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- a continuous area or expanse which is free, available, or unoccupied. (implicitly refers to objects ie “space as stage for things”)
- the dimensions (degrees of freedom) of height, depth, and width (refers implicitly to measurement) within which all things exist and move (“space as a stage for motion”)
- a blank between printed words, characters, numbers, etc. (ie a stage is a stage because it is empty. to communicate information, a prerequisite is a low entropy (ie highly ordered) channel. we would say “space as a channel for information flow”
sets of mathematical abstractions capture the role of space as a stage for things, and a stage for motion.
euclidean space R^n is what we are most familiar with. more specific/exotic conceptions of space could be said to be obtained by simply abstracting part of the structure possessed by euclidean space eg:
- vector spaces: abstract the operations on R^n of addition and scalar multiplication (to model motion, displacement and so on)
- metric spaces: abstract the function d:R^nxR^n -> R which computes the distance between two points (ie measurement)
- topological spaces: abstract set of open balls (locality, proximity)
- normed vector spaces: abstract addition, scalar multiplication and the norm ||v||=(∑^ni=1Vi^2)^1/2 (motion and measurement)
- inner product spaces: abstract addition and scalar multiplication and the dot product. (motion and angles)
hilbert spaces: abstract addition, scalar multiplication