It'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…
The Octonion Math That Could Underpin Physics
111–120 of 192 posts
Re: The Octonion Math That Could Underpin Physics
#112Earlier quoted context omitted.
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.)
Thanks for the correction. I'm sure I've heard biographers (Jessica Goldestein perhaps?) describe his faith as having had a strong influence, but I must have misconstrued it quite a bit as memory faded. You're right, he was raised Lutheran. But his father was Catholic.
Re: The Octonion Math That Could Underpin Physics
#113Re: The Octonion Math That Could Underpin Physics
#114I 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.”
Eh, there's a balance to be struck, and that's obvious to anyone that's seriously thought about this issue.
There's no shortage of specious ideas. From outright quackery, the naive, or the simply misguided, there are countless ways to waste a lot of time and money on ideas that will lead nowhere. Especially when all you're guided by is intuition.
That's not to say that intuition isn't valuable; rather, intuition shouldn't be the only guiding principle. If an idea is 'real' in the sense that it will produce substantive research findings, it's reasonable to expect some kind of evidence for this. Maybe you're trying to show that A -> D. Well, showing a bit about B or C can go a long way to convincing people it's worth looking at the link between A and D.
The counter-argument is that it's possible there is not B or C, that a large and courageous leap is required to get to D. That's indeed possible, and arguably has been demonstrated with some famous results. But it doesn't follow that you simply must take everyone's giant leaps seriously and give them funding.
Ultimately, we go by proxies. If A -> D is required, well, at least show us that you got from A to B in some other issue. Give us evidence, however imprecise, that you might be that 1 in a 1,000 (or 1,000,000?) leaper that lands somewhere successfully.
So yes, there's an art to knowing when an idea is 'ripe' for a wider audience, and when it's time to stake your career on it. There's no reason you can't work on something in the background or during a sabbatical. The notion that 'academic incrementalism' is so destructive is a tenuous one, and certainly isn't being demonstrated here. To argue this is to say that Dixon would have been successful, if only he had gotten this or that position. Yet is appears that his approach was the issue, not financial or departmental support. I'd argue that the academic system correctly identified that his idea wasn't ready yet. Now that some demonstrable progress is being made, even if it falls far short of 'D', it's attracting attention and enthusiasm.
Finally, the notion that academia produces no actual research is farcical.
Re: The Octonion Math That Could Underpin Physics
#115I 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.”
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.
> Einstein never accepted quantum mechanics
Wait? Since when did Einstein not accept Quantum Mechanics? He won the Nobel Prize for his work in discovering an important part of Quantum Mechanics.
Einstein rejected the Copenhagen Interpretation of Quantum Mechanics. That's not the same thing as rejecting Quantum Mechanics.
I've seen no claims that he would object to either the Everett or Bohm Interpretations of QM. (But then again, he died before they were invented.)
Einstein, by the way, is far from the only critic of the Copenhagen Interpretation. Quite a few physicists are drawn to the Everett Interpretation instead, for instance.
Re: The Octonion Math That Could Underpin Physics
#116> “Because while it’s very easy to imagine noncommutative situations — putting on shoes then socks is different from socks then shoes — it’s very difficult to think of a nonassociative situation.” If, instead of putting on socks then shoes, you first put your socks into your shoes, technically you should still then be able to put your feet into both and get the same result. “The parentheses feel artificial.” > The oc…
If you're looking for a deep look at time -- a Total Perspective Vortex, I'd recommend "Spontaneous Inflation and the Origin of the Arrow of Time". https://arxiv.org/abs/hep-th/0410270
I don't know. Unidirectionality of time just feels like something that should be everywhere. Nonassociativity of basic math that governs all of matter seems for me to be a better source for it than inflation or entropy or gravity or any one specific thing.
Re: The Octonion Math That Could Underpin Physics
#117If anyone can make octonions cool it would that physicist
Re: The Octonion Math That Could Underpin Physics
#118I think this is the first article I've ever read about the octonions that didn't include the following John Baez quip: "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…
Is there an easy explanation of what problems quaternion or octonions solve?
Imaginary numbers are needed to take the square root of a negative number, and complex numbers result from combining the new numbers with the real numbers. Complex numbers also allow solving roots that aren't found in just real numbers.
But I have no similar comparison of what I can do with a quaternion or octonion that I can't do with a complex number. I remember seeing some w based number system derived from the cube root of either 1 or -1 (forget which, but w and -w were the solutions that weren't 1 or -1), but it did all the same things that complex numbers do and was considered mostly uninteresting.
It also seems like there is a pattern to go infinitely beyond octonions, but they all behave identical to octonions, but are the octonions even needed in the same way complex numbers are needed, or do they just make some math problems easier to work with?
Re: The Octonion Math That Could Underpin Physics
#119Earlier quoted context omitted.
As a current PhD student, I'd say the vast majority of researchers aren't anywhere near that enthusiastic about their research.
Many marriages fail in the time it takes to complete a phd. If academia can’t find a way to fund people with actual interest then somethings wrong.
many marriage fail while trying to get your PhD, imo
Re: The Octonion Math That Could Underpin Physics
#120Earlier 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.
This is very false. Einstein's reputation was built on explaining known phenomena such as the photoelectric effect and Brownian motion. Special relativity was heavily motivated by a pile of puzzling evidence and a bunch of existing mathematics.
> This is very false.
But didn't Einstein write the following in his own book?
“I believe in intuition and inspiration. … At times I feel certain I am right while not knowing the reason. When the eclipse of 1919 confirmed my intuition, I was not in the least surprised. In fact I would have been astonished that it turned out otherwise. Imagination is more important than knowledge. For knowledge is limited, whereas imagination embraces the entire world, stimulating progress, giving birth to evolution. It is, strictly speaking, a real factor in scientific research.”