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    In a continuous manifold, the concept of the manifold and... — Carmelics
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    Home/Modality & Possibility
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    In a continuous manifold, the concept of the manifold and its continuity properties can be separated from its metrical 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.In a discrete manifold, the determination of a set necessarily implies the determination of its quantity or cardinal number.
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    • 2.A continuous manifold does not share this necessary coupling between set determination and quantitative/metrical determination.
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    • 3.Riemann demonstrated that local differential topological structure and metrical structure are logically independent in a continuous manifold.
      ?

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    Reasons Against

    2 perspectives
    Reason against 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.
      ?

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    • 2.The topological notion of a neighborhood, necessary to define continuity, cannot be specified without some measure of proximity that smuggles in metrical commitments.
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    • 3.Therefore, the separation of manifold structure from metrical structure is a post-hoc rational reconstruction, not a genuine metaphysical independence.
      ?

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    Reason against 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.
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    • 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.
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    Related

    A continuous manifold does not share this necessary coupling between set determi...If topological properties of a manifold are only determinate relative to a broad...In a discrete manifold, the determination of a set necessarily implies the deter...Poincaré argued that geometry, including the topology of continuous space, is ch...
    +4 moreShow less
    Riemann demonstrated that local differential topological structure and metrical ...The topological notion of a neighborhood, necessary to define continuity, cannot...Therefore, the separation of manifold structure from metrical structure is a pos...Weyl's own constructive program in 'Das Kontinuum' shows that defining continuit...

    Similar

    Riemann's separation thesis shows that a continuous manifold's topolog...86%If metrical structure is not entailed by the manifold's continuity pro...84%Riemann demonstrated that local differential topological structure and...83%The space problem (das Raumproblem) arises as a genuine philosophical ...80%

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    Under the influence of Gauss and Grassmann, Riemann’s great philosophical contribution consisted in the demonstration that, unlike the case of a discrete manifold, where the determination of a set necessarily implies the determination of its quantity or cardinal number, in the case of a continuous manifold, the concept of such a manifold and of its continuity properties, can be separated form its metrical structure. Using modern terminology, Riemann separated a manifold’s local differentia
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    Details

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