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Coherence in Movement

Preliminary Remarks

We must immediately distinguish motion of physical, material objects from motion of geometric elements.

Material objects appear to move continuously and to cohere as they do so – which is to say that a physical object appears to smoothly transport all its properties at once, in one go, while apparently maintaining some sort of distribution about some single place, its so-called centre. This centre, somehow, even as it moves, represents the whole object's place, or location. As motion proceeds, this centre is supposed, somehow, to describe—literally, to scribe (that is, to write, as a pen writes)—a continuous “track”, called a locus.

Let us call this "Newtonian motion".  It is certainly not projective motion.

It is not immediately obvious what, exactly, this track is, or how, exactly, it is made.

It cannot be made by “joining up” points, like stringing beads:  points cannot be joined up, because they have no size. So, the track is not a concatenation of points. Ink flows from a pen while it writes, and is left behind. What is it that flows from a place, while it writes, and how is it left behind?

It is hard to see how to move a place from place to place. It is harder still to see how this may be done continuously, smoothly, like a puck gliding on ice.

To reiterate, a point is a place. That is all it is. It is, ‘a somewhere’. How is, ‘a somewhere’, moved somewhere else? Can it be so moved? Does a moved place leave a hole where it had been? If a point were a pea or a particle, moving it would be really just a matter of dis-placing it: it goes from place-cum-point A, to place-cum-point B.

However, places A and B, alias points A and B, are not the items being moved in such an operation. The peas or particles are. So, what if points A and B, as places, are moved, in some other style of operation?

It is pretty obvious I suppose what the differences here are: particles or peas are not points. They consist of something other than the elements of geometry—or so we suppose. If we can pick it up and put it down, it is probably physical. It will have qualities - such as, possibly, mass - which we suppose to be somewhat indifferent to place.

Where it is, is not expected to greatly affect what it is.

The physical items translate, but the places do not. We would be intensely surprised, and, I imagine, not a little perturbed, if the inverse occurred.

Yet, when we “do” geometry, we move its elements happily and all the time, for all that they are not physical, and for all that we have no purchase on them whatsoever. We can't pick them up or put them down.  What do we suppose we are doing?

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