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    Differentials must be zeros, and the derivative dy/dx mus... — Carmelics
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    Differentials must be zeros, and the derivative dy/dx must be the quotient 0/0

    SkepticismTruth & Knowledge
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    2 reasons for
    1 reason against

    Reasons For

    2 perspectives
    Reason for 1 of 2
    ?
    • 1.Berkeley's critique in 'The Analyst' concedes that calculus yields correct results, implying the 0/0 interpretation captures a genuine mathematical operation.
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    • 2.Leibniz himself treated dy and dx as formal symbols subject to algebraic cancellation, making 0/0 a legitimate ratio under the law of continuity.
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    • 3.A ratio of two vanishing quantities preserves a determinate finite value as both approach zero simultaneously, making 0/0 well-defined in context.
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    Reason for 2 of 2
    ?
    • 1.Carnot's 'Réflexions sur la métaphysique du calcul infinitésimal' defends compensating errors, implying zeros in numerator and denominator systematically cancel.
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    • 2.If the derivative is understood as a limit of a process rather than a static quotient, then 0/0 names the indeterminate form whose resolution is the derivative's very definition.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.The concept of an infinitesimal as a quantity less than any assignable magnitude yet unequal to zero is incoherent
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    • 2.If infinitesimals are not valid non-zero quantities, then the increments in a ratio of evanescent quantities must themselves be zero
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    Related

    A ratio of two vanishing quantities preserves a determinate finite value as both...Berkeley's critique in 'The Analyst' concedes that calculus yields correct resul...Carnot's 'Réflexions sur la métaphysique du calcul infinitésimal' defends compen...If infinitesimals are not valid non-zero quantities, then the increments in a ra...
    +3 moreShow less
    If the derivative is understood as a limit of a process rather than a static quo...Leibniz himself treated dy and dx as formal symbols subject to algebraic cancell...The concept of an infinitesimal as a quantity less than any assignable magnitude...

    Similar

    The quotient 0/0 is indeterminate and requires a procedure to assign i...76%Infinitesimals must be non-zero to avoid division by zero in the diffe...74%The quotient 0/0 can represent any number whatsoever73%Therefore any number α satisfies the equation α · 0 = 0, making α a va...70%

    Source

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    SEP: continuity
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    Euler rejected the concept of infinitesimal in its sense as a quantity less than any assignable magnitude and yet unequal to 0, arguing: that differentials must be zeros, and \(\Dy/\Dx\) the quotient \(0/0\). Since for any number \(\alpha\), \(\alpha \cdot 0 = 0\), Euler maintained that the quotient \(0/0\) could represent any number whatsoever.[23] For Euler qua formalist the calculus was essentially a procedure for determining the value of the expression \(0/0\) in the manifold situations it
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    Validity: Extracted via Max plan + API grounding/validity checks

    Details

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