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
    Metric properties of a geometry can be defined intrinsica... — Carmelics
    Home/Modality & Possibility
    HistoryEditSee Inverse

    Metric properties of a geometry can be defined intrinsically using projective invariants rather than by convention from a numerical manifold.

    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.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

    Reasons Against

    2 perspectives
    Reason against 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 against 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

    Sign in or register to share your perspective on this statement.

    Next step

    Based on where you are in your exploration

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

    Topics

    Modality & PossibilityTruth & Knowledge

    Connections

    1 topic

    Causation2 linked

    Related

    A certain function of this cross-ratio behaves like an ordinary distance functio...By fixing a conic κ and ranging point pairs over the region bounded by κ, the cr...Frege's analysis of implicit definition shows that deriving metric notions from ...Poincaré argued in 'Science and Hypothesis' that geometric axioms are convention...
    +6 moreShow less
    Projective geometry presupposes an ambient space with enough structure to define...The collineations mapping a given conic onto itself form a group under which thi...The cross-ratio of collinear point quadruples is an invariant of the projective ...The selection of the absolute conic κ is itself a conventional act that imports ...Therefore the claim of intrinsic, convention-free metric definition conceals a r...Without an independent criterion for selecting κ, the Klein construction relocat...

    Similar

    Each classical geometry is uniquely characterized by a particular grou...77%Euclidean geometry is characterized by the group of translations and r...74%Each of these three geometries is internally consistent and derivable ...74%Alternative conceptions of vector properties on which they cannot be i...74%

    Source

    AI-extracted1/3 agreementValid
    SEP: geometry-19th
    View source passageHide passage
    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
    Extraction notes

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

    Details

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