. I think I said somewhere in the text that x touches y *in virtue* of x’s location touching y’s location. I really don’t know how else to think about things in contact unless it is at least equivalent to some property of the spatio-temporal regions they occupy.


  3. It’s been a couple of months since I read Parts and Places, so I don’t remember the context of this example, but could it be that they are simply *not* viewing the Cantor bar as something composed of points, but rather as something indefinitely divisible – i.e. they assuming or at least allowing for atomlessness? (So, expressed in traditional language, parts of the Cantor bar would be exactly the intersections of non-trivial intervals with the Cantor bar.)

    It seems plausible to me, given that one of the possible motivations for doing mereotopology is not having to view objects as sets of points. At any rate, even if this is not what the authors meant, it provides an example of an object without self-connected parts.


  4. If we have a fusion axiom, we have to admit the fusions of such intersections, too, which seems to introduce a further problem: if we want to maintain atomlessness we shouldn’t be able to express the point 1/3, for example, as the complement of the fusions of parts of the bar which are wholly above or wholly below 1/3. As long as we don’t admit such a predicate, though, I think we shouldn’t have problems even with unrestricted fusion and complementation.


  5. Sorry. This morning I didn’t read this all that carefully. I find it rather strange that they consider points to be self connected. You seem to run into paradoxes (or at least unwanted theorems) in this case:

    Thm: Everything consists of a fusion of selfconnected parts. (I know absolutely nothing about mereotopology, so I think those terms work). Proof: Just look at the object as a fusion of the points that make it up.

    My guess is that by “part” of the Cantor bar, they mean the intersection of any interval (no matter how small) with the Cantor bar. That way you pick up more than one point, none of which touch each other. i.e. A part is any subset of the Cantor bar that is not a single point.

    I think I’ve just confused myself more, though. On another note, C8 definitely sounds like a path connected condition. A fix to your counterexamples would be to make C8 read: if x and y are touching, then there must be a selfconnected object which overlaps both x and y and is completely within their closure. This may have been what they meant?


  6. Hi Preno,

    Thanks for your comment. So on your interpretation only intersections of the cantor bar with an interval of non-zero length will be a part. If you mean closed intervals, you still get point parts: intersect the interval [1/2, 2/3]. That intersects the cantor bar at only 2/3 and is a point.

    But I think the problem is general, and applies to Hilbertthm90’s comment too. Tell me what your domain is, and what your parthood relation is, and I’m pretty sure you’ll find it violates the mereological axioms.

    The set of non-zero intervals clearly isn’t a model of mereology, because fusions of disconnected intervals aren’t intervals, and complements of intervals aren’t either. If you close this set under arbitrary unions and complements you get pointy things again. (You can union intervals together to get R\{0}, then take the complement to get the point {0}.) So basically, if the mereology contains the non-zero length intervals and you can take fusions and complements, you end up with pointy things again.


  7. Woops. I meant “open interval,” since that is the basic unit in topology.

    So it has to be closed under fusions and complements? This is seeming much more measure theoretic than topological (in more mathy terminology). Topology we care about arbitrary unions and finite intersections, whereas measure theory we care about countable unions and complements. Mereology seems to be a mix of the two.

    I’m going to read more before posting, since my intuition is a bit off from what I was thinking.


  8. Well, yeah, as I said, if you admit fusion conditions like “everything not equal to x”, then you trivially get a self-connected part, namely the atom {x}. But if we want to opt for atomlessness, then I don’t think it would make sense to admit such conditions anyway.

    I’m more curious whether it’s possible to maintain that the Cantor set has no self-connected parts if we disregard fusions but accept complements. It seems pretty difficult (as A-B always has to be infinite), but it just occured to me that we could try admitting as parts only those sets which are similar to the whole. If A and B are on the “same level”, then A-B is trivially A, if A is on a “higher level” than B, then A-B is infinite and consists of a finite number of copies of B. It should also be possible to broaden our domain of parts to finite unions of such building blocks. (Mereology of fractals sounds like a cool topic to study.)

    (Re path-connectedness, C&V’s equivalent would seem to be transitive connectedness: TCxy iff there is a finite sequence x,a,…,n,y where the neighbours touch each other. Path-connected can’t be the proper interpretation of touching because (0,1) is path-connected to (2,3) via (0.5, 2.5), but obviously not connected to it. Of course, I dont remember all that, I just read it here.)


