Motion, like magnitude and time, is a continuum and therefore implies the notion of the infinite. This notion was prominent in the speculations of the early Greek philosophers, both among the Physicists and the Pythagoreans and Platonists. Aristotle, in consequence, could not ignore it. In the Physics he devotes five chapters to it. Since these chapters are rather involved, we shall limit ourselves to sketching the contents.
1. Reasons in Support of the Infinite 2
2. The Infinite Does Not Exist in Act 3
First of all, if the infinite exists, where is it likely found? Or, what kind of thing is it? Aristotle begins with some remarks to the effect that the infinite, as the physicist speaks of it, cannot be separate from sensible things, in the manner of Plato's ideas or the Pythagorean numbers. Hence, if there is an infinite we must look for it in the world of bodily things.
The question, then, comes to this: Are there infinite bodies? Aristotle produces an array of arguments, both of the logical and the physical order, to show the impossibility of such bodies. One of these, the only one we shall rehearse, revolves on his theory of place. Every body has a place, but place is necessarily determined and finite. Up and down, for example, are determined and demarcated positions, and so are other regions of space. Since place is limited, the bodies it encompasses must also be limited.
Nor can there be an actually infinite number of bodies, since number is by definition numerable or measurable, whereas the infinite cannot be actually numbered.
3. The Infinite Does in Some Way Exist
Granted that the infinite does not exist in act, does it exist at all? The answer is that it does, since three of the arguments mentioned above in support of it are well founded. The first one relates to time, which, on Aristotle's supposition of eternal motion, must have neither beginning nor end.5 Secondly, number is infinite, that is, may be increased without end. Thirdly, and most important of all, magnitudes are divisible ad infinitum. Still, the actual, or actually realized, infinite is impossible. Yet, as we have just seen, the infinite does in some way exist. But since its existence cannot be actual, it will be potential only. In a word, there is an infinite, not actually but potentially.
But what does this mean? The potentiality of the infinite, like that of motion, is of a special kind. Ordinarily, a thing in potency can be actually realized. Socrates in potency in a block of marble can become a Socrates in act in the same marble. But the infinite can never make the transition to act. To say, therefore, that something is potentially infinite can only mean that a given process can be carried on in it ad infinitum or endlessly. Thus, magnitudes can always be further divided (infinity of division); numbers can always be added to (infinity of composition); and time can always be increased or divided (infinity of composition and division). Consequently, the infinite, far from being a perfection, denotes incompleteness or imperfection, and one would be in grave error to conceive it here as something perfect or absolute.
There is, however, another kind of infinite, an infinite that is actual, and utter perfection. This is the infinity of Pure Act. If the one is extensive and quantitative, the other, in a manner of speaking, is intensive and qualitative. But the infinity of Pure Act is not our present business, and we mentioned it only to caution the unwary.
4. The Infinitely Divisible, or the Continuum
If something is infinitely divisible, it is also continuous or a continuum. Hence, we shall next speak of the continuum, even though Aristotle does not expressly treat of it in connection with his discussion of infinity, but defers it to Books V and VI. 6
The continuum is opposed to the discontinuous, or the consecutive, as well as to mere contact. Contact approaches continuity but is still not it. These three terms - consecutive, contact, continuous - show, therefore, a certain progression, from the utter absence of continuity to its complete presence. They are defined as follows:
With these definitions in mind it is readily seen why the continuum cannot be composed of actual parts. If these parts are distinct, they have their real and distinct limits or extremities, in which case they may be in contact but are not continuous. If these parts are conceived as truly continuous, they are no longer absolutely distinct, hence no longer actual parts. Besides, in any continuum one can always multiply parts indefinitely; thus the continuum is endlessly or infinitely divisible. To put it squarely, the continuum is not composed of actual parts, but it is potentially divisible ad infinitum. So, the line is not composed of points, time is not composed of instants, and motion is not composed of rests. But - and this is the import of the potentially divisible - at all points of a line, or time, or motion, or any continuum, we may arbitrarily mark divisions and thereby assign parts.
This, then, is Aristotle's conception of the continuum, and it served him well in refuting Zeno's sophistical arguments against motion. These arguments had assumed that the continuum is actually composed of parts or, what is the same, composed of actual parts. Aristotle's idea of the continuum was, clearly now, quite different, and the difference provided him the tool to unmask and undo Zeno's reasoning.