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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Replying to @AnnKSterzinger
"Proof surfaced in 1898 that the reals, complex numbers, quaternions and octonions are the only kinds of numbers that can be added, subtracted, multiplied and divided." Quanta's math coverage is usually very good but this is ouch wrong
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Replying to @St_Rev @AnnKSterzinger
The correct statement is insanely fiddly and I don't know how you'd do it in a popular science article, but Quanta usually finds a way and this is disappointing.
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Replying to @St_Rev @AnnKSterzinger
Roughly, they're the only finite-dimensional division algebras over the real numbers. Unpacking that would take a while, but basically they're the only ones if: a) you want your system to include everyday numbers like 2, 31/7 and pi b) you don't want your system to be too big
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Replying to @St_Rev @AnnKSterzinger
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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Replying to @TWakalix
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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Replying to @St_Rev
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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Replying to @TWakalix
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.