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    A genuine number must measure in and of itself — Carmelics
    Statements
    321,452
    Perspectives
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    42
    Home/Philosophy of Language
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    A genuine number must measure in and of itself

    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
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    • If a purported 'number' leaves the task of measuring to the rationals, then we have no need of it as a distinct number
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    Reasons Against

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    Reason against 1 of 2
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    • 1.Real numbers like √2 are defined by Dedekind cuts, which partition the rationals without requiring any independent measuring capacity.
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    • 2.A number's legitimacy derives from its structural role in a complete ordered field, not from autonomous metric function.
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    • 3.Completeness of ℝ—filling every gap in the rationals—is itself a mathematically productive property Wittgenstein's criterion cannot capture.
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    Reason against 2 of 2
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    • 1.Frege argued in Grundgesetze that numbers are logical objects whose validity is grounded in consistent formal definition, not operational utility.
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    • 2.Excluding irrational numbers for failing to 'measure in themselves' would eliminate π and e, crippling analysis and physics without principled justification.
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    Related

    A number's legitimacy derives from its structural role in a complete ordered fie...Completeness of ℝ—filling every gap in the rationals—is itself a mathematically ...Excluding irrational numbers for failing to 'measure in themselves' would elimin...Frege argued in Grundgesetze that numbers are logical objects whose validity is ...
    +2 moreShow less
    If a purported 'number' leaves the task of measuring to the rationals, then we h...Real numbers like √2 are defined by Dedekind cuts, which partition the rationals...

    Similar

    For a real number to be a genuine number, it must be possible to estab...83%Hume's Principle (HP) is not sufficient to ensure that terms of the fo...79%Strict finitists hold that only numbers constructible by counting or e...78%If a purported 'number' leaves the task of measuring to the rationals,...78%

    Source

    AI-extracted1/3 agreementValid
    SEP: wittgenstein-mathematics
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    To this end, Wittgenstein demands (a) that a real number must be “compar[able] with any rational number taken at random” (i.e., “it can be established whether it is greater than, less than, or equal to a rational number” (PR §191)) and (b) that “[a] number must measure in and of itself” and if a ‘number’ “leaves it to the rationals, we have no need of it” (PR §191) (Frascolla 1980: 242–243; Shanker 1987: 186–192; Da Silva 1993: 93–94; Marion 1995a: 162, 164; Rodych 1999b, 281–291; Lampert 2009).
    Extraction notes

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

    Details

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