11–20 of 192 posts
I think the following quote is the QED for academia ruining any chance at actual research: “What I had was an out-of-control intuition that these algebras were key to understanding particle physics, and I was willing to follow this intuition off a cliff if need be. Some might say I did.”
"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. " Not true. Any field (in the algebra sense) has these properties. [1] Quaternions and octnonions also have weirder properties: quaternions are non-commutatve (j k=-k j) and octnonions are non-associative: a(bc) != (ab)c. I think the article meant thes…
"There are exactly four normed division algebras: the real numbers ($\R$), complex numbers ($\C$), quaternions ($\H$), and octonions ($\O$). The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are a slightly flashier but still respectable younger brother: not ordered, but algebraically complete. The quaternions, being noncommutative, are the eccentric cousin who is shunned at important family gatherings. But the octonions are the crazy old uncle nobody lets out of the attic: they are /nonassociative/."
>In mathematics, the octonions are a normed division algebra over the real numbers .
wtf is a normed division algebra??
>In mathematics, Hurwitz's theorem is a theorem [...] solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic form.
... right.
I have the same problem when I try to understand anything statistics related, I get hit by a barrage of unknown words and my brain just melts.
Is there any place that explains mathematical concepts in ... different ways?
I think the following quote is the QED for academia ruining any chance at actual research: “What I had was an out-of-control intuition that these algebras were key to understanding particle physics, and I was willing to follow this intuition off a cliff if need be. Some might say I did.”
"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. " Not true. Any field (in the algebra sense) has these properties. [1] Quaternions and octnonions also have weirder properties: quaternions are non-commutatve (j k=-k j) and octnonions are non-associative: a(bc) != (ab)c. I think the article meant thes…
Example: rational numbers
> There the game stops. 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. 1. I think they meant "the only kinds of numbers constructed in this way ". 2. Sedenions can still be added, multiplied, subtracted and divided. it's just that multiplication and division lose most of their useful properties. Wit…
>it's just that multiplication and division lose most of their useful properties. Specifically they lose the property of not having zero divisors. There exists sedonions a,b != 0 such that ab = 0
Of course associativity doesn't hold in the octonions either, but it holds just enough for cancellation to work.
> There the game stops. 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. 1. I think they meant "the only kinds of numbers constructed in this way ". 2. Sedenions can still be added, multiplied, subtracted and divided. it's just that multiplication and division lose most of their useful properties. Wit…
N(xy) = N(x)N(y) - N is called norm.
Without this property, you have zero divisors.
edit: throwawaymath uses better notation: |xy| = |x| • |y|
I think that maybe if you lock a mathematical physicist in a box forever with an x-TeV collider without letting them upgrade it, they will eventually find a theory that hits all of the datapoints and depending on their philosophical weakness then declare it "final."