Surplus Structure from the Standpoint of Transcendental Idealism: The “World Geometries” of Weyl and Eddington

Abstract

Recent discussions of “structural realism” in philosophy of science, jointly with a newly burgeoning philosophical interest in gauge aeld theories, have resuscitated in narrow compass the perennial puzzle attending the role, or roles, of mathematical representation in the formulation of physical theories. Structural realists urge that the continuity across theory change deemed characteristic of the “mature sciences” occurs only at the formal mathematical level, rather than of any particular entities posited by either preceding or successor theory. The realist intuition that science yields closer and closer “approximations” to the truth about nature is then redeemed as a growing accumulation of mathematical form or structure (Worrall 1989). The sought continuity between older and newer theory hence lies in a correspondence between their fundamental equations. In this way, mature sciences are regarded as tracking the relational structure of physical reality, the real relations among unobservable entities and not the content or nature of those entities. However, precisely which parts of a theory’s mathematical representation describe physical reality or, more modestly, could be considered to have a physical correlate, is a paramount issue in the interpretation of the gauge theories of modern particle physics. Here the presence of additional, or “surplus,” mathematical degrees of freedom introduce ambiguity into the physical system’s mathematical characterization. Famously, the four coordinate degrees of freedom im-