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
    The term 'irrational number' should be extended to includ... — Carmelics
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Home/Philosophy of Language
    HistoryEditSee Inverse

    The term 'irrational number' should be extended to include lawless and pseudo-irrationals

    Philosophy of LanguageProof of definition segments
    ?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.Lawless and pseudo-irrationals are needed for the mathematical continuum
      ?

      Think about whether this reason is strong or weak

    • 2.Lawless and pseudo-irrationals are more like rule-governed irrationals than like rationals
      ?

      Think about whether this reason is strong or weak

    • 3.Mathematical terms may be extended to cover conceivable numbers that fit more naturally within an existing category than outside it
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Mathematical terms gain their meaning from the rules and techniques governing their use, not from phenomenological similarity to other objects.
      ?

      Think about whether this reason is strong or weak

    • 2.Lawless sequences, lacking any rule of extension, cannot be the subject of proofs or calculations in the way rule-governed irrationals can.
      ?

      Think about whether this reason is strong or weak

    • 3.Extending 'irrational number' to cover lawless sequences conflates a grammatical category with a mere analogy, producing conceptual confusion rather than mathematical insight.
      ?

      Think about whether this reason is strong or weak

    Reason against 2 of 2
    ?
    • 1.Brouwer and the intuitionists argued that mathematical objects exist only insofar as they are constructible by finite mental operations.
      ?

      Think about whether this reason is strong or weak

    • 2.A 'lawless' sequence is by definition not constructible through any specifiable procedure, making it mathematically unintelligible on constructivist grounds.
      ?

      Think about whether this reason is strong or weak

    • 3.Extending a defined mathematical term to cover entities that violate the constructive basis of that definition undermines the integrity of the original definition.
      ?

      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

    Philosophy of LanguageProof of definition segments

    Connections

    1 topic

    Modality & Possibility1 linked

    Related

    A 'lawless' sequence is by definition not constructible through any specifiable ...Brouwer and the intuitionists argued that mathematical objects exist only insofa...Extending 'irrational number' to cover lawless sequences conflates a grammatical...Extending a defined mathematical term to cover entities that violate the constru...
    +5 moreShow less
    Lawless and pseudo-irrationals are more like rule-governed irrationals than like...Lawless and pseudo-irrationals are needed for the mathematical continuumLawless sequences, lacking any rule of extension, cannot be the subject of proof...Mathematical terms gain their meaning from the rules and techniques governing th...Mathematical terms may be extended to cover conceivable numbers that fit more na...

    Similar

    An irrational number is only an extension insofar as it is a sign (a n...84%An irrational number is not a unique infinite expansion, but rather a ...84%Lawless and pseudo-irrationals are more like rule-governed irrationals...83%Lawless and pseudo-irrationals are needed for the mathematical continu...81%

    Source

    AI-extracted1/3 agreementValid
    SEP: wittgenstein-mathematics
    View source passageHide passage
    Superficially, at least, it seems as if Wittgenstein is offering an essentialist argument for the conclusion that real number arithmetic should not be extended in such-and-such a way. Such an essentialist account of real and irrational numbers seems to conflict with the actual freedom mathematicians have to extend and invent, with Wittgenstein’s intermediate claim (PG 334) that “[f]or [him] one calculus is as good as another”, and with Wittgenstein’s acceptance of complex and imaginary numbers.
    Extraction notes

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

    Details

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