  9. Hmm…now I’m thoroughly confused. Certainly path connected implies connected…unless again there is an overlap of terminology being used in different ways. There is no path from 0.5 to 2.5, and in the sense I know it, path connected means that between any to points there is a path connecting the two that lies entirely inside the space.

    The sense that you seem to have just used it means that two spaces are path connected if there exists a path connecting them. This is trivially the case for any spaces if you don’t require the path to be completely inside of the space.


  10. Hi Preno,

    I’m not sure I did admit fusion conditions like ‘not equal to x’, where x is a point (this requires reference to the particular model.)

    By the way, there are perfectly consistent mereologies that have unrestricted fusions and are still atomless. It sounded to me like you were saying you had to restrict the fusion axiom to get atomlessness. (Two examples: the regular open sets in Euclidean space, and equivalence classes of sets with positive Lebesgue measure, that differ only by a null set.)

    Also, why did you say (0,1) is path connected to (2,3)? I don’t think transitive connectedness is the same as path connected.


  11. We must have posted simultaneously!

    In reply to your earlier post, Hilbertthm90, the ‘mereo’ in ‘mereotopology’ refers to the mereological bit (which just requires the domain to be a complete Boolean algebra under parthood.) The ‘topology’ is the topological bit which tells us which objects touch each other. The interesting bit is the principles that concern both primitives, for example “if x is a part of y, the everything connected to x, is connected to y”, and the C8 principle discussed.


  12. Hello! I wandered onto your blog through the Philosophers Carnival and couldn’t help but notice that you had a post about Varzi…. I was wondering whether you’ve ever gone to one of his talks?

    He was my metaphysics professor at Columbia University last year and he’s funny, receptive to questions, and quite possibly the best philo teacher I’ve ever had.

    Anyways, I’m working on getting my own philo blog started so I guess I’ll share the link when I’ve got some things started. Keep up the philosophizing….

    🙂


  13. I’ve never been to one of his talks actually.

    Good luck with your blog though – let me know once you’ve got it started.


  14. Sorry, I forgot about this discussion until I read your new post.

    Fusion itself, of course, doesn’t force you to admit atoms (because it’s a purely synthetic principle), but unrestricted fusion with complementation (which you used to disprove my original example) seems inconsistent with atomlessness, at least in a set-theoretic setting, if you can take the fusion of all objects (sets) which don’t include c. Or, in this particular case, fusions of objects wholly/partly larger than c.

    Transitive connectedness is indeed not the same as path-connectedness, but C&V define path-connectednesss of x as transitive connectedness of all of its parts. So in their sense, the fusion of (0,1) and (2,3) is path-connected. Which just makes my comment about the interpretation of “touching” irrelevant (“path-connected can’t be the proper interpretation of touching”), as you had, afaict, the normal notion of path-connectedness in mind.


  15. Come to think of it, that’s a pretty weird definition of path-connectedness, as any part x is always connected to all other parts of x via x (assuming parthood entails connection).


  16. Hi Preno,

    We need to get clear on some background issues.

    What is ‘c’? It’s not a mereological primitive – the only mereological primitive is parthood. If you add c as a new primitive, it still doesn’t follow that the complement of the fusion of the things c is not a part of is a point – what if c itself is gunky?

    I guess you could add a non-mereological axiom saying ‘c has no proper parts’. But obviously ‘c is an atom and everything is gunky’ is inconsistent.

    If you think that atomlessness is inconsistent with unrestricted composition and complementation, how did I just prove their consistency (I gave you two models – they are both models of classical mereology, including complementation and unrestricted composition.)


  17. Regarding the second point – I’m afraid I don’t have Parts and Places with me at the moment. It would be suprising if they used ‘path-connected’ for that notion.

    I was using ‘path-connected’ in, what I believe, is the convential way (e.g. as used in topology.) See the wiki article.


  18. By ‘c’ I mean some fixed element (say, 0) of the set (say, R) some subsets of which we are defining parthood on, not a thing in the mereological sense. If we take U to be complement of the fusion of all things that don’t include c, then U cannot have any proper parts (because then, thanks to complementation, it would also have proper parts which don’t include c). Assuming, that is, that the subset relation coincides with parthood.

