The Mathematical Overdetermination of Physics

Abstract

You are fully entitled not to know what I mean by the “mathematical overdetermination of physics”. As a first approximation to such an understanding I would like to remind you of the related though different so-called theoretical overdetermination of a corpus of observational data. Just as a physical theory often exhibits an unnecessary rich structure when compared with the observational data to be explained by it, so the mathematics introduced to formulate a physical theory frequently brings a wealth of structures into play that cannot be matched by the physical elements of that theory. One might even be tempted to identify the two cases. But the distinction between theoretical and observational terms, on which the second overdetermination is based, is generally different from the distinction between mathematical and physical terms. Theoretical terms may be intended to have physical referents, though unobservable ones. By contrast, mathematical terms occurring in a physical theory may be meant to have no physical interpretation within this theory. Yet there are structural similarities between the two cases. In both we find ourselves deluded in the expectation that for a precise reformulation of an informally given corpus of statements only two things have to be considered: (1) the concepts characteristic for the corpus in question, and (2) the logical notions binding together those concepts. Rather a third component has to be taken into account.