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    Carmelics

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    Home/Original/inverse
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    Inverse View

    It is not the case that The cross-ratio of four collinear points P, Q, R, S is uniquely determined

    ?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 formula AB·CD / AD·CB presupposes a metric structure, yet projective geometry is defined precisely by its independence from metric relations.
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      Think about whether this reason is strong or weak

    • 2.Importing Euclidean length ratios into a projective proof smuggles in assumptions that are unavailable within the projective framework being described.
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      Think about whether this reason is strong or weak

    • 3.A genuinely projective derivation must ground cross-ratio invariance in incidence relations alone, not in coordinate distances assigned after an arbitrary mapping.
      ?

      Think about whether this reason is strong or weak

    Reason for 2 of 2
    ?
    • 1.The mapping of C to the point at infinity is a conventional stipulation, not a necessary geometric fact, making the 'uniqueness' result dependent on a freely chosen normalization.
      ?

      Think about whether this reason is strong or weak

    • 2.Poincaré argued in 'La Science et l'Hypothèse' that geometric conventions are adopted for convenience, so results derived via conventional coordinate assignments carry no stronger necessity than the convention itself.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.The four points can be mapped onto points A, B, C, D on the real line with A at the origin, C at infinity, and D at 1
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      Think about whether this reason is strong or weak

    • 2.Given those three fixed positions, the position of B is uniquely determined
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      Think about whether this reason is strong or weak

    • 3.The cross-ratio equals AB·CD / AD·CB, which equals x when AB has length x
      ?

      Think about whether this reason is strong or weak

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    Strongest counterpoint
    Explore the most compelling reason on the other side.