I recently drove a thread about NSA and ECC off-topic, by way of speculating what breakthroughs NSA might have hidden away among its secrets. In the process found my general interest in mathematics, topology and physics re-awakening.
This led me to go out and find yet another paper that suggests Joy Christian's purported refutation of Bell's Theorem is in error, and this time the paper is more general and, it seems to me, powerful than others I have read in the past, especially ones that basically just rely on there being errors in some of Christian's equations. This one instead profers a wide field of error, into which it claims Christian's approach falls.
Meanwhile however the NSA and ECC had already led me into an interesting area of speculation and that speculation might not actually involve locality thus might not really care whether locality is broken or not. Which means for its purpose Bell's theorem might not be particularly interesting.
The thread had led me from the number four, as the four colour theorem, through to octonians, as seemed to be called for by Joy Christian's writings. I already know that quaternions can be used to do a whole bunch of interesting transformations upon three dimensional images and such. In the thread the idea had come to me (from the term 7-sphere) that maybe seven dimensions could be similarly worked with using octonions.
I also had long long long ago some ideas about how one could use a space-sheet or a time-sheet to represent the world, with one being the complement of the other. I don't think I have anymore the paper on which I wrote those ideas and I am fairly sure i never transcribed them into any computer so I think what i recall of them is all that i have left of them now.
The idea though had been that any infinitely tiny slice of time is static, so that we usually need two slices in order to derive notions such as movement and velocity. Part of the enquiry was whether one really does need two slices or whether one instant of time could contain all the information necessary for dynamics - the arrow of time, movement, accelleration and so on - to be derived.
The idea of using a time sheet and a space sheet was, I think, the idea that the space sheet would show all the static "things", and the time sheet would show all the "forces" or whatever it would be that would cause the configurations of those "things" to be changing. It would have things like potential energy depicted on it, for example, if potential energy is not something implicit in the static configuration of the "things". It would show momentum, if momentum is not something implicit in the static "things". It would encode the dynamics.
Possibly one might not need both in order to depict the universe, if one such representation actually implied, in its details, all the information represented in the other.
The off the cuff idea I came up with in the course of the thread, that maybe trying to relate a three dimensional situation to another three dimensional situation through an arrow of time, reminded me of those olden days when i had thought about a space sheet and a timesheet, because I also back then considered to space sheets, that is, two static slices of instantaneous time, and how forces, dynamics, could be cast as basically all the changes needed to get from one moment to another moment.
Another thing I had done long ago was to try some of those clever fractals one can make by iterating affine transformations across points of a space. From that I learned that a general equation or transformation can be applied to points "from outside", it need not be some kind of behavior encoded into the point itself. One can come up with tree-like shapes, for example, without needing a cellular automaton; indeed without any cells at all in a sense. The rules generate the shapes, directly from an empty coordinate-space, without referencing values each point or cell of that space already contains.
So. What I had said in the thread was that maybe taking a three dimensional situation and relating it to another three dimensional situation by means of a one dimensional arrow of time might add up to seven dimensions. Later I have thought "hmm" about that because of course usually when one is relating a past situation to a present or future situation (for example) one has in mind three dimensional situations that are situated in the same three dimensions. So why would one need two sets of three dimensions related by a one-dimensional time? Wouldn't one, rather, want the same three dimensions to contain different situations at different points along a timeline?
Could that be where the idea of four-dimensional spacetime goes wrong? If it does go wrong that is, of course. ;) According to wikipedia there are only four normed division algebras. The quaternions are wonderful for doing transformations of three dimensional space but what about a fourth dimension? If we wanted to do similar things to a four-dimensional space wouldn't we need something a little larger than quaternions? Maybe we would only need what one might term "penternions", that is, arrays/vectors of five numbers? But those are not normed division algebras, it seems. So maybe we need to jump all the way to octonians? Maybe octonians could even be used as some kind of "four dimensional space related to four dimensional transformations"?
What would happen if we used octonians to generate seven-dimensional images similarly to the way we can use quaternions to represent affine transformations which we can use to generate three-dimensional images?
Maybe eight dimensions gives us enough numbers per point to be able to represent each point as not only having a position in three dimensions of space and one of time but also to have a momentum along each of those dimensions?
All this stuff about numbers of dimensions always of course brings back to mind Crowley's "Naples arrangement" (see his "Book of Thoth") in which one point is a point, two is a line, three is area, four is volume, five is time and so on. (Six is a point, located in space and time; seven is the point's idea of bliss; eight is the point's idea of thought; nine is the point's idea of being; ten is the whole thing: the point, with all of the above, and all around it - all the points, the whole shebang, the actual world.) I am now thinking maybe one could use the term "agency" for five, "agent" (or maybe even "self"? but I am second-guessing myself) for six, "motive" for seven, "method" for eight, "behavior" for nine. Or something like that. But enough; Crowley is, after all, not really famous for being much of a physicist nor even mathematician. :)
But back to the number four again. It only takes four colours to colour any two-dimensional map. According to the holographic universe theory all the information about a volume might be able to be represented on its surface, so much so that, for example, the surface of a black hole might contain all the information "inside it"; so much so that it maybe need not even have any inside, just a surface. Do we only need four fundamental forces to "explain" the universe because we only need the surface of a universe in order to code all information about that universe? Is the fact that we have so far only needed four forces to explain the universe basically just the four colour theorem with forces as the colours? That seems like as good a place as any to stop writing and go do some more thinking... ;)