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    Leibniz's argument that Δx² = 0 depends crucially on the ... — Carmelics
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    Leibniz's argument that Δx² = 0 depends crucially on the assumption that the portion of the curve between abscissae 0 and Δx is straight.

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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
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    • 1.If the portion of the curve between abscissae 0 and Δx is not assumed to be straight, it does not follow that Δx² = 0.
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    • 2.Denying the straightness assumption undermines the derivation of Δx² = 0.
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Leibniz's method of infinitesimals operates within a formal calculus where Δx² vanishes by the law of homogeneity, not by geometric straightness assumptions.
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    • 2.Henk Bos's historical analysis shows Leibniz grounded the omission of higher-order infinitesimals in algebraic order relations, not local linearity of curves.
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    • 3.The straightness assumption conflates Newton's method of first and last ratios with Leibniz's distinct algebraic framework for differential calculus.
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    Reason against 2 of 2
    ?
    • 1.Robinson's non-standard analysis vindicates Leibniz's infinitesimal reasoning by showing Δx² = 0 relative to Δx follows from rigorous hyperreal arithmetic, not geometric assumptions.
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    • 2.If Δx is a non-zero infinitesimal, then Δx² is infinitesimal of higher order, making Δx²/Δx standard-part zero without presupposing any straightness of the curve.
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    Truth & Knowledge2 linkedModality & Possibility1 linked

    Related

    Denying the straightness assumption undermines the derivation of Δx² = 0.Henk Bos's historical analysis shows Leibniz grounded the omission of higher-ord...If the portion of the curve between abscissae 0 and Δx is not assumed to be stra...If Δx is a non-zero infinitesimal, then Δx² is infinitesimal of higher order, ma...
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    Leibniz's method of infinitesimals operates within a formal calculus where Δx² v...Robinson's non-standard analysis vindicates Leibniz's infinitesimal reasoning by...The straightness assumption conflates Newton's method of first and last ratios w...

    Similar

    Denying the straightness assumption undermines the derivation of Δx² =...87%The argument shows that e² = 0 given the assumption of an infinitesima...79%Leibniz grants that there is an infinitesimal straight stretch of the ...79%Once a curve reaches H⁺(Σ₀), it can be continued endlessly into the pa...73%

    Source

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    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

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