The Continuous and the Discrete

Abstract

THE RELATIONSHIP BETWEEN THE IDEAS of continuity and discreteness has played no less important a role in the development of mathematics than it has in science and philosophy. Continuous entities are characterized by the fact that they can be divided indefinitely without altering their essential nature. So, for instance, the water in a bucket may be continually halved and yet remain water¹. Discrete entities, on the other hand, typically cannot be divided without effecting a change in their nature: half a wheel is plainly no longer a wheel. 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.