4.7

 ------- Uncertainty

Parallelisms : Business - Economy - Quantum mechanics - Quantum field - Relativity

Business - Economy

For being complete, the three possible cases are represented in the adjacent figure: at the bottom the concurrent case that has been introduced at the previous page; in the middle the oblique case with uncertainties as introduced in the above figure and at the top, the last case where the two groups are walking in parallel like they would never meet.

The "concurrent" case illustrates the situation where we comply with our environment - say we go at the same target like when we share common goals with a client at executing an order or a project on his behalf. 

The parallel case might be like two companies that are not concurrent - i.e. one in IT and one in logistic - and that are targeting the same sectors. One may say - like in quantum mechanics - that they are compatible.

The oblique case may illustrate an uncertainty that can also be exemplified in the economic or business filed.

Let's consider two competitors that responded to a demand of quotation from a customer who only needs one instance of the requested product.

Before the customer decision, both competitors may account for a possible sale as they quoted their offers and one need to balance each of their writings by an uncertainty because they do not know the issue (see top figure - left balance).

After the customer decision, obviously one offer will end up real and the other will vanish but still they both remain competitors so that the potential to be confronted again to a similar case still exist - hence the uncertainty remain at the same level (see top figure - right balance).

An other frequent case in business is an intangible that is at the same time an asset and a product - i.e. a software application is a product when you sell it but also an asset because you still own it after the sale. Similarly a knowledge and a know how is incorporated in a product at the production time but it is also an asset as it does not quite the company with the product.

It will be many occasions where a choice will be required - i.e. of course at writing the associated costs and expenses in the books but also at fixing prices: when acting as a service company you will tend to charge the customer will the total development costs of a software application while when acting as a publisher you will tend to consider a development as an asset and to distribute the charges on several product.

Many intangibles and associated cost are such - i.e. like customers acquisition costs, marketing expenses, reputation, brand awareness, tangible and intangible information, knowledge, education, acquired attitudes management, professional and social network and so on.

In economy, non rival products and public goods are frequently of owning multiple natures.

Classical mechanics - Quantum mechanics

Let's say that we are now going to finalize the rules such that our "camera" works always at giving separable and complete records sets.

1. When the observed records are naturally balanced - say respecting the "debit = debit" rule of the double-entry accounting, the camera will do nothing but close the snapshot "as it appeared".

When it is not the case - say the observations are not naturally balanced -, the "camera" will make itself the balance - at closing the snapshot - by complying with the following rules:

2. An "uncertainty" will never "be alone": conversely to what has been said in the upper frame of this page, accounting for one uncertainty will always generate accounting for a second one. Uncertainties will go by pair - say that an uncertainty worth is always at the minimum 2 bits.

3. At the opposite - and also conversely to the upper frame of this page - "void" will only be "single" - say as "the disappearance" of one of two co-existing cases out of which one survived and the other one vanished.

4. The camera will eventually upgrade the number of expressed dimensions so that their total is 3 - in reference to the table below.

Before to slightly longer explain this last point, let's first illustrate here below how our records table - shown at the page 3.4 - is transformed:

The dimension referred in the 4th rule is an item-cluster and an "expressed dimension" is when something is observed in an item cluster.

Of course [q], [p] and [t] that we introduced are dimensions and since we introduced them, we know that to be complete, a description must own at least 3 dimensions.

So the camera's rules might be summarized by saying: when an observation set shows up naturally balanced, don't do anything as the records already show up like complete and that the addition of two more dimensions will only add balanced records; when an observation shows up non balanced, make it balance and complete - say make it owning 3 dimensions by completing with u-and-v's.

To understand the left column of the table - that is "expressed dimensions" - care must be reminded that an isomorphism - say alike an equivalence - exists between the left and the right eye so that the dimension number can be lower than the imprinted records number - say when a same record is "duplicated" in both eyes, it is accounted only for one dimension.

The "expressed dimensions" of line 16 is noted "2 - (2 x 1)" only to remind that this case may sometimes be seen as a "true two" or as a "twice one" - i.e. see the parallelisms of the page 3.4 .

- "Bell theorem" - Out of the trivial case where no records are imprinted, let's say that we convene to assimilate classical mechanics at "commutativity and uniqueness" and quantum mechanics at "non commutativity"  and at "commutativity and multiplicity".

