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Could There Be Exactly Two Things?

April 23, 2008

Lewis’s theory of worlds is sometimes criticized because it invalidates certain true modal claims. Could there be no objects what so ever? According to Lewis no, if worlds are maximally self-connected regions of spacetime the smallest world consists of one spacetime point. But intuitively, yes: we can keep subtracting objects until we don’t have anything left. Or to put it otherwise, once you’ve gotten to the last two objects you can either delete the first, or delete the second. Since both operation are possible, why aren’t both together?

Similarly, one might have thought it possible that there be two disconnected regions of spacetime – this idea is common in science fiction. But again according to Lewis there is no such world, if they are truly spatiotemporally disconnected, they are two worlds.

What I find slightly weird is that not only is there no world with no objects, but there is no world with exactly two objects1 (or indeed any finite or countable number.) First note it is not possible for two closed sets to be touching without overlapping. Secondly note that our mereological atoms (spacetime points in this case) are closed. A world w is a maximally self connected region, which means if x and y fuse to w, then x and y must be connected. So if there was a world containing exactly two points and nothing else, they would have to be touching. Since they are closed they must be overlapping, and since they are atomic they must be identical (if x and y are atomic and overlap, then x = y.) This contradicts the assumption that there were two objects.

Now of course Lewis later refines his view in the Plurality of Worlds. There he says that possible worlds aren’t all the possibilities – possible individuals are possibilities as are worlds supplemented with information about the de re representation of various objects (see here.) Since the fusion of two worlds is a possible object we can get our spatially disconnected epochs. What about our two disconnected atoms? Well, a similar story can be told – but its not quite that simple. We need to decide a principle that modal realism is silent about – namely that one can have two qualitatively indistinguishable worlds. If you can, there should be no problem with helping yourself to two point worlds and taking their fusion. Much more controversial is the possibility of no things whatsoever. If the null fusion (the fusion of just those things which are not self-identical) is legitimate, there is a possible individual corresponding to this possibility. My point is that Lewis can account for these problems, but only at the expense of some controversial metaphysics.

1I’m ignoring their fusion for now, technically I mean there is no world with exactly three objects but this obscures my point. That there is no world with exactly two objects for the mereological reason is also worrisome, but not relevant.

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7 comments

  1. This set of problems also gives a trivializing answer to the age-old question, “why is there something rather than nothing” – after all, Lewis says there couldn’t have been nothing!

    As for there not being the possibility of exactly two objects, why must the mereological atoms be closed? In the actual world they are, but why couldn’t things have been such that atoms were open, or neither closed nor open? Of course, we’d need some specific version of the latter possibility in order for a fusion of finitely many of them to be connected.


  2. Hmm, that’s a good point. (Although I don’t think it would help if both atoms where open, because then they’d both be closed and we’d be in trouble again.) Sometimes people talk about the possibility of spacetime being discrete in the sense that there is a ‘smallest possible length’. I guess that would (misleadingly) correspond to the indiscrete topology over the two atoms. In this case all objects would be connected to each other.

    Of course it gets a bit vague which mathematical structures are to count as possible space-time isomorphs. In fact, he has to postulate *some* cut-off point to avoid the Forrest-Armstrong paradox.


  3. But why think it is possible that nothing exist, or that there be only two objects? You say some find these claims intuitively true. Maybe so (I don’t, BTW), but at least Lewis gives arguments for his view. I just curious why someone would accord intuitions so much weight in this case. Again, if there were good reasons to think that it is possible that nothing exist, etc. then that’s fine. I would love to hear those reasons.


  4. Well I thought I did give reasons to think that its possible that nothing (concrete) exists. We can keep subtracting objects until nothing is left. What is puzzling is that once you’ve gotten to the last two objects there’s a possibility in which the first is deleted, and similarly for the second, but deleting both at once isn’t a possiblity.

    When your theory starts telling you you can’t travel faster than the speed of light, there can’t be a prime number of things, or whatever, there comes a point where it stops being a theory of modality. What exactly do you think a theory of modality is supposed to be accounting for if not these kinds of things?


  5. Maybe I misunderstood. If you’re taking possibility here to be merely logical possibility, then I agree that it seems that a world where nothing exists is possible in this thin sense. As I faintly recall, Lewis takes propositions to be the sets of worlds where they’re true. But since the proposition that nothing exists is true in none of his worlds, that proposition is the empty set. But then that proposition is the same as any other expressed by a claim that comes out impossible on his view. Intuitively, these propositions are different. That’s a problem. I suppose this is related to your worry.

    I took you to be talking of metaphysical possibility. You can conceive of a scenario where objects are subtracted until nothing is left, but I know of no reason to think that this tells us anything about whether such a scenario is metaphysically possible.


  6. I am talking about metaphysical possibility.

    As far as we can know anything about metaphysical possibility, conceivability is a pretty good test. In fact, I’d say it is our primary means for seeing if something is metaphysically possible.

    You have given me no reason to doubt that conceivability has led us astray here. If it were a water h20 case, sure, but its not. If you want to deprive us of our primary source of modal knowledge, what would you suggest instead? On what basis am I supposed to decide which theory of modality is true?


  7. When is it appropriate to appeal to intuition sans argument? I think this is an interesting question. I’m inclined to say that we sometimes have no choice but to do so, but in general it seems bad.

    In this case, you say conceivability is some kind of reliable guide to possibility. I say we need reasons to accept this. You (seem to) say rather that we can accept this unless there are reasons to reject it. Here is the general case: Someone asserts some proposition P. Is it rational to (1) accept P until countervailing considerations come to light, or (2) accept P only if we have reasons to think it’s true? I think (2) is the way to go, unless there is some other reason mitigating against (2). Do you think the conceivability/possibility thesis is importantly different than the general case I described, or do you disagree with my assessment of the so-called general case?

    So, I think it is incumbent on proponents of the conceivability thesis to defend their view. But even if you disagree with this, here is a reason to doubt the thesis. It seems to me that we can conceive of scenarios that are obviously impossible. Consider the scenario according to which all there is is a necessarily existing baseball. This scenario does not obtain in the actual world, which means it does not obtain in all possible worlds, which it must if it is possible (possibly necessarily P implies necessarily P). For most conceivings, I know of no way to check whether they are or are not possible, but cases like the one I just described give us a determinate negative answer for at least some conceivings. So, I have no reasons to accept the conceivability thesis, and the only conceivings I can check come out impossible.

    You ask how we’re supposed to decide which theory of modality is true. I guess we have to evaluate the arguments for and against each. I just don’t think it’s a good move to reject Lewis’ theory and all his arguments because some of the consequences clash with our unsupported intuitions about modality.



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