Why are complex numbers not called duonions then?
And why are the real numbers not called "onions"?
I went upstairs to find the machine where I was already logged in so I could hunt down and upvote this comment.
The Octonion Math That Could Underpin Physics
81–90 of 192 posts
Re: The Octonion Math That Could Underpin Physics
#82In the approach from the article it seems as you are picking a mathematical structure in the middle of nowhere with the universe faintly visible at the horizon and then you start wandering around hoping that you will stumble across a path leading to the horizon. But there don't seem to be many reason to believe that such a path exists, there are countless mathematical objects you could pick as a starting point and al…
> In the approach from the article it seems as you are picking a mathematical structure in the middle of nowhere Quaternions were very popular way of expressing the "classical" physics around the 19th century (and the vector algebra we know today is in some ways just a derivative of quaternion algebra). Complex numbers are extremely useful in many fields even today. It's hardly in the middle of nowhere.
But the article is not really talking about quaternions - which are surly a useful tool, probably best known for the nice way in which they can describe rotations - but about R⊗C⊗H⊗O. And it's at the very least not obvious that this thing is anywhere close to where they journey is hopped to lead to.
Re: The Octonion Math That Could Underpin Physics
#83It's rather amusing that the author assumes that non-associtiave objects are "weird" for physicists (or at least that was my reading), since the velocity addition formula is in general non-associative and that has been extensively studied. (I remember three separate occasions in my undergrad particle physics class where we actually went through all the calculations involved with the velocity addition formula and fina…
Not to mention operator calculus. Operators do not, in general, associate. Feynman developed his own notation for operator calculus.
How do operators in general fail to associate? Usually the "multiplication" operation for operators is function application, which is a paradigmatic example of a thing that is associative.
Re: The Octonion Math That Could Underpin Physics
#84"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…
It's in the article towards the end, but these are the only kinds of ... "over real numbers," which constraint Furey believes may be only an approximation.
Re: The Octonion Math That Could Underpin Physics
#85Earlier quoted context omitted.
That happens in general for matrices too.
Yes, and this is a bit obvious, but reals, complex numbers, split complex numbers, quaternions, octonions, sedenions, can all be represented as matrices of the appropriate form.
That's at most sort-of-true. It's not possible to represent octonions by matrices of numbers in such a way that multiplication of matrices corresponds to multiplication of octonions, because matrix multiplication is associative and octonion multiplication isn't.
Re: The Octonion Math That Could Underpin Physics
#86So, did Furey get a faculty job or can she be viewed with her accordion somewhere?
Re: The Octonion Math That Could Underpin Physics
#87Earlier quoted context omitted.
It's my understanding that most of Einstein's theory was the product of intuition, backed up after-the-fact with mathematics and experimentation. Intuition isn't a bad compass, as long as you can set it aside if reality measurably contradicts it. In fairness, Einstein never accepted quantum mechanics because they flew in the face of his intuition, but it still got him pretty far.
Gödel was motivated by convictions about meta-mathematics, motivated by his Catholic fatih. Perhaps a purely mechanistic world view would have eventually generated the same result, e.g. Turing's halting problem, but who knows.
Gödel's background was Lutheran, not Catholic, and I don't think he was particularly orthodox. (I have no idea how his religious convictions influenced his metamathematical ones; you might well be right about that.)
Re: The Octonion Math That Could Underpin Physics
#88A layman's quest to understand wtf this is... >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 any…
Well yeah, math books... Jokes aside, i too think we need a new framework for divulgating math that actually tells what you need to know, without just handwaving at it, BUT without the amount of technical details of a mathematics class. What do you know, i think this is Possible, too.. You can communicate a surprising amount of information if you use words properly. Of course, since this has never been done except fr…
I'm still hoping for an ELI5 version of Wikipedia (e.g. as a language option).
Re: The Octonion Math That Could Underpin Physics
#89Earlier quoted context omitted.
Well yeah, math books... Jokes aside, i too think we need a new framework for divulgating math that actually tells what you need to know, without just handwaving at it, BUT without the amount of technical details of a mathematics class. What do you know, i think this is Possible, too.. You can communicate a surprising amount of information if you use words properly. Of course, since this has never been done except fr…
I'm still hoping for an ELI5 version of Wikipedia (e.g. as a language option).
I don’t know if it’s a language option, but a Simple English version of Wikipedia does exist. https://en.wikipedia.org/wiki/Simple_English_Wikipedia
Re: The Octonion Math That Could Underpin Physics
#90"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…
It's in the article towards the end, but these are the only kinds of ... "over real numbers," which constraint Furey believes may be only an approximation.
Even when adding "over the real number", the claim in the article is wrong: R(x), the algebra of rational functions of one variable with real coefficients is a field, and a module over R containing a isomorphic image of R as a subfield.
You really do need the additional constraint of the Hurwitz form to restrict the possibilities to R, C, H, O ...
(Of course this has nothing to do with the work of Furey - I'm just seconding that the claim in the article is incomplete and inexact as worded.)