Rather than counting objects/distances, one way I like to think about the definition of space is by translation symmetry. You do get into symmetry in your post but it’s mixed together with a bunch of other themes.
Like, you are in your cave and drop a ball. You then walk out of the cave and look back in. The ball is still there, but it looks smaller and you can’t touch it anymore. You walk in, pick up the ball, and walk out again, and then drop the ball outside. The ball falls down the same way outside the cave as it does inside.
If you think of what you observe from a single position as being a first-person perspective, then you can conceive of transformation that take one first-person perspective to a different one, but for such a transformation to make sense, objects need to have positions in space so they can be transformed.
Notably, you don’t need a collection of symmetric objects, or a volume with limited capacity for containing things, in order for space to make sense (and you can make up alternate mathematical rules that have limited capacity and similar objects but have no space). On the other hand, if you don’t have something like translational symmetry, it feels like you’re working with something that’s not “space” in a conventional sense? Like it might still be derived from space, but it means you can’t talk about “what if stuff was elsewhere?” within the model, which seems like the basic thing space does.
(I guess one could further distinguish global translation symmetry vs local translation symmetry, with the former being the assertion that ~you have a location, and the latter being the assertion that ~everything has a location. Or, well, obviously the latter is an insanely exaggerated version of locality which asserts that Nothing Ever Interacts, but I feel like this is where the physics-as-the-study-of-exceptions stuff goes.)
I also like to think that something similar applies to other symmetries, e.g. symmetry to boosts are basically asserting velocity is a sensible concept (and quantum mechanics provides a reductionistic explanation of how they function).
Be careful. Physics seems to be translation invariant, but space is not. You can drop the ball in and out of the cave and its displacement over time will be the same, but you can definitely tell whether it is in the cave or out of the cave. You can set your zero point anywhere, but that doesn’t mean that objects in space move when you change your zero point. Space is isotropic. There’s no discernible difference between upward, sideways, or diagonal, but if you measure the sideways distance between two houses to be 40 meters, a person who called your “sideways” their “up” will measure the distance between the houses to be 40 meters up and down. You can do everything here as you can do there, but here is not there. In the absence of any reference point, no point in space is different from any other point, but in the absence of any reference point there’s no need for physics, because if there was anything to describe with physics, you could use it as a reference point.
I suppose you could try to define space as the thing you can move around in without changing your physics, but the usual strategy is to define physics and derive conservation of momentum from the fact that your physics is translation invariant.
Rather than counting objects/distances, one way I like to think about the definition of space is by translation symmetry. You do get into symmetry in your post but it’s mixed together with a bunch of other themes.
Like, you are in your cave and drop a ball. You then walk out of the cave and look back in. The ball is still there, but it looks smaller and you can’t touch it anymore. You walk in, pick up the ball, and walk out again, and then drop the ball outside. The ball falls down the same way outside the cave as it does inside.
If you think of what you observe from a single position as being a first-person perspective, then you can conceive of transformation that take one first-person perspective to a different one, but for such a transformation to make sense, objects need to have positions in space so they can be transformed.
Notably, you don’t need a collection of symmetric objects, or a volume with limited capacity for containing things, in order for space to make sense (and you can make up alternate mathematical rules that have limited capacity and similar objects but have no space). On the other hand, if you don’t have something like translational symmetry, it feels like you’re working with something that’s not “space” in a conventional sense? Like it might still be derived from space, but it means you can’t talk about “what if stuff was elsewhere?” within the model, which seems like the basic thing space does.
(I guess one could further distinguish global translation symmetry vs local translation symmetry, with the former being the assertion that ~you have a location, and the latter being the assertion that ~everything has a location. Or, well, obviously the latter is an insanely exaggerated version of locality which asserts that Nothing Ever Interacts, but I feel like this is where the physics-as-the-study-of-exceptions stuff goes.)
I also like to think that something similar applies to other symmetries, e.g. symmetry to boosts are basically asserting velocity is a sensible concept (and quantum mechanics provides a reductionistic explanation of how they function).
Be careful. Physics seems to be translation invariant, but space is not. You can drop the ball in and out of the cave and its displacement over time will be the same, but you can definitely tell whether it is in the cave or out of the cave. You can set your zero point anywhere, but that doesn’t mean that objects in space move when you change your zero point. Space is isotropic. There’s no discernible difference between upward, sideways, or diagonal, but if you measure the sideways distance between two houses to be 40 meters, a person who called your “sideways” their “up” will measure the distance between the houses to be 40 meters up and down. You can do everything here as you can do there, but here is not there. In the absence of any reference point, no point in space is different from any other point, but in the absence of any reference point there’s no need for physics, because if there was anything to describe with physics, you could use it as a reference point.
I suppose you could try to define space as the thing you can move around in without changing your physics, but the usual strategy is to define physics and derive conservation of momentum from the fact that your physics is translation invariant.
Formally, I mean that translation commutes with time-evolution. (Maybe “translation-equivariant” would be a better term? Idk, am not a physicist.)
I guess my story could have been better written to emphasize the commutativity aspect.