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    The infinitesimal concept should be retained in the found... — Carmelics
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    The infinitesimal concept should be retained in the foundations of the calculus

    Skepticism
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.Infinitesimal methods are more efficient than alternative approaches
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    • 2.Infinitesimals serve as the 'glue' that causes points on a continuous line to lose their individual identity, which is essential to a proper conception of continuity
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Weierstrass, Cauchy, and Dedekind successfully rebuilt calculus on rigorous epsilon-delta foundations without invoking infinitesimals.
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    • 2.A foundational concept that can be eliminated without expressive or inferential loss fails the criterion of theoretical indispensability.
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    • 3.The standard real analysis framework achieves greater ontological parsimony by grounding limits in quantified inequalities over real numbers alone.
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    Reason against 2 of 2
    ?
    • 1.Berkeley's critique in 'The Analyst' demonstrated that 17th-century infinitesimals involved contradictory reasoning: treated as nonzero in division, then discarded as zero.
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    • 2.Non-standard analysis (Robinson, 1966) rescues infinitesimals only by embedding them in a hyperreal system requiring the ultrafilter lemma, a strong set-theoretic axiom.
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    • 3.Introducing ontologically exotic entities requiring powerful axioms to avoid contradiction imposes a foundational cost that outweighs alleged gains in geometric intuition.
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    No other argument is better1 linkedCausation1 linked

    Related

    A foundational concept that can be eliminated without expressive or inferential ...Berkeley's critique in 'The Analyst' demonstrated that 17th-century infinitesima...Infinitesimal methods are more efficient than alternative approachesInfinitesimals serve as the 'glue' that causes points on a continuous line to lo...
    +4 moreShow less
    Introducing ontologically exotic entities requiring powerful axioms to avoid con...Non-standard analysis (Robinson, 1966) rescues infinitesimals only by embedding ...The standard real analysis framework achieves greater ontological parsimony by g...

    Similar

    Early infinitesimal calculus involved a logical inconsistency in the t...88%To do full justice to both Leibniz's and Nieuwentijdt's conceptions of...84%The word 'infinite' ought to be avoided in mathematics wherever it see...83%The concept of an infinitesimal as a quantity less than any assignable...81%

    Source

    AI-extracted1/3 agreementValid
    SEP: continuity
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    Peirce’s conception of the number continuum is also notable for the presence in it of an abundance of infinitesimals, Peirce championed the retention of the infinitesimal concept in the foundations of the calculus, both because of what he saw as the efficiency of infinitesimal methods, and because he regarded infinitesimals as constituting the “glue” causing points on a continuous line to lose their individual identity.
    Extraction notes

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

    Details

    Weierstrass, Cauchy, and Dedekind successfully rebuilt calculus on rigorous epsi...
    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit