Aristotle's Infinity

Alexis Cohen
February 15, 2006

Infinity in the Physics, Book III4-8

Aristotle begins his explanation of infinity with an appeal to scientific knowledge. If the Physics undertakes the task of illuminating scientific knowledge, then Aristotle is obliged to discuss infinity to this end[1]. For Aristotle, “[s] cientific knowledge of nature involves taking magnitudes and change and time into consideration, and each of them is bound to be either infinite or finite”(202b30). Yet, not everything is either infinite or finite, e.g. qualities and points. Thus, a discussion of infinity does not construct the whole of scientific knowledge.
The Greek word for ‘infinite’, apeiron, is the same word for ‘unlimited’. In its limitlessness, those whom Aristotle calls the “natural scientists”, consider the infinite as neither an origin nor an end. Instead, they think the infinite “is taken not to have an origin, but to be the origin of everything else—to contain everything and steer everything, as has been said by those thinkers who do not recognize any other causes…apart from the infinite”(203b7)[2]. Unlike the natural scientists, both Plato and the Pythagoreans used the word infinite to describe unstructured stuff awaiting an imposed limit in the guise of either a shape or a form[3]. Aristotle, however, differs from all those mentioned above in that he considers the infinite as that “which cannot be traversed”(204a3).
Before attempting to define infinity, Aristotle lays five popular considerations that generally lead people “to infer that something infinite does exist”(203b15). The first two are the infinity of time and the division of magnitudes in mathematics. The last three considerations, however, are less straightforward. Thirdly, Aristotle finds that there is a notion that the persistence of generation and destruction is due to an infinite source “from which anything which is generated is subtracted”(203b17). Aristotle also lists the common notion that there must be a limit to the finite and, hence, no ultimate limit. Lastly, people believe that “number and mathematical magnitudes and the region beyond the heavens seem to be infinite because they do not give out in our thought. And if the region beyond the heavens is infinite, then it seems that body must be infinite too…”(203b22). Aristotle waits till end of the Book III to refute these arguments, however, with the exception that consideration one, that of time, is partly admissible. Aristotle refutes these considerations because they entertain the notion of actual infinity, which only later is proven impossible.
Before refuting the five popular reasons for infinity, Aristotle is compelled to ask whether or not infinity is a substance, an attribute, or neither a substance nor an attribute. In order to answer his own question, Aristotle discusses two different possible meanings of infinity. As mentioned earlier, infinity is that which cannot be traversed or that which “whose nature is such that it might be traversed, but is not in fact traversed or bounded”(204a7). Aristotle then quickly adds one more thought: infinity is either infinite by addition, by division, or by neither.
After posing all these questions, Aristotle declares in his heading for Book III.5 that the infinite is not itself a substance. Also, the infinite cannot stand separation from perceptible things and be just itself. To further defend his views Aristotle asks, “how could there be an independent infinite, if there cannot be independent number and magnitude?”(204a17). Thus, infinity is a property of number and magnitude[4]. Aristotle also declares that it is impossible for infinity to exist both in actuality and “to have substantial existence as a principle”(204a20)[5]. It is here that Aristotle first alludes to the distinction between actual and potential infinity that he will later expound. Before articulating this difference, which is essential for Cantor and his set theory, Aristotle declares infinity as “a coincidental attribute of things”(204a29). Accordingly, the infinite is not a principle. Instead, infinity is an attribute of principles.
In order to later refute the idea of actual infinity, Aristotle debates whether or not there is an infinitely extended body. Ultimately, Aristotle denies the existence of an infinitely extended body because he defines body as “‘that which is bounded by surface’”(20b4). That which is bounded cannot be that which is unlimited, or infinite. Furthermore, Aristotle makes a mathematical claim and states that “[n] or can any number that exists apart from perceptible things be infinite either, because number, and anything which has a number, is countable; anything countable can be counted, and it follows that it would be possible to traverse the infinite”(204b6). It is also important to note that, for Aristotle, it is impossible to simultaneously have an infinitely extended body and a place for this body[6]. Hence, there “is no actually infinite body”(206a7).
Although there is no actually infinite body, Aristotle consents to potential infinity. The verb ‘to be’, here, has two different meanings. ‘To be’ means either ‘to be potentially’ or ‘to be actually’ (206a14). However, ‘being potentially’ does not always refer to a future actuality. In terms of potential infinity, Aristotle uses the ‘to be’ analogously “in the same way that we use it of a day or a contest—that is, because one thing happens after another”(206a21). Here, ‘to be’ can be understood as moving successively as long as that which has passed, qua part, is understood as gone. Consequently, infinity is not “‘that which has nothing beyond itself’”, but rather “‘that which always has something beyond itself’”(206b33). The infinite is seemingly dynamic and, because of this dynamism, the infinite is not whole.
Aristotle defines whole “as that which has no part missing”(207a8). Suggested in the definition of “wholeness” is completeness, and what is complete has reached its end. End, in this sense, is a limit. What is whole is limited and, thus, not infinite. Yet, the whole is similar to the infinite. The infinite, unlike the whole, does not contain all things within it. Instead, “the infinite is matter for the completion of a magnitude and is potentially (not actually) the completed whole…and it takes something else to make it whole and finite, which it is not in its own right; and in so far as it is infinite, it does not contain but is contained”(207a21). The idea that the infinite can be contained, however, is contestable. After all, what is a container other than a type of limit?
Both matter and infinity are contained within things and it is form that does the containing. Accordingly, in terms of the fourfold division of causes, infinity is a material cause. Infinity is also not stable “but is being generated, as time is, and as the number of time is”(207b15). Time is infinite, but only in the sense that the parts of time do not persist. Time, in its successive movement, is always changing and, for Aristotle, time is infinite only “if change is infinite”(207b25).

Question for Consideration:
How can we draw together Aristotle’s definition for infinity with becoming and nature (physis)?
Do we agree that the infinite is matter? What are the consequences of this theory?

[1] The first two sentences of the Physics allude towards Aristotle’s goal of gaining scientific knowledge: “In any subject which has principles, causes, and elements, scientific knowledge and understanding stems from a grasp of these…It obviously follows that if we are to gain scientific knowledge of nature as well, we should begin by trying to decide about its principles”(184a10).
[2] Here, Aristotle is drawing reference to Heraclitus’s 41st Fragment which states: “Wisdom is one thing: to understand with true judgment how all things are steered through all” (Richard Hooker ©1995).
[3] Translator Robin Waterfield alerts readers to this fact in a footnote for 203a4 in the 1996 Oxford World’s Classics addition of the Physics.
[4] “In the second place, how could there be an independent infinite, if there cannot be independent number and magnitude? Infinity is in its own right a property of number and magnitude, so, of the three, an independently existing infinite is the least necessary”(204a17).
[5] Translator Robin Waterfield argues that this argument is a weak one, page 250.
[6] “In short, if place cannot be infinite, and if every body occupies place, then there cannot be an infinite body”(205b35).