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    The only four division algebras built on the real numbers... — Carmelics
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    The only four division algebras built on the real numbers are the reals (ℝ), complex numbers (ℂ), quaternions (ℍ), and octonions (𝕆).

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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
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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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    Reasons Against

    2 perspectives
    Reason against 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 against 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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    Related

    A fictionalist or formalist about mathematics, following Hartry Field or Hilbert...Bott and Milnor (1958) independently proved the same result.Infinite-dimensional normed division algebras, such as those explored in non-sta...Kervaire (1958) independently proved that only four division algebras built on t...
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    Mathematical existence proofs establish formal consistency within axiomatic syst...The Kervaire-Bott-Milnor result presupposes the standard definition of 'division...The theorem thus characterizes a structural feature of a human-constructed forma...Therefore the claim's scope is restricted to a conventional definitional choice,...

    Similar

    Kervaire (1958) independently proved that only four division algebras ...86%The algebras ℝ, ℂ, ℍ, and 𝕆 play an important role in descriptions of...69%Internal Set Theory resolves the asymmetry between standard and non-st...69%The equational theory of the integers contains no law of the form x+x+...68%

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    More complicated numbers systems with generalizations of this type of multiplication in 4 and 8 dimensions can be defined. Kervaire (1958) and Bott & Milnor (1958) independently proved that the only four division algebras built on the reals are \(\mathbb{R}\), \(\mathbb{C}\), \(\mathbb{H}\) and \(\mathbb{O}\), so the table gives a comprehensive view of all possible algebra’s that define a notion of extensiveness. For each of the number classes in the table a separate theory of information me
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    Details

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