Hank, we are so far apart on our understanding of dimensionality, I really have nothing to contribute your discussion.
But for those who are interested in geometric physics and extra time dimensions, Cumrun Vafa of Harvard appears to be the name to watch. Here are a few of his articles:
SPARE TIME. Descartes gave us co-ordinate geometry, with its three spatial dimensions. Einstein put time on an equivalent footing, creating 4-dimensional spacetime. String theory added six more spatial dimensions, and M theory added yet one more for a total of eleven (see Update 329). Now Cumrun Vafa of Harvard has added still another----an extra element of time---to make the existing theories more compatible with each other. Because of possible side effects, such as faster-than-light travel or questionable causality (time would not be measured sequentially along an axis but would spread out into a plane), Vafa's "F Theory" has not found many adherents yet. (New Scientist, 1 November.)
Dualities as Geometric Transitions (pdf)
Doctor Stochastic, you said:
Sorry to be so scarce lately, A-G. Work life is frantic and may stay that way a while.
Personally, I think the question of dimensionality sets up the next frontier for the human mind.
More or less intuitively, I’ve proposed that, if there is a dimension beyond three of space and one of time -- 3D[s] + 1D[t] = 4D -- it may be a t-dimension, not an s-dimension.
But there is huge resistance to this proposition. If the fifth dimension is a time dimension, then all known laws of causality are thrown into doubt. The 4D world guarantees you a linear succession in time, past to present to future. Add another dimension, then what does the world look like?
It doesn’t matter, I suppose, whether the “next” (5D) dimension is of space or of time. Either way, bizarre results result. Consider the following, from the article you bumped me to (“The Curse of Dimensionality,” by Mario Koppen), which suggests that spatial dimensions are basically a “mental product”:
“...Poincare also considered the question about the three dimensional nature of our world, and he gives a surprisingly modern-sounding idea. According to him, a thinking entity, as ... human beings are, could assign to their world [whatever number of] dimension[s] they like, since all of them are mathematically equivalent. Thereby, the world appears to be an abstract premise for sensoric perception and muscular activity, and spatial dimensions are basically a mental product. The choice for three depends on the configuration of the human nervous system and the so-acquired evolutionary advantage. By using two 2-dimensional retinas, the movement of the two hands has to be monitored. A dimension of 2 would not suffice to perform such a control, and a dimension of 4 would allow for movements during which the hands may even shortly be disconnected from the remaining body. So, a dimension of 3 seems to be a good compromise....”
But if it were possible to have access to a fourth spatial dimension, a few “weird” things would become possible:
“-- It would be possible to directly perceive the interior of a body;
-- The content of a box could be taken without opening the box;
-- A node could be removed from a string without moving the ends of the string;
-- A body could be lifted without external forces;
-- A body could be moved into its mirror form….”
Koppen appears to argue that if there is a fifth dimension accessible to the mind, it is likely to be spatial, not temporal:
“…considering time as a fourth dimension is not the same as considering a fourth spatial dimension. While it is a good trick for describing relativity, it is [just not] more: We cannot move [back and forth] in time….”
Further he notes that “Up to now, there is no experiment known which demonstrates the fact of a merely 3-dimensional world, and also there is no fact known for which a 3-dimensional world would be optimal…. [And yet] simultaneous perception of different aspects of the same thing…would be a simple act for a four-dimensional (thinking) being….”
So Koppen basically rules out the possibility of any “extra” dimension being time (because of the “problems” that it introduces with respect to causality, the invariance of the time order); he suggests that if there is an extra dimension, it is spatial – which would then allow for the possibility of “simultaneous” perception “of different aspects of the same thing.”
But I can’t help but feel that this sort of thing, if anything, is an argument for an extra time dimension – the word “simultaneous” is the clue here.
Yet to visualize extra dimensions is extraordinarily difficult – perhaps because Poincare was right, that perception of dimensions is conditioned by the “configuration of the human nervous system and the so-acquired evolutionary advantage.”
Maybe it’s time for an expansion of consciousness, an “emergent property” that may provide a new “evolutionary advantage” for the human race.
Some day. :^)