I don't think these examples get to the heart of the issue with betweenness. This kind of thing is not why people like Hilbert formalised betweenness, I believe. (I think it had more to do with diagrammatic reasoning issues, such as why bisector of angle ABC must intersect AC.)
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The problems mentioned do not occur if two points determine a unique line, which Euclid was clearly highly aware of assuming (e.g. in Prop. I.4), so they do not demonstrate any need for treating betweenness within Euclid's system.
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Looking forward to this book.
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Thanks for the vote of confidence! It will be in production very soon.
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The concept of 'between' does break down in higher dimension. But isn't it well-defined in 1-D, where Euclid seems to have used it?
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Axioms of betweenness appear in the geometries of Hilbert and Tarski, which can be seen as corrections of Euclid, who had used betweenness implicitly, without any axioms governing it. But it is not just about dimension, since betweenness is problematic on a circle, in 1D.
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Awesome. (Kids are smart!!!)
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A non-spatial reinterpretation of geometry quickly exposes many implicit assumptions. Example: Interpret point as person and line as people sharing a mutation. Most proofs no longer apply.
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