Infinitely large objects

Note: This blog post has been retracted, since I no longer think of it as a good representation of how I think about its topic. I may, or may not, have written a better post about the same topic since; check the full list of posts.

Thomas Aquinas considers (S.T. Iª,7,3) whether an infinitely large body may exist:

On the contrary, Every body has a surface. But every body which has a surface is finite; because surface is the term of a finite body. Therefore all bodies are finite. The same applies both to surface and to a line. Therefore nothing is infinite in magnitude.

This argument, from the nature of surfaces, is repeated in the corpus of the article and in the reply to the second objection:

The same applies to a mathematical body. For if we imagine a mathematical body actually existing, we must imagine it under some form, because nothing is actual except by its form; hence, since the form of quantity as such is figure, such a body must have some figure, and so would be finite; for figure is confined by a term or boundary.

[...] 

Reply to Objection 2. Although the infinite is not against the nature of magnitude in general, still it is against the nature of any species of it; thus, for instance, it is against the nature of a bicubical or tricubical magnitude, whether circular or triangular, and so on. Now what is not possible in any species cannot exist in the genus; hence there cannot be any infinite magnitude, since no species of magnitude is infinite.

This argument seems to me pretty stupid. There are many figures which have infinite magnitude; they are simply open in one of their directions. One of them would be the parabola, or, in three dimensions, a paraboloid.

Another argument is made in the article from the nature of motion; he argues that such a body could not move. This assumes that an immovable body cannot exist, which I agree with:

The same appears from movement; because every natural body has some natural movement; whereas an infinite body could not have any natural movement; neither direct, because nothing moves naturally by a direct movement unless it is out of its place; and this could not happen to an infinite body, for it would occupy every place, and thus every place would be indifferently its own place. Neither could it move circularly; forasmuch as circular motion requires that one part of the body is necessarily transferred to a place occupied by another part, and this could not happen as regards an infinite circular body: for if two lines be drawn from the centre, the farther they extend from the centre, the farther they are from each other; therefore, if a body were infinite, the lines would be infinitely distant from each other; and thus one could never occupy the place belonging to any other.

Suppose there were a paraboloid extending infinitely into space, and we were to rotate it about its center only a little bit, similarly to how a conical drill rotates. It would seem that, while the points near to the center would move only by a little bit, as we wanted, the points farther from the center would rotate by proportionally larger amounts, to the point where, although an infinite speed is not reached at any point, in any event light-speed would seem to have to be reached at some point, which is absurd.

I chose to consider a case that does not seem to have this problem: an infinite rope. At first, I thought of an infinite chain, but then the distinctness of its links would seem to make it more nearly akin to an infinite multitude than to a single large object.

Imagine there were an infinite rope. We can see one end of it, and there is no other end because it goes infinitely in one direction. 

What happens if you pull the rope? You can’t get more of the rope, because then you would have pulled all of its infinite weight. But you also can’t pull it tighter, because then it would have to have a definite other end point, just like how, with any other rope, you could travel while holding it tight so as to circle its other endpoint.

The rope cannot be moved in the direction opposite to the infinite direction, and it can also not be pulled tight, or so it seems to me. So it seems such a rope cannot exist.

Anton objected that the rope could be pulled by an infinite force, or it could have zero mass.

Now, I reject the notion of infinite force for a reason similar to the reason I reject the notion of an immovable object: admitting both leads to a paradox, and discussions of this paradox have led me to think neither notion is plausible. But let us admit the notion of an infinite force for the sake of argument.

Supposing that the rope, existing as I described it, were pulled by an infinite force, it would be unclear with what speed it would move. It seems that the rope could, in theory, move with any of different speeds; but there is only one infinity. So moving the rope in this way does not seem plausible.

And a similar problem arises supposing that the rope has no mass: any kind of tug on the rope would seem sufficient to pull “the whole” rope, which is absurd.

To the former problem, Anton answered that there are multiple quantitative infinities. I have heard of such notions in mathematics, but I do not think any current notion of multiple infinities would apply to a force; and, in any event, I was already straining to admit an infinitely strong force of any kind.

Calvin gave this suggestion:

The rope could have infinite length and finite mass if the density exponentially decreases.

Everyone liked Calvin’s suggestion, myself included. It seems very apt and plausible.

Upon closer examination, someone could think of how density of matter is a measure of how close together, or far apart, its particles are. After decreasing in density by some amount, the rope would be a liquid and then a gas, and then it is unclear how it would remain a cohesive rope. But we could suppose that it is possible, nevertheless, that it could somehow remain a cohesive rope throughout all of these decreases; it is not as implausible to me as the earlier rejected notions.

Another alternative is that it is not the density that decreases exponentially, but the thickness of the rope. We could then seem to be supposing a different strange concept, for the rope would at some point become thinner than the diameter of the smallest particles we know. But this is not so bad either, I think. Besides, I think our smallest particles currently known are now often supposed to be “point-like”, which I take to mean that they are like the volumeless points of Euclidean geometry. If so, there is no lower limit on the rope’s decreasing thickness. So, such an infinite rope seems to me to be plausible.