The History of the Continuum

The continuum is a term used in mathematics and physics. It has a variety of meanings, but is most commonly used to refer to a compact connected metric space (Kuratowski 1968; Lewis 1983).

The term “continuum” first appeared in the late 17th century and was defined as a system of points in which each point is associated with exactly as many other points. This is the basic idea behind the ‘continuum hypothesis’, a proposition that asserts that there is no set intermediate in cardinality between the continuum and the natural numbers.

This ‘hypothesis’ has an impact on modern set theory and has been debated since the early 1900s. But the question of whether it is true or not has never been solved and remains an unresolved issue.

A few millennia before Descartes and Newton, place – the topos of Aristotle, according to which bodies move within an absolute space – was still the most important concept. It was, however, an elusive entity, a sort of ether devoid of materiality, and it was impossible to explore directly.

Two millennia after, however, place – and the topos it represented – was transformed into space: that is, the spatial concept of the absolute, or, better, the simple location in a space of infinite extent. This was a very important change, because it allowed to investigate the nature of the continuum in which things happen and exist.

It also changed the focus with respect to which analysing that concept was done: a focus that tended especially to put on the dimensional character of the continuum.

The continuum has a long and interesting history in the world of philosophy and science. There have been several theories, ranging from the classical to the modern, which try to define its nature.

These theories mainly fall into two categories: those that are realist and those that are antirealist.

A realist theory tries to describe the continuum as a physical entity that contains both mass and energy; it also claims that it is not an abstract concept, but something that can be investigated directly, by mathematical techniques.

Its main rival is the antirealist theory which tries to explain it as an abstract, non-physical entity, without any materiality, such as a void or, in some ways, a plenum.

This ‘antirealist’ theory, however, is far from a universal consensus: it can be divided into two groups: the first group is represented by Rene Thom and the second by Kronecker.

Thom, in his book ‘The Physics of the Continuum’, describes a’reverse continuum’ as an alternative way of understanding the continuum that involves both qualitative and quantitative heterogeneity.

He explains this as being an extension of the continuous, but in the sense that it is divisible but factually not divided: a ‘continuum of extended parts’, which he compares with water.

He argues that the experience of physical extension was a key to the perception of the continuum in the human mind. It is, therefore, a secondary analogy that cannot be distinguished from the geometric concept of a continuum, but which is at the basis of man’s ability to comprehend this notion.