Recursion, Incursion, and Hyper-incursion

Recursion, Incursion, and Hyper-incursion

 

How do Past and Future inform the present?

What happens in the Present is not only determined by the Past but also by the Future.  Karma and Destiny both play a role as to what is going on in your life Now.

Key Terms

  • Recursion
  • Incursion
  • Hyper Incursion
  • Discrete Processes
  • Cellular Automata
  • Fractal Machine
  • Hypersets
  • Interpenetration
  • Turing Machine
  • Symmetry
  • Non Well Founded Set Theory
  • Sets as Graphs
  • Leela
  • Predetermined Future
  • Bhagya
  • Fate
  • Destiny
  • Karma
  • Anticipation
  • Four Causes of Aristotle
  • Material Cause
  • Formal Cause
  • Efficient Cause
  • Final Cause
  • Left Computer
  • Right Computer
  • Parallel Computing
  • Fifth and the Fourth in Music Theory
  • Bicameral Brain
  • Hemispheric Division of Brain
  • One, Two, Three.  Where is the Fourth?

From GENERATION OF FRACTALS FROM INCURSIVE AUTOMATA, DIGITAL DIFFUSION AND WAVE EQUATION SYSTEMS

The recursion consists of the computation of the future value of the variable vector X(t+l) at time t+l from the values of these variables at present and/or past times, t, t-l, t-2 ….by a recursive function :

X (t+ 1) =f(X(t), X(t-1) …p..)

where p is a command parameter vector. So, the past always determines the future, the present being the separation line between the past and the future.

Starting from cellular automata, the concept of Fractal Machines was proposed in which composition rules were propagated along paths in the machine frame. The computation is based on what I called “INclusive reCURSION”, i.e. INCURSION (Dubois, 1992a- b). An incursive relation is defined by:

X(t+l) =f(…, X (t+l), X(t), X(t-1) ..p..).

which consists in the computation of the values of the vector X(t+l) at time t+l from the values X(t-i) at time t-i, i=1, 2 …. , the value X(t) at time t and the value X(t+j) at time t+j, j=l, 2, …. in function of a command vector p. This incursive relation is not trivial because future values of the variable vector at time steps t+l, t+2 …. must be known to compute them at the time step t+ 1.

In a similar way to that in which we define hyper recursion when each recursive step generates multiple solutions, I define HYPERINCURSION. Recursive computational transformations of such incursive relations are given in Dubois and Resconi (1992, 1993a-b).

I have decided to do this for three reasons. First, in relativity theory space and time are considered as a four-vector where time plays a role similar to space. If time t is replaced by space s in the above definition of incursion, we obtain

X(s+ l) =f( …, X(s+ 1), X(s), X (s-l) …p.).

and nobody is astonished: a Laplacean operator looks like this. Second, in control theory, the engineers control engineering systems by defining goals in the future to compute their present state, similarly to our haman anticipative behaviour (Dubois, 1996a-b). Third, I wanted to try to do a generalisation of the recursive and sequential Turing Machine in looking at space-time cellular automata where the order in which the computations are made is taken into account with an inclusive recursion.

We have already proposed some methods to realise the design of any discrete systems with an extension of the recursion by the concept of incursion and hyperincursion based on the Fractal Machine, a new type of Cellular Automata, where time plays a central role. In this framework, the design of the model of any discrete system is based on incursion relations where past, present and future states variables are mixed in such a way that they define an indivisible wholeness invariant. Most incursive relations can be transformed in different sets of recursive algorithms for computation. In the same way, the hyperincursion is an extension of the hyper recursion in which several different solutions can be generated at each time step. By the hyperincursion, the Fractal Machine could compute beyond the theoretical limits of the Turing Machine (Dubois and Resconi, 1993a-b). Holistic properties of the hyperincursion are related to the Golden Ratio with the Fibonacci Series and the Fractal Golden Matrix (Dubois and Resconi, 1992). An incursive method was developed for the inverse problem, the Newton- Raphson method and an application in robotics (Dubois and Resconi, 1995). Control by incursion was applied to feedback systems (Dubois and Resconi, 1994). Chaotic recursions can be synchronised by incursion (1993b). An incursive control of linear, non- linear and chaotic systems was proposed (Dubois, 1995a, Dubois and Resconi, 1994, 1995). The hyperincursive discrete Lotka-Voiterra equations have orbital stability and show the emergence of chaos (Dubois, 1992). By linearisation of this non-linear system, hyperincursive discrete harmonic oscillator equations give stable oscillations and discrete solutions (Dubois, 1995). A general theory of stability by incursion of discrete equations systems was developed with applications to the control of the numerical instabilities of the difference equations of the Lotka-Volterra differential equations as well as the control of the fractal chaos in the Pearl-Verhulst equation (Dubois and Resconi, 1995). The incursion harmonic oscillator shows eigenvalues and wave packet like in quantum mechanics. Backward and forward velocities are defined in this incursion harmonic oscillator. A connection is made between incursion and relativity as well as the electromagnetic field. The foundation of a hyperincursive discrete mechanics was proposed in relation to the quantum mechanics (Dubois and Resconi, 1993b, 1995).

