John L. Bell - The Continuous, the Discrete and the Infinitesimal in Philosophy and Mathematics

We are all familiar with the idea of continuity . To be continuous is to be separated , like the scattered pebbles on a beach or the leaves on a tree. Continuity connotes unity, while discreteness, plurality.

The continuous and the discrete form a pair of archetypal oppositions.

The domain of the discrete is a model of orderliness, a realm in which quality is entirely reduced to quantity and over which the concept of number reigns supreme. Since the units populating the realm of the discrete do not possess intrinsic qualities which might serve to distinguish them from one another, their difference is manifested through plurality alone. The simplicity of the principles governing discreteness has recommended it as a paragon of intelligibility, a realm within which reason can be realized to its fullest extent.

By contrast, the realm of the continuous is a jungle, a labyrinth. It teems with such exotic and intractable entities as incommensurable lines, horn angles, space curves, and one-sided surfaces. The taming of this jungle by reduction to the discrete has been a principal task, if not the principal task, of mathematics. The origins of this effort lie in the practical procedure of mensuration, of comparing continuous magnitudes in a useful way.

If the constituency of the continuous is complex, it was founded in apparent simplicity. Space and time have traditionally been taken as founding members of that constituency. Certain philosophers have maintained that all natural processes occur continuously: witness, for example, Leibnizs famous apothegm natura non facit saltus nature makes no jump. In mathematics, the word is used in the same general sense but has come to be furnished with increasingly precise definitions. So, for instance, in the later eighteenth century, continuity of a function was taken to mean that infinitesimal changes in the value of the argument induce infinitesimal changes in the value of the function. With the abandonment of infinitesimals in the nineteenth century, this definition gave way to one employing the more precise concept of limit .

While it is the fundamental nature of a continuum to be undivided , it is nevertheless generally (although not invariably) held that any continuum admits of repeated or successive division , namely, that the process of dividing a continuum into ever smaller parts will never terminate in an indivisible or an