The above table shows that our camera will always exhibit a clear cut between classical and quantum mechanics - which in turn is only at reducing the Bell theorem to the upper bound values of those two domains.   

- "Other set up" - The above table and associated content is a good illustration of the manner that an holotomial analysis can only be utilized to produce an image that fits a given observer wish or view point.

We already passed through simpler sets of rules - i.e. a simple-entry single-eye system - and we may still infer more sophisticate ones - i.e. we may ask the camera to upgrade the dimension of the naturally balanced observation also to 3 (this possibility will be discussed in the parallelisms of the page 5.6).

In principle there isn't any best set of set up rules but there are different holotomial set up for external requirements or given aims - i.e. as already mentioned, we may like to develop a set up so that the total dimension of any record set of the above table is 3.

We are so in line with our experiential learning experience that provided with reified solutions only at responding to external needs. The same will be true with an holotomial set up and we may state - conversely to our understanding of the work of M. Callon and B. Latour - that a configuration may not own "per se" an intrinsic value but that it only owns a value with regards to the usage that is done of it.

More close to a day life experience, it is equivalent to say that we need a camera that is able to take the picture that we like to have in a comfortable manner.

Quantum mechanics - Quantum field

If we get back on the first paragraph of this frame, one of the resolution of the oblique case can be that one company receive the client's purchase order while the other one does not receive any.  So the case resolves in a concurrent case (one competitor executes the order in compliance with the client action) and a parallel case (the two competitors remain looking at the same sector but with temporarily no interaction.

The transition from the oblique case to the case of concurrent-parallel pair can be accounted by a suite of motions - for which accounting has been  presented in the page 4.3 - so that the accounting system pass conversely from one situation to an other.

This in turn will give a cost of the transition that reflects the efforts or forces that have been necessary for the transition.

Recollecting again all the previous concepts, the above means that the holotomial analysis may provide with a relationship between the fields, the forces and the particles - say in the sense that it may calculus estimations that are compliant with quantum like cases.

Relativity

As already said - at page 4.5 - the concept of a spherical manifold has previously been introduced by Einstein from totally different motivations.

In short we can say that Einstein analyzed that a universe completely submitted a Newton's type of laws would require instantaneous links between all its components - say that the earth would anytime know instantaneously how the sun is attracting it.

The relativity having demonstrated that information could not travel faster than the speed of light, it is clear that extremely remote planets may not know about sun's attraction information in an instantaneous manner. So the Newton's model is not an accurate one in any case.

Having introduced the concept of spherical and closed manifold for the space-time (x, y, z, t), Einstein introduced also the concept that forces - or fields - would be generated by manifolds curvature - and further that matter - say also energy - would also be able to locally modify the manifold curvature.

To some extend, we may translate the concept of manifold by saying that it is a geometrical being that in a sense allows local properties being described - or perceived - in a different way than the global ones with a compliant coherence between local and global ones - and so this local difference may be related to fields or acting forces.

This in turn may suggest that an alternative resolution of our "oblique case" may be investigated via a change of the curvature - or at least of the local metric - on our spherical manifold.

Our working media being a geometrical observation space for computing records, we can not argue that this track may no lead to results that own advantageous characteristics. We don't see however - at least at this stage - how such a solution may be enrolled without additional assumption - i.e. like an additional constrain on the metric of our spherical manifold or a assumption having an other nature that purely geometric.

We prefer to propose first - in the next paragraph - a manner that unifies in a seeming rich fashion quantum like calculus to relativistic aspects.

One of the motivation of our choice is that at this stage of our development, we are still mainly - and comfortably - working within a soundly abstract space at using only simple tools - like T-accounts - and we do not see any reason - or advantage - that would suggest us to introduce curvature variations - which in turn would require to introduce a more elaborated and more constraining metric than the very simple one that has been so far required.

In example, the resolution of the oblique case like presented in the previous paragraph seems quite simple to handle via a quantum like solution - say it infers only balancing T-accounts. Then why to develop a second solution for a same case when it is at the probable cost to have it more complex.

