Missing the point in noncommutative geometry

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

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 (a) the metaphysics of such a geometry, and (b) how the appearance of smooth manifold might be recovered as an approximation to a fundamental noncommutative geometry.

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Nick Huggett
University of Illinois, Chicago

Citations of this work

On algebraic naturalism and metaphysical indeterminacy in quantum mechanics.Tushar Menon - 2024 - Studies in History and Philosophy of Science Part A 105 (C):1-16.
Towards noncommutative quantum reality.Otto C. W. Kong - 2022 - Studies in History and Philosophy of Science Part A 92 (C):186-195.

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A determinable-based account of metaphysical indeterminacy.Jessica M. Wilson - 2013 - Inquiry: An Interdisciplinary Journal of Philosophy 56 (4):359-385.
What price spacetime substantivalism? The hole story.John Earman & John Norton - 1987 - British Journal for the Philosophy of Science 38 (4):515-525.

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