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    The choice of conic κ determines which geometry — Euclide... — Carmelics
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Home/Modality & Possibility
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    The choice of conic κ determines which geometry — Euclidean, Lobachevskian, or elliptic — is realized by the resulting metric structure.

    Modality & PossibilityTruth & 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.Klein's distance function on region R is determined by the nature of the conic κ.
      ?

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    • 2.Different conics yield metric structures satisfying either Euclidean, Lobachevskian, or elliptic theorems.
      ?

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    • 3.Each of these three geometries is internally consistent and derivable from the projective construction.
      ?

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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.The projective plane lacks an intrinsic metric, so the choice of κ imposes rather than discovers geometrical structure.
      ?

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    • 2.Poincaré's conventionalism holds that no empirical or formal fact compels one metrical interpretation over another.
      ?

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    • 3.If geometry is conventional, κ selects a descriptive framework, not a realized geometry in any mind-independent sense.
      ?

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    Reason against 2 of 2
    ?
    • 1.Klein's construction presupposes a background absolute conic defined over the real projective plane, which is not itself projectively invariant.
      ?

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    • 2.Cayley's original complaint stands: metric concepts smuggled into projective foundations undermine the claim that metric geometry is grounded in projective geometry alone.
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    Related

    Cayley's original complaint stands: metric concepts smuggled into projective fou...Different conics yield metric structures satisfying either Euclidean, Lobachevsk...Each of these three geometries is internally consistent and derivable from the p...If geometry is conventional, κ selects a descriptive framework, not a realized g...
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    Klein's construction presupposes a background absolute conic defined over the re...Klein's distance function on region R is determined by the nature of the conic κ...Poincaré's conventionalism holds that no empirical or formal fact compels one me...The projective plane lacks an intrinsic metric, so the choice of κ imposes rathe...

    Similar

    Different conics yield metric structures satisfying either Euclidean, ...87%Klein's distance function on region R is determined by the nature of t...74%Weyl showed mathematically that the conformal structure can be derived...72%The Riemannian metric structure underlying Einstein's general theory i...71%

    Source

    AI-extracted1/3 agreementValid
    SEP: geometry-19th
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    Can metric properties be fixed in this way? Traditionally one defines the distance between two points (x1, … ,xn) and (y1, … ,yn) of a numerical manifold as the positive square root of (x1 − y1) 2 + … + (xn − y n)2. The group of isometries consists of the transformations that preserve this function. However, this is just a convention, adopted to ensure that the geometry is Euclidean. Using projective geometry, Klein thought of something better. No real-valued function of point pairs, defined on
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    Validity: Extracted via Max plan + API grounding/validity checks

    Details

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