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Re: Theoretical basis for physics: pure theory versus empirical 'theory'

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On Sunday, October 8, 2023 at 2:59:37 PM UTC-4, Tom Capizzi wrote:
> On Sunday, October 8, 2023 at 10:42:04 AM UTC-4, Tom Capizzi wrote:
> > On Wednesday, August 23, 2023 at 9:41:11 AM UTC-4, Timothy Golden wrote: 
> > > This gambit occurs in hindsight, yet here I can present it with foresight. 
> > > To what degree does a basis for physics imply pure mathematics? 
> > > I believe that this statement is acceptable and that it exposes a flaw in modern theoretical physics, for their basis is done empirically. They readily bow to the mathematics of the real number, and since space exhibits a three dimensional nature they happily go to RxRxR as a sensible workspace of, let's say, a first physical form. Yet the three was plucked from thin air and is a matter of empirical correspondence, and so we see a bleak theoretical claim here for all theory that works from such a basis, which is modern physics. While most of us will admit that the experimental physicists have the upper hand, few will admit that it started way back here. 
> > > 
> > > To explain that all of theory is empirical is to explain that we lack a clean theory. To expose the problem as a lacking in mathematics is where I care to shine the light. This notion of 'basis' and what exactly it means... does the ideal mathematics then provide emergent spacetime without any physics? In effect these are the details which make a basis, or at least which give a basis correspondence. 
> > > 
> > > To claim that you will require three copies of your basis for your basis, and never have any theory for the three, or four for that matter... and to claim that identical copies are effective: now here I am attacking the Cartesian product and it would seem impossible to do so, yet from this theoretical stance it can be done. Still, this is not the start of the journey that I would take you on on this thread. It is more a stop along the way. 
> > Physics is mathematics. Certainly, that's true of relativity. Physics has no explanation for the existence of a cosmic speed limit. Physics has no explanation for why relativistic momentum diverges from the Newtonian formula. Physics expects us to believe that solid objects shrink for the benefit of a moving observer, even when the object is motionless. Then, they have the gall to argue that there are no contradictions in relativity by changing the rules of logic. All of these are logically explained by mathematics. 
> > 
> > In general terms, here's the problem with physics. Newton started the ball rolling with his invention of physics. In his physics, the universe was rectilinear and real. And at the velocities which he had data for, this was good enough. Even NASA used Newtonian physics to get to the Moon, because it was good enough. The Lorentz factor, which determines the degree of time dilation and length contraction, for all speeds up to escape velocity, differs from unity by less than 1 part per billion. In the distance to the Moon, this is less than about 1 foot, less than the wobble of the planet. But at high speeds, Newtonian physics breaks down. Einstein tried to patch this with a universe that was hyperbolic and real. He made an educated guess about the invariance of lightspeed, and asserted the contradictions associated with time dilation and length contraction, as they were necessary to support his flawed logic. The problem is that the universe is both hyperbolic and hypercomplex. Relativity does not incorporate the hypercomplex feature. And physics will never find these features by their obsession with experiments. 
> > 
> > Hypercomplex measurements are not possible, because they are perpendicular to all real dimensions, and cannot contribute to their magnitudes. Hypercomplex rotations are even perpendicular to complex rotations. This is best illustrated on the surface of a sphere, where an ordinary complex rotation is a longitude shift, and a hypercomplex rotation is a latitude shift. Before relativity was a gleam in Einstein's eye, before Newton invented physics and calculus, even before Galileo proposed his relativity, the basis of relativity was used to construct the Mercator Projection. That basis is the gudermannian function. Applying hyperbolic trigonometry to this function establishes the existence of a cosmic speed limit. It is the basis of a new postulate that incorporates the hypercomplex nature of the universe. It is simply this: the laws of the universe do not allow any observer to measure any more than what is real to the observer. This supersedes Einstein's 2nd Postulate, and accounts for other features like time dilation and length contraction as geometrical projections. 
> > 
> > It is well-known that the Lorentz transformation is a pure, hyperbolic rotation. What is not so well-known is that every hyperbolic rotation is associated with a hypercomplex phase shift as well. This phase shift is the gudermannian of the hyperbolic rotation angle, which we call rapidity. And this phase angle determines what is real to all observers, the real, cosine projection of the gudermannian. Since the gudermannian varies with relative velocity, reality is different for every observer, and the contradictions of relativity vanish. Physics uses the gudermannian without naming it. When they assert that relative velocity can be expressed as c sin(θ), that defines the gudermannian angle, θ. Physics defines the Lorentz factor empirically, based on countless observations. But it is still empirical. The gudermannian defines it analytically. Given an arbitrary hyperbolic angle, η, and its gudermannian, θ, dη/d
> Sorry. This post accidentally got cut off before it was finished. To continue: 
> 
> dη/dθ = γ, the Lorentz factor. This is an analytical definition. The solution to the diffeq can be expressed in numerous ways. Here is a list: 
> cosh(η) = sec(θ) = γ, the Lorentz factor 
> coth(η) = csc(θ) 
> csch(η) = cot(θ) 
> sech(η) = cos(θ) = 1/γ = α, the projection cosine that determines what portion is real 
> tanh(η) = sin(θ) = v/c 
> sinh(η) = tan(θ) = u/c = p/mc, celerity/c and relativistic momentum/mc 
> 
> Differentiate any one of these forms with respect to either angle, and the result is the above diffeq. These solutions are pure math, yet they predict nearly all of special relativity. The math does not claim to be physics, but an impartial comparison of the mathematical relationships shows that they are in complete agreement with all the experimental evidence. Closer analysis of the diffeq shows that it predicts a cosmic speed limit and the geometric reason why momentum does not follow Newton's law (as well as why relativistic mass does not exist). The projection cosine also explains time dilation and length contraction, as well as a universal definition of simultaneity. They are, in fact, not necessary to prove the invariance of c, but as all the above functions are members of a 6-group, they are all intimately related, and any one can be derived from any of the others. The myriad of applications of hyperbolic trigonometry to physics would take a book to list. That kind of space is not available here. Wait for the book.

