For the sake of clarity, it would seem, the temporal aspect of things is conveniently put to the side in discussions that use Flatland-derived techniques to help us imagine higher spatial dimensions as accurately as we can. It would only add an unnecessary element of confusion.
Anyway, if time is a dimension it makes perfect sense to regard it is a “temporal” dimension with qualities distinct from the three spatial ones. After all, if the temporal dimension were actually a spatial dimension, why would human consciousness experience it in a seamless sequence of three-dimensional cross-sections woven together with causal associations?
Well, those two-dimensional Flatland inhabitants only experience their world through one-dimensional cross-sections. Looking down on their sheet-of-paper universe, we can clearly see a circle, but do the other shapes see the circle? Yes, but not completely, at least not all at once. They would see the circle from the side. They could travel around the circle, see it from all sides and piece it all together in their minds, but they still would have only ever seen sides, and even all sides would not be equivalent to our swift gaze from above — a direction no one embedded into the 2-D Flatland could look at or point to and would only come to suspect and strive to imagine if they developed physics.
There is a side of them we can only see, and it brings together all the sides they can see together into a cohesive whole they are incapable of conceiving.
Similarly, we can only visually perceive 2-D cross-sections of our 3-D universe. 3-D creatures have never seen all three dimensions at once; such a perception would require a creature of four dimensions gazing down on our “flat” 3-D universe. There is a side to 3-D you have to be 4-D to see.
Again, time is ignored: but if it is a spatial dimension, might our interactions with it be similar?
We would have to be a fifth-dimensional creatures traveling through a four-dimensional universe we experience in a seamless sequence of three-dimensional cross-sections. The movement we call time requires the fifth dimension.
The curvature of three-dimensional space implies an additional spatial dimension that is not equivalent to time — again, a fifth dimension offering the necessary direction for 3-D space to warp and fold.
We experience 4-D space by communing with an 4-D object embedded in that space and experiencing it, and through it the detectable 4-D universe, in 3-D cross-sections.
Why are these cross sections woven together with causal associations? For the same reason that the shading on either end of a wall up ahead “suggests” its curvature, and so that this apparent wall is in actuality our necessarily incomplete perspective of a circular boundary: causality “implies” the fourth dimension.
We are always moving forward as we travel through it, though we may do so at different rates — and, it would appear, not towards a fixed and deterministic target but only a probabilistic direction.