4.5

 ----- Time-Space

Parallelisms : Classical mechanics - Quantum mechanics - Accounting - SpaceTime - Relativity - Riemannian geometry

Classical mechanics - Quantum mechanics - Accounting

The sphere allows to list the different projections that can be made. They are listed in the following table:

The bottom line indicates the views that consider only one dynamical variable at the time. Of course when we combine them all, we may recover like a complete view in the format [q] x [p] x [t].

It is what classical mechanics does and we understand now that it will only be valid in cases where the three dynamical variables are conversely like three independent worlds, and so they describe the complete picture in those cases and they commute because they independency.

If we like to maintain a complete view with only one of those mono-dimensional view, we have seen in the section devoted to the accounting foundations - at page 3.4 - that the general usage of would be handle only at the expense of adding the two accounts [u] and [v]. We understand now that those two extra accounts are only like upgrading the description at the 3 dimensions required to be complete - say owning the 3 dimensions of [q] x [p] x [t].

In turn we see here that this necessary "closure" seems now like nothing but adding the two necessary additional dimensions that are required to have a complete view owning three dimensions.

The middle line exhibits the projections where two of the three dynamical variables are coupled. Of course those views are incomplete and conversely to what we said in the mono-dimensional views, we will need to add one "closing" dimension to each view if we like to handle them as complete view.

The "closing" dimensions reflect our unknown or our uncertainty - in sense of Shannon which is the information capability - about the rest of the system. The possible values are in general 1 or 2 bits - also in the sense of Shannon.

One sees here that the "uncertainty principle" does not come as an acceptable or required principle but as a consequence of our choice to handle only complete views for any case but at handling only records that does not own the necessary 3 dimensions.

This of course recalls for a parallelism with quantum physics and the Heisenberg principle but it may also suggest at the accountants that their system may look complete only at accounting for individualized tangibles goods. In the case they would like to keep their system complete in general - i.e. also when they like accounting for intangibles - they may need to add the additional dimensions - say the additional views that we presented at the previous page 4.4 .

It is also a question to us that the incredible and endless complexity or incompleteness that is own by some theories - i.e. like quantum mechanics and string theories - might be only like handling one of the partial views of the above table within an incomplete formalization - say not a complete double-entry accounting owning the necessary dimensions number.

SpaceTime - Relativity

We could name this paragraph an "anti-parallelism" as it will highlight more a difference than a similitude with physics.

It is well known that when he introduced the relativity, Einstein introduced the "Space-Time" - say a space made on the set of variables (x, y, z, t) - as a spherical manifold. Einstein did so on physical motivations at the quest of laws that may govern the real world.

We also came up here above with a spherical manifold - see the top of this page - but care must be taken that its nature and usages are completely different - with regards to completely distinct motivations.

We remind first that we are not working "within the real world" to possibly discover laws of nature but our aim is to work within an observation space where we like to gain the ability to record observations made on the apparent real world and/or induced by observers subjectivity.

For this reason, we came up effectively with a spherical manifold but of completely different in nature when compared to the one introduced in physics by Einstein - say that we remain at this stage with a completely abstract metric and our manifold embed the variables [q], [p] and [t] while the Einstein one introduced a physically defined metric - i.e. the world curvature - and only embed variables that may correspond to [q] and [t].

As expressed at the page 4.1 - we remember that we considered the 3 variables [q], [p], [t] and that we chose to expose our development from the view point of [t] - in mentioning that similar developments from the view points of [q] and [p] could be obtained by permutations of the symbols.

In this sense we adopted the denomination time-space to differentiate from the Einstein denomination - say space-time - and to indicate that it is part of a suite - say t-space, q-space, p-space - where the first part of the denomination indicates the choice of the view point and the second part only indicates that we look at the space "in its most general sense".

We remind that the aim of the development of holotomial analysis is to provide observers with an analytical tool that enables them to record observations of any nature and eventually to develop further analysis and possible calculus from them.

At the reverse of Einstein that inferred its manifold construction from physical arguments, noticeable is the fact that we came up at building a spherical world with no need of any argument related to any particular domain.

Say that our experiential learning originated within business and regional economic development to extend further at other domains like marketing, communication, visual interface design, coaching or psychology - see section 1 - but when it came at building our methodological foundations, no one specific domain has been invoked to ground our construction from the start to the point of defining a spherical manifold.

One can check that no other argument than arguments of geometrical nature have been invoked - so that our development should still a priori apply to any domain.

In this regard, the people liking to observe the world as Einstein suggested should be able to  perform their observation and calculus within the observation space that we are developing.

It is not possible yet at this stage to expose a manner of how this problem can be handled and this aspect will be developed later in the parallelisms of the page 4.7.

Riemannian geometry

The above figure illustrates that we can now see the maps that we discovered by experiential learning as axonometric projections of portions of a sphere - hence also as planes that are tangent to a sphere.

In this sense, the sphere of the figure at the top of this page is actually a manifold - say that it owns the property of a plane in the neighborhood of any point of its surface.

In this sense, one can say also that our sphere and our maps constitute an abstract variety of a Riemannian closed geometry. Say that the combination of those two components - the sphere and the plane - explains now why we have been able to see our maps as a space in which we could locate any observation.

As we did not invoke so far any specific domain of activity at building our variety, it is understandable that our manifold remains with an abstract metric - say in example that the radius of our sphere is not having any value.