Thursday, April 6, 2017


Self-colocation is weird. An easy way to generate it is with time travel. You take a ghost or other aethereal object who time travels to meet his past self, and then walks into the space occupied by his past self--ghosts can walk into space occupied by themselves--so that he is exactly colocated with himself. If you don't like ghosts, time travel a photon--or any other boson--into the past and make it occupy the same place as itself. But time travel is controversial.

However, it occurs to me that one can get something a bit like self-colocation with an aethereal snake and no time travel. An aetherial snake can overlap itself. First, arrange the snake in spiral with two loops. Then gradually tighten the ring, so that the outer ring of the spiral overlaps the inner one, until the result looks like a single ring. Suppose that the snake exhibits no variation in cross-section. So we have a snake that is wound twice in the same volume of space. The whole snake occupies the same region as two proper parts of itself. [I'm not the only person in this room generating odd examples: Precisely as I write this, I hear our four-year-old remarking out of the blue that she wished she had two bodies, so she could be in two places at once. A minute or so later she is talking of twenty bodies.]

(The animation was generated with OpenSCAD using this simple code.)

So far it's not hard to describe this setup metaphysically: the whole overlaps two proper parts. But now imagine that our snake ghost is an extended simple. We can no longer say that the snake as a whole occupies the same region as a proper part of it does, as the snake no longer has any proper parts. But there seems to be a difference between the aethereal snake being wound twice around the loop and its being wound only once around it.

If we accept the possibility of aethereal objects that can self-overlap and extended simples, we need a way to describe the above situation. A nice way uses the concept of internal space and internal geometry. The snake's internal geometry does not change significantly as the spiral tightens. But the relationship between the internal space and the external space changes a lot, so that two different internal coordinates come to correspond to a each external coordinate. That's basically how my animation code works: there is an internal coordinate that ranges from 0 to 720 as one moves along the snake's centerline (backbone?), which is then converted to external xyz-coordinates. Initially, the map from the internal coordinate to the external one is one-to-one, but once things are completed, it becomes two-to-one (neglecting end effects).

The idea of internal and external space allows for many complex forms of self-intersection of extended simples. And all this is great for Aristotelians who are suspicious of parts of substances.

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