We remind again that we are not at the quest of discovering some kind of nature laws but in the duty to propose analytical tools to provide operational solutions. So we like first to test how far goes the most simple coherent instance of the holotomial analysis prior to infer additional calculus complexity - say when one sees where  theoretical physics went up since the last 50 years, we are not a priori asking to embrace a similar track if not strongly required.

Quantum mechanics - Relativity

When Einstein introduced the concept of a spherical manifold for representing the universe, he also introduce the concept that the universe can be view as isotropic - say homogeneous - on the ground that a sphere is isotropic - say that no point "a" can be distinguished from a point "b" on the surface of a sphere.

In our spherical manifold, we may say that the Einstein view point is true in "general" - say that all the things that we do not consider via a given limited configuration - or otherwise say anything that we don't see or look at - can be equivalently located anywhere on the sphere say a consequence that we do not mind where they are.

So at the exception of our given and consider map configuration, most of the sphere can be seen as isotropic it embed anything that we do not consider or see at given time - say that we are currently looking at only a very small part of the total universe.

Of course, this isotropy does not stands true "locally - say that we consider that a given maps or configurations means a local scale of a complete space - like in the sense of the definition of a manifold.

So it become clear that a local scale does not exhibit isotropy and homogeneity while the sphere "in general" may. In this sense, we will name a "local scale" every maps or configurations that we can handle - and say local relative to us as a map is done and watched by "a given person".

As such it appears that we can locally observe anything - by construction of successive holotomial configurations - but that all those configurations are not compatible - say that they can not be utilized all at once at the same time.

Non compatibility being a typical manifestation of quantum behavior, we can investigate an observation structure - say additional camera set up rules - such that our spherical manifold exhibits a quantum nature for a "local field" and a continuous nature for a "far field" - indicating by "continuous" that we associate isotropy-homogeneity with the distribution characteristics of the far fields variables.

Practically it means that the far field variables look distributed everywhere - say that they are affecting and applicable equivalently on everyone everywhere - i.e. like the Euro-Dollar exchange rate, the low level of Chinese wages seen from Europe or US, the oil price or the Wall street stock market.

An individual actor - i.e. like in example a business owner of an SME in Belgium - will see a certain number of those far fields variables like shaping - say sizing - its sphere like in fixed instance that is similar for him and his neighborhood: when he is involved in an "oblique case" like above - say quoting a mid-size customer request in his neighborhood - the far field variables are the above examples or any other goods that he may need to buy on the international markets to fulfill an eventual order.

All those parameters are the same for his own submission and for his competitors. Say that those parameter reifies a background "curvature" that is the same for everyone in its neighborhood..

The far field being treated, at his local scale the business owner will be confronted at a quantum case as previously described with regards to its direct competitors - say at competing with other supplying SME's in case which issue is uncertain.

He of course knows about the international variables but as they generate identical conditions for any one and that he has little chance to make them change only for himself, one can understand by this simple example that the SME owner is mainly focused on the parameters that can generate a differentiation with his competitors - say that he feels mostly the quantum aspect of his situation which is a potential reification dependant upon a neighborhood contextualization.

To pursue on this example of the SME, the purchase decision of the customer will change the system state in his neighborhood but there is a very small chance that it induces a significant influence on the far field variable like the Euro/Dollar exchange rate or the oil price.

Hence because the sphere curvature will not be changed by the local change of state, it is so coherent to say that the sphere is homogeneous in general but that the local scale may exhibit a discrete character - in the sense of bifurcations from one time-cycle to an other one.

Then we will join the quantum and the Einstein view point by saying that the local scale provided by an holotomial space - say the "small scale" which is the particular problem or configuration on which we focus via a plane projection - will appear quantum and that the far fields - which are the large majority of our global spherical space - will appear as continuously distributed.

A coherence of the above with the holotomial space construction requires that it stands true if we exchange the actors from the small scale and the far field - which can easily make sense as "continuously distributed" is in a sense equivalent to "non localized", say in turn that it also own some quantum nature.

The role inversion allows us to more precisely illustrate that a local scale does not correspond to short physical distances but to a representation defined by an holotomic mapping.

Let's design an holotomy that focused on the large far field variables that we mentioned above.

Those variables become so a "small scale" that embeds quantum like fundamentally hazardous choices - say that this suggests to see exchange rates or stock markets as owning hazardous variations.

