Geometry reflects the presupposition that space can be considered outside of the context of time. This is the first presupposition in the list. Is this an accurate assumption?
Geometry considers it meaningful to consider space outside of the context of time. Geometry completely ignores time. More modern models of space do not deem this useful, and represent time to lesser or greater degree. In modern physics, there is even question as to whether space has any existence at all devoid of time. All modern models of science abandon completely the notion that space exists independently from time, or that space can be meaningfully considered outside of the context of time. It is now recognized that all of space is always in motion. For example, all atoms and their constituent subatomic particles are constantly rotating about their axes and as well that they are constantly moving through space to new positions relative to everything else in space. Motion through space requires time. It is not possible to consider a snapshot of space that is completely devoid of all motion, as envisioned in geometry, except within the imagination, because space cannot exist devoid of motion. Geometry can ignore motion, but only because it ignores time.
Let us reconsider, and integrate time within our model. Let us begin slowly, and consider time to be another fundamental component of nature, in addition to the space component, yet a component that is completely distinct from space. How shall we represent time?
Time is commonly recognized as constituting a single dimension. How is this single dimension of time commonly symbolized? Geometry has a concept known as the ray. Whereas a line represents an extension of points in a pair of directions, a ray represents an extension of points in one direction only. In geometry, as a model of space only, the ray and the line are considered unified into a single dimension.
How shall we consider time? The most common metaphor for time for speakers of English is the notion that “time flies like an arrow.” And how does an arrow fly? Like an extension of points that flow in one direction only. Time is well represented by the geometric concept of a ray. To use a term that better reflects the finite nature of space, let us call it a ray segment.
Although the geometric model of space does not distinguish the ray from the line, in terms of dimensions, once we include time within our model of space, we recognize that our culture clearly distinguishes the ray from the line, as the ray symbolizes our understanding of time, whereas the line is one symbol of our understanding of space.
In our model that represents both time and space, we now have 5 total dimensions of time or space; these are the dimensions of the point, the ray segment, the line segment, the area, and the volume.
However, if we look at the most recent models of nature, it is commonly considered that time is in fact not distinct from space, but that time and space exist only in an integrated form, as space-time. Without going into details at this point, such a consideration would yield not 5 dimensions of time plus space, but 5 dimensions of space-time.