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    A position number system with zero as a placeholder allow... — Carmelics
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    Home/Modality & Possibility
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    A position number system with zero as a placeholder allows an infinite set of natural numbers to be represented using only a finite set of symbols

    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.Zero functions as a placeholder that encodes positional magnitude
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    • 2.Any finite set of symbols combined with zero can represent arbitrarily large numbers by varying digit positions
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Representing a number and instantiating it are distinct: finite symbols can denote infinite numbers only if an infinite medium (time, space, or computation) is available for arbitrarily long strings.
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    • 2.Wittgenstein's rule-following considerations show that no finite set of symbols autonomously determines its own extension to infinitely many cases without an interpreting practice.
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    Reason against 2 of 2
    ?
    • 1.The claim conflates potential infinity (always being able to add another digit) with actual representability, a distinction Aristotle and later Cantor took to be philosophically significant.
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    • 2.A number requiring n digits demands a physical token of length n, so unbounded natural numbers are not representable by finite means but only by a finite scheme applied indefinitely.
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    Modality & PossibilityTruth & Knowledge

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    Philosophy of Language1 linked

    Related

    A number requiring n digits demands a physical token of length n, so unbounded n...Any finite set of symbols combined with zero can represent arbitrarily large num...Representing a number and instantiating it are distinct: finite symbols can deno...The claim conflates potential infinity (always being able to add another digit) ...
    +2 moreShow less
    Wittgenstein's rule-following considerations show that no finite set of symbols ...Zero functions as a placeholder that encodes positional magnitude

    Similar

    Any finite set of symbols combined with zero can represent arbitrarily...83%In principle an infinite number of symbols would be needed to represen...79%In a position system, the same symbolic rules govern addition, subtrac...77%The Roman numeral system is inferior to position systems for general a...75%

    Source

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    The simplest way of representing numbers is via a unary system. Here the length of the representation of a number is equal to the size of the number itself, i.e., the number “ten” is represented as “\\\\\\\\\\”. The classical Roman number system is an improvement since it contains different symbols for different orders of magnitude (one = I, ten = X, hundred = C, thousand = M). This system has enormous drawbacks since in principle one needs an infinite amount of symbols to code the natural numbe
    Extraction notes

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

    Details

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