This is your first bout with physics, isn't it? Thus far you've remained purely mathematical as I recall. 
At some level the naming criteria for the trig functions is suspect. Let's say at least that it is not compact.
I have to admit as well that the hyperbolic system still has a foreign feel for me, and it may always be so.
I have done some sailing as well, and owned a plastic sextant, and learned some of the twenty brightest objects in the Northern sky, and am pretty good at math, and still the damn thing doesn't quite click. To confess that I am 3D challenged; mechanically challenged; I have to own this even as I can lay claim to the discovery of general dimensional algebra in its balanced form.

I really am blown away by all this content. It seems very fresh. You've been busy. But allow me to be accusatory at the same time, which I know you pretty well expect... does the obnoxious nomenclature of the trigonometry system somewhat commence from the commitment to the right angle as some sort of guidance, whereby these uniquely named individuals crop up whose identities are one and the same and merely shifted by that magical phase angle you so humbly land in toward the end of your vent? Of course those phase angles are right angles, and ought we then to come up with some more of them since we were working in quarters the whole time why aren't there four instead of two? That the inverse happens to be two stages around here... and that the minus sign suffices for this need; in the Cartesian plane, anyways. Good gravy, if you can work a P4 generic solution you'll have your hyperbolic trig, and eat it too? 

Generally speaking, the spherical surface is a theoretical assumption. Working at sea level all the time, and yet upon introducing elevation once, which is our mechanical limit, what happens if we introduce it again? I have little doubt that your own version will be  more nuanced, but the idea that we are going general dimensional seems to be held back. Back that way lays Olariu's cosexp(), and I wouldn't doubt if it is down there in the bibliography. Then too, what hope is there of begetting the general dimensional from a series of one dimensional spaces? Arguably you could put the same to polysign and claim that they are built from zero dimensional spaces, but this interpretation can be falsified. P1 are but an instance, and the raw magnitude devoid of sign does track like an angle possibly. Honestly it is of cosmological relevance. The pursuit of infinity on a 128 bit machine will not need the carry if we get it our way. We'll simply scrape off the top 64 bits and throw them away, ready for another product operation. This may be the op that we are after. Of course if you'd like you can have them back, but that is for you to carry. The idea that there must be room for products; in order that we ensure closure; an ambiguity arises that is underappreciated by the math cretins of old. I guess it's a sidetrack, but there it is. It's sort of like when we throw away the imaginary part of our value in engineering; concerned only by the magnitude, by which the phase incidentally automatically follows, supposedly, in our control systems. It is a unitized system; a normalized system. https://www.physicsforums.com/threads/why-3db-frequency-shows-45degree-phase-shift.164970/
All the same inversions, trigonometry, and so forth, and then you'll find the physicists doing it yet another way in terms of absorption and reflectance, which is interesting and effective appartently, and this craziness is upon a single dimensional signal... let alone a three dimensional substrate, or four, or so. Having gotten your y parameters straight, let's try another way. Along the way B and H disappeared. V and I will have to do, and the sort of z they make that once seemed simple gains new levels of difficulty under the Laplace transform, where infinities rear their head, and that damn exponential, too.

