2.2

 -- Holotomy

Parallelisms : Mathematics - Quantum logic - Quantum mechanics - Psychology - Riemann hypothesis

Mathematics

The next table underlines how the holotomial analysis compare with the differential analysis:

In a sense, the holotomial and differential analysis looks complementary.

Both can be considered as abstract observation spaces with "opposite complementary properties. Say that - at the opposite of the the holotomial analysis - the differential analysis:

- may never embed a complete description of the "world" - say that even when a integral is extended at the whole space, the description remains partial as the variable set is usually finite - while the holotomial analysis is always complete

- does not contain per se a "quantum" component - say that the "quantum" concept has been introduced via an intention to extend the usage of the differential analysis at describing process owning such a property - while a "quantum" is naturally embodied by the holotomial analysis.

We will also demonstrate later that the holotomial analysis necessarily embeds an uncertainty principle - as well as a property that can be acquainted with the concept of entropy.

Those concepts received also echoes with the differential analysis but it only happened afterwards the physical observations of those properties - and not because they were embodied in the differential analysis - while they are in the holotomial analysis.

Quantum logic - Quantum mechanics

The concept of "complete space" entered our progression very early after we created our first maps - like reported at the page 1.3 .

We discovered that for any additional company name - located anywhere in the world - we could always indicate a plot for its location on a map - say like if we were ourselves at the center of the map and that the map border was the border of the universe.

It was actually like a space and like a one space where surprisingly any actor could in principle find a place even though the space had finite dimensions - hence came with us the concept of "complete" space or view.

As said in the section 1 - page 1.4 - we realized that we could infer a lot of usages and actions plan from the maps. The concept of "complete space" made incidentally a step ahead when we referred at the proposal of Stephen Covey to organize our priorities - ref. "The Seven Habits of effective People" - S. Covey - 1989.

In short S. Covey illustrated that we can represent the set of our concerns by a circle and circumscribe inside of it - by a smaller circle - the concerns on which we may have an influence.

We sharply translated it by asking people to look at the world via an "eye" - see our "Stephen-Covey-eye" version in the adjacent figure - where the center was the "actions that you can perform yourself", while we - simplistically - named the rest "all the rest".

It was obvious that this eye was also a "complete space", but the fact that we mentioned "all the rest" suggested that any sort of records could be embodied within this one space.

When we say anything, it is actually anything of any nature - i.e. objects, ideas, suggestions, concepts, people, company, equations, whatever, but also coordinates, momentum and time - say the basic variables of system dynamics.

From this point, we generalized the item-cluster constructions and the item-clusters superposition as introduced in the upper text frame of this page.

Of course by assembling adequately the item-clusters on the display, we easily recovered our maps with the additional conviction that we may extend their usages on the base of dynamical variables.

Because we already found a parallelism between our maps and quantum mechanics - say via similitude to a mix of Hilbert spaces -, we naturally came back on this field and in particular on the superposition of orthocomplemented lattices as presented in the seminal work of G. Birkhoff and J. Von Neumann in "The Logic of Quantum Mechanics" published in The Annals of Mathematics - 2nd Ser., Vol. 37, N°4. (Oct., 1936), pp. 823-843.

(warning: the rest of the present parallelism on quantum logic is mainly constructed as a discussion related to the seminal work G. Birkhoff and J. Von Neumann. Being so, the discussion is essentially related to mathematical logics and per se no easily tractable for people who are not familiar with those aspects and specific terminologies. At the opposite the next parallelism - at the bottom of the page - is discussing a similar aspect but in terms that relate to human being issues - in turn in terms which are more commonly accessible).

In a first aspect, our approach is similar to theirs because our single holotomies are made of an item-cluster and of its absolute complement.

One can verify that if we take an item-cluster "a" and an item-cluster "b" as defined in the upper text frame of this page, we comply with their definition of the complemented lattices - say with their relations L71, L72 and L73 that defines the absolute complementation (i.e. ref to "complement set theory" on Wikipedia ) - and one can also verify that we also comply with the modular relation - namely the relation L5 reproduced further down on this page.

Our procedure does own a slight difference with quantum logic - and quantum mechanics - because we always handle the holotomies as complete "world views" while quantum logic - and quantum mechanics - only consider subsets of "world views" when the combine or superpose several experimental proposition.

Namely - in the language introduced in the upper text frame of this page - they "forget" the part that we named the "rest of the world".

