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
    Projective geometry presupposes an ambient space with eno... — Carmelics
    Home
    HistoryEditSee Inverse

    Part of a larger discussion

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

    Projective geometry presupposes an ambient space with enough structure to define collinearity, harmonic conjugates, and incidence relations.

    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    1 reason against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.Harmonic conjugates and cross-ratios require a field structure that presupposes an underlying ambient space with metric or affine properties.
      ?

      Think about whether this reason is strong or weak

    • 2.Incidence relations (point-line containment) are meaningless without a topological or set-theoretic framework defining distinct geometric objects.
      ?

      Think about whether this reason is strong or weak

    • 3.Collinearity presupposes a notion of 'betweenness' or alignment that only emerges from an ordered or structured ambient continuum.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Projective geometry can be defined purely axiomatically (Desargues' theorem, axioms of incidence) without reference to any ambient space whatsoever.
      ?

      Think about whether this reason is strong or weak

    • 2.Harmonic conjugates and cross-ratios are definable through synthetic constructions (complete quadrilaterals) that don't require metric or field structure.
      ?

      Think about whether this reason is strong or weak

    • 3.Modern category-theoretic formulations treat projective geometry as a structure arising from abstract incidence axioms, not presupposing ambient space.
      ?

      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.

    Connections

    2 topics

    Truth & Knowledge1 linkedModality & Possibility1 linked

    Related

    Collinearity presupposes a notion of 'betweenness' or alignment that only emerge...Harmonic conjugates and cross-ratios are definable through synthetic constructio...Harmonic conjugates and cross-ratios require a field structure that presupposes ...Incidence relations (point-line containment) are meaningless without a topologic...
    +3 moreShow less
    Metric properties of a geometry can be defined intrinsically using projective in...Modern category-theoretic formulations treat projective geometry as a structure ...Projective geometry can be defined purely axiomatically (Desargues' theorem, axi...

    Details

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