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    Carmelics

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    Made withinDC&Austin
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    Home/Original/inverse
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    Inverse View

    It is not the case that Cross-ratio can be defined intrinsically in projective geometry using quadruples of collinear points

    ?Set your confidence on the premises below to see your aggregate.

    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 1.The cross-ratio requires coordinatization by real numbers, and this numerical structure is not derivable from purely projective incidence axioms alone.
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    • 2.Pappus's theorem must be assumed as an independent axiom to guarantee commutativity of the coordinate field, revealing an algebraic presupposition external to projective geometry.
      ?

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    • 3.Therefore the cross-ratio is not intrinsic to projective geometry but depends on importing Euclidean or arithmetic structure through the back door.
      ?

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    Reason for 2 of 2
    ?
    • 1.Von Staudt's construction of the cross-ratio from the algebra of throws presupposes a continuous ordered field, which is a topological rather than purely projective datum.
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    • 2.Klein and Pasch recognized that completeness assumptions about the real line smuggle in metric concepts that projective geometry was meant to precede foundationally.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.The projective group preserves straight lines
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    • 2.Any ordered triple of distinct collinear points can be mapped uniquely to any other ordered triple of distinct collinear points
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    • 3.No transformation in the projective group can map an arbitrary quadruple of collinear points onto an arbitrary such quadruple
      ?

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