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Smooth Infinitesimals in the Metaphysical Foundation of Spacetime Theories

Chen, Lu and Chen, Lu (2022) Smooth Infinitesimals in the Metaphysical Foundation of Spacetime Theories. [Preprint]

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Abstract

I propose a theory of space with infinitesimal regions called smooth infinitesimal geometry (SIG) based on certain algebraic objects (i.e., rings), which regiments a mode of reasoning heuristically used by geometricists and physicists (e.g., circle is composed of infinitely many straight lines). I argue that SIG has the following utilities. (1) It provides a simple metaphysics of vector fields and tangent space
that are otherwise perplexing. A tangent space can be considered an infinitesimal region of space. (2) It generalizes a standard implementation of spacetime algebraicism (according to which physical fields exist fundamentally without an underlying manifold) called Einstein algebras. (3) It solves the long-standing problem of interpreting smooth infinitesimal analysis (SIA) realistically, an alternative foundation of spacetime theories to real analysis (Lawvere 1980). SIA is formulated in intuitionistic logic and is thought to have no classical reformulations (Hellman 2006). Against this, I argue that SIG is (part of) such a reformulation. But SIG has an unorthodox mereology, in which the principle of supplementation fails.


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Item Type: Preprint
Creators:
CreatorsEmailORCID
Chen, Luluc@umass.edu
Chen, Luluchen@ku.edu.tr
Keywords: Continuum; smooth infinitesimal geometry; smooth infinitesimal analysis; vectorial quantity; tangent space; Einstein algebras; nonclassical mereology.
Subjects: Specific Sciences > Mathematics > Logic
Specific Sciences > Physics > Classical Physics
Depositing User: Dr. Lu Chen
Date Deposited: 06 Mar 2022 20:24
Last Modified: 06 Mar 2022 20:24
Item ID: 20118
Subjects: Specific Sciences > Mathematics > Logic
Specific Sciences > Physics > Classical Physics
Date: 2022
URI: http://philsci-archive.pitt.edu/id/eprint/20118

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