Aristotle and Modern Mathematical Theories of the Continuum

The mathematical structure of the continuum, in the guise of the domain of continuous, differentiable functions, has proved immensely useful in the study of nature. However, we have learned to be sceptical of any claim to the effect that our current favourite mathematical theory necessarily describes the actual structure of the physical universe. The continuous manifold of space-time may be no more than a helpful idealisation, when in fact space-time has a minutely grainy or quantised structure. Nonetheless, the question of whether the classical continuum is an accurate representation of the structure of space-time is a separate question from the one which we have to answer today. We are interested in the mathematical concept of the continuum itself. In saying that we may develop a mathematical theory of the continuum regardless of whether such a continuum is actually to be found in the universe, we are relying on the premise that there is such a thing as pure mathematics, a body of knowledge whose evidential basis rests on something other than observation of the physical world. Consider, for example, the proposition that the tangent to a point P on a circle's circumference is perpendicular to the radius connecting P and the centre of the circle. As a proposition of pure mathematics, this proposition is true independently of whether there really are entities in the physical world that meet the mathematical definition of a circle, namely, that of a figure all