Dissenting voices

Continuous entities are accordingly distinguished by the feature that—in principle at least— they can be divided indefinitely without altering their essential nature. So, for instance, the water in a bucket may be indefinitely halved and yet remain water. Aristotle nowhere to my knowledge defines discreteness as such but we may take the notion as signifying the opposite of continuity—that is, incapable of being indefinitely divided into parts. Thus discrete entities, typically, cannot be divided without effecting a change in their nature: half a wheel is plainly no longer a wheel1. Thus we have two contrasting properties: on the one hand, the property of being indivisible, separate or discrete, and, on the other, the property of being indefinitely divisible and continuous although not actually divided into parts. Still, one and the same object can, in a sense, possess both of these properties. For example, if the wheel is regarded simply as a piece of matter, it remains so on being divided in half. In other words, the wheel qua wheel is discrete, but as a piece of matter, it is continuous. Examples like this show that continuity and discreteness are complementary attributes originating through the mind's ability to perform acts of abstraction, the one arising by abstracting an object’s divisibility and the other its self-identity. In mathematics it is the concept of whole number, later elaborated into the set concept, that provides an embodiment of the idea of pure discreteness, that is, of the idea of a collection of separate individual objects, all of whose properties—apart from their distinctness—have been refined away. The basic mathematical representation of the idea of continuity, on the other hand, is the geometric figure, and more particularly the straight line. By their very nature geometric figures are continuous; discreteness is injected into geometry, the realm of the..
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