ABSTRACT
This chapter uses 'infinity' to describe the endlessly nested possibilities that finite things afford. The language of infinity however literally or metaphorically understood. And it means that, when we consider the paradoxes of the infinitely small, we must not be seduced by consistent mathematical formulae in to thinking that the relevant super task stories are really coherent. The formulae need to be interpreted. The chapter discusses the larger scale that is, to reassess the history in the light of the ideas that have begun to emerge and, at the same time, to develop the ideas. It assumes that the truly mathematically infinite is something that resists mathematical scrutiny. This follows the further assumption that they are susceptible to mathematical scrutiny. It is a problem for anyone such as Aristotle, L. Wittgenstein, and perhaps M. Dummett who believes that there are infinitely many natural numbers, and that there is therefore a sense in which they can never all be constructed.
