Aristotle (384–322 B.C.)
Physics: Books V to VIII
There are three kinds of change: generation, where something comes into being; destruction, where something is destroyed; and variation, where some attribute of a thing is changed while the thing itself remains constant. Of the ten categories Aristotle describes in the Categories (see previous summary of the Organon), change can take place only in respect of quality, quantity, or location. Change itself is not a substance and so it cannot itself have any properties. Among other things, this means that changes themselves cannot change. Aristotle discusses the ways in which two changes may be the same or different and argues also that no two changes are opposites, but rather that rest is the opposite of change.
Time, space, and movement are all continuous, and there are no fundamental units beyond which they cannot be divided. Aristotle reasons that movement must be continuous because the alternative—that objects make infinitesimally small jumps from one place to another without occupying the intermediate space—is absurd and counterintuitive. If an object moves from point A to point B, there must be a time at which it is moving from point A to point B. If it is simply at point A at one instant and point B at the next, it cannot properly be said to have moved from the one to the other. If movement is continuous, then time and space must also be continuous, because continuous movement would not be possible if time and space consisted of discrete, indivisible atoms.
Among the connected discussions of change, rest, and continuity, Aristotle considers Zeno’s four famous paradoxes. The first is the dichotomy paradox: to get to any point, we must first travel halfway, and to get to that halfway point, we must travel half of that halfway, and to get to half of that halfway, we must first travel a half of the half of that halfway, and so on infinitely, so that, for any given distance, there is always a smaller distance to be covered first, and so we can never start moving at all. Aristotle answers that time can be divided just as infinitely as space, so that it would take infinitely little time to cover the infinitely little space needed to get started.
The second paradox is called the Achilles paradox: supposing Achilles is racing a tortoise and gives the tortoise a head start. Then by the time Achilles reaches the point the tortoise started from, the tortoise will have advanced a certain distance, and by the point Achilles advances that certain distance, the tortoise will have advanced a bit farther, and so on, so that it seems Achilles will never be able to catch up with, let alone pass, the tortoise. Aristotle responds that the paradox assumes the existence of an actual infinity of points between Achilles and the tortoise. If there were an actual infinity—that is, if Achilles had to take account of all the infinite points he passed in catching up with the tortoise—it would indeed take an infinite amount of time for Achilles to pass the tortoise. However, there is only a potential infinity of points between Achilles and the tortoise, meaning that Achilles can cover the infinitely many points between him and the tortoise in a finite amount of time so long as he does not take account of each point along the way.
The third and fourth paradoxes, called the arrow paradox and the stadium paradox, respectively, are more obscure, but they seem to aim at proving that time and space cannot be divided into atoms. This is a position that Aristotle already agrees with, so he takes less trouble over these paradoxes.
Aristotle argues that change is eternal because there cannot be a first cause of change without assuming that that cause was itself uncaused. Living things can cause change without something external acting on them, but the source of this change is internal thoughts and desires, and these thoughts and desires are provoked by external stimuli. Arguing that time is infinite, Aristotle reasons that there cannot be a last cause, since time cannot exist without change. Next, Aristotle argues that everything that changes is changed by something external to itself. Even changes within a single animal consist of one part of the animal changing another part.
Aristotle’s reflections on cause and change lead him ultimately to posit the existence of a divine unmoved mover. If we were to follow a series of causes to its source, we would find a first cause that is either an unchanged changer or a self-changing changer. Animals are the best examples of self-changers, but they constantly come into being and pass away. If there is an eternal succession of causes, there needs to be a first cause that is also eternal, so it cannot be a self-changing animal. Since change is eternal, there must be a single cause of change that is itself eternal and continuous. The primary kind of change is movement and the primary kind of movement is circular, so this first cause must cause circular movement. This circular movement is the movement of the heavens, and it is caused by some first cause of infinite power that is above the material world. The circular movement of the heavens is then in turn the cause of all other change in the sublunary world.
The problems associated with time, change, continuity, and infinity are all related. If space and time are continuous, that implies that there are an infinite number of points in space or moments in time between any two given points or moments. As Zeno’s paradoxes sharply illustrate, assuming continuity in space and time then raises the problem of how we can ever cross an infinite number of points in space or pass an infinite number of moments in time. If they are infinite they have no end by definition. How we make sense of the concepts of infinity and continuity, then, are not simply mathematical questions but questions that have real bearing on how the world is put together. One solution, proposed by philosophers known as the Atomists, is that time and space are not continuous but consist rather of very small, indivisible units. Aristotle rejects this position on the grounds that it makes nonsense of the idea of change: something can only be in a state of change if it makes a continuous transition from one state to another. Aristotle wants to hold on to change, but to do so, he must also uphold the continuity of time and space, which puts him into trouble with Zeno’s paradoxes.
Aristotle’s distinction between potential and actual infinities is an ingenious means of maintaining the continuity of space and time without falling victim to Zeno’s paradoxes. Denying out of principle the very idea of infinity would raise all sorts of complicated mathematical problems, so Aristotle does not want to rule infinity out entirely. However, he is steadfast in denying the actuality of infinity: he says that the universe is not infinitely large, that there is not an infinite amount of matter in it, and so on. However, he grants, it is in theory possible that we could count up to infinity or measure an infinite number of points on a ruler, and so on. Because we could potentially divide up time or space infinitely, we can accept the continuity of space and time as well as the existence of a state of change. However, because neither space nor time can ever actually be divided up infinitely, Zeno’s paradoxes do not hold muster.
Aristotle’s distinction between actual and potential infinities has been the topic of a great deal of debate and has ultimately been proved false. In the nineteenth century, mathematicians developed a rigorous means of expressing concepts such as continuity and infinity that renders Aristotle’s distinction between two kinds of infinity unnecessary. It turns out Zeno was right, at least in a limited sense: though change is possible, there is no such thing as a state of change. We can accept that space is continuous and accept that an object moving through space passes through an infinite number of points so long as we do not insist that it is in a continual state of change. For example, we can easily show how Achilles overtakes the tortoise by showing in a table the relative positions of Achilles and the tortoise at different moments in time. This table will show position and time but will say nothing about the motion of the two bodies. Motion is something we can infer from the fact that Achilles is at one place at one moment and at another the next, but Achilles is not in a “state of motion” at any of those given moments. Not all mathematicians would agree to the solution outlined here, and the Zeno’s paradoxes remain a subject of debate even now.
Aristotle returns to his idea of an unmoved mover in greater detail in the Metaphysics, but it is worth noting here the role this divine figure plays relative to the rest of the cosmos. Aristotle places the earth at the center of the cosmos, orbited by a number of concentric spheres holding the sun, the moon, the planets, and ultimately the stars. The movement of all heavenly bodies, then, is circular, and the earth itself is a sphere at the center of other spheres. Aristotle explains that all these spheres are in motion because of a divine figure beyond the outer sphere of stars. This unmoved mover can himself only move the sphere of the stars, and the movement of the stars in turn influences the movement of all the other spheres and hence of life on earth. We can see in this conception of the cosmos why astrology has had such a grip on the Western mind: everything that happens on the earth in Aristotle’s conception is ultimately a reaction to the movement of the heavens.
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