Because they work with closed spaces which do not own a closure to make them all similar, quantum logic - and quantum mechanics - end up with "intractable" questions that do not appear in our holotomial constructions.

The paper of G. Birkhoff and J. Von Neumann enable an illustration of this aspects as they closed their paper with two suggested questions that receive simple answers in the holotomial procedure.

The adjacent illustration answers at their first question: "What experimental meaning can one attach to the meet and join of two given experimental propositions ?".  On the illustration, our two experimental propositions "Is this an item 1 ?" and "Is this an Item 2 ?" are each represented by an holotomy, say that each proposition is completely represented within a similar big square.

The rule of "the rest of the world" makes that both the joint and meet of our two propositions are both the same space - say a complete world view - hence the joint and the meet of our two holotomies are equivalent and they define an other unique holotomy - say the proposition 3 - from two different "view points".

On an experimental view point, it means that the holotomial analysis is able - even in the quantum case - "to easily express the joint and the meet of two experimental propositions - simply by having independent observers read off the measurements which either propositions involves, and combining the results logically" (the former quoted " " sentence is from G. Birkhoff and J. Von Neumann who underlined that in their approach, it is not possible in general in quantum mechanics, hence their question about the meaning of the meet and the join).

We will show in a further section - see section 3 - that this only requires that we infer a complete formalization that is similar to the double-entry accounting system as introduced by Luc Pacioli in the 15th century.

Namely, one can illustrate this by saying that - in this actual formalization - quantum logic and mechanics (and classical mechanics) are like companies that would account for their financial statement only with the left part of their balance sheets.

This will not cause a problem in the cases there is a one-to-one correspondence between the assets and the liabilities (say like in classical mechanics) but as soon as the liabilities can distribute in different and partial manners between several left side accounts - or when they are intangible - it may become intractable to obtain a coherent view of a company with only a partial accounting system. In particular a double-entry accounting procedure will become a necessity to recover "the two independent observers" mentioned by G. Birkhoff and J. Von Neumann.

We may more simply say that "liabilities" are "potentials" and "assets" are "observations" and say that both mechanics and business accounting are looking at a similar problem. So in the case of mechanics - in particular quantum - a part of the problem comes from the fact that one only utilizes a simple entry accounting system which in turn infers an incompleteness of the description***.

Noticeable is in any case the fact that in the case of holotomial analysis, we comply by construction with the lattices definition in usage in quantum logic and - also by construction - that we maintain a completeness - say a coherence for the description -  like in the double-entry accounting system. So it is not so surprising that we have been able to describe quantum like cases - see the reference to Hilbert space in the parallelism of the page 1.3 - within a frame that remained understandable at business men being.

To get back more precisely on the paper of  G. Birkhoff and J. Von Neumann, we will show that the holotomial analysis does not either encounter major problems with the second question that they left at the end of their work: "What simple and plausible physical motivation is there for the condition L5 ?" (The condition L5 - that is represented in next the figure - is named "the modular identity").

In their analysis,  G. Birkhoff and J. Von Neumann illustrated graphically the "modular identity" by a modular lattice - see the lower right figure - and they stated that this modular lattice does not contain any sublattice that is isomorphic to the lattice graphically represented - say that we understand this statement by "the lattice is unique".

Also here, the unicity is an artifact due to an incomplete description. The introduction of our second observer - see our double eye system at the page 3.3 - shows that if they would have utilized a superposition of complete holotomies - say of "complete world views" - they would have conclude that it always exist at least one isomorphism of their experimental proposition.

For drawing the relation L5 within our "item-clusters system", let's take "c" as an item cluster and "b" as "the rest of the world" ("b" being taken as an other item cluster would lead to the same conclusion).

Next we add an item-cluster "a" and we assemble the superposition such that "a" is included in "c". We so obtain a figure that satisfies the modular identity and the relation L5 resumes to "a = a ".

(note: we ask the reader to forgive us about the fact that the upper frame of this page only partially introduces the construction rules of the holotomial superpositions for the sake capturing the basic procedure in an easy manner. The page 2.4 introduces those rules more precisely with the particular restrictions of no overlap neither inheritance which are applied in the construction of the above illustration).

Labeling variables with specific names being with no implication on their physical nature, we would not change any physical reality by permuting "c" and "b" within the above relation L5.

Conversely we will assemble the superposition such that "a" is now included in "b" and still we will obtain a figure that satisfies the modular identity - say the relation L5 will resume to "a = a".

