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    Extending 'irrational number' to cover lawless sequences ... — Carmelics
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    Challenges→The term 'irrational number' should be extended to include lawless and pseudo-irrationals

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

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    1 reason for
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    Reasons For

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    Reason for
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    • 1.Traditional irrationals satisfy algebraic or analytic definitions; lawless sequences lack constructive specification, making them fundamentally different kinds of objects.
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    • 2.Using 'irrational' for both √2 and unspecifiable sequences obscures their distinct mathematical roles and obstructs rigorous proof and pedagogy.
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    • 3.The extension rests on superficial analogy (non-rational-like behavior) rather than structural homology, violating standards for conceptual expansion in mathematics.
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    Reasons Against

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    • 1.Mathematical categories are pragmatic tools, not natural kinds; if lawless sequences behave like irrationals in relevant contexts, the extension clarifies rather than confuses.
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    • 2.Constructivism and classical mathematics disagree on what constitutes legitimate objects; dismissing one tradition's extensions as merely analogical begs foundational questions.
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    • 3.Historical mathematics routinely extends concepts (negative numbers, imaginary units) by analogy; productive insight often emerges from such analogies, not despite them.
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    Proof of definition segments1 linkedPhilosophy of Language1 linked

    Related

    Constructivism and classical mathematics disagree on what constitutes legitimate...Historical mathematics routinely extends concepts (negative numbers, imaginary u...Mathematical categories are pragmatic tools, not natural kinds; if lawless seque...The extension rests on superficial analogy (non-rational-like behavior) rather t...
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    The term 'irrational number' should be extended to include lawless and pseudo-ir...Traditional irrationals satisfy algebraic or analytic definitions; lawless seque...Using 'irrational' for both √2 and unspecifiable sequences obscures their distin...

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    2 (1 for, 1 against)
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