Cohl Furey sounds like a goddamn new Marvel character, but actually she's a mathematical physicist. Can that really be her name? As a novelist, I feel reality is cheating by being so fucking dramatic:https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720/ …
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This may be a dumb question, but what if this system HAS to be big? I know, the right answer is usually the simplest (reduce your fractions, kids!), but we ARE looking at shit that may or may not be in a place depending on whether you're looking at it...
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The real numbers are already really big, though! Mathematicians and mathematical physicists often have intuitions like, a system should be big enough to maneuver in, but not so big that you get completely lost in it.
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If it's too big it's useless anyway, in other words?
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Right.
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Would “continuous” capture “over the real numbers, as in we still have your standard numbers”, or is there some finite continuous field that doesn’t have a subset that acts like the reals?
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Continuity isn't the right concept here, finite fields have the discrete topology by default. And every finite field is an extension of Z/p for some prime p -- such a field is said to have 'characteristic p' whereas R, C etc. are 'characteristic 0'.
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Generally a structure with addition, subtraction, and multiplication is called a 'ring'; a ring that's also a vector space over a field F is called 'an algebra over the base field F', where F can be R or a finite field or whatever.
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The original assertion condenses to 'the only finite-dimensional, possibly nonassociative, division algebras over the real numbers are R, C, H, O' Once you get down into the weeds you have to be super careful about what you say, it gets exhausting sometimes.
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If algebras are rings, and rings have to be associative in their multiplication, then what do you mean by a “possibly non-associative division algebra”?
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I mean "we're weakening this specific requirement", ie 'algebraic structure that is a division algebra except we give up associativity'.
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It's a theme: R -> C loses linear ordering C -> H loses commutativity H -> O loses associativity
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Which of course is a general phenomenon -- drop an axiom and the potential structures explode. Part of what makes this progression interesting to a lot of people is that it *doesn't* explode, you get just one new structure at each step.
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. Banned in Sweden. SubGenius, Zhuangist, white-hat troll. Defrocked mathematician. Brain problems.