This paper will present new developments and will show that the incursion and hyper-incursion could be a new tool of research and development for describing systems where the present state of such systems is also a function of their future states. The anticipatory property of incursion is an incremental final cause which could be related to the Aristotelian Final Cause.

Aristotle identified four explicit categories of causation: 1. Material cause; 2. Formal cause; 3. Efficient cause; 4. Final cause. Classically, it is considered that modem physics and mechanics only deal with efficient cause and biology with material cause. Robert Rosen (1986) gives another interpretation and asks why a certain Newtonian mechanical system is in the state (phase) Ix(t) (position), v(t) (velocity)]:

1. Aristotle’s “material cause” corresponds to the initial conditions of the system [x(0), v(0)] at time t=0.

2. The current cause at the present time is the set of constraints which convey to the system an “identity”, allowing it to go by recursion from the given initial phase to the latter phase, which corresponds to what Aristotle called formal cause.

3. What we call inputs or boundary conditions are the impressed forces by the environment, called efficient cause by Aristotle.

As pointed out by Robert Rosen, the first three of Aristotle’s causal categories are tacit in the Newtonian formalism: “the introduction of a notion of final cause into the Newtonian picture would amount to allowing a future state or future environment to affect change of state in the present, and this would be incompatible with the whole Newtonian picture. This is one of the main reasons that the concept of Aristotelian finality is considered incompatible with modern science.

In modern physics, Aristotelian ideas of causality are confused with determinism, which is quite different…. That is, determinism is merely a mathematical statement of functional dependence or linkage. As Russell points out, such mathematical relations, in themselves, carry no hint as to which of their variables are dependent and which are independent.”

The final cause could impress the present state of evolving systems, which seems a key phenomenon in biological systems so that the classical mathematical models are unable to explain many of these biological systems. An interesting analysis of the Final Causation was made by Emst von Glasersfeld (1990). The self-referential fractal machine shows that the hyperincursive field dealing with the final cause could be also very important in physical and computational systems. The concepts of incursion and hyper-incursion deal with an extension of the recursive processes for which future states can determine present states of evolving systems. Incursion is defined as invariant functional relations from which several recursive models with interacting variables can be constructed in terms of diverse physical structures (Dubois & Resconi, 1992, 1993b). Anticipation, viewed as an Aristotelian final cause, is of great importance to explain the dynamics of systems and the semantic information (Dubois, 1996a-b). Information is related to the meaning of data. It is important to note that what is usually called Information Theory is only a communication theory dealing with the communication of coded data in channels between a sender and a receptor without any reference to the semantic aspect of the messages. The meaning of the message can only be understood by the receiver if he has the same cultural reference as the sender of the message and even in this case, nobody can be sure that the receiver understands the message exactly as the sender. Because the message is only a sequential explanation of a non-communicable meaning of an idea in the mind of the sender which can be communicated to the receiver so that a certain meaning emerges in his mind. The meaning is relative or subjective in the sense that it depends on the experiential life or imagination of each of us. It is well- known that the semantic information of signs (like the coding of the signals for traffic) are the same for everybody (like having to stop at the red light at a cross roads) due to a collective agreement of their meaning in relation to actions. But the semantic information of an idea, for example, is more difficult to codify. This is perhaps the origin of creativity for which a meaning of something new emerges from a trial to find a meaning for something which has no a priori meaning or a void meaning.

