Summary and Discussion

We have isolated three uses of infinities in physics: dense, range-of-values, and useful-fiction. Range-of-values seems generally unproblematic, but illustrative of the idea that our understanding of the universe can involve infinities of some sort. These are not directly measured, but rather are supported by a mix of inductive reasoning and evidence; and they do not seem to present major difficulties.

The infinities in the Benardete example perhaps lie in between range-of-values and dense: many location values are posited, yet they come close to representing an infinite density of something—but it is not clear just what. And there is no outright paradox if we apply ordinary physics and a little commonsense.

But when we replace imagined barriers with actual physical entities such as fields, things can quickly get bizarre, as in infinite values for charge and force and mass densities. While our discussion focused on a point-model of the electron, any point- source field will do. There are standard tools for representing this—for instance the delta function—but these are usually seen as merely useful calculational devices and not as possible models of what the universe is like. I am arguing that the great success of such tools speaks to the strong possibility of an underlying phenomenon well-worth trying to model.

I am not urging that NSA need be the mathematical physics of the future. There are certainly other directions to consider, such as the surreal numbers studied by Conway, Kruskal and others (see Wikipedia entry [29]). In addition, Bell [3] presents an approach to infinitesimals (but not infinities) based on “smooth worlds” where logic (and geometry) gets even stranger than in NSA yet where physics again comes into play. And indeed infinity (of the dense kind) might happen not to be physically sensible at all. But the idea should not be discarded out of hand.