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    Berkeley's critique in 'The Analyst' established that inf... — Carmelics
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    Challenges→If curves are infinilateral polygons, then the lengths of the sides of those polygons must be nilsquare infinitesimals.

    Berkeley's critique in 'The Analyst' established that infinitesimals treated as nonzero when convenient and zero when convenient commit a logical fallacy of inconsistent supposition.

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    Reasons For

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    Reason for
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    • 1.A quantity cannot logically be both zero and nonzero simultaneously in the same derivation without violating the law of non-contradiction.
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    • 2.Early calculus texts explicitly treated infinitesimals as nonzero for division, then discarded them as zero for final answers, lacking rigorous justification.
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    • 3.Berkeley's critique forced mathematicians to develop rigorous limits theory, ultimately vindicating his demand for logical consistency in foundations.
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    Reasons Against

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    • 1.Infinitesimals need not be real numbers; modern nonstandard analysis treats them rigorously as elements of hyperreal fields with consistent logical status.
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    • 2.Berkeley conflated informal heuristic procedures with formal logical claims; useful computational methods need not satisfy strict philosophical rigor simultaneously.
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    • 3.The infinitesimal's varying role reflects context-dependent reasoning (algebraic vs. limiting), not inconsistency—similar to how 'open' differs in topology and everyday language.
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    Key Terms

    Berkeley(as the author being discussed)
    George Berkeley was an Irish philosopher from the 1600s-1700s who argued that physical objects don't exist independently of being perceived—they only exist because someone is thinking about or observing them.
    Inconsistent supposition(as the specific fallacy Berkeley accused mathematicians of committing)
    Assuming something is true in one situation but false in another situation without any good reason for changing your position.
    Logical fallacy(as what Berkeley claimed calculus contained)
    A mistake in reasoning where the logic doesn't actually hold up, even if it might sound convincing at first.
    The Analyst(as the specific work being referenced)
    A famous philosophical essay Berkeley wrote in 1734 that criticized the mathematical methods of calculus, arguing they contained logical inconsistencies.
    infinitesimals(Peirce's philosophy of mathematics and foundations of calculus)
    Quantities that constitute the 'glue' causing points on a continuous line to lose their individual identity, thereby grounding the concept of a true continuum

    Connections

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    Truth & Knowledge1 linkedModality & Possibility1 linked

    Related

    A quantity cannot logically be both zero and nonzero simultaneously in the same ...Berkeley conflated informal heuristic procedures with formal logical claims; use...

    Details

    Type
    claim
    Perspectives
    2 (1 for, 1 against)
    Edits
    1 edit
    Berkeley's critique forced mathematicians to develop rigorous limits theory, ult...
    Early calculus texts explicitly treated infinitesimals as nonzero for division, ...
    +3 moreShow less
    If curves are infinilateral polygons, then the lengths of the sides of those pol...Infinitesimals need not be real numbers; modern nonstandard analysis treats them...The infinitesimal's varying role reflects context-dependent reasoning (algebraic...