Mind dynamics seems to be a parallel process and the way we express ideas by language is sequential. Is the sequential information the same as the parallel information? Let us explain this by considering the atoms or molecules in a liquid. We can calculate the average velocity of the particles from in two ways. The first way is to consider one particular particle and to measure its velocity during a certain time. One obtains its mean velocity which corresponds to the mean velocity of any particle of the liquid. The sec- ond way is to consider a certain number of particles at a given time and to measure the velocity of each of them. This mean velocity is equal to the first mean velocity. So there are two ways to obtain the same information. One by looking at one particular element along the time dimension and the other by looking at many elements at the same time. For me, explanation corresponds to the sequential measure and understanding to the parallel measure. Notice that ergodicity is only available with simple physical systems, so in general we can say that there are distortions between the sequential and the parallel view of any phenomenon. Perhaps the brain processes are based on ergodicity: the left hemisphere works in a sequential mode while the right hemisphere works in a parallel mode. The left brain explains while the right brain understands. The two brains arecomplementary and necessary.

Today computer science deals with the “left computer”. Fortunately, the informaticians have invented parallel computers which are based on complex multiplication of Turing Machines. It is now the time to reconsider the problem of looking at the “right computer”. Perhaps it will be an extension of the Fractal Machine (Dubois & Resconi, 1993a).

I think that the sequential way deals with the causality principle while the parallel way deals with a finality principle. There is a paradox: causality is related to the successive events in time while finality is related to a collection of events at a simultaneous time, i.e. out of time.Causality is related to recursive computations which give rise to the local generation of patterns in a synchronic way. Finality is related to incursive or hyperincursive symmetry invariance which gives rise to an indivisible wholeness, a holistic property in a diachronic way. Recursion (and Hyper recursion) is defined in the Sets Theory and Incursion (and Hyperincursion) could be defined in the new framework of the Hypersets Theory (Aczel, 1987; Barwise, Moss, 1991).

If the causality principle is rather well acknowledged, a finality principle is still controversial. It would be interesting to re-define these principles. Causality is defined for sequential events. If x(t) represents a variable at time t, a causal rule x(t+l) = f(x(t)) gives the successive states of the variable x at the successive time steps t, t+l, t+2, … from the recursive functionf(x(t)), starting with an initial state x(0) at time t=0. Defined like this, the system has no degrees of freedom: it is completely determined by the function and the initial condition. No new things can happen for such a system: the whole future is completely determined by its past. It is not an evolutionary system but a developmental system. If the system tends to a stable point, x(t+l) = x(t) and it remains in this state for ever. The variable x can represent a vector of states as a generalisation.

In the same way, I think that determinism is confused with predictability, in modern physics. The recent fractal and deterministic chaos theory (Mandeibrot, 1982; Peitgen, Jurgens, Saupe, 1992) is a step beyond classical concepts in physics. If the function is non-linear, chaotic behaviour can appear, what is called (deterministic) chaos. In this case, determinism does not give an accurate prediction of the future of the system from its initial conditions, what is called sensitivity to initial conditions. A chaotic system loses the memory of its past by finite computation. But it is important to point out that an average value, or bounds within which the variable can take its values, can be known;

it is only the precise values at the successive steps which are not predictable. The local information is unpredictable while the global symmetry is predictable. Chaos can presents a fractai geometry which shows a self-similarity of patterns at any scale.

A well-known fractal is the Sierpinski napkin. The self-similarity of pattems at any scale can be viewed as a symmetry invariance at any scale. An interesting property of such fractals is the fact that the final global pattern symmetry can be completely independent of the local pattern symmetry given as the initial condition of the process from which the fractal is built. The symmetry of the fractal structure, a final cause, can be independent of the initial conditions, a material cause. The formal cause is the local symmetry of the generator of the fractal, independently of its material elements and the efficient cause can be related to the recursive process to generate the fractal. In this particular fractal geometry, the final cause is identical to the final cause. The efficient cause is the making of the fractal and the material cause is just a substrate from which the fractal emerges but this substrate doesn’t play a role in the making.

