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    It is not necessary to conjure up the picture of the infi... — Carmelics
    Home/Philosophy of Language
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    It is not necessary to conjure up the picture of the infinite (of the enormously big) in mathematical reasoning.

    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
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    • 1.The calculus contains nothing infinite.
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    • 2.Once we see that the calculus contains nothing infinite, there is no mathematical deficit that requires supplementation by a picture of the infinite.
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.Gödel's incompleteness theorems were discovered through reasoning about the totality of all provable sentences, an explicitly infinite domain.
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    • 2.Mathematical reasoning that produces genuine results via infinite pictures cannot be dismissed as mere psychological scaffolding without mathematical content.
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    • 3.Wittgenstein's finitist criterion for mathematical meaningfulness would, if applied consistently, eliminate large portions of classical number theory.
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    Reason against 2 of 2
    ?
    • 1.Cantor's transfinite arithmetic generates theorems (e.g., |ℕ| < |ℝ|) that are unintelligible without a picture of completed infinite totalities.
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    • 2.If a mathematical calculus generates results that require infinite structures for their interpretation, then pictures of the infinite are mathematically indispensable.
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    Related

    Cantor's transfinite arithmetic generates theorems (e.g., |ℕ| < |ℝ|) that are un...Gödel's incompleteness theorems were discovered through reasoning about the tota...If a mathematical calculus generates results that require infinite structures fo...Mathematical reasoning that produces genuine results via infinite pictures canno...
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    Once we see that the calculus contains nothing infinite, there is no mathematica...The calculus contains nothing infinite.Wittgenstein's finitist criterion for mathematical meaningfulness would, if appl...

    Similar

    The word 'infinite' ought to be avoided in mathematics wherever it see...80%Once we see that the calculus contains nothing infinite, there is no m...78%There is no such thing as an infinite mathematical extension.77%Arithmetic theories such as Peano Arithmetic can express mathematicall...76%

    Source

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    SEP: wittgenstein-mathematics
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    An infinite sequence, for example, is not a gigantic extension because it is not an extension, and ‘\(\aleph_0\)’ is not a cardinal number, for “how is this picture connected with the calculus”, given that “its connexion is not that of the picture | | | | with 4” (i.e., given that ‘\(\aleph_0\)’ is not connected to a (finite) extension)? This shows, says Wittgenstein (RFM II, §58), that we ought to avoid the word ‘infinite’ in mathematics wherever it seems to give a meaning to the calculus, rath
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    Details

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