On the other side, one can understand that someone caring about interconnected parameters like Euro/Dollar exchange rate and oil price will not take care of private SME's owners and consumers in an individualized manner within the holotomial view of his concerns.

He will group those actors within global population segments representing worldwide distributions: the SME's and population segment - say the former "small scale" - will become embodied in far field variables from the stand point of this new holotomic vision embedding the variables that were the former far fields from the stand point of an SME.

As such we can now understand that a "local scale" is not defined per se by "km" or "physical distances" on the earth but by an holotomial coherent view embedding actors that effectively interact. That is, the "local scale" defined in this sense behaves quantum like - while the complementary "large scale" of this view - say the "far fields" - behaves as evenly distributed.

From the above we can then infer an alternative geometrical configuration for our "camera" and capitalize some additional properties for this configuration:

- Homogeneity: our spherical manifold is homogeneous "in general" not in the sense that the physical components that may "live" on are all the same but in the sense that the dynamical structure is everywhere the same: a local discrete complemented by a continuous global.

This is compliant with the fact that we defined the sphere as [q] x [p] x [t] "in one block". Say that it highlights that we could not define individualized axis like [q], [p] and [t] for the sphere that so remain abstract.

- Quantum appearance: the above implies also the sense that when we look "in general", [q], [p] and [t] co-exist simultaneously everywhere - say can be known everywhere at the same time - while if we look "in particular" they appear discretely individualized - like in an holotomic view.

One can see in this property a complementarity between continuous and discrete, between classic and quantum, between continuous differential and discrete holotomial analysis - hence the denomination "holotomial" or between the homogeneity of a sphere and the discrete individualization of an holotomography - say then also between the "real world" and "the observer".

Say that in the aim of producing a tool that enables to compute a large variety of visions, the holotomial analysis looks like it can be an interesting candidate.

- Co-existence of opposite symmetries: out of the previous paragraph, we already introduced several times the co-existence of pairs of opposite complemented complete views - i.e. with the double eyes system and all the companioning double entry balances - starting from page 3.3 - or with the anticipation implying a pair of opposite quantum like fields - page 4.6.

All those dual and opposite symmetries are in fact only the consequence of the fact that we constructed a single-item holotomy such that it is a complete view that necessarily generates an isomorphic "twin" - see page 3.3 - and that their join and meet are both similar - see the parallelisms of the page 2.2 .

As we demonstrated that any object or concept can be represented by a single item holotomy - no matter whether the object or the concept is single or composite - it means that anything represented by an holotomy will own this fundamental opposite dual symmetry.

So we know that the dual symmetries are a consequence of the construction of our smallest scale that is a item-cluster and the above that this property could also be seen as induced by our highest scale the dual symmetry is also own by our complete holotomial space, namely also by the spherical manifold completing the plane projection.

Otherwise said, we started by constructing the discrete portion of this complete set , next we ended here at completing it by a continuous complement and we demonstrated that we can design its opposite symmetric by conversely inversing the roles between discrete and continuous.

If we go even further out our "highest" manifold scale to include the real world and the observer,  the dual symmetric oppositions can be also seen in this set - i.e. at saying that our observation space is the meet and the join of the reality and the subjectivity or that in our observation the vision of the observer is the closure of the real world and vice versa the real world is the closure of the vision of observer.

The first part of the sentence in bold can be illustrated by saying that our brain is like a dynamic closure for our perceptions of the real world and the second part by saying that a project that we elaborate in our imagination will be closed by its execution in the real world.

- a fixed curvature for the sphere: the far field - say the components on which the individual component of an holotomy can not have an direct influence - are fixing the curvature of the sphere - say its radius - at a same value for every one.

This aspect of "a same sphere for everyone" is a significant shift with regards to our initial experiential learning. The users experiences that have been related in the section 1 illustrate that the organic mapping can provide with advantages in several domains but not that one can have a common base for all those domains.

Our though at this stage was that anyone could build his own view of the world independently - so it leaves the sphere like an abstract space with no more constrain than being a sphere - and everything was fine with this view point - say a kind of "many worlds" or "many minds" - in a plain sense - that eases the communication of a given view point.

The importance that we may have an advantage configure our "camera" with a same sphere for everyone appeared later - i.e. like in the example previously described with the SME business owner quoting an offer.

