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    Carmelics

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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Home/Original/inverse
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    Inverse View

    It is not the case that The only four division algebras built on the real numbers are the reals (ℝ), complex numbers (ℂ), quaternions (ℍ), and octonions (𝕆).

    ?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.The Kervaire-Bott-Milnor result presupposes the standard definition of 'division algebra' requiring finite dimensionality over ℝ.
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    • 2.Infinite-dimensional normed division algebras, such as those explored in non-standard analysis, are not ruled out by this theorem.
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    • 3.Therefore the claim's scope is restricted to a conventional definitional choice, not a metaphysically necessary boundary on algebraic structure.
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    Reason for 2 of 2
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    • 1.Mathematical existence proofs establish formal consistency within axiomatic systems, not mind-independent Platonic facts about what algebras 'exist'.
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    • 2.A fictionalist or formalist about mathematics, following Hartry Field or Hilbert, would deny the claim describes a discovery about abstract objects.
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    • 3.The theorem thus characterizes a structural feature of a human-constructed formal system, making 'only four exist' a category error if read realistically.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.Kervaire (1958) independently proved that only four division algebras built on the reals exist.
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    • 2.Bott and Milnor (1958) independently proved the same result.
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