Octonions

February 25, 2019 Martin K. Jones 1 Comment

Octonions are the fourth, the last, and the greatest of the so-called normed division algebras based on the Real numbers. When I first learned about the Octonions way back when, I didn’t like them because they weren’t associative like the Quaternions, the Complex numbers, and the Reals. But now I’m fine with that, and they may be important for new theories of physics!

Octonions have the general form:

a₀e₀ + a₁e₁ + a₂e₂ + a₃e₃ + a₄e₄ + a₅e₅ + a₆e₆ + a₇e₇

Where the coefficients a_i are Real and the bases e_i have (something like) the following relations:

e₀ = 1 (and -1 = e₁² = e₂² = …)

e₁ = I = e₂e₃ = e₇e₆ = e₄e₅

e₂ = J = e₅e₇ = e₃e₁ = e₄e₆

e₃ = IJ = e₁e₂ = e₆e₅ = e₄e₇

e₄ = K = e₅e₁ = e₆e₂ = e₇e₃

e₅ = IK = e₇e₂ = e₁e₄ = e₃e₆

e₆ = JK =e₅e₃ = e₁e₇ = e₂e₄

e₇ = IJK = e₆e₁ = e₃e₄ = e₂e₅

In addition, if any of the products like e₂e₃ = e₁ are reversed you get the negative, so e₃e₂ = -e₁.

Non-associativity is demonstrated by going through the list of triples:

(e₁e₂)e₃ = e₃² = -1

e₁(e₂e₃) = e₁² = -1

(e₁e₂)e₄ = e₃e₄ = e₇

e₁(e₂e₄) = e₁e₆ = -e₇

(e₁e₂)e₅ = e₃e₅ = -e₆

e₁(e₂e₅) = e₁e₇ = e₆

so it is hit or miss I guess. Also note that e₇ = (IJ)K = -I(JK). And for all (e_ie_j)e_k and e_i(e_je_k), if they are not equal, is one equal to the negative of the other? And do I have to multiply them all out to find out?

Further Reading:

https://en.wikipedia.org/wiki/Octonion

http://math.ucr.edu/home/baez/octonions/

https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720/

http://mathworld.wolfram.com/Octonion.html

https://ncatlab.org/nlab/show/normed+division+algebra

[*11.33]

fourfolds, Mathematics

Quaternions

February 22, 2019 Martin K. Jones 1 Comment

The Quaternions are a number system that enlarges the Complex numbers, just as the Complex numbers enlarge the Real numbers. In fact, Quaternions can be thought as special pairs of Complex numbers, just as Complex numbers can be thought as special pairs of Real numbers.

Quaternions can be used for all sorts of wonderful things, such as rotations in 3D space, instead of using matrices. Above is a pitiful diagram (although better than my last one) of the Quaternion units 1, i, j, and k used in the typical representation a + b i + c j + d k. Please read about them in the links below and be amazed!

Further Reading:

https://en.wikipedia.org/wiki/Quaternion

https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation

http://mathworld.wolfram.com/Quaternion.html

http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/

https://www.quantamagazine.org/the-strange-numbers-that-birthed-modern-algebra-20180906/

[*10.160]

eightfolds, fourfolds, Psychology, sociology

Spiral Dynamics

February 21, 2019 Martin K. Jones Leave a comment

Spiral Dynamics is an intriguing model for the different levels of development and interaction that individuals (and even cultures) may exhibit towards each other and the world. It has interesting parallels to Erikson’s Eight Stages of Psychosocial Development and even Leary’s Eight Brain Circuit Model of Consciousness.

Below is a list of these levels by (Catchy) Name, Social Structure, Motives, and Means of Coping:

SurvivalSense, Loose Bands, Survival, Instinctive
KinSpirits, Tribes, Magic/Safety, Animistic
PowerGods, Empires, Power/Dominance, Egocentric
TruthForce, Pyramidal, Order/Morality, Absolutistic
StriveDrive, Delegative, Autonomy/Achievement, Multiplistic
HumanBond, Egalitarian, Approval/Equality/Community, Relativistic
FlexFlow, Interactive, Adaptability/Integration, Systemic
GlobalView, Global, Compassion/Harmony, Holistic

And maybe another, further step if we don’t destroy ourselves and the Earth by just being stupid.