Finally, the concepts of incursion and hyperincursion can be related to the theory of hypersets which are defined as sets containing themselves. This theory of hypersets is an alternative theory to the classical set theory which presents some problems as the in- completeness of G6del: a formal system cannot explain all about itself and some propositions cannot be demonstrated as true or false (undecidability). Fundamental entities of systems which are considered as ontological could be explain in a non-ontological way by self-referential systems.

Please see my related posts

On Anticipation: Going Beyond Forecasts and Scenarios

Autocatalysis, Autopoiesis and Relational Biology

Key sources of Research

 

Computing Anticipatory Systems with Incursion and Hyperincursion

Daniel M. DUBOIS

 

Click to access cd554835f0ae367c3d3e3fa40f3e5e5f5f11.pdf

 

 

 

Anticipation in Social Systems:

the Incursion and Communication of Meaning

Loet Leydesdorff 

Daniel M. Dubois

Click to access casys03.pdf

 

 

 

GENERATION OF FRACTALS FROM INCURSIVE AUTOMATA, DIGITAL DIFFUSION AND WAVE EQUATION SYSTEMS

Daniel M. Dubois

 

Click to access dubois.pdf

 

 

 

Non-wellfounded Set Theory

https://plato.stanford.edu/entries/nonwellfounded-set-theory/

Hypersets

  • Jon Barwise &
  • Larry Moss

https://link.springer.com/article/10.1007/BF03028340

Non-well-founded set theory

https://en.wikipedia.org/wiki/Non-well-founded_set_theory

On Synchronicity

On Synchronicity

There are invisible ties between us.

 

Click to access Synchronicity2010.pdf

Causality has to do with events that happen in sequence, a cause producing an effect, whereas synchronicity has to do with events that happen together.

Synchronicism is the prejudice of the East, causality is the modern prejudice of the West.  – Carl Jung, 1929

 

 

Key Terms

  • Synchronicity
  • Serendipity
  • Interconnected Universe
  • Quantum Entanglements
  • Fractal Universe
  • Recursive Universe
  • Platonic Solids
  • Carl G Jung
  • Events in Sequence
  • Events in Parallel
  • Space Structure
  • Ether
  • Geometry of space
  • Complex Numbers
  • Shri Yantra Geometry
  • Mind and Matter
  • Brain and Mind
  • Parapsychology
  • Occult
  • Esoteric

 

https://coincider.com/about-coincidences/history/ 

History

The history of coincidence studies can be told through the stories of thefour people who coined words for the types of coincidences they noticed in their lives. The most famous is Carl Jung and synchronicity, but he was not the first.

Serendipity: Horace Walpole (1717-1797)

Horace_Walpole_by_John_Giles_Eccardt

Horace Walpole, a member of the British House of Commons in the 18th century, recognized in himself a talent for finding what he needed just when he needed it.A gift in the form of a portrait of a Grand Duchess whom Walpole had long admired arrived from his distant cousin in Florence, Italy. Walpole needed a coat of arms to decorate the new picture frame and just happened to find it an old book. On January 28, 1754, Walpole, thrilled with this coincidence, wrote to his cousin Horace, giving a name to his ability to find things unexpectedly—serendipity.

He got the name from a fairy tale called “The Travels and Adventures of Three Princes of Sarendip.” Sarendip (or Serendib) is an ancient name for the island nation Sri Lanka off India’s southern coast. The king of the fable recognized that education requires more than learning from books, so he sent his sons out of the country to broaden their experience. Throughout the story, the clever princes carefully observed their surroundings, and then used their observations in ways that saved them from danger and death.

For Walpole, serendipity meant finding something by informed observation (sagacity as he called it) and by accident. The main ingredients of serendipity include luck, chance, active searching, and informed observation.

