## Tuesday, November 28, 2023

### Relativistic Aristotelian beginnings

From purely geometrical facts, it follows that every spatially extended entity is arbitrarily small at its beginning and at its end in almost every reference frame.

A stronger result is possible in the special case of the beginnings of substances in simple Aristotelian substantial change. In simple Aristotelian substantial change, substance A wholly changes into a new substance B by having all of the terminal matter of A be the proximate matter of B without any temporal gap. I claim that then substance B comes into existence at a single point in every reference frame (i.e., the temporal bottom of B fits into a light cone).

For suppose that in some frame F, substance B comes into existence at two F-simultaneous and distinct spacetime points z1 and z2 (these could be points at which there is still A but arbitrarily close to points of B or these could be points at which there is B but that are arbitrarily close to points of A). Then there is another frame F′ at which z1 is earlier than z2. Let ti be the time of zi in F′. Because of how the terminal matter of A is the proximate matter of B, there is matter of A arbitrarily close to z2. Hence, arbitrarily close to time t2, we will have A still existing. However, arbitrarily close to time t1 we will have B already existing. Since t1 < t2, it follows that according to F, we will have A still existing after B has already come to exist. So the cause is partly later than the effect, which is absurd.

Maybe there is a way around this in more complex cases where multiple substances result in one new substance. I am not sure.

But here is another way to see the pointiness of the beginnings of substances if one accepts the Aristotelian idea that substances are individuated by their initial matter and we take matter to be infinitely subdivisible. Suppose in frame F, substance B begins at distinct and simultaneous points z1 and z2. Let F as before be a frame where z1 is earlier than z2. Then according to frame F, substance B already exists before its matter close to z2 exists (I am assuming matter is infinitely subdivisible, and in this case a relevant division happened). So its matter close to z2 cannot be essential to its individuation. And there is a third frame, F, where z1 is later than z2, and so the matter close to z1 cannot be essential to B’s individuation. It follows that none of the matter can be essential to the individuation of the substance if the substance starts at two or more places at once. Thus, a substance must start at a single point.

What happens in substantial change then? It seems that if we are to preserve relativity, we have to say that the new substance comes into existence at a single point z out of one or more preceding substances. If matter individuates (which I am dubious of), then the matter immediately around z is what does the individuating. The substance’s form then spreads out from z, perhaps incorporating more and more of the stuff around z, at the speed of light or less.

Of course, all these problems disappear if we allow for faster-than-light causation in substantial change. But that should be a last resort.