The Infinite

The infinite is either:

• that which is incapable of being traversed because it is not its nature to be traversed, or

• that which admits only of incomplete traverse or scarcely admits of traverse, or
• that which, though it naturally admits of traverse, is not traversed or limited.

A thing may be infinite in respect of addition or of subtraction, or both.

The infinite cannot be a separate, independent thing.

This is because it is neither a size nor number.

Infinity itself is its substance and not an accident of it, it will be indivisible. This is because the divisible is either:

• magnitude, or
• plurality.

But if infinity is indivisible then it is not infinite, except as the voice is invisible.

But we are not examining this sort of infinite, but the infinite as untraversable.

How can an infinite exist by itself, unless number and magnitude also exist by themselvess-since infinity is an attribute of these?

If the infinite is an accident of something else, it cannot be qua infinite an element in things, as the invisible is not an element in speech, though the voice is invisible.

Evidently, the infinite cannot exist actually. For then, any part of it that might be taken would be infinite. This is because ’to be infinite’ and ’the infinite’ are the same, if the infinite is substance and not predicated of a subject.

Therefore infinity is either:

• indivisible, or
• divisible into infinites.

But infinity cannot be many infinites. Each infinite-part would also be infinite, if the infinite is substance and a principle.

Therefore, infinity must be indivisible.

But if so, then actually-infinite must be of a certain quantity. This cannot be.

Therefore, infinity belongs to its subject incidentally.

But if so, then it cannot be it that is a principle, but that of which it is an accident-the air or the even number.

This inquiry is universal.

The infinite is not among sensible things.

If a body is ’that which is bounded by planes’, there cannot be:

• an infinite body either sensible or intelligible nor
• a separate and infinite number

This is because number, or anything which has a number, is numerable.

Concretely, the infinite can neither be composite nor simple.

1. It cannot be a composite body

This is because the elements are limited in multitude.

The contraries must be equal.

No one of them must be infinite. If one of the 2 bodies falls short of the other in potency, the finite will be destroyed by the infinite.

But it is impossible for each to be infinite.

This is because body is that which has extension in all directions.

But the infinite is the boundlessly extended. If the infinite is a body, then it will be infinite in every direction.

2. The infinite body cannot be one and simple.

It cannot be something apart from the elements, from which they generate these. There is no such body apart from the elements.

Nor is it fire or any other of the elements. None of them is infinite. It is impossible for the whole—even if it were finite—to be or to become one of them, as Heraclitus says that all things at some time become fire.

The same argument applies also to the “One” which some natural philosophers posit over and above the elements. For everything changes from its contrary, e.g., from hot to cold.

A sensible body is somewhere. The place of the whole and of the part is the same, e.g., of the earth.

Consequently, if the whole is homogeneous, it will either be immovable or always in motion.

But this is impossible.

For why should it move down rather than up, or anywhere else?

If it were a clod of earth, where will it move or remain at rest?

For the place of the body akin to it is infinite.

Will it, then, occupy the whole place? And how?

What, then, is its state of rest and motion?

Either it will be at rest everywhere—so it will not move—or it will move everywhere—so it will not stand still.

But if the whole is heterogeneous, the places are also heterogeneous, and first of all, the body of the whole is not one except by contact.

Then, these parts will be either finite or infinite in kind. They cannot be finite; for if the whole is infinite, some of them will be infinite and others not (e.g., fire or water); but such a thing causes destruction to the contraries. If, however, they are infinite and simple, the places will also be infinite, and there will be infinite elements. But if this is impossible and the places are finite, the whole must also be finite.

In general, it is impossible for there to be an infinite body and place for bodies, if every sensible body has either weight or lightness. For it must move either toward the center or upward, and it is impossible for the infinite, either as a whole or half or any other part, to undergo either motion. For how would you divide it? Or how can the infinite have a part below and another above, or an extremity and a center?

Every sensible body is in a place, and there are six kinds of place, but these cannot exist in an infinite body. In general, if it is impossible for place to be infinite, it is also impossible for body to be so; for that which is in place is somewhere, and this means either up or down or one of the other directions, each of which is a limit.

The infinite is not the same in magnitude, motion, and time as if it were a single nature; but the latter terms are spoken of in reference to the former. Thus, motion is called infinite with reference to the magnitude on which it moves, or is altered, or increases.

Time is called infinite because of motion.