    You seemed to commit yourself to this kind of fusions when you said “you can union intervals together to get R\{0}, then take the complement to get the point {0}”. My point was that this doesn’t seem to be a problem with the particular system of sets one admits as things, but with admitting “everything that doesn’t contain 0” as a fusion condition.

    About your models, can’t I just take the complement of the fusion of regular open sets that don’t include zero? Again, this seems to be the construction you used yourself. As for the other example, it appears to be consistent because you don’t use the subset relation as the parthood relation (so we can’t distinguish R and R\{0}).

    About their notion of path-connectedness, it’s not from Parts and Places but from Spatial Reasoning and Ontology: Parts, Whole and Relations, page 46. (I think I linked to it in some previous comment, too.) I realize my previous comment about path-connectedness was not relevant as it had nothing to do with the (normal) way you were using it.


  19. Ah, I see my mistake now. You said “union together”, I must have read “fuse”. You’re right.


    • How could any of this be better stated? It coduln’t.


  20. OK.

    I think the problem is I’m talking about a theory – a theory that has various models, some of which make atomlessness true, some that don’t. You appear to be talking about a particular model of that theory. If the model you are thinking of is just the power set of some set, interpreting parthood as subsethood, then its not going to make atomlessness true. Any singleton will be an atom, and every object will be a union (which *in this case* is the same as fusion) of atoms.

    If you had been thinking of a different model, none of what you said would have made sense. The model might not have contained 0. Or it might have contained 0, but 0 could have had proper parts according to the model (interpret proper parthood not as subsethood but \{ \langle 1, 0\rangle\}.) Just because, intuitively, an object doesn’t have any parts doesn’t mean it can’t be used to represent an object with parts in a model – you have to remember they’re formal representations.


  21. Actually, you say it here

    “By ‘c’ I mean some fixed element (say, 0) of the set (say, R) some subsets of which we are defining parthood on, not a thing in the mereological sense.”

    The subset lattice is just one of many models of mereology – all you have shown is that these kinds of models don’t have gunk.

    “can’t I just take the complement of the fusion of regular open sets that don’t include zero? ”

    The *union* of the regular open sets not containing 0 is R\{0}. This is not itself a regular open set so is not in the model (it’s not equal to the interior of its closure, so its not regular.) However, look at the definition of ‘fusion’ (see e.g. wikipedia) and you’ll see in this model fusions don’t correspond to unions – they correspond to the *interior of the closure* of the union. Thus the fusion you gave me is just R, the only object that doesn’t have a complement.

    (By the way, {0} isn’t regular open either, so its not in the domain of the model and its not obvious you can define this fusion in the language. Obviously it doesn’t matter if you use a second order fusion axiom.)


  22. Yes, I was talking about systems of subsets of some set, where parthood = subsethood, which is the kind of models we were talking about originally.

    My point is still true but not really relevant, as you didn’t rely on any fusion axiom to create R\{0}. I don’t know why I thought you did, since you clearly said “union”.


  23. “Yes, I was talking about systems of subsets of some set, where parthood = subsethood, which is the kind of models we were talking about originally.”

    In which case, you are still wrong unfortunately. By Stone’s representation theorem, both the gunky models I gave you are isomorphic to a collection of subsets of some set with subsethood as the order (not the full powerset though.)


  24. Ah, you’re right, thanks for the clarification. Looks like I was hopelessly confused between fusions and unions. (My last comment was crossposted with yours.)


  25. No worries.

    Frank Arntzenius has a good discussion of the two models I mentioned. He also addresses the (potentially confusing) point that the models look like they’re made from atoms (this is really just an artefact of these models, that doesn’t show up in the thing being modelled.)


  26. […] Philosopher’s Carnival 82. Sorry I’ve been too busy to contribute. I liked the entry “when do two objects touch?” […]


  27. Out of curiosity have you read Cartwright’s “Scattered Objects”? (The link goes to a version of the paper in a metaphysics reader available on Google Books – my copy is from Cartwright’s book but I assume it’s the same)


  28. I’m afraid I haven’t, although I’ve been told to read it several times! I’ll check it out.



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