Well, using a argument in where a dimension is provable, a axiom, then there exist no two dimensional subjects (and objects) inside SpaceTime. That as they should 'disappear', observed from a angle of that third, 'non existing', dimension they lack. If you exchange dimension for degrees of freedom though, then the way something can move is what define its dimensions. Matter can move (in itself, as well as in space) in, or if you like :) call it, 'defining' three room dimensions and one 'time' dimension.

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Defining it from degrees of freedom it becomes perfectly acceptable, to me, defining two dimensional lattices laid upon each other, becoming a three dimensional piece of matter to our three (four) dimensional observer, that is. (but it sure as h** took me quite some time reaching this conclusion:)

Or maybe not? It depends on how they move, doesn't it :) The point may be that one lattice has only two directions which in its constituents can move, but as you join two together, there might become a new direction through which they can interact. At least it solves the question of how to prove a two dimensional lattice for me. It's not as perfect as I would like it though.

You could possibly also relate it to 'Matter tells space how to curve. Space tells matter how to move' which then inevitably will lead us to gravity and geometry those days. Einstein didn't define relativity from geometry though, that is a later concept as I've understood it.

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The most important thing to me is that you can't have it both ways. Either dimensions preexist, to then gets filled with stuff, or it is degrees of freedom that will define what we call dimensions. I prefer the later myself as it fits relativity better (locality and observer dependencies). But you need co-existing frames of reference for the later to become a measurable SpaceTime, as it is those frames of reference interacting that creates our possibility of measurements. Any which way one prefer it one can't have a string 'vibrate' though, without both introducing a arrow and one 'room dimension', or degree of freedom, more than the one defining a string 'at rest'. A tension is slightly different possibly? But not really, from where I look at it. Better to call it a property, like 'spin', if so. You could imagine a string that is stressed in the one dimension it exist in possibly, as compressing and decompressing, but that still need to introduce a arrow in which it can do so. From a degree of freedom it is the same, you need that arrow.

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There is a difference between what we call room dimensions and a arrow. The arrow has one direction in which it takes everything with it. For a room dimensionality, or degree of freedom, it doesn't matter what direction you measure a distance in, a meter is still a meter. So when speaking of time reversibility, it may make theoretical sense, but as we all know we are going to die, it practically doesn't. And btw, as I measure that distance, the arrow I do it in will have a same direction, no matter that 'meters' orientation inside a SpaceTime.