With everyone having a different sphere, the business owner never know if a competitor may not have a more advantageous sphere providing him with better prices for goods available on the international market.

(note: one can raise the objection that it might be frequent cases that a competitor tries and obtains a better price for some supplies on the international market. This is in fact not an objection to the holotomial analysis but the way it is utilized and understood. First, the holotomial analysis sufficiently demonstrates the presence of dual opposite non reducible fields - see the previous property - to understand that it exists permanently a trends that some actors tries to change the far field conditions. Second, as long as no one succeed to change the far field, the sphere does not change and when someone manages to change one of his far field conditions, either he brings this far field parameter in the neighborhood - say this variable becomes by the fact "quantumly negotiable" - either his special conditions becomes available for everyone and so he changes the radius of the sphere for all.

In turn this allows to snapshot dynamics of evolution phenomena liaised to populations statistical nature - say like democracy or market trends: when you are nearly alone for an attitude, this attitude does not influence the whole but is only appearing like individualized spots in relationship wit specific - say local - contextualization, while at the reverse your attitude is largely shared by a large amount of other people, it may influence the far field but in a similar manner for everyone. This note confirms so that there no need for invoking a sphere curvature change to snapshot the case raised as an objection).

Important also is to understand that if the sphere curvature - say the radius - is fixed at a same value for every one, there is no necessity - even not meant - to evaluate this common value. Say that the sphere remains abstract or that the only property that is utilized is the spherical nature (at the point that even drawing a radius or a referential of any sort do not own any sense).

One can imagine that at talking with its accountant, a business owner may come up with a sort radius number - as he only handles numbers - but when he goes to his psychologist - where the sphere remains the same - the psychologist will not be able - with his renown competence - to handle any number - at the exception of his check.

(note: a few people - like those handling mechanics - may raise the objection that if the is set up with an abstract radius it does not correspond to their cases - as they can measure a radius - and - worse - they not be able to use the camera within the frame of their job.

To understand the case of those people, it is enough to consider a system made only of physical bodies and where the total amount of energy is fixed - without entering into details say that it is true that a conservative system where the total energy is fixed will apparently give a tangible value to the radius of the sphere.

The answer to this valuable objection allows to illustrate the difference - and the possible (dual opposite ?) complementarity - between the differential calculus and the holotomial calculus.

Say that you start to describe your bodies problem in the holotomial discrete tangent plane and that your trajectories are accounted so in discrete manners - say as presented in the pages 4.2 and 4.3.

You will reach the conditions of the above objection by inferring a "limit" transition that render the discrete step as small as you wish - say at creating a continuous description in the tangent plane which in turn will fix the trajectories to a fix metric and consequently also provides with the appearance that we may have a sphere - or say a manifold - with a tangible metric associated to those limit conditions.

If one like to give - shortly - a perception of why this manifold appear with the transition at a continuous limit, one can refer anticipatively to the page 5.7 where it is shown the total available energy was represented in the discrete plane by a stock of discrete particles. When we pass at the  continuous limit, only the stock associated with the actual motion will remain in the plane and we will need to grow an storage place outside of the plane for eventual stocking of the unused quantities.

Alternatively said, the manifold that appears associated to the plane is the results of the action of passing from discrete to continuous - which is an action that only happen in the plane and does not affect our abstract manifold.

The resulting continuous differential calculus and manifold may perfectly suites those particular cases - i.e. like raised by the above objection - but not all the cases that can be envisaged with the holotomial analysis.

It is known that differential calculus - that is based on an assumption of local continuity - suites correctly problems where this local continuity appears like a limit and that we keep working far above this limit - i.e. electrical circuits and vibration equations equation for waves lengths that are significantly greater that the system size - but that in case the existence if this local limit is required - or that we work below the existence of this local continuity - it is our experience that the differential analysis hardly giving an affordable meant to avoid chaotic investigations - it is also our experience that this aspect of local continuity or continuity as a limit is very often shaded if not ignored however to our souvenir it has been already highlighted by Maupertuis in the 18th century).

A very simple operational illustration of this local continuity requirements is often found in functions created on base of daily records - i.e. like sales or stock exchange records. When you estimate daily variations on the base of daily records they often don't anything but fuzzy variations, while if you base them on moving averages which encompass several days, you may obtain continuous like variations - say that the moving averages create like a wave length that equates several days.