Seriality: Paul Kammerer (1880-1926)

PaulKammerer

Biologist Paul Kammerer spent hours sitting on benches in various public parks in Vienna noting repetitions among the people who passed by. He classified them by sex, age, dress, whether they carried umbrellas or parcels, and by many other details. He did the same during the long train rides from his home to his office in Vienna. Kammerer was not particularly interested in meaning—only repeated sequences of numbers, names, words, and letters. Two examples can illustrate his thinking: His wife was in a waiting room reading about a painter named Schwalbach when a patient named Mrs. Schwalbach was called into the consultation room.A second example involved his friend Prince Rohan. On the train his wife was reading a novel with a character “Mrs. Rohan.” She then saw a man get on the train who looked like Prince Rohan. Later that night the Prince himself unexpectedly dropped by their house for a visit.

He defined “seriality” as “a recurrence of the same or similar things or events in time and space” which, “are not connected by the same acting cause.”  To him these repetitions were simply natural phenomena.

Kammerer thought these similarities were part of the structure of natural law, and in his 1919 book Das Gesetz Der Serie outlined what he thought these laws to be along with a broad set of classifications of their types and qualities.

Synchronicity: Carl Jung (1875-1961)

Psychologist Carl G. Jung

Carl Jung grew up in Swiss family that, on his mother’s side, embraced the paranormal. His personal experiences included apparitions (the disembodied figure of another person) and poltergeists (troublesome ghosts), spiritualistic communications (communication with people after their deaths), and materializations (creation of matter from unknown sources). His experiences also included telepathic, clairvoyant, and precognitive dreams, prophetic visions, psychokinetic events, and out-of-body and near-death experiences.

He invented the word synchronicity from the Greek syn—with, together—andchronos—time. Synchronicity means moving-together-in-time. Its fundamental characteristic is the surprise that occurs when a thought in the mind is mirrored by an external event to which it has no apparent causal connection. He also used the word synchronicity to refer to “an acausal connecting principle” that he placed on equal status with causation.

He included many strange events under the synchronicity umbrella including telepathy, precognition, and clairvoyance, along with poltergeists, apparitions, divination (e.g. the I Ching), and astrology. The definition of synchronicity has been stretched in many different directions.

Simulpathity: Bernard Beitman (1942–) Founder of Coincidence Studies

BernardBeitman

The term “simulpathity” defines a specific subclass of meaningful coincidences: the simultaneous experience at a distance by one person of another person’s distress. The experience occurs without the two people being together in the same place and sometimes without conscious awareness of its source. One person is in pain and another person feels distress for no apparent reason. Sometimes the distress is very similar to the other person’s pain. Often, the two people share a strong emotional bond. The largest number of simulpathity reports comes from twins, although reports involving mothers and their children are also prominent.

Simulpathity suggests that the individuals are more closely bonded than current scientific thought holds possible.

Simulpathity — from the Latin simul (simultaneous) and the Greek pathos (suffering) — differs from “sympathy.” The sympathetic person is aware of the suffering of the other but does not usually feel it. In the experience of simulpathity, one person suffers along with the other person and can experiences some form of that suffering. Only later is the simultaneity of the distress recognized, although some twins know just why they are feeling pain—the other twin is now feeling it.

Please see my related posts:

Interconnected Pythagorean Triples using Central Squares Theory

Indra’s Net: On Interconnectedness

The Great Chain of Being

On Holons and Holarchy

Mind, Consciousness and Quantum Entanglement

Geometry of Consciousness

Systems View of Life: A Synthesis by Fritjof Capra

Consciousness of Cosmos: A Fractal, Recursive, Holographic Universe

Myth of Invariance: Sound, Music, and Recurrent Events and Structures

Shape of the Universe

Reflexivity, Recursion, and Self Reference

Key Sources of Research

 

Synchronicity: Nature and Psyche in an Interconnected Universe

·
Joseph Cambray

https://www.researchgate.net/publication/26905461_Synchronicity_Nature_and_Psyche_in_an_Interconnected_Universe

 

 

Synchronicity and Healing

BERNARD D. BEITMAN, ELIF CELEBI, AND STEPHANIE L. COLEMAN

 

Click to access 18-Beitman-Chap18.pdf

 

 

 

SYNCHRONICITY
An Acausal Connecting Principle

CG Jung

https://archive.org/details/223463118SYNCHRONICITYAnAcausalConnectingPrincipleJung

 

 

 

CHANGING VIEWS OF SYNCHRONICITY-
FROM CARL JUNG TO ROBERT PERRY

Christopher Jargodzki

 

Click to access Synchronicity2010.pdf

Synchronicity, Mind, and Matter

Wlodzislaw Duch

 

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.79.8651&rep=rep1&type=pdf

 

 

 

Synchronicity: did Jung have it right?

