Earman and Norton's hole argument demonstrates that substantival geometric fields underdetermine physical reality, undermining the claim that such a field is a well-defined real entity.

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Earman and Norton(as the authors being referenced): Two philosophers (John Earman and John D. Norton) who wrote influential papers about 'supertasks'—thought experiments involving infinite sequences of actions happening in finite time.
Physical reality(as used in metaphysics): The material world we can observe and interact with—all the concrete stuff like objects, energy, and matter.
Substantival geometric fields(as what the argument challenges): The idea that space and time are real, concrete things (like a substance) that exist independently, rather than just being ways we measure distances between objects.
The hole argument(as used in philosophy of physics and metaphysics): A thought experiment in philosophy of physics (developed by John Earman and David Norton) that asks whether two physical situations that differ only in how we label points in spacetime are actually the same or different; it challenges whether our descriptions of reality are truly capturing what's real versus just how we're representing it.
Underdetermine(in epistemology (theory of knowledge)): When something doesn't give you enough information to figure out the complete answer—it leaves multiple possibilities still open.
Well-defined real entity(describing what a geometric field may or may not be): Something that clearly exists as a definite, concrete thing rather than being vague, ambiguous, or merely a useful tool for calculation.

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