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 the definition of 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, for number or that which has a number is numerable.

Concretely, the infinite can neither be composite nor simple.

For (a) it cannot be a composite body, since the elements are limited in multitude. For the contraries must be equal and no one of them must be infinite; for if one of the two bodies falls at all short of the other in potency, the finite will be destroyed by the infinite. And that each should be infinite is impossible.

For body is that which has extension in all directions, and the infinite is the boundlessly extended, so that if the infinite is a body it will be infinite in every direction.

Nor (b) can the infinite body be one and simple-neither, as some say, something apart from the elements, from which they generate these (for there is no such body apart from the elements; for everything can be resolved into that of which it consists, but no such product of analysis is observed except the simple bodies), nor fire nor any other of the elements. For apart from the question how any of them could be infinite, the All, even if it is finite, cannot either be or become any one of them, as Heraclitus says all things sometime become fire. The same argument applies to this as to the One which the natural philosophers posit besides the elements. For everything changes from contrary to contrary, e.g. from hot to cold.

Further, a sensible body is somewhere, and whole and part have the same proper place, e.g. the whole earth and part of the earth.

Therefore if (a) the infinite body is homogeneous, it will be unmovable or it will be always moving. But this is impossible; for why should it rather rest, or move, down, up, or anywhere, rather than anywhere else? E.g. if there were a clod which were part of an infinite body, where will this move or rest? The proper place of the body which is homogeneous with it is infinite. Will the clod occupy the whole place, then? And how? (This is impossible.)

What then is its rest or its movement? It will either rest everywhere, and then it cannot move; or it will move everywhere, and then it cannot be still. But (b) if the All has unlike parts, the proper places of the parts are unlike also, and, firstly, the body of the All is not one except by contact, and, secondly, the parts will be either finite or infinite in variety of kind.

Finite they cannot be; for then those of one kind will be infinite in quantity and those of another will not (if the All is infinite), e.g. fire or water would be infinite, but such an infinite element would be destruction to the contrary elements. But if the parts are infinite and simple, their places also are infinite and there will be an infinite number of elements; and if this is impossible, and the places are finite, the All also must be limited.

In general, there cannot be an infinite body and also a proper place for bodies, if every sensible body has either weight or lightness. For it must move either towards the middle or upwards, and the infinite either the whole or the half of it-cannot do either; for how will you divide it? Or how will part of the infinite be down and part up, or part extreme and part middle? Further, 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 there cannot be an infinite place, there cannot be an infinite body; (and there cannot be an infinite place,) for that which is in a place is somewhere, and this means either up or down or in one of the other directions, and each of these is a limit.

“The infinite is not the same in the sense that it is a single thing whether exhibited in distance or in movement or in time, but the posterior among these is called infinite in virtue of its relation to the prior; i.e. a movement is called infinite in virtue of the distance covered by the spatial movement or alteration or growth, and a time is called infinite because of the movement which occupies it.