Then the motivation of the relation L5 is to ensure that "a" can remain equivalent to itself independently of where we observe it in the complete space. Say that if "a" is observed via a given holotomial configuration, the relation L5 ensures that "a" can potentially be observed identical everywhere within the holotomial configuration. Say that "a" is a particle.

Noticeable is that this property allows to utilize the observation space to describe particle motions or the presence of identical particles in several places - say like an electron may move and we may have several particles that are electrons, objects can move and exist in several instance in our observation holotomial space.

By such the relation L5 leads also indirectly to other significant consequences in the frame of the holotomial analysis.

- If we remember that "a", "b" and "c" are all single-item holotomies and - as already said - that the labeling of variables can not have any implication upon any particular meaning, so "a", "b" and "c" can be anytime arbitrarily permuted and still have an equivalent sense.

It means in turn that any holotomy - or holotomies superposition - can be taken as an observation space and that any holotomy - or holotomy superposition - can be taken as a particle.

(note: in turn it also means that the holotomial analysis is a possible receptacle for a unification between the forces and the particles as both can a priori receive a description that might be "at the convenience" either a configuration either a particle).

- It also means equivalently that - in the observation space - at any holotomy or holotomies superposition we can always associate a number of isomorphisms that correspond to the assembly-permutation number that we can afford with a given set of holotomies.

In practice, this number is accessible in general only within the observation space and it is an opportunity now to remind that this space is surrounded by the real world and the observer.

The observer may agree about arbitrary permutations when we keep on variables like a, b, c, ... but when we use words or give a particular meaning to a set of symbols, the observer will only retain the configuration that make sense to him.

We do not know what the real is exactly doing and basically we may never know as long as we stand handling an observation space.

So, we may accept that the real world may not agree to access at all the configuration locations that are offered by the permutations of the observation space, that the world may own configuration permutations in extra of those known by "the observation-space manager" and also that he may not like to fill the configuration that are accepted by the observer in a way that make sense to him;

Visually, those discrepancies between the world, the observation space and the observer will not change anything - one can analyze that any normal drawing and any human vision are holotomies as they owns generally a compliance - or a "limit" compliance - with the configuration rules of the page 2.4 .

When we will infer an accounting formalization - at the section 3 - the "observation-space manager" will need to take in account with those discrepancies and we can introduce in advance that he will need to account for:

- real - say [1] like for observations actually made of the real world
- imaginary - say [i] like for imaginary reality provided by the brain of the observer
- void - say [v] like for permutations that could exist in the world but that can not be perceived
- uncertainty - say [u] like for permutations that can be perceive by the observer and for which he remain uncertain of about their nature

We will demonstrate in the following sections that the introduction of a double-entry formalism to support the usage in general of an observation space will require the usage of those "four" type of account.

Before finalizing this comment page on quantum logic and quantum mechanics, we will mention that we found a possibility of confusion in scientific and rational analysis that originates from liaisons that are considered as intrinsic characteristics.

This in turn infers concepts like transitivity, properties, inheritances - and so on - as being intrinsic part of a system which is not a true case in general or a priori.

To stay on a safe way and keep building our observation space on a general sense, it must be clear that in the foundations of our holotomial observation space, we will consider any holotomy and "any combinations of" as single and individualized entities.

As an example in semantic, a letter - like a, b, c, ... - are individualized entities, but also words, sentences, pages, books, libraries and so on.

As an example in quantum mechanics, we will not adopt in the observation space that the spin of an electron might be seen as a property of that electron as well as the values +1/2 and -1/2 being like properties of the spin.

By using an holotomy like we introduced, the electron, the electron-spin and an observed value of it - i.e. +1/2 - will all be taken as independent entities by "our camera".

In turn, all the following experimental propositions are independent:

- is this an electron ?
- is this a spin of the electron ?
- is this a spin - of the electron - that values +1/2 ?
- is this a spin - of the electron - that values -1/2 ?

At each proposition, the holotomic analysis will make correspond an item-cluster in which we may imprint "1" when the system gives a positive answer at the experimental proposition.

The converse conventions are expressed in the forthcoming page 2.4 that describes the observation-space configuration rules - and where the above example with the spin of the electron will be reminded within the parallelisms of the bottom frame.

Of course in an operational environment, we will never require that a complete individualization would be maintained when it may exist an operational translation that genders more simple manners to handle a given situation - i.e. apples and peers exchanges in a grocery does not require that all the clients maintain a double-entry accounting system for their family budgets.

