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    If curves are infinilateral polygons, then the lengths of... — Carmelics
    Home/Modality & Possibility
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    If curves are infinilateral polygons, then the lengths of the sides of those polygons must be nilsquare infinitesimals.

    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.Leibniz grants that there is an infinitesimal straight stretch of the curve (a side of an infinilateral polygon coinciding with the curve) between abscissae 0 and e, which does not reduce to a single point.
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    • 2.If such a stretch exists, then e cannot be equated to 0.
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    • 3.The argument shows that e² = 0 given the assumption of an infinitesimal straight stretch.
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Robinson's non-standard analysis rigorously formalizes infinitesimals without nilsquare conditions, using hyperreals where e²≠0 for any nonzero e.
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    • 2.If a consistent alternative infinitesimal framework assigns nonzero squares to infinitesimals, then nilsquareness is not a necessary condition for infinilateral polygon sides.
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    • 3.The Leibnizian geometric intuition of infinitesimal curve segments can be preserved under hyperreal arithmetic without invoking nilsquare constraints.
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    Reason against 2 of 2
    ?
    • 1.Berkeley's critique in 'The Analyst' established that infinitesimals treated as nonzero when convenient and zero when convenient commit a logical fallacy of inconsistent supposition.
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    • 2.The derivation of e²=0 from e≠0 in the supporting argument relies on the same equivocation Berkeley identified: e is nonzero to generate a segment, then its square is annihilated.
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    • 3.A quantity derived through inconsistent suppositional reasoning cannot serve as a well-defined mathematical object, undermining nilsquare infinitesimals as legitimate curve-side lengths.
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    Topics

    Modality & PossibilityTruth & Knowledge

    Connections

    2 topics

    Causation2 linkedProof of definition segments1 linked

    Related

    A quantity derived through inconsistent suppositional reasoning cannot serve as ...A quantity that is not zero but whose square is zero is a nilsquare infinitesima...Berkeley's critique in 'The Analyst' established that infinitesimals treated as ...If a consistent alternative infinitesimal framework assigns nonzero squares to i...
    +6 moreShow less
    If such a stretch exists, then e cannot be equated to 0.Leibniz grants that there is an infinitesimal straight stretch of the curve (a s...Robinson's non-standard analysis rigorously formalizes infinitesimals without ni...

    Similar

    Nieuwentijdt's conception requires nilsquare infinitesimals that measu...88%Leibniz held that curves may be considered as infinilateral polygons.85%If the curve y = x² is an infinilateral polygon, then the infinitesima...83%Leibniz's own principle that curves are infinilateral polygons entails...82%

    Source

    AI-extracted1/3 agreementValid
    SEP: continuity
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    Now Leibniz could retort that that this argument depends crucially on the assumption that the portion of the curve between abscissae 0 and \(\Dx\) is indeed straight. If this be denied, then of course it does not follow that \(\Dx ^2 = 0\). But if one grants, as Leibniz does, that that there is an infinitesimal straight stretch of the curve (a side, that is, of an infinilateral polygon coinciding with the curve) between abscissae 0 and \(e\), say, which does not reduce to a single point then \(e
    Extraction notes

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

    Details

    The Leibnizian geometric intuition of infinitesimal curve segments can be preser...
    The argument shows that e² = 0 given the assumption of an infinitesimal straight...
    The derivation of e²=0 from e≠0 in the supporting argument relies on the same eq...
    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit