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
    There must be gaps between the rational numbers that requ... — Carmelics
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Home/Modality & Possibility
    HistoryEditSee Inverse

    There must be gaps between the rational numbers that require filling with irrational numbers to account for continuous motion.

    CausationModality & Possibility
    ?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.If an object moves continuously from A to B, it traverses all distances along that path.
      ?

      Think about whether this reason is strong or weak

    • 2.The rational numbers are only dense, not continuous — an object traveling only rational distances would skip some intervals.
      ?

      Think about whether this reason is strong or weak

    • 3.Continuous motion therefore traverses distances that cannot be commensurately measured by rational numbers alone.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Continuous motion is a physical phenomenon whose mathematical description is a modeling choice, not a metaphysical necessity.
      ?

      Think about whether this reason is strong or weak

    • 2.A model using only rational-valued coordinates can approximate any physical trajectory to within any finite precision demanded by measurement.
      ?

      Think about whether this reason is strong or weak

    • 3.The alleged 'gaps' are artifacts of conflating the mathematical real line with physical space, not discovered features of motion itself.
      ?

      Think about whether this reason is strong or weak

    Reason against 2 of 2
    ?
    • 1.Wittgenstein argued that irrational numbers are not discovered by revealing pre-existing gaps but are constructed through extending mathematical grammar via new rules.
      ?

      Think about whether this reason is strong or weak

    • 2.The supporting argument illicitly treats the density-of-rationals as a spatial deficiency rather than a feature of how we apply number concepts to extension.
      ?

      Think about whether this reason is strong or weak

    • 3.No physical experiment can detect a trajectory passing through an irrational rather than a nearby rational coordinate, undermining the causal-explanatory claim.
      ?

      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 & PossibilityCausation

    Connections

    1 topic

    Truth & Knowledge2 linked

    Related

    A model using only rational-valued coordinates can approximate any physical traj...Continuous motion is a physical phenomenon whose mathematical description is a m...Continuous motion therefore traverses distances that cannot be commensurately me...Distances not measurable by rationals correspond to gaps between rational points...
    +6 moreShow less
    If an object moves continuously from A to B, it traverses all distances along th...No physical experiment can detect a trajectory passing through an irrational rat...The alleged 'gaps' are artifacts of conflating the mathematical real line with p...The rational numbers are only dense, not continuous — an object traveling only r...The supporting argument illicitly treats the density-of-rationals as a spatial d...Wittgenstein argued that irrational numbers are not discovered by revealing pre-...

    Similar

    Recursive irrationals alone are insufficient to fill all gaps in the r...84%The rational numbers are only dense, not continuous — an object travel...80%Distances not measurable by rationals correspond to gaps between ratio...79%Adding recursive irrationals to the rationals still leaves gaps in the...78%

    Source

    AI-extracted1/3 agreementValid
    SEP: wittgenstein-mathematics
    View source passageHide passage
    The problem, as Wittgenstein sees it, is that mathematicians, especially foundationalists (e.g., set theorists), have sought to accommodate physical continuity by a theory that ‘describes’ the mathematical continuum (PR §171). When, for example, we think of continuous motion and the (mere) density of the rationals, we reason that if an object moves continuously from A to B, and it travels only the distances marked by “rational points”, then it must skip some distances (intervals, or points) not
    Extraction notes

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

    Details

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