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    π' is not a genuine irrational number. — Carmelics
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    Home/Philosophy of Language
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    π' is not a genuine irrational number.

    Philosophy of LanguageTruth & 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.π belongs to all base-notational systems, while π' belongs only to one.
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    • 2.There can't be irrational numbers of different types.
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.The distinction between base-notation systems is a feature of representation, not of mathematical objects themselves.
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    • 2.If π' has a well-defined decimal expansion and satisfies the algebraic definition of irrationality, it qualifies as irrational by standard criteria.
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    • 3.Wittgenstein's criterion conflates epistemic accessibility across notations with ontological status, a category error Frege warned against in Grundlagen.
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    Reason against 2 of 2
    ?
    • 1.Benacerraf's insight that mathematical identity cannot depend on representational accidents applies directly: no number essentially belongs to one notation.
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    • 2.π' can be systematically translated into any positional base, meaning its confinement to one notation reflects our current knowledge, not an intrinsic property.
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    Philosophy of LanguageTruth & Knowledge

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    Related

    Benacerraf's insight that mathematical identity cannot depend on representationa...If π' has a well-defined decimal expansion and satisfies the algebraic definitio...The distinction between base-notation systems is a feature of representation, no...There can't be irrational numbers of different types.
    +3 moreShow less
    Wittgenstein's criterion conflates epistemic accessibility across notations with...π belongs to all base-notational systems, while π' belongs only to one.π' can be systematically translated into any positional base, meaning its confin...

    Similar

    Pseudo-irrationals like π' are 'homeless' numbers.82%An irrational number is only an extension insofar as it is a sign (a n...81%The term 'irrational number' should be extended to include lawless and...78%There can't be irrational numbers of different types.75%

    Source

    AI-extracted1/3 agreementValid
    SEP: wittgenstein-mathematics
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    Although a pseudo-irrational such as \(\pi '\) (on either definition) is “as unambiguous as … \(\pi\) or \(\sqrt{2}\)” (PG 476), it is ‘homeless’ according to Wittgenstein because, instead of using “the idioms of arithmetic” (PR §186), it is dependent upon the particular ‘incidental’ notation of a particular system (i.e., in some particular base) (PR §188; PR §182; and PG 475). If we speak of various base-notational systems, we might say that \(\pi\) belongs to all systems, while \(\pi '\) belon
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    Details

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