Grassmann gives me hope, but i am still troubled.
The notion of a reference frame is only a basis for constructing a reference frame system, It is the reference frame system that holds the information about reality in invariant forms. In other words we have a system of reference frames and the invariant interference of their influence on our subjective experience is what we recognise or regard as eternal verity, truth, reality. Grassmann's linear algebra method and praxis makes it so straightforward that it is hard to believe that we have had to struggle to comprehend it. It is an apprehension of the interaction i have with space, the relationship i have in deriving a model (FS) of notFS from interacting with notFS.
It is at once far more general than any other conception while being far more specific. It convolutes with the wau i have to perceive things remarkably well, so that for the first time i feel i can notate my experience concisely but transparently. The reason why i believe is that Grassmann looked at the convolution and realised that we had a way of assessing it through combinatorics. Doing the work made clear to him the limits of the applicability of his conceptio. He then designed rules to stay safely within these limits and in that way provided us with a map of a land he had already explored in general . We are now able as he attempted , to fill in the detail.
That it is a map and not reality, and not the last word on reality, nor even the most sensitive map is shown by Cliffords immediate generalisation of Grassmann into Bi quaternions. linking immediatelyy Graswsmann's "map" with Hamilton Sketch map. I put it that way because Hamilton derived an abstract notion and applied it as much as he could to geometry, whereas Grassman Started with the full Euclidean geometry and "streched it". And before doing it he worked out all the ways he could sensibly stretch it. Thus his notion of a "vector" is indeed a elastic geometrical fo, a dynamic euclidean form. What his genius is was to take the "found" or inherent vectors in Euclidean geometry, the natural ones geometers us without realising and to make them explicit and accessible. by doing that he revealed the language we use when we manipulate space as being notable: that is we can denote a spatial manipulation action or intention explicitly and so write down precisely what we are doing or intend to do concisely and what that results in. "Numbers" or rather trig ratios then become coefficient or adjectives and adverbs of these new verbs or s we call them vectors.
This is Grassmann's conception, and this is why he was not understood in his time. His 'Zweig" was in fact not a twig or a branch but a foundational trunk of geometrical thinking specifically but logical and linguistic thinking generally.
For example it took millenia to end up with 26 letters in the english alphabet, but combinatorially and including punctuation and rules of grammar we can describe every aspect of our experiential continuum. However , we still know that this facility as flexible and fluid and essential as it is is still only providing approximate communication information, and different rules provide different maps of the same thing, ie the different languages!
The link to language is fundamental and deep, but as i have explained the Logos Summetria response undelies it all, so that it is not what is a subset of what but rather what is the relativistic structural foundation our neurology supplies in which we may begin to compare and contrast and distinguish? For it is that which leads to both language and geometrical mathematics, or as i refer to it spaciometry.
I have discussed before that in a reference frame parallelism is the notion of independent reference , not orthogonality, which carries the same idea but relates it to our perpendicularity sensory organ: it senses perpendicularity. Cartesian coordinates rely on the independence of the measurements and it is was natural to think that was related to the unique orthogonality of the 3 axes, but in fact Grassmann demonstrates powerfully that it is in fact parallelism that confers independence, and thus any reference frame can be established using that as an ordering rule for measurement. Thus we establish any number of non parallel directions, take the point of intersection of these orientations as a centre of rotation and an origin, and then order the measurements by rotation writing them down in an ordered set. This set of n measurements is now a generalised coordinate system relative to that centre of rotation and origin, We can then make independent measurement anywhere in space by measuring parallel to any or all of these reference orientations
This is why Grassmann is so straight forward, and why his method is so general, and of course there are geometrical relationships based on trigonometric ratios between all of the orientations.
Now the ordering of the set of coefficients/coordinates of the orientations actually follows a locus. In the plane it is a closed boundary,usually but not always a circle. It could also be an open boundary like a spiral and the interplay between the 2 is precisely what defines modularity. Now in 3d space we need 2 things a surface closed or open that surrounds the centre of rotation/origin and a path closed or open on that surface. Thus we could have a spheroidal surface on which a circle is drawn through the poles. but in that case we need other such loci to ensure we can reference all of space. The choice then is usually either great circles hat intersect at the poles ,ie radials from the poles or parallel circles parallel to a reference great circle. These choices section the coordinates into ordered sets which themselves are relatively ordered reflecting the geometry and the reference system chosen.
There is another alternative set of loci and that is the set of trochoids. One in particular the cornu spiral or the spherical helix going from pole o pole provides the possibility of an infinite set as a coordinate system: that is one long string of coefficients as it orders all the radials. Of course it seems clear that we could only ever approximate such system, and it remains to be shown that a single helix could order all the radials. Peanno's curve could do it and so that is another possible locus for a coordinate system.
Its funny, but Peanno was one of the first mathematicians to really grasp what Grassmann was expositing.
So parallelism and postulate 5 in Euclid really have a bearing on Reference systems that hold information about reality.
The minimum such system consists of my subjective gnomon and a local gnomon/ Although a reference frame requires a second arbitrarily small region to establish an axis in point of fact reference frames are mutual things and that is what is meant by relativity: my subjective gnomon and the local gnomon could equally well be referenced by each other,BUT the subjective gnomon is unique in that every reference gnomon relates to it and has a relative position that moves as it moves. This is the problem of parallax and incidentally why i think Einstein has got part of his theory wrong because he excludes the subjective gnomon. I tend to use gnomon because it relates directly to euclid and indeed carries the implication of trigonometric ratios in the reference frame.
So the problem with postulate 5 is parallax and perspective. I am sure that Euclid had not intended to say anything about infinity, but rather about an iterative process, Thus if the local measurement is that the lines are parallel that was good enough, He was well aware that lines which are locally parallel appear to meet in a point in the distance, his point was that they may meet, but they do not cross. Thus postulate 5 really states that any lines that meet bur do not cross are parallel. This applies equally well on a spherical surface and with regard to a straigtht line and a curve. The notion of parallel like the notion of an"angle" is much more general in Euclid than we are commonly taught. The tangent was devised just for this purpose of dealing with parallelism in curves, and the behaviour of tangents define the limit of parallelism. Thus if a curve has a tangent that never crosses a another line or tangent then that curve is parallel at the pont of that tangent.
The sophistication of the thought of the greek fathers is often underestimated. Very very often Newton borrowed ideas directly from the greeks in his formulation of the calculus, and indeed the calculus is applied trigonometry, and extension to trigonometry, just as Grassmann's Ausdehnungen are his elastic forms,stretchable geometry,bendable Euclid!
Thus the issue is that to apprehend reality we need not just one vector reference frame but a system of vector reference frames which define the notion of relativity and, most important this ,include the effect of the subjective gnomon.
Grassmann seems to me to be the best and safest way forward. As Clifford has combined it with Hamilton it seems Clifford algebras will light the path.However the light shines more brightly from Grassmann.