Consideration of the structure of spacetime does not exhaust discussion of physical infinities. There are cases in which physical quantities take on infinite values that have nothing to do with either the small-scale or large-scale structure of spacetime. For example, following the discussion of Ehrlich (1982), we note that modern physics legitimates the notions of negative, infinite, and hotter than infinite temperatures. Understanding how this can be and yet not be problematic requires an understanding of the distinction between extensive and intensive magnitudes; so, after a brief discussion of thermodynamics, we shall conduct an exploration of the relevant parts of the theory of measurement.

Next, we turn our attention to what is, perhaps, the most bewildering role for the concept of the infinite in modern physics: namely, the discussion of the role of renormalisation in quantum field theory. At least initially, it seems that quantum field theories use methods of approximation that produce infinite terms that are then “thrown out” – though only after finite parts of these terms are removed – prior to the production of answers that are in better agreement with observation than are answers in any other branch of science. If there is any part of science that justifies belief in physical infinities, then at least, prima facie, it seems that it will be quantum field theory.