Time Gunk and Zeno

30 05 2005

I suppose this is an odd topic to post about immediately after FEW, having nothing to do with what went on there. But I talked to several people about this at lunches and in the evening, and they don’t think anyone has written up this idea before.

It seems that some objects may be “metaphysically simple” in the sense that they have no proper parts. For instance, it was once thought that atoms had this property (hence their name) and it may still be thought that quarks and electrons do. However, some people believe it’s at least possible that some objects are atomless, in that every part of the object can be further subdivided into more parts. Any such object is said to be made up of “atomless gunk”.

Of course, all this is talking about the spatial parts of some object. But as Ted Sider points out (in his chapter on Temporal Parts), it seems natural to consider the temporal parts of some objects as well. Some examples he gives are “Ted when he had long hair”, “Ted when he had short hair”, “the piece of clay while it was shaped into a statue”, and “the current time slice of the Eiffel Tower”.

But the talk of time slices (Sider’s “stage view”, I think) seems to me to make the presumption that he didn’t want to make in the case of spatial parts. If we think that some spatial parts of objects may be atomless, in that they have no part without a proper spatial subpart, then why not think that some temporal parts of some objects might have a similar divisibility property? In the spatial case, it seems possible that some physical objects have atomic parts and some not. In the temporal case this would seem weirder, but there’s obviously no logical contradiction with this idea.

I think Sider wants to allow that space and time both be coordinatized by real numbers. Thus, it could still make sense to talk about points of space even if they are located in the region spanned by some gunky object – there’s no need to suppose that the object and space have any parts in common, especially if one is a relationalist about space rather than a substantivalist. Similarly, an object could be temporally gunky even if it makes sense to talk about instantaneous moments of time. This would allow a temporally gunky object to exist at the same time as some non-gunky one.

If two objects cannot spatiotemporally overlap without sharing some parts, then this would also mean that no matter how you break the universe up into parts, there are only countably many disjoint ones. (Assuming the coordinatization of space-time is Archimedean.)

If all objects are temporally gunky, then this would provide a nice resolution of Zeno’s paradox of the arrow. The paradox says that at any moment, the arrow has a specific location. Thus, there is no moment at which the arrow moves. So the arrow must be stationary. However, if the arrow is temporally gunky, then it doesn’t make sense to talk about the arrow at any particular instant. It may only make sense to talk about the arrow extended over a (perhaps extremely small) interval of time. Any such part of the arrow occupies slightly different spatial regions at each moment, and thus every part of the arrow is moving. The “time slices” don’t move, but they also don’t exist on this picture.

For someone who believes what I imagine Sider to believe, that every collection of objects forms a further object, no matter how the parts are arranged in space and time, the paradox also goes away even if time is atomic. However, the explanation seems to miss something about the intuitive idea of motion. The theory suggests that it doesn’t make sense to talk about the motion of a temporally non-extended object, and I would agree. But this would mean that there are some objects (temporally extended ones) of which it makes sense to ask if they are in motion, and some objects (the stages) of which it doesn’t. Perhaps this is no worse than saying that an object must be spatially extended in order to ask whether it has a direction, but it does seem at least minorly more troubling.

And to say that there are some times at which the arrow is moving and some times at which it isn’t would only make sense (if at all) when talking about the whole arrow, and not its stage, because the stage is also part of many other objects that don’t move at all. But if it has no temporally atomic parts, then in the interval when the arrow moves from the bow to the target, it has no parts which are not moving. In some larger intervals it may have some parts that move and some that don’t. But at any rate, Zeno’s paradox disappears

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4 responses

1 06 2005
Brian Weatherson

I don’t know whether anyone has addressed Zeno’s paradox from this perspective, but I know Frank Arntzenius at Rutgers is working on issues about the gunkiness or otherwise of time. And I remember there being some literature on this, but I can’t find the reference. Frank or someone working on this stuff should know the reference.

1 06 2005
Kenny

Thanks for the tip! Arntzenius’ paper “Gunk, Topology, and Measure” has a bunch of this stuff, including a discussion of Zeno’s arrow at the beginning, and a development of the mathematics that might underwrite such a description of spacetime. And a bibliography I’ll check out. (It sounds like he wrote about it as early as 2000 in “Are there really instantaneous velocities?”, which I haven’t had a chance to check out yet.)

It’s also quite exciting that he makes use of a theorem that shocked me the first time I had to prove it as a homework assignment, but I’ve wanted to teach to mathcamp students since then. This is the result that there exists a dense open subset of [0,1] that has measure strictly less than 1. It shows that you can’t put a measure on the algebra of regular closed subsets of the real line, which is the first attempt to provide a “pointless” structure for spacetime.

The structure he ends up developing is the “up to measure 0” structure, which remarkably enough, is exactly what I need for the functions I described in my paper at FEW, so it all relates back in the end!

15 06 2005
Kenny

Looks like someone else has already written this up as an objection to Sider’s stage view. Joshua Stuchlik, “Not All Worlds are Stages”

3 10 2006
Flots Masriach

Are you sure 56452 about this?!?

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