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.
Mr. Hunt,
ReplyDeleteWhile it may be that you have in fact divined my conclusion, I would suggest that you have not sufficiently examined the possibility that the units are "at no distance from each other." Suppose we take a random smallest possible unit of space. This unit will be surrounded (most likely) by other equally small units. Since each unit abuts each other unit the distance between each unit from the edges will be zero. From the center the distance between any other unit's center will be exactly one unit, assuming that the units are all of the same shape and size.
Thus the discreteness of space will still be preserved.
The classical problem of the incommensurability of the hypotenuse of a triangle to the sides clearly refutes your explanation.
ReplyDeleteNo single three-dimensional shape reiterated through space could possibly answer to the description you give. Perhaps you will say that cubes serve the purpose? They do indeed, provided we only ever ask about the distance between two units of space which can be found on a line perpendicular to their adjacent plane surfaces. However, if I ask for the distance between two units in space arranged "around a corner" from each other, it will be impossible to give their distance in terms of the same basic unit. Instead, a further division of space will be necessary.
Sebastian, please excuse the polemical tone in my reply to your comment.
ReplyDeleteYes, but Amos, the fact that you cannot measure the distance without resorting to some smaller unit does not prove that the unit you are using to measure exists. Suppose I presented you with the following argument:
ReplyDeleteThere are 31,556,926 seconds in a year. Let us say that I have dividend income from investments in the amount of 30,000. This means that every second I make about 9/100 of a penny. Therefore, pennies must in fact be divided into hundredths.
I think you will agree that this argument is flawed. The fact that a discrete quantity can be theoretically measured at a smaller level does not make the quantity any less discrete or the division "discovered" any more real. Thus it is with space. Granted that the hypotenuse is not a whole unit--all this means is that we must resort to fiction to quantitatively measure in the first place.
Therefore I must conclude that your objection, clever as it is, happens to be wrong.
Also, no apologies are necessary as to your tone. I hardly found it polemical, and I am (at any rate) a fan of polemics.
I shall endeavor then to be more genuinely polemical in the future. What I was apologizing for really was my pretense that I had foreseen your explanation, which is to say that perhaps I was apologizing only to myself, since I don't believe anyone else was aware that I was making any such pretense.
ReplyDeleteBut do you honestly mean to say, Sebastian, that there is no motion except in six cardinal directions?