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    Inconsistent non-standard analysis has computational adva... — Carmelics
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    Inconsistent non-standard analysis has computational advantages over classical non-standard analysis in the theory of differentiation

    Modality & PossibilityTruth & Knowledge
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.Robinson's non-standard analysis is based on infinitesimals and their reciprocals, the infinite numbers
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    • 2.An inconsistent version of non-standard analysis permits discarding higher-order infinitesimals
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    • 3.The ability to discard higher-order infinitesimals yields computational advantages in the theory of differentiation
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Classical non-standard analysis already permits neglecting higher-order infinitesimals via the standard part function without logical inconsistency.
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    • 2.A computational advantage gained by tolerating contradiction is undermined if the same results are derivable in a consistent framework by equivalent means.
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    • 3.Robinson's transfer principle provides a rigorous and conservative method for discarding infinitesimal remainders, rendering the inconsistent approach redundant rather than advantageous.
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    Reason against 2 of 2
    ?
    • 1.Priest's dialetheist frameworks permitting true contradictions require paraconsistent logic, which weakens the inferential power needed to derive reliable computational results.
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    • 2.In paraconsistent arithmetic, the loss of ex contradictione quodlibet comes at the cost of blocking valid classical inferences that underpin standard differentiation techniques.
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    • 3.A system offering computational shortcuts through inconsistency trades formal reliability for convenience, violating the mathematician's minimal criterion of trustworthy proof.
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    Related

    A computational advantage gained by tolerating contradiction is undermined if th...A system offering computational shortcuts through inconsistency trades formal re...An inconsistent version of non-standard analysis permits discarding higher-order...Classical non-standard analysis already permits neglecting higher-order infinite...
    +5 moreShow less
    In paraconsistent arithmetic, the loss of ex contradictione quodlibet comes at t...Priest's dialetheist frameworks permitting true contradictions require paraconsi...Robinson's non-standard analysis is based on infinitesimals and their reciprocal...Robinson's transfer principle provides a rigorous and conservative method for di...The ability to discard higher-order infinitesimals yields computational advantag...

    Similar

    Non-standard analysis provides simpler and more intuitive proofs of ma...80%Robinson's non-standard analysis is based on infinitesimals and their ...78%Non-standard real analysis can prove results in real analysis that wer...78%An inconsistent version of non-standard analysis permits discarding hi...76%

    Source

    AI-extracted1/3 agreementValid
    SEP: mathematics-inconsistent
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    There have proved to be many places throughout analysis where there are distinctive inconsistent insights. The examples in the remainder of this section are drawn from Mortensen (1995). For example: (1) Robinson’s (1974) non-standard analysis was based on infinitesimals, quantities smaller than any real number, as well as their reciprocals, the infinite numbers. This has an inconsistent version, which has some advantages for calculation in being able to discard higher-order infinitesimals. Inter
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    Details

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