"The mind is a compact, multiply connected thought mass with internal connections of the most intimate kind. It grows continuously as new thought masses enter it, and this is the means by which it continues to develop."

Bernhard Riemann On Psychology and Metaphysics ca. 1860

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Saturday, January 27, 2007

The problem with the concept of time as dimensionality

If you review Riemann's famous Hypotheses which underlie geometry habilitation paper, it strikes you that via the concept of an n dimensional manifold there exists a specific manner in which n dimensions carry over or evolve into n+1 dimensions. In spatial dimensions, we can continuously move without affecting or disturbing the quality of the dimension per se. This is what Riemann signified as a continuous manifold. For the purposes of functional space, it is convenient to treat of other characteristics of physical reality such as pressure or temperature as added such dimensions which are likewise continuous. (However, it is the case that this may not quite be true absolutely, e.g. there is an absolute zero for temperature, which is why at or near that region the "normal" laws of physics are seemingly violated.)

Now consider whether this is true of time. How do we move in time? We do so mentally in two directions. Therefore, it appears to the naive first impression to be of a similar quality of dimension as a spatial dimension. In fact, we mark time as a cyclical function through the use of space. However, can we actually, not merely in the imagination or virtually, move backward in time? Is it in any way truly reversible in the sense that the three spatial dimensions are? Obviously and self evidently not. Why then do we blandly accept the concept of physical space-time? Further, why do we wish to indulge in the idea of n dimensionality as anything other than as a useful fiction that serves us to create a sort of functional “space.” Which fiction, however, it must be cautioned, must not be confused with the three “real” physical dimensions. On this score, none other than Riemann himself weighed in. In his philosophical remnants, he discusses time. Its cyclical nature is relative to the system or life form at issue. For example, the scale of time for a tree is far different that that say of a butterfly. However, we do see that biologically there is a unifying metric that is indeed quite physical, i.e. a circadian rhythm determined by the rotation of the earth. Does anyone in his or her right mind suppose that this rotation or cycle may actually be reversed? On the other hand, it is extremely useful and a uniquely human power that we may reverse and move ahead in time in our imagination. Without that power, our judgment in changing our course of action would be impossible. (Sort of like George Bush and company vis-à-vis the war in Iraq.)

The question is then, does the mathematics of relativity permit us to think of time (or the speed of light) as a dimension in the same sense as the real physical continuum? How is the mind immanent in these dimensions? What is the limit of the transmission of ideas over physical space and time? Faster than the speed of light? Can an idea be truly transmitted effectually ?

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