Tiny Spaces20 January 2021
Famously, the Euclidean axiomata for space seemed necessary to many, so that various philosophers concluded or argued that some knowledge or something playing a rôle like that of knowledge derived from something other than experience. Yet there were doubters of one of these axiomata — that parallel lines would never intersect — and eventually physicists concluded that the universe would be better described were this axiom regarded as incorrect. Once one axiom was abandoned, the presumption of necessity of the others evaporated.
I think that our concept of space is built upon an experience of an object sometimes affecting another in ways that it sometimes does not, with the first being classified as near when it does and not near when it does; which ways are associated in the concept of near-ness are selected by experience. The concept of distance — variability of near-ness — develops from the variability of how one object affects another; and it is experience that selects which variabilities are associated with distance. Our concept of space is that of potential (realized or not) of near-ness.
The axiomata of Euclid were, implicitly, an attempted codification of observed properties of distance; in the adoption of this codification or of another, one might revise which variabilities one associated with distance. One might, in fact, hold onto those axiomata exactly by revising which variabilities are associated with distance. In saying that space is non-Euclidean, one ought to mean that the Euclidean axiomata are not the best suited to physics.
Just as the axiomata of Euclid become ill-suited to physics when distances become very large, they may be ill-suited when distances become very small.
Space might not even be divisible without limit. The mathematical construct of continuity may not apply to the physical world. At least some physical quantities that were once imagined potentially to have measures corresponding to any real number are now regarded as having measures corresponding only to integer multiples of quanta; perhaps distance cannot be reduced below some minimum.
And, at some sub-atomic level, any useable rules of distance might be more complex. On a larger scale, non-Euclidean spaces are sometimes imagined to have worm-holes, which is really to say that some spaces would have near-ness by peculiar paths. Perhaps worm-holes or some discontinuous analogue thereöf are pervasive at a sub-atomic level, making space into something of a rat's nest.