Monday, January 18, 2010

How Chunky is Space?

Speaking of chunks, how about Sebastian's argument "that space, in its actuality, must be finitely subdivided?"
Now, it would seem that it would take an infinite amount of time to move through an infinite amount of space, unless one has the capacity to move at an infinite velocity. Since human beings do not seem to possess the means to means to travel at an infinite velocity, it follows that no one would be able to cross even a finite distance, since to do so they would have to move through the infinite subdivisions of the finite amount of space. In fact, nothing unable to move at an infinite velocity would be able to move at all, since it would have to cross infinite space to reach a point even infinitesimally farther away from where it started.

At first I resisted, but now I am reconciled to the logic of this reasoning. However, I am making this concession with no idea in mind but to show that the same argument also proves that space is infinitely subdivided. The argument says that it would take an infinite amount of time to move through an infinite amount of space at a finite velocity. But how far apart should we say the discrete units of space are, which according to the argument are supposed to divide space? At no distance from each other, or at some distance? If at no distance, then all space is in the same place, and no motion is possible. If at some distance, then there is either a measure for the distance between them, which is a further division of space, or the distance between them is infinite, and motion from one to the other would require infinite time and/or infinite velocity, neither of which is available to humans or to anything with whose motion we are familiar.

The only possible conclusion of this argument, from which two contradictory conclusions follow, is that motion is not what the argument takes it to be, namely, a traversal of a certain kind of topological extension, which unlike other such extensions is perfectly blank and indifferent. I have the feeling this is what Sebastian has been trying to get at anyway, but only he can say.