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    Carmelics

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    Inverse View

    It is not the case that It is not immediately evident that all functions in Brouwer's sense must be continuous.

    ?Set your confidence on the premises below to see your aggregate.

    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 1.Brouwer's continuity theorem, proven within intuitionistic mathematics, entails that every total function from the continuum to naturals is continuous.
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    • 2.Any function defined on a spread in Brouwer's sense must be determined by finite initial segments of choice sequences, which structurally necessitates continuity.
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    • 3.The bar theorem and fan theorem, central to Brouwer's own mathematical practice, collectively enforce continuity on all functions defined over the full continuum.
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    Reason for 2 of 2
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    • 1.Troelstra and van Dalen demonstrate in 'Constructivism in Mathematics' that intuitionistic function-existence requires lawlike or choice-sequence definability, ruling out discontinuous cases.
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    • 2.A function on Brouwer's continuum that were discontinuous would require completed infinite information at a point, contradicting the temporal, constructive nature of choice sequences.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Brouwer's concept of function is less restrictive than Weyl's.
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    • 2.Brouwer did not dismiss the possibility that discontinuous functions could be defined on proper parts of a continuum.
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