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    The cross-ratio of four collinear points P, Q, R, S is un... — Carmelics
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    The cross-ratio of four collinear points P, Q, R, S is uniquely determined

    Truth & Knowledge
    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 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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    • 2.Given those three fixed positions, the position of B is uniquely determined
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    • 3.The cross-ratio equals AB·CD / AD·CB, which equals x when AB has length x
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 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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    • 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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    • 3.A genuinely projective derivation must ground cross-ratio invariance in incidence relations alone, not in coordinate distances assigned after an arbitrary mapping.
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    Reason against 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.
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    • 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.
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    Related

    A genuinely projective derivation must ground cross-ratio invariance in incidenc...Given those three fixed positions, the position of B is uniquely determinedImporting Euclidean length ratios into a projective proof smuggles in assumption...Poincaré argued in 'La Science et l'Hypothèse' that geometric conventions are ad...
    +4 moreShow less
    The cross-ratio equals AB·CD / AD·CB, which equals x when AB has length xThe cross-ratio formula AB·CD / AD·CB presupposes a metric structure, yet projec...The four points can be mapped onto points A, B, C, D on the real line with A at ...The mapping of C to the point at infinity is a conventional stipulation, not a n...

    Similar

    Cross-ratio can be defined intrinsically in projective geometry using ...80%The cross-ratio of collinear point quadruples is an invariant of the p...79%Any ordered triple of distinct collinear points can be mapped uniquely...75%The cross-ratio equals AB·CD / AD·CB, which equals x when AB has lengt...74%

    Source

    AI-extracted1/3 agreementValid
    SEP: epistemology-geometry
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    Klein’s insight, following von Staudt, was that an exactly similar argument involving quadruples of collinear points can be used to define cross-ratio in projective geometry. The projective group preserves straight lines, and any ordered triple of collinear points can be mapped to any ordered triple of collinear points, and the map that sends a given ordered triple of distinct points to another ordered triple of distinct points is unique, but there is no transformation in the group that can map
    Extraction notes

    Validity: Extracted via Max plan + API grounding/validity checks

    Details

    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit