Aristotle confirms his definition of change and examines alternative explanations. Some thinkers define change as difference, inequality, or non-being. Aristotle rejects these proposals because something does not necessarily change merely by being different or unequal. Change cannot be adequately classified under such categories. Instead, he explains that change appears difficult to define because it is indeterminate. It cannot be identified simply with potentiality, actuality, or privation. Change is an actuality but it is incomplete, since it is the realization of a potential that has not yet reached its final state.

For this reason, change must be understood as the special kind of actuality he previously described: The actuality of what exists potentially insofar as it is potential. Change is the “actuality of the changeable” (60). Aristotle also explains that change involves interaction between what causes change and what is capable of being changed. Through contact, the agent introduces a form into the subject, bringing about change, such as when an actual human being generates a human being from what is potentially human.

Aristotle addresses the question of where change occurs. He argues that change “takes place in the thing” (60) that is capable of being changed, rather than in the agent that produces the change. Change is the actuality of what is potentially changeable, and it is brought about by something that has the capacity to cause change. The actuality involved in causing change and the actuality involved in being changed are, in a sense, the same. A single process connects the agent and the subject, just as a single distance can be described from two directions.

However, although acting and being acted upon refer to the same process, they differ in definition. Teaching and learning, Aristotle suggests, illustrate this relationship. The process is one, but it is described differently depending on whether it is considered from the perspective of the teacher or the learner. Aristotle concludes that change is a single actuality occurring in the subject that is being changed, even though it involves both the agent and the patient.

Aristotle introduces the problem of infinity as an essential topic for natural science, because magnitudes, change, and time may appear—like “everything” (62)—either finite or infinite.

Earlier philosophers frequently treated the infinite as a fundamental principle. The Pythagoreans and Plato regarded the infinite as a distinct entity, while other natural philosophers treated infinity as a property of basic elements such as air or water. Thinkers such as Anaxagoras and Democritus explained the world through infinitely many elements or parts. Several considerations led philosophers to affirm the existence of infinity. Time appears endless, magnitudes can be divided without limit, and processes of generation and destruction seem to require an unlimited source. Some thinkers also believed the region “beyond the heavens” (65) to be infinite.

Aristotle notes that the issue is difficult, since both accepting and denying infinity leads to problems. He instead begins by clarifying the different meanings of the term, noting that something may be infinite through endless extension, division, or addition.

Aristotle argues that the infinite cannot exist as an independent substance. If the infinite were a substance rather than an attribute, it would have to be indivisible, yet anything that is actually infinite must have magnitude and therefore be divisible. If it were divisible, each part would also have to be infinite, which leads to contradictions. Aristotle concludes that the infinite cannot exist as a substance, but only as an attribute of something else.

He then considers whether an infinitely extended body could exist among perceptible things. Aristotle argues that this is “impossible” (68). A body is defined as something bounded by surfaces, so an infinite body would contradict this definition. Natural bodies also occupy places and move according to directions such as up and down. In an infinite body, there would be no determinate place or direction, making natural motion impossible. Aristotle thus concludes that “there is no actually infinite body” (71).

Aristotle argues that infinity does exist, but only in a qualified sense. If infinity did not exist at all, several consequences would follow: Time would have a beginning and end; magnitudes would not be divisible indefinitely; and numbers would not extend without limit.

To avoid these problems, Aristotle distinguishes between potential and actual existence. He maintains that infinity does not exist as an actual magnitude. No body or magnitude is actually infinite. Instead, infinity exists potentially. A magnitude can be divided without limit and this endless divisibility constitutes the sense in which infinity exists.

Infinity also appears in processes such as time, where one moment follows another without final completion. In this way, infinity is not a completed whole but an ongoing process. Aristotle believes that the infinite exists only potentially, through continuous division or succession, rather than as a fully realized and independent entity.

Aristotle explains several consequences of his account of infinity. Infinity exists through division, but not as an unlimited increase in magnitude. Magnitudes can be divided indefinitely, but there is no perceptible magnitude that exceeds every definite size. Matter and the infinite are contained within things, while form provides the limiting structure. He contrasts number and magnitude. In numbers, there is a smallest unit—the one—which prevents infinite decrease, yet numbers can always increase beyond any given quantity.

Magnitudes show the opposite pattern. They can be divided without limit, but they cannot increase to an infinite size. Aristotle also notes that infinity in movement and time depends on magnitude. Time is infinite because change continues indefinitely. Mathematicians do not require an actual infinite magnitude, since any finite magnitude can be divided according to the same proportions. Aristotle concludes that the infinite functions as a kind of “material cause” (76) and is associated with the continuous.

Aristotle responds to arguments that claim an actual infinite exists. He argues that these arguments do not demonstrate the existence of a distinct infinite entity. First, the persistence of generation does not require an infinite body, because in a finite world the destruction of one thing can produce another. Second, the claim that finite things must be limited by something else does not imply infinity, since contact is always relative and occurs only between particular things.

Third, Aristotle says, the ability of the mind to imagine something infinitely large does not prove that such an infinite exists in reality. Aristotle also explains that time appears infinite because successive parts follow one another, though each part passes away. Finally, neither continual subtraction nor imagined increase creates an actually infinite magnitude. He concludes that infinity exists only in the limited sense already described, not as an actual, independent entity.