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    Early infinitesimal calculus involved a logical inconsist... — Carmelics
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    Home/Skepticism
    HistoryEditSee Inverse

    Early infinitesimal calculus involved a logical inconsistency in the treatment of infinitesimals

    Skepticism
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
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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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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 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 against 2 of 2
    ?
    • 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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    Related

    A procedure that reliably generates true conclusions from true premises constitu...Anachronistically attributing logical inconsistency to early calculus conflates ...Berkeley's 'ghosts of departed quantities' objection targets rhetorical inconsis...If the same mathematical objects early calculus employed now admit full logical ...
    +5 moreShow less
    Infinitesimals must be non-zero to avoid division by zero in the difference quot...Newton's and Leibniz's methods consistently produced correct results, suggesting...Robinson's non-standard analysis (1966) demonstrated infinitesimals can be rigor...The same infinitesimals are then treated as zero when taking the limitThis slippage between non-zero and zero is logically unsound

    Similar

    The infinitesimal concept should be retained in the foundations of the...88%To do full justice to both Leibniz's and Nieuwentijdt's conceptions of...83%No infinitesimal is an upper bound for all other infinitesimals.80%The concept of an infinitesimal as a quantity less than any assignable...79%

    Source

    AI-extracted1/3 agreementValid
    SEP: infinity
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    This slope is equal to \(\frac{(x+\epsilon)^2-x^2}{\epsilon}\). In order for this fraction to make sense, \(\epsilon\) must be non-zero. However, we can calculate that this value is \(\frac{2x\epsilon+\epsilon^2}{\epsilon}\), or \(2x+\epsilon\). At this point, we no longer need \(\epsilon\) to be non-zero, so the slope can be said to be just \(2x\). This sort of slippage between non-zero and zero for these infinitesimals is what made Berkeley refer to them as “the ghosts of departed quantities”.
    Extraction notes

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

    Details

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