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    Carmelics

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    Home/Original/inverse
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    Inverse View

    It is not the case that Early infinitesimal calculus involved a logical inconsistency in the treatment of infinitesimals

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    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 1.Berkeley's 'ghosts of departed quantities' objection targets rhetorical inconsistency, not logical invalidity in the formal derivations themselves.
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    • 2.Newton's and Leibniz's methods consistently produced correct results, suggesting their procedures tracked mathematical truth despite informal exposition.
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    • 3.A procedure that reliably generates true conclusions from true premises constitutes a functionally valid inference pattern, even absent rigorous formalization.
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    Reason for 2 of 2
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    • 1.Robinson's non-standard analysis (1966) demonstrated infinitesimals can be rigorously formalized within a consistent extension of the real numbers.
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    • 2.If the same mathematical objects early calculus employed now admit full logical consistency, the original inconsistency was in the metalanguage, not the mathematics.
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    • 3.Anachronistically attributing logical inconsistency to early calculus conflates absence of rigor with presence of contradiction—a category error in historical assessment.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.Infinitesimals must be non-zero to avoid division by zero in the difference quotient
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    • 2.The same infinitesimals are then treated as zero when taking the limit
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    • 3.This slippage between non-zero and zero is logically unsound
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