Immanuel Kant wrote in Prolegomena to Any Future Metaphysics (1783):
“That everywhere space (which is not itself the boundary of another space) has three dimensions and that space in general cannot have more dimensions is based on the proposition that not more than three lines can intersect at right angles in one point. This proposition cannot at all be shown from concepts, but rests immediately on intuition and indeed on pure intuition a priori because it is apodictically (demonstrably) certain.”
The notion that multi-dimensionality would be based on more than three lines intersecting at right angles at one point is not currently assumed to be true, but this is a classic case of theoreticians being out of their depth notwithstanding convincing reasons for belief in the existence of dimensions that we cannot perceive with our senses and can just barely conceive with our intellect.
Dimensions can be grouped in terms of how we perceive them, either with our senses or conceptually. (It is best to approach the matter from a phenomenological standpoint.) We are most immediately aware of the three spatial dimensions that Kant had in mind. Length, width and depth, X, Y and Z axes, apply to objects situated in ordinary space. Time may be considered as measured along a fourth dimension. It is perceived to be different in that whereas we control our direction and rate of travel through the spatial dimensions, our direction of travel is fixed in the temporal dimension and our rate of travel is for the most part beyond our control.
Cosmologists discern a theoretical basis for additional dimensions, which can be individually described if not directly experienced like the four “primary” dimensions.
The fifth dimension consists of a means of measuring the differences and similarities between our reality and possible other ones. The sixth dimension, hypothetical but none the less real on an intellectual level, would, if we could experience it, display for us the attributes of all possible worlds that, like this one, would begin with a Big Bang.
The seventh dimension would provide access for all worlds starting with different (non-Big Bang) conditions. The eighth dimension would provide a planar view of all universe pasts beginning with different initial conditions diverging into an infinite number of branches. The ninth dimension compares the histories of all possible universes beginning with different laws of physics that could be conceived along with various initial conditions.
The tenth dimension, in this scheme of things the final one, images everything possible. By its nature this dimension implies an endpoint in terms of what humans could conceive. Notwithstanding, mathematicians true to form feel obliged to press on, and in that spirit they are able to speak of n dimensions where n equals infinity.
String Theory as currently proposed requires these extra dimensions that we can imagine but not sense. Why are we unable to perceive them? A number of models have been proposed. One is that the excess dimensions have been “compactified” on a very small scale. Another is that our cosmological unit exists on a three-dimensional submanifold, known in String Theory as a “brane”, on which all known particles other than gravity are restricted. That may be why gravity remains partly outside our understanding.
It may be that at some time in the future our formulations and terminologies will seem quaint and unrealistic, similar to our current view of pre-Galilean thinking. For the present, however, we shall probably continue in our accustomed mode to speculate on the infinities that surround us.
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