Skip to content
Carmelics
TopicsThinkersChangesContributorsLoading account…

    Carmelics

    A reasoning platform. Break down any belief into clear reasons, explore both sides, and weigh the evidence honestly.

    Navigate

    • Topics
    • Search
    • Recent Changes
    • Contribute
    • How It Works
    • Glossary
    • Thinkers
    • Contributors
    • About
    • Statistics
    • Terms
    • Privacy

    Database

    Statements
    —
    Perspectives
    —
    Topics
    —

    Press ? for keyboard shortcuts

    LoyalLoyalJusticeJustice
    Made withinDC&Austin
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Home/Original/inverse
    See Original
    Inverse View

    It is not the case that Metric properties of a geometry can be defined intrinsically using projective invariants rather than by convention from a numerical manifold.

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

    Reasons For

    2 perspectives
    Reason for 1 of 2
    ?
    • 1.The selection of the absolute conic κ is itself a conventional act that imports metric structure prior to the projective derivation.
      ?

      Think about whether this reason is strong or weak

    • 2.Poincaré argued in 'Science and Hypothesis' that geometric axioms are conventions, not truths, so privileging one conic over others requires extra-projective justification.
      ?

      Think about whether this reason is strong or weak

    • 3.Without an independent criterion for selecting κ, the Klein construction relocates rather than eliminates the conventional element in metric geometry.
      ?

      Think about whether this reason is strong or weak

    Reason for 2 of 2
    ?
    • 1.Projective geometry presupposes an ambient space with enough structure to define collinearity, harmonic conjugates, and incidence relations.
      ?

      Think about whether this reason is strong or weak

    • 2.Frege's analysis of implicit definition shows that deriving metric notions from projective ones does not reduce ontological commitment if projective space itself harbors metric presuppositions.
      ?

      Think about whether this reason is strong or weak

    • 3.Therefore the claim of intrinsic, convention-free metric definition conceals a regress in which projectivity already encodes the structure it purports to ground.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.The cross-ratio of collinear point quadruples is an invariant of the projective group.
      ?

      Think about whether this reason is strong or weak

    • 2.By fixing a conic κ and ranging point pairs over the region bounded by κ, the cross-ratio becomes a function of point pairs alone.
      ?

      Think about whether this reason is strong or weak

    • 3.A certain function of this cross-ratio behaves like an ordinary distance function on the region R.
      ?

      Think about whether this reason is strong or weak

    Next step

    Based on where you are in your exploration

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