The fact that we more clearly stated here above that the holotomial analysis is a calculus means help us to clarify also that is cam be utilized at observing cases but more certainly at handling operational - day to day - cases by providing a manageable vision on it.

Let's take in illustration the case of two persons trying to communicate on a project. If they have communication problems, they may now refer to the holotomial vision to verify if they own a similar or at least the common background - say the far field - required for understanding the project components - say the past - but also to verify if they do own a similar understanding of the aim the project - say of future.

It will be shown in the next section - particularly in pages 5.2 and 5.3 - that the holotomial analysis allows to handle all those possibly complex questions via visual constructions accessible on the ground of simple geometrical arguments - say in turn with an alternative manner to numerical solutions that may sometimes required significant efforts however they are only based on handling T-accounts.

It will be shown that both approaches - say geometrical-visual and numerical - are equivalent which in turn provides a very useful advantage to the teams which are made of a mix of people where not all understand or handle numbers.

- Entanglement and completeness: to frame this point, we shortly remind that quantum mechanics has generated a lot of philosophical discussions among which a renown one has been the confrontation between the Copenhagen interpretation - say physics is describing but not explaining the nature - and Einstein who was recalling to understand underlying motivations for such a reality.

Einstein was strongly against the theoretical aspect of quantum mechanics that he was considering being non complete.

In a famous attempt to prove his view point - renown as the EPR paradox - Einstein, Podolsky and Rosen invoked mental experiences of producing entangled particles - say particles which behavior is correlated - and they explained that if we next transfer them very far away from each other, according to quantum mechanics those particles would remain instantaneously correlated.

Because the information transmission was upper bounded by the speed of light, Einstein and his colleagues argued that this was either physically wrong - and so quantum mechanics validity was not universal as claimed - either it was physically correct - and so quantum mechanics was missing a part to explain how the entangling information would be transmitted or simultaneous owned.

However this question is soundly embodied in actual dynamics - we have not enter dynamics per se yet - we will shortly illustrate in the table below an aspect that can be underlined at this stage within the concern of holotomial observation space.

If we named our continuous spherical manifold the "Einstein field" - to recall that the continuous aspect of the [q] x [p] x [t] spherical manifold has been originated from the Einstein statement that a spherical manifold should be continuous - and if we recall the data of the table in the parallelisms of the page 4.5 , we may suggest that our holotomial calculus may offer a complete structure that is summarized by the following table where the last column characterizes the distributions of the particles:
 

A translation of the table in current language would be: "continuous" means that everything co-exist simultaneously everywhere - say that one can not individualize items but one can always draw everywhere on the sphere a grid - i.e. real and imaginary meridians and parallels - that will find - at any location on the sphere - the item it visualizes from a complete plane view.

"Discrete" means that any object can be observed independently of any other object and "multiple" means a mix of both the "discrete" and "continuous" modes - say that a partial items simultaneity exists every time and everywhere and items might be individualized but not all simultaneously. 

The two above paragraphs are illustrated in the two following figures that suggest in an alternative fashion how the coordinates, actions and times are successively distinguishable, partially superposed and totally superposed in the different level of the holotomial analysis:

In turn, we suggest also that the differential calculus would give a similar table - but not similar illustrations - for the distributions of metrics.

However, this resemblance is only a parallelism senso stricto. The differential analysis and the holotomial analysis own in common a kind of methodological similitude but they do not own a common nature - say that the holotomial analysis "surrounds" the differential analysis - like it surrounds any other type of analysis as shown in the figure of the page 3.5 .

In our stand point of being working in an observation space, both holotomial and differential analysis can express the meet and the join of the objective and the subjective worlds.

Differential analysis is however only applicable at domains where the observer can define a specific metrics - say like physics and mechanics - while holotomial analysis may handle those same domains plus - in principle - any other. Say that the holotomial analysis is independent of the definition of a specific metrics.

Noticeable also is that the holotomial analysis calculus is based on a very common double-entry accounting system - that has been introduced by a mathematician of the 15th century. Because this accounting system is easily accessible and widely practiced in the world, it may provide people wit a much easier access for the holotomial analysis  than for the differential analysis.