- Surplus Structure from the Standpoint of Transcendental Idealism:The "World Geometries" of Weyl and Eddington

Recent discussions of "structural realism" in philosophy of science, jointly with a newly burgeoning philosophical interest in gauge field 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 imposed **[End Page 76]** by requirement of general covariance in Einstein's gravitational theory raised the specter of an *indeterministic* evolution of its dynamical variables from a given initial data set (Hilbert 1915). Modern Yang-Mills theories, exhibiting a different kind of "gauge freedom" arising from a stipulation that certain "internal" quantities (not associated with a space-time coordinate transformation) vary independently from space-time point to point, similarly fail to have well-posed initial value problems. As it turns out, in the later case the degrees of local gauge freedom are exactly "compensated" by the simultaneous introduction of a new physical ("gauge") field into the mathematical description. In several important cases, the particles ("conserved currents") of such fields have actually been observed, whereas the theory's dynamics (its Lagrangian) remains invariant under the corresponding "gauge transformations" removing the gauge degrees of freedom (e.g., Mills 1989; O'Raifeartaigh and Straumann 2000; Teller 2000). Phenomenologically this "gauge argument" has been extraordinarily successful in the electroweak theory and in quantum chromodynamics even though there is no immediate physical counterpart for the demanded local invariance. In all these cases, the physical system in question is controlled by requirements imposed within the "surplus" mathematical structure. In itself, this is a rather astonishing fact. When considered as different species of gauge field theories, the similarity between general relativity and Yang-Mills theories is brought out even more strikingly in the suggestion, by Yang and others, that gauge theories have an intrinsically geometrical character within the principal fiber bundle formalism of modern differential geometry (Yang 1977; Drechsler and Mayer 1977). Thus gauge theories appear to furnish grist for the mill of those who fancy "a mysterious, even mystical, Platonist-Pythagorean role for purely mathematical considerations in theoretical physics," a situation recently described as "quite congenial to most practicing physicists" (Redhead 2001b, p. 13).

Obviously, these are deep issues that will not be resolved here. Yet the perceptibly growing view among physicists and philosophers that nature is inherently structural or mathematical perhaps testifies not so much to the power of argument but rather to the paucity of alternative accounts of the nature and meaning of mathematical representation in physics. We must go back to the early days of general relativity to find one such alternative in the bold attempts to extend Einstein's geometrization of gravitation to electromagnetism, the only other known fundamental interaction before the discovery of the "weak" nuclear weak force in the 1930s. Significantly, this is also the locus of the origin of the idea of gauge invariance, though not quite in the modern sense. The mathematician **[End Page 77]** Hermann Weyl is, of course, the central figure, but for our purposes, the clearest application and extension of Weyl's initial idea lies in a little...