The generalized individualization has been introduced for the significant simplifications that are induced in the theoretical foundations - in particular the noticeable fact that the global phase space may be defined as a simple manifold having the shape of a sphere - see page 4.5 .

Once this stage will be achieved, most of the theoretical aspects that serve had building the foundations will be replaced by "handling the properties of a sphere". It is a lot more simple and still it will remain natively compliant with most of the constrains that have been set before, with the experience advantage that a sphere and its associated projections are a lot more simple and natural to handle that pure abstract and relational mathematics - i.e. when we look at a red chair, we perceive the red chair as a single entity - say a cluster on the sphere or on the map - and not as a chair which owns the property to be red - like in most of the actual Boolean and computerized taxonomies.

Psychology - Quantum logic

The psychologists frequently utilize psychometric tests which are based on a suite of questions enforcing - or alternatively only suggesting - the choice of one - and possibly only one - answer among a set of proposed suggestions - which for most look extreme and exclusive or say like creating a tension that one can not often resume in a satisfactory manner - say that when the choices are forced like obligatory, one may have difficulty to adhere to them anyway.

One renown family of those tests has been constructed on the work of K. Myers and I. Briggs who based personality assessments on pairs of opposite psychological characteristics - i.e. Extraversion/Introversion or the Jungian set of psychological types.

Questions associated with the type example Extraversion/Introversion might be of the style:

- during your working time,
          choice 1: you always like to talk to any external people at anytime
          choice 2: you never like to talk to any external people while at completing a work

Otherwise said, those questions are like exclusive extremes. For people who fit those extremes, they easily demonstrate significant trends while for people who are moderate either compliant with a diversity of contextualization, none of the proposed answers ever actually fits - so that the results of the suite of answers are in a sense like hazard driven - say in a very similar manner that quantum uncertainty is seen to infer fundamental hazard in the answer of an experimental proposal in quantum mechanics.

The following figure here below illustrates how the personality test illustrated by our above example is viewed respectively via the view point of quantum logic and holotomial analysis:

In quantum logic, we have two pairs of equivalent experimental propositions - each being made of two exclusive parts - say a characteristic and its complement or its opposite. In the above figure, this is represented by a lattice with two exclusive branches - note that "0" means prior the test and "1" after the test.

In an holotomial analysis, we have four experimental propositions: each opposite pairs proposed in the quantum logic mode are split in single-item clusters each complemented by "the rest of the world" - as presented in the upper frame at the top of this page. All those four propositions own so a location for any type of system's answer - say any sign that the world would answer. So they are all equivalent and they can be superposed like in the right part of the above figure.

One can also see on the above figure that the lattices on the left do not consider the case of "no answer" while a room for this eventuality is always represented in the holotomial superposition.

Obviously, quantum logic is only considering a "tiny" part of the world possible configurations - say it restricts the world at only two spots like Extraversion and Introversion - while the holotomial analysis always considers the entire world.

Consequently, quantum logic presuppose other world's restrictions - i.e. extraversion is equivalent to "talking" and introversion to "not talking" - and so quantum logic may ignore - or find strange or abnormal - behaviors for which it has not been set up for a potential identification - i.e. an introversion that may however not restrain someone to like talking with external people - or more frequent, someone that is able to adapt his behavior and is compliant with a variety of contextualization may appear strange within an observation space which is not configured for such a diversity of attitudes.

In consequence, quantum logic may only see correctly things that are compliant with his set up in a intrinsic nature - while the holotomial analysis can correctly see both things that are intrinsically like "quantum logic compliant" plus also things that are not and that depend or vary according to a range of contextualizations.

We underline that the "forced answers" cases may even be worse as quantum logic may so translate - conclude - things like being intrinsic - and may not anytime have the capability to recognize when this enforcement is not accurate - say like the common wish that one have to stick a unique label on human beings. At the opposite, the holotomial analysis may always let the world to show up the way "he wants" as no a priori is made to restrict the system answers.

It is our understanding that the initial work of Myers-Briggs was looking at measuring probabilities of workers being comfortable with - or say enjoying - a job. In this sense their approach looks initially acquainted with an holotomial analysis - say measuring a probability that an event possibly happen when someone is in contact with a given type of externality.

This kind of analysis set up - which is estimating probabilities of reifications induced by given contextualizations - may accept the "no answer" cases and may cope with co-existing positive probability values for any - say even opposite - experiential propositions. This is kind of test set up is very similar to the one which raised the parallelism between our mapping procedures and the Hilbert space concept - see the parallelism of the page 1.3 .

