Pitts, J. Brian (2022) On Two Slights to Noether's First Theorem: Mental Causation and General Relativity. Rethinking the Concept of Laws of Nature: Natural Order in the Light of Contemporary Science.

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Abstract
It is widely held among philosophers that the conservation of energy is true and important, and widely held among philosophers of science that conservation laws and symmetries are tied together by Noether's first theorem (that rigid symmetries yield conservation). However, beneath the surface of such consensus lie two slights to Noether's first theorem.
First, there is a 325+year controversy about mindbody interaction in relation to the conservation of energy and momentum, with occasional reversals of opinion. The currently popular Leibnizian view, dominant since the late 19th century, claims to find an objection to broadly Cartesian views (and nonepiphenomenalist property dualism) in their implication of energy nonconservation. Here energy conservation is viewed as an oracle, an unchallengeable black box. But Noether's first theorem and its converse show that conservation and symmetry of the laws stand or fall together. Absent some basis for expecting conservation in brains that has a claim on the Cartesian (whose view implies the absence of law symmetries in brains), the objection is circular. An empirically based argument is possible, but is a different argument with little force except insofar as it is rooted in neuroscience.
Second, General Relativity has a 100+yearlong controversy about whether gravitational energy exists and is objectively localized. The usual view is that gravitational energy exists but is not objectively localized, though some deny its existence. Without positive answers to both questions, generally applicable conservation laws do not exist: energy is not conserved. This conclusion is startling in itself and a problem for conserved quantity theories of causation. Yet Noether's first theorem applies to General Relativity, which has uncountably many symmetries of its laws and so has conservation laws, indeed uncountably many of them. Many authors downplay these laws due to their quirky properties; some authors even attempt to explain the laws' supposed nonexistence in terms of an absence of symmetries of the geometry, which is a distraction. Thus Noether's first theorem is widely ignored, left uninterpreted, or distorted in relation to General Relativity. Taking the theorem seriously seems possible, however, restoring the conservation of energy, or rather, energies.
How do these controversies relate? One sometimes finds claims that General Relativity's supposed lack of conservation laws answers Leibniz on behalf of Descartes. Taking seriously the superabundance of formal conservation laws in General Relativity, however, suggests that General Relativity resists (not facilitates) mindtobody causation. This conclusion can be proven apart from interpretive controversies. The resistance is, however, finite and tends to be swamped by larger worldview considerations.
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