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.