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 In a continuous manifold, the concept of the manifold and its continuity properties can be separated from its metrical structure.

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

    Reasons For

    2 perspectives
    Reason for 1 of 2
    ?
    • 1.Weyl's own constructive program in 'Das Kontinuum' shows that defining continuity requires analytic resources that already presuppose metric-like structure.
      ?

      Think about whether this reason is strong or weak

    • 2.The topological notion of a neighborhood, necessary to define continuity, cannot be specified without some measure of proximity that smuggles in metrical commitments.
      ?

      Think about whether this reason is strong or weak

    • 3.Therefore, the separation of manifold structure from metrical structure is a post-hoc rational reconstruction, not a genuine metaphysical independence.
      ?

      Think about whether this reason is strong or weak

    Reason for 2 of 2
    ?
    • 1.Poincaré argued that geometry, including the topology of continuous space, is chosen by convention relative to coordinated physical laws, making pure topological facts theory-laden.
      ?

      Think about whether this reason is strong or weak

    • 2.If topological properties of a manifold are only determinate relative to a broader physical framework, then metric and manifold structure are co-determined, not separable.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.In a discrete manifold, the determination of a set necessarily implies the determination of its quantity or cardinal number.
      ?

      Think about whether this reason is strong or weak

    • 2.A continuous manifold does not share this necessary coupling between set determination and quantitative/metrical determination.
      ?

      Think about whether this reason is strong or weak

    • 3.Riemann demonstrated that local differential topological structure and metrical structure are logically independent in a continuous manifold.
      ?

      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.