Regarding your article "polysigned_t12_and_three_flies.pdf" I had written a bit last night, but why call it t12 instead of P12? T12 to me is the tatrix, which is 
   T12 = P1P2P3P4...P11P12

Well, here it is. Sorry to be so long winded.
Gotta say, I'm really enjoying the section "Goldeyson's Predicaments". It's fun reading. Thanks, Tom.
Am I to understand that you are taking hyperspheres quite seriously here?
Can we imbue that spherical surface with a varying depth component?
Of course we can. That the result is a contortion of the original spherical assumption is acceptable. Were we to assign an ideal sphere to the Earth for instance all sorts of measures would incur and accrue having to do with the variances from that ideal form. 

Alright, I've found https://en.wikipedia.org/wiki/Gudermannian_function, so I guess it's no joke.

It would be fascinating if trigonometry took on new light from such. I did have some simple circle tracing research: Back in P3 graphing a circle essentially involves a series of increment operations like: *1, -0.001, +0.0002, *0.00014, -0.0.001, etc., where these differential values are tracing out a unit radius circle whose center is at '0', starting at *1, and progressing toward -1 from that position. In effect, however you care to code these things, and one of my favorite ways is as z^n from say z=*1-0.00001 , and if you want to unitize the results that is fine, but these will be pretty clean. Anyway, each sign component can act as a nondecreasing function, and they will be the same function shifted by 120 degrees, or using ordinary revolutionary language one third (of the revolution). I still wonder if there is a way to do it in P4 as there is in P3, but they are like tractor beams tuned to the nature of P4, P5, and so forth. They go planar, the z^n do. Mu really does have the special way around.

Such clever fun you are having. Great. So happy to know somebody else is working on it. You surely do know the long way around, and so many stops along the way. I am amazed and impressed, and just the sound of Goldeyson's Predicaments makes me want to read on. I still don't get it, though. This is all really your work. I guess I do see the Golden name working though for its suggestive value. Bravo, sir. Pat me on the back again, please. Honestly people, Tom here has five times the smarts that I do. I just got lucky on my one smart lot. Covering the ground I can speak highly of. Following the ground that Tom covers is quite another thing.

Oh good; I did not post yet. Off on one of your tangents I find:
 "They introduced the concept of Discretely Straight Walkability"
 - https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=ac961197fd6fde8c186987b8f6082047ea4b6dd3

A unit lattice of signons may not have linear addressing, but they do suggest a sort of putPixel(), or even putGroup(), but in the context of the signon. The simplex traverse, perhaps followed by a signon? In other words I believe there is a fine and a coarse, but that was already available from these unit mappings. Ultimately anyways, for now, the final address will in fact be an (x,y) coordinate structure to this display technology, unless you happen to have a vector driven engine... which might still have some artifact of this same. The raw form versus the final rendered form: are they the same thing? What is a basis? Did it construct the space or was it merely an adequate representation? Even an inadequate representation may have passed muster for prepolysign humans. After all, the lead they drank out of is the lead we breathed as children of the leaded gasoline generation. Bicyclists beware: as you are huffing and puffing up that hill, and a big diesel truck downshifts next to you, are you really going to hold your breath?
Been there, done it many times. Foul. This day even more foul high tech smells along the roadway... is inhaling synthetic oil lubing our lungs forever?
Got any teflon in there? 

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Re: Theoretical basis for physics: pure theory versus empirical 'theory' Tom Capizzi <tgcapizzi@gmail.com> - 2023-10-08 07:42 -0700
  Re: Theoretical basis for physics: pure theory versus empirical 'theory' Physfitfreak <Physfitfreak@gmail.com> - 2023-10-08 11:52 -0500
    Re: Theoretical basis for physics: pure theory versus empirical 'theory' Tom Capizzi <tgcapizzi@gmail.com> - 2023-10-08 12:09 -0700
  Re: Theoretical basis for physics: pure theory versus empirical 'theory' Timothy Golden <timbandtech@gmail.com> - 2023-10-08 11:09 -0700
  Re: Theoretical basis for physics: pure theory versus empirical 'theory' Tom Capizzi <tgcapizzi@gmail.com> - 2023-10-08 11:59 -0700
    Re: Theoretical basis for physics: pure theory versus empirical 'theory' Timothy Golden <timbandtech@gmail.com> - 2023-10-09 09:23 -0700
    Re: Theoretical basis for physics: pure theory versus empirical 'theory' Timothy Golden <timbandtech@gmail.com> - 2023-10-10 13:46 -0700
      Re: Theoretical basis for physics: pure theory versus empirical 'theory' Timothy Golden <timbandtech@gmail.com> - 2023-10-10 15:01 -0700

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