Synthese 199 (1-2):4695-4728 ()

Nick Huggett
University of Illinois, Chicago
Noncommutative geometries generalize standard smooth geometries, parametrizing the noncommutativity of dimensions with a fundamental quantity with the dimensions of area. The question arises then of whether the concept of a region smaller than the scale—and ultimately the concept of a point—makes sense in such a theory. We argue that it does not, in two interrelated ways. In the context of Connes’ spectral triple approach, we show that arbitrarily small regions are not definable in the formal sense. While in the scalar field Moyal–Weyl approach, we show that they cannot be given an operational definition. We conclude that points do not exist in such geometries. We therefore investigate the metaphysics of such a geometry, and how the appearance of smooth manifold might be recovered as an approximation to a fundamental noncommutative geometry.
Keywords spacetime  geometry   noncommutative geometry  quantum gravity
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Reprint years 2021
DOI 10.1007/s11229-020-02998-1
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References found in this work BETA

The Metaphysics Within Physics.Tim Maudlin - 2007 - Oxford University Press.
Emergent Spacetime and Empirical (in) Coherence.Nick Huggett & Christian Wüthrich - 2013 - Studies in History and Philosophy of Modern Physics 44 (3):276-285.
What Price Spacetime Substantivalism? The Hole Story.John Earman & John Norton - 1987 - British Journal for the Philosophy of Science 38 (4):515-525.
Quantum Metaphysical Indeterminacy.Claudio Calosi & Jessica Wilson - 2019 - Philosophical Studies 176 (10):2599–2627.

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Towards Noncommutative Quantum Reality.Otto C. W. Kong - 2022 - Studies in History and Philosophy of Science Part A 92:186-195.

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