Kurt Forrer

 

Click to access bb26804cdc465b355e2f2e09c574d50dc4bb.pdf

 

 

 

C.G. Jung’s Synchronicity and Quantum Entanglement:  Schrodinger’s Cat ‘Wanders’ Between Chromosomes

 

Click to access Limar.Synchronicity.pdf

Synchronicity: The Bridge Between Matter and Mind

F. David Peat
Bantam Books (1987)

Emotions Over Time: Synchronicity and Development of Subjective, Physiological, and Facial Affective Reactions to Music

C. G. Jung’s Psychology of Religion and Synchronicity

By Robert Aziz

Jung on Synchronicity and the Paranormal

By Carl Gustav Jung

 

 

Synchronicity and Emergence

JOSEPH CAMBRAY

Click to access Cambray.pdf

Jung, synchronicity, & human destiny: Noncausal dimensions of human experience.

Synchronicity: Science, Myth, and the Trickster

Allan Combs
Marlowe & Co. (1996)

Synchronicity, Science and Soul-Making: Understanding Jungian Synchronicity …

By Victor Mansfield

Synchronicity: Nature and Psyche in an Interconnected Universe

By Joseph Cambray

Geometry of Consciousness

Geometry of Consciousness

 

From Synergetics : Geometry of Thinking

 

0.1 “Synergetics is the geometry of thinking. How we think is epistemology, and epistemology is modelable; which is to say that knowledge organizes itself geometrically…” (I, 905.01)

0.2 “Any conceptual thought is a system and is structured tetrahedrally. This is because all conceptuality is polyhedral.” (I, 501.101) “By tetrahedron, we mean the minimum thinkable set that would subdivide Universe and have interconnectedness where it comes back upon itself.” (I, 620.03)

0.3 “All systems are polyhedra: All polyhedra are systems.” (II, 400.56)

 

Key People:

  • Ralph Abraham
  • BuckMinster Fuller
  • Stafford Beer
  • Arthur M. Young
  • Anthony Judge
  • Hermann Haken

 

 

Key Sources of Research:

 

Dynamics: Geometry of Behavior

Ralph Abraham

Click to access ms22.pdf

Click to access ms149.pdf

Click to access ms148.pdf

Click to access ms147.pdf

Click to access ms146.pdf

Click to access ms145.pdf

Click to access ms144.pdf

Click to access ms143.pdf

Click to access ms140.pdf

Click to access ms139.pdf

Click to access ms137.pdf

Click to access ms134.pdf

Click to access ms134b.pdf

Click to access rmic-pub.pdf

Click to access rmkplusfigs.pdf

 

Synergetics : Geometry of Thinking

BuckMinster Fuller

http://www.rwgrayprojects.com/synergetics/intro/scenario.html

Click to access folding.great.circles.2008.pdf

 

Anthony Judge

https://www.laetusinpraesens.org/docs00s/synerg.php

https://www.laetusinpraesens.org/docs80s/84nsetsx/x17.php

 

Team Syntegrity

Stafford Beer

Click to access TSI-Artikel.pdf

Click to access Stafford_Beer_-_Origins_Team_Syntegrity.pdf

 

Synergetics

Haken

Click to access Kroeger.pdf

 

Geometry of Meaning

Arthur Young

http://www.arthuryoung.com/theory.html

 

The Reflexive Universe: Evolution of Consciousness

Young, Arthur M.

1976

 

Consciousness: The Last Frontier of Geometry

Catherine A. Gorini

 

Click to access gorini_frontier91594.pdf

 

Mereon Matrix

http://www.mereon.org

https://www.cymascope.com/cyma_research/mereon_research.html

Click to access 756449251.pdf

http://www.frankchester.com

Click to access Quaternions-Phoenix-Bird-presentation-v14.pdf