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    Without an independent criterion for selecting κ, the Kle... — Carmelics
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    Challenges→Metric properties of a geometry can be defined intrinsically using projective invariants rather than by convention from a numerical manifold.

    Without an independent criterion for selecting κ, the Klein construction relocates rather than eliminates the conventional element in metric geometry.

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    1 reason for
    1 reason against

    Reasons For

    1 perspective
    Reason for
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    • 1.Klein's approach defines geometry through group-theoretic invariants, but selecting which group κ to use requires prior geometric intuition, not derivation.
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    • 2.Different choices of κ (Euclidean, hyperbolic, projective groups) yield different geometric properties, yet no principled method selects among them a priori.
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    • 3.The foundation merely shifts from axioms about space to stipulations about group structure—the conventional element persists rather than disappears.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.Mathematical convention (choosing κ) differs categorically from geometric convention; once κ is fixed, all geometric properties follow deductively without further choice.
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    • 2.Relocating conventionality to group selection is methodologically preferable because it isolates and clarifies where human choice enters, improving transparency.
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    • 3.Physical experiments or consistency requirements can constrain κ selection empirically, showing the relocation enables rather than obscures rigorous justification.
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    Related

    Different choices of κ (Euclidean, hyperbolic, projective groups) yield differen...Klein's approach defines geometry through group-theoretic invariants, but select...Mathematical convention (choosing κ) differs categorically from geometric conven...Metric properties of a geometry can be defined intrinsically using projective in...
    +3 moreShow less
    Physical experiments or consistency requirements can constrain κ selection empir...Relocating conventionality to group selection is methodologically preferable bec...The foundation merely shifts from axioms about space to stipulations about group...

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