Abstract
It is standardly assumed that space and time consist of extensionless points. It is also a fairly standard assumption that all matter in the universe has point-sized parts. We are not often explicitly reminded of these very basic assumptions. But they are there. For instance, one standardly assumes that one can represent the states of material objects, and of fields, by functions from points in space and time to the relevant point values. Electric fields, mass densities, gravitational potentials, etc. … are standardly represented as functions from points in space and time to point values. This practice would seem to make no sense if time and space did not have points as parts.
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Notes
- 1.
Well, it could still be deterministic if the equations of motion were first-order, as they are in quantum mechanics. Still, one might like to think that even if the equations of motion are second-order, as they are in classical mechanics, the world could be deterministic.
- 2.
- 3.
Borel regions of the real line: start with the collection of all open intervals, then close this collection up under countable union and intersection, and complementation.
- 4.
This doesn’t quite mean that it has to be a point particle, since it could still be a line, or an infinitely thin surface. But one can define a notion of “a converging set of regions” in such a way that the particle does indeed have to be a point particle if it is entirely within each of the regions in the “converging set of regions.”
- 5.
This is so because the algebra of closed regular regions of the real line is not “weakly distributive,” and one cannot have a “semi-finite” countably additive measure that is defined on every element of an algebra that is not weakly distributive. A measure is said to be “semi-finite” if every element that has infinite measure has a part that has finite measure. For the definition of weak distributivity and a proof of the fact that one cannot put a semi-finite measure on an algebra that is not weakly distributive, see Chapters 32 and 33 of Fremlin (2002).
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Arntzenius, F. (2011). Gunk, Topology and Measure. In: DeVidi, D., Hallett, M., Clarke, P. (eds) Logic, Mathematics, Philosophy, Vintage Enthusiasms. The Western Ontario Series in Philosophy of Science, vol 75. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0214-1_16
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