Geometric Objects and Perspectivalism

Abstract

The purpose of this article is to consider the metaphysics of geometric and non-geometric objects as they appear in physical theories such as general relativity, and the interactions between these considerations and the contemporary doctrines of perspectivalism and fragmentalism in the philosophy of science. I argue for the following: (i) Taking (following Quine) a kind's being associated with a projectable predicate as a necessary condition for its being natural, there is a sense in which geometric objects can be assimilated to natural kinds, but non-geometric objects cannot; this affords a rational reconstruction of philosophers' and physicists' suspicion of the latter (although this verdict can also be questioned). (ii) Even granting this, nongeometric objects can nevertheless represent real quantities in a perspectival sense-this is one way in which the doctrine of perspectival realism can be endorsed. (iii) More than this: the recognition that non-geometric objects can represent real quantities in a perspectival sense affords support for fragmentalism: the view (at least in part) that frame-dependent effects are physically real. That being said, there are arguments to be made that perspectivalism is superior to this fragmentalism. (iv) There is a certain sense in which perspectivalism should be congenial to proponents of the 'dynamical approach' to spacetime theories-however, the pairing is, in fact, imperfect. (v) Endorsing perspectivalism/fragmentalism in this sense does not commit one to endorsing related-but arguably more opaque-'structuralist' views.

What represents spacetime? And what follows for substantivalism vs. relationalism and gravitational energy?

“…I have explained previously how asking Noether's first theorem how many conserved energies to expect-one for each rigid symmetry of the action, hence infinitely many [Bergmann, 1958]-resolves Schrödinger's false-negative objection [Pitts, 2010]. Lacking a coordinate transformation is not a bug [Read, 2022]. In fact it is a feature: it permits the expression of infinitely many energies with only 10 or 16 components [Pitts, 2010].…”

“…Differential geometry generalized from tensors to "geometric objects" [Nijenhuis, 1952, Schouten, 1954, Yano, 1955, Aczél and Go lab, 1960, Anderson, 1967, Friedman, 1973, Pitts, 2006, Pitts, 2012, Read, 2022, a literature that made interesting progress into the 1960s. While more general notions existed, the basic idea was to generalize the tensor transformation law to any local algebraic transformation law built using derivatives of one coordinate system with respect to another.…”

Causation and the conservation of energy in general relativity

“…The literature has focused on two problems with local conservation laws of the type just discussed: (a) pseudotensorial objects such as t µν are not 'geometric objects', which is to say that they do not have well-defined transformation rules relating their coordinate representations. 14 This appears to preclude the possibility of ascertaining the coordinate-independent reality associated with said coordinate representations (via appeal to, say, the Kleinian approach to geometry-see (Wallace [2019])-which proceeds by identifying the invariants associated with such representations and the transformations between them), and thereby to raise the threat that, if such objects were to be taken seriously, reality would be fundamentally perspectival (for further discussion of this issue, see (Read [2022])).…”

“…Regarding (a), why should geometric objects be the sine qua non of physical theorising? On the one hand, one could embrace the perspectivalism which appears to go hand-in-hand with the use of such objects (Read [2022]). On the other handand more conventionally-one could point out that non-geometric objects abound in physics (most notably, consider the case of spinors), and thereby ask: if one is willing to be a realist in the case of those other quantities and objects, why not be a realist here also?…”

On Two Slights to Noether’s First Theorem: Mental Causation and General Relativity

“…Presumably nothing real can be created or destroyed through mere descriptive choice. The idea that all coordinate systems are admissible and equally good in GR seemed to imply that anything worth discussing in the theory should be tensorial (broadly construed), that is, should admit changing its components in one coordinate system into another coordinate system in an algorithmic way-which is to say, should be a "geometric object" in the sense of classical differential geometry (Nijenhuis, 1952;Anderson, 1967;Pitts, 2006;Duerr, 2021;Read, 2022). (A geometric object's transformation law can be affine rather than linear, can be nonlinear, or can involve higher derivatives, for example.…”