Skip to content
Carmelics
TopicsThinkersChangesContributorsLoading account…

    Carmelics

    A reasoning platform. Break down any belief into clear reasons, explore both sides, and weigh the evidence honestly.

    Navigate

    • Topics
    • Search
    • Recent Changes
    • Contribute
    • How It Works
    • Glossary
    • Thinkers
    • Contributors
    • About
    • Statistics
    • Terms
    • Privacy

    Database

    Statements
    —
    Perspectives
    —
    Topics
    —

    Press ? for keyboard shortcuts

    LoyalLoyalJusticeJustice
    Made withinDC&Austin
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    To do full justice to both Leibniz's and Nieuwentijdt's c... — Carmelics
    Home/Philosophy of Language
    HistoryEditSee Inverse

    To do full justice to both Leibniz's and Nieuwentijdt's conceptions of infinitesimals, two distinct sorts of infinitesimals are required.

    Philosophy of LanguageTruth & Knowledge
    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.Leibniz's conception requires differentials that obey the same algebraic laws as finite quantities.
      ?

      Think about whether this reason is strong or weak

    • 2.Nieuwentijdt's conception requires nilsquare infinitesimals that measure the lengths of the sides of infinilateral polygons.
      ?

      Think about whether this reason is strong or weak

    • 3.Nilsquare infinitesimals are necessarily smaller than Leibnizian differentials.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Leibniz himself denied that infinitesimals had any fixed ontological status, treating them as useful fictions governed by the law of continuity rather than as genuine magnitudes of a specific kind.
      ?

      Think about whether this reason is strong or weak

    • 2.If Leibnizian differentials are fictions rather than objects, the contrast with Nieuwentijdt's nilsquare infinitesimals is a difference in formal role, not a difference in kind requiring two distinct sorts of entity.
      ?

      Think about whether this reason is strong or weak

    • 3.Nieuwentijdt's nilsquare condition can be reinterpreted as a constraint on the order of approximation within a single fictional calculus, dissolving the purported need for ontological dualism.
      ?

      Think about whether this reason is strong or weak

    Reason against 2 of 2
    ?
    • 1.Abraham Robinson's non-standard analysis provides a single coherent framework in which both Leibnizian differentials and nilsquare-like infinitesimals are expressible as distinct elements within one number system.
      ?

      Think about whether this reason is strong or weak

    • 2.If a unified formal system can accommodate both conceptions without positing ontologically distinct kinds, the claim that two *sorts* of infinitesimals are required overstates the metaphysical need.
      ?

      Think about whether this reason is strong or weak

    Sign in or register to share your perspective on this statement.

    Next step

    Based on where you are in your exploration

    Strongest counterpoint
    Explore the most compelling reason on the other side.

    Topics

    Philosophy of LanguageTruth & Knowledge

    Connections

    1 topic

    Modality & Possibility2 linked

    Related

    Abraham Robinson's non-standard analysis provides a single coherent framework in...If Leibnizian differentials are fictions rather than objects, the contrast with ...If a unified formal system can accommodate both conceptions without positing ont...Leibniz himself denied that infinitesimals had any fixed ontological status, tre...
    +5 moreShow less
    Leibniz recognized the need for differentials but not nilsquare infinitesimals, ...Leibniz's conception requires differentials that obey the same algebraic laws as...Nieuwentijdt's conception requires nilsquare infinitesimals that measure the len...Nieuwentijdt's nilsquare condition can be reinterpreted as a constraint on the o...Nilsquare infinitesimals are necessarily smaller than Leibnizian differentials.

    Similar

    The infinitesimal concept should be retained in the foundations of the...84%Early infinitesimal calculus involved a logical inconsistency in the t...83%Weyl's reformulation of the space problem should incorporate Riemann's...82%No infinitesimal is an upper bound for all other infinitesimals.81%

    Source

    AI-extracted1/3 agreementValid
    SEP: continuity
    View source passageHide passage
    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