Ketland, Jeffrey (2023) Length Abstraction in Euclidean Geometry. [Preprint]

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Abstract

I define abstract lengths in Euclidean geometry, by introducing an abstraction axiom: $\lambda(a,b) = \lambda(c,d) \iff ab \equiv cd$. By geometric constructions and explicit definitions, one may define the \emph{Length structure}: $\Len = (\Len, \oplus, \preceq, \bigcdot)$, ``instantiated by Euclidean geometry'', so to speak. I define the notion of a ``(continuous) positive extensive quantity'' and prove that $\Len$ is such a (continuous) positive extensive quantity. The main results given provide the general characterization of $\Len$ and its symmetry group (the multiplicative group of the positive reals); along with the relevant mathematical relationships between (abstract) lengths and \emph{coordinate} lengths (relative to a coordinate system); and also between lengths, measurement scales and units for length.

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Item Type:	Preprint
Creators:	Creators Email ORCID Ketland, Jeffrey jeffreyketland@gmail.com 0000-0002-5128-4387
Keywords:	Geometry; quantities; measurement; abstraction principle; representation theorem; length
Subjects:	Specific Sciences > Physics General Issues > Structure of Theories
Depositing User:	Dr Jeffrey Ketland
Date Deposited:	05 Feb 2023 14:01
Last Modified:	05 Feb 2023 14:01
Item ID:	21714
Subjects:	Specific Sciences > Physics General Issues > Structure of Theories
Date:	3 February 2023
URI:	http://philsci-archive.pitt.edu/id/eprint/21714

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