You know you’re in the Twilight Zone when you witness a set of parents discover that their daughter is missing, and they don’t call the police. They call a physicist. Oddly, that turns out to be a good move, because Betina, their daughter has wandered away from three dimensional space, and is stuck in “a fourth dimension.” She is followed by her dog, of course.
I’ve suggested that the value of The Twilight Zone in philosophical inquiry comes from its rich fund of exercises in philosophical imagination. These exercises put stress on our concepts; they force us to attempt to apply them in new and unusual ways that we simply don’t get to try out in our ordinary experience. A seemingly minimal constraint on the the stories we tell is that the possible states of affairs they contain must really be possible. For example, if we described a ball as being completely blue on its exterior surface and also completely white on its exterior surface, we would not be describing a possible state of affairs. Our attempt at the description of a possible world would fail, since our attempted description is logically inconsistent.
Some inconsistencies are obvious, such as ones involving an all blue and all white ball, while other inconsistencies are only revealed by an analysis a set of propositions. Examples of a non-obvious inconsistency include some descriptions of time travel. We hinted at the inconsistency in “Walking Distance,” where Martin Sloan doesn’t have a limp at time t and does have a limp when he returns to time t after time-traveling to the past and causing the accident to his childhood self that results in the limp.
We can distinguish between logical possibility and physical possibility. A physically possible state of affairs is one that is logically possible, but is also consistent with the laws of nature. To describe Betina as lost in another dimension is to propose what is logically possible, that space has other than, either fewer or more than, three dimensions, but it is not consistent with our best physical theories, which take space to be three dimensional, or spacetime, with a dimension for time, to be four dimensional.
So the episode is inconsistent with our best physical theory. But it’s logically possible that our physics could be wrong, and it’s possible that space has more than three dimensions, and that a little girl and her dog could enter that dimension through “a gap opening” to that dimension. The difficulty is that it we really can’t do more than postulate this possibility. We lack the resources to represent the possibility, since our representational resources are spatially three dimensional and our spatial concepts are drawn from our physical theory.
“Little Girl Lost” doesn’t just announce that Betina is lost in the fourth (spatial) dimension. It attempts to show this to us, and this is problematic in several ways. The first problem is that her parents, in their home, can still hear Betina and the dog. So sound travels across the dimensions, which we can represent in our space, but not in the posited extra dimension. If it did, we would be able to follow it. The second problem has to do with the hole or gap through which Betina has passed to enter the special dimension, discovered by the physicist friend on the case . What can we say about a hole? Does a hole have a size? Is it in space, or is it the absence of space? Is it in our three-dimensional space or in the fourth spatial dimension, or both? When Betina’s father places his upper body through the hole, how can he be half in the fourth dimension and half out of it? When he sees Betina and the dog, they are represented as being in three dimensions. They have height, width, and distance from him. The fourth dimension is just a foggy three dimensional space, at least as it is represented. And how could it be otherwise? Finally, it isn’t clear how an extra dimension would be incompatible with the first three dimensions. In contemporary physics, space and time are represented as the four dimensional spacetime. Representing the location of objects is a matter of assigning values in each of the four dimensions. While an n-dimensional spacetime has n-dimenions, it’s not clear what it would mean for an object to have just one of the n dimensions.
Lewis, D. K., and Lewis, S. R., 1970, ‘Holes’, Australasian Journal of Philosophy, 48: 206–212; reprinted in D. K. Lewis, Philosophical Papers. Volume 1, New York: Oxford University Press, 1983, pp. 3–9.