At the opposite, when a test is set up as the lattices on the left side of the above figure - i.e. it forces exclusive choices - say "the no answer" is not accepted and/or it ends up with only a value on one of the two branches of the lattices i.e. at the cost of retaining only the choice with the highest scores average. It is our experience that a test handle in this way often tends to infer an interpretation like the measured properties are intrinsic.

This might be accurate "in some case" - i.e. like in physics when an interaction looks after the highest average - but it is not accurate "in general" - say that a psychologist who like to refer such results as intrinsic personality types should companion his conclusions by additional informations ensuring  that other alternatives are unlikely to happen.

The above allows to highlight in advance that the holotomial analysis is only a frame that one may set up over experimental propositions and results so that one can see the results within a "complete" frame - say complete in the sense that any interpretation mode might be investigated. From there, one may either keep the interpretation within the complete view point or eventually restrict the interpretation at the aim of serving specific goals.

This property of the holotomial analysis being a "frame over" existing propositions or practices will be highlighted by the fact that the holotomial metrics will be defined as an "add-on" that is juxtaposed sidewise existing informations or other metrics - like shown in the figure of the page 3.5 .

Quantum logic - Quantum mechanics - Riemann hypothesis

We like to briefly underline that the sketch at the right of the above figure can be seen as an Hilbert space - say a complete Hilbert space in the sense that the whole world can be embodied in it - while the lattices like on the left are often recalled as a subspace of an Hilbert space - i.e. in quantum mechanics.

That is, a subspace of an Hilbertspace is usually not a complete description in the sense introduced by the construction of an holotomy and it may always be incorporated within an holotomial description by writing the expression

[subspace of Hilbert space].[1] x [subspace of Hilbert space].[i]              (1)

that is written in compliance with the conventions of formalization which will be introduced for the holotomial analysis - say in the along next section and in the page 3.5 .

The probability that an observation - say any observation - is embodied in the expression (1) is equal to 1 by construction - we may say that P(1) = 1.

Say that if the probability of the outcomes of an experimental proposition are described by a subspace of an Hilbert space and that we set the probability to have an outcome equal to 1, we come in a situation similar to the lattices of the above figure - say the total probability for an outcome is 1 and the probability of the different possible outcomes will reflect the probability of occurrence for each branch.

By considering what is said in the two previous paragraphs and the relation (1), one can create the mathematical artifact that the total probability of an outcome might be expressed by the following relation

P(1) = 1/2 + i[H] = 1            (2)

where 1/2 correspond to the lattice drawn in the above image and where i[H] correspond to its imaginary image as "the rest of the world". Say that i[H] is an imaginary an subspace of an Hilbert space - say an image which embeds the an imaginary image of the distributions of the probabilities for the different outcomes of the experimental proposal.

This expression (2) is reminding the Riemann hypothesis via the similitude that the central part of the expression owns with the operator (1/2 + iH) proposed by M. Berry and J.P. Keating in "H = xp and the Riemann zeros".

Say that if a Zeta function transform

At the first problem they mentioned in their conclusions, the properties of the holotomic space - which are introduced in the sections 3 and 4 - suggest that we may propose that the Hamiltonian "xp" considered by M. Berry and J.P. Keating corresponds to a [q] x [p] holotomy, that it applies to the holotomic space [q] x [p] x [t] and that the "quantallization" is given by the common and unique metric owned by [q] and [p] - say the "bit" like introduced at the page 3.5 and 4.1 .

Seen is this view - say in the view of the mathematical artifact providing with the relation (2) and the view point of the holotomial analysis given in the previous paragraph - the non imaginary solutions E_n of the equation

Zeta(1/2 + iE_n) = 0      (3) - (M. Berry and J.P. Keating)

 might be provided by a system for which the bit-stock - see page 5.7 - only owns real values.

By so it suggests that the speculative view proposed by M. Berry and J.P. Keating - that E_n are energy levels - seems to stand true in some instances of an holotomial description. 

*** We may conversely say that a part of the problem of business is in the fact that one does try to account for "intangibles" as "tangibles" - say that one accounts for "imaginary" as "real".

Out of the scope of this introduction to holotomial analysis, one can show that a correction would lead to follow an accounting extension in a fashion as shown at the page 4.4 , which in turn would lead accounting to embrace concepts alike "quantum fields".