We are used to thinking about reality as a 3-dimensional space, perhaps with an added 4th dimension of time. But is this realistic?
3-dimensional space is commonly assumed to be composed from lower-dimensional spaces, consisting of 0, 1 and 2 dimensions. So a 0-dimensional space is a singular point that does not extend in any direction. Extend this point along one axis as you get a 1-dimensional space — a line. Extend this line at right angles to its length and you create a 2-dimensional space — a plane. Extend this plane at right angles to its surface and you get a 3-dimensional space — a cube. The process can continue, by extending the cube across all of its surfaces you get a 4-dimensional space — a tesseract.
While all this is mathematically consistent, how does is relate to reality, i.e. the world beyond our mathematical conception? In the first place, there can be no real (non-imaginary) 0-dimensional space. If the point has any existence at all (other than as an imaginary proposition) then it will have some spatial dimension — however small. The same would be true of the line and plane, which, however thin, would still have to have some depth and width, and therefore would really be 3-dimensional.
So there are no 'real', substantial objects that exist solely in 0, 1 or 2 dimensions. It would seem obvious, though, that substantial things really do exist in 3-dimensional space, but is this so? If the 0, 1, or 2 dimensions are useful conceptual conventions but ultimately insubstantial why should the 3rd-dimension be any different? Could it be that the 3rd dimension has only conceptual existence too? I would argue yes, that what we take to be the 'realness' of 3-dimensional space is just as conceptual as 0, 1 or 2-dimensional space, which is to say it does not really exist outside of our conception.
To look at it in one way: in order to establish 3-dimensional space you need axes in fixed relation to one another. You need a 1-dimensional x-axis, a 1-dimensional y-axis and a 1-dimensional z-axis (using the standard notation). Since none of these 1-dimensional axes have any substantial existence outside conception (as already shown) then we can say 3 insubstantial things added together cannot produce something substantial: three zeros make zero.
To look at it another way: a line cannot be extruded from a point with no extension; a plane cannot be extruded from a line with no extension; a cube cannot be extruded from a plane with no extension, and so on.
To look at it another way: The traditional 3-axes of space require a fixed viewing position. Each moves away from a fixed origin, and in fixed relation to the observer. In order to see a cube in the way we are used to we adopt a particular viewing position which looks at it from one angle, the spatial co-ordinates being fixed in relation to that. But this singular viewing position is a very limited view of the cube, which in fact can potentially be viewed from all angles at once, including all internal as well as external viewpoints. When considered from all its potential viewpoints simultaneously the standard axes no longer apply since there is no fixed viewing position to which they relate. When viewed from all directions at once there is no 'up', 'down', 'forward' or 'back'.
Consequently I would argue that 3-dimensional space is just as much an ideal conception of reality conceived from a single viewpoint as the lower dimensions. It has no more substance than these.