Further Reading:

Don Edward Beck, Christopher C. Cowan / Spiral Dynamics: mastering values, leadership and change

Don Edward Beck, et. al. / Spiral Dynamics in Action: humanity’s master code

https://en.wikipedia.org/wiki/Don_Edward_Beck

Fixing our world: Understanding Spiral Dynamics will help us get back on track

Spiral Dynamics Integral: How to Use Grave’s Values Model for Psychological Development

https://en.wikipedia.org/wiki/Integral_theory_(Ken_Wilber)

http://www.awaken.com/2014/04/ken-wilber-summary-of-spiral-dynamics-model/

[*11.41, *11.45]

eightfolds, fourfolds, Psychology

Howard Gardner’s Theory of Multiple Intelligences

February 16, 2019 Martin K. Jones Leave a comment

Next we examine (oh so superficially!) Howard Gardner’s Theory of Eight(-ish) Multiple Intelligences. To wit (ha-ha):

Intra-personal
Interpersonal
Bodily-kinesthetic
Nature-existential
Visual-spatial
Verbal-linguistic
Musical-rhythmic
Logical-mathematical

Further Reading:

https://en.wikipedia.org/wiki/Theory_of_multiple_intelligences

https://www.tecweb.org/styles/gardner.html

Multiple Intelligences

Howard Gardner / Frames of Mind: The Theory of Multiple Intelligences (1983)

Howard Gardner / Multiple Intelligences: The Theory in Practice (1993)

Howard Gardner / Multiple Intelligences: New Horizons in Theory and Practice (2006)

[*11.41]

eightfolds, fourfolds, Psychology, sociology

Erik Erikson’s Eight Stages of Psychosocial Development

February 12, 2019 Martin K. Jones 1 Comment

Erik Erikson developed an eight stage developmental model of the psyche for the full life of the human individual. Each stage has a key conflict (with both positive and negative sides), its resolution into a “virtue,” and the typical age range for that conflict.

Trust vs. Mistrust, “Hope”, Infancy (0-1 years)
Autonomy vs. Shame, “Will”, Early Childhood (1-3 years)
Initiative vs. Guilt, “Purpose”, Play Age (3-5 years)
Industry vs. Inferiority, “Competency” (or “Ability” or “Skill”), Schooling (5-12 years)
Ego identity vs. Role confusion, “Fidelity”, Adolescence (12-18 years)
Intimacy vs. Isolation, “Love”, Young Adult (18-40 years)
Generativity vs. Stagnation, “Care”, Adulthood (40-65 years)
Ego integrity vs. Despair, “Wisdom”, Old Age (65-? years)

I would think that these ages would depend significantly on cultural norms.

Further Reading:

https://en.wikipedia.org/wiki/Erikson%27s_stages_of_psychosocial_development

https://courses.lumenlearning.com/teachereducationx92x1/chapter/eriksons-stages-of-psychosocial-development/

Erik Erikson’s Stages of Psychosocial Development

[*11.44]

Consciousness, eightfolds, fourfolds, Psychology

The Eight Circuit Model of Consciousness

February 7, 2019 Martin K. Jones 3 Comments

Timothy Leary’s dead…
No, no-no, he’s outside, looking in…

— The Moody Blues

Timothy Leary came up with this scheme of eight circuits of consciousness and Robert Anton Wilson expanded and renamed most of them. It’s basically a “great-chain” of evolutionary brain circuitry, leading from lower to higher states of consciousness.

labeled my diagram with RAW’s names, but below I also list Leary’s names in parentheses.

Oral Bio-survival (Vegetative-invertebrate)
Anal Territorial (Emotional-locomotion)
Semantic Time-binding (Laryngeal-manual Symbolic)
Socio-sexual (Socio-sexual Domestication)
Neurosomatic (Neurosomatic)
Metaprogramming (Neuro-electric)
Morphogenetic (Neurogenetic)
Non-local Quantum (Neuro-atomic Metaphysiological)

Pretty wild stuff!

Further Reading:

https://en.wikipedia.org/wiki/Eight-circuit_model_of_consciousness

http://www.phinnweb.org/neuro/8-circuit/

https://en.wikipedia.org/wiki/Timothy_Leary

https://en.wikipedia.org/wiki/Robert_Anton_Wilson

[*11.42]

eightfolds, fourfolds, Psychology

John Beebe’s Eight Function Model

February 5, 2019 Martin K. Jones Leave a comment

After my last post about an ad-hoc eightfold metaphysics, I spent a few minutes trying to create an eightfold psychology. Feeling insufficient to the task, I quickly put this aside and instead wondered if there was any prior work for this, and lo and behold, I found a wealth of such things. Eight seems to be a popular number for the enumeration of the psyche and related subjects.

I’ve already mentioned Robert Plutchik’s Emotions and Mihaly Csikszentmihalyi’s Flow, which are both sort of psychological dissections. If you’re into Jung and related psychologies, Jungian analyst John Beebe developed his eight-function model of personality types that blends Jungian concepts together into an interesting system. The eight types are split into two sets of four: for every type in the first “positive” set (ego-syntonic) there is a corresponding type in a “shadow” set (ego-dystonic). Each set of four has a superior, an auxiliary, a tertiary, and an inferior function.

Hero/Heroine (superior function)
Father/Mother (auxiliary function) (or good parent)
Puer/Puella (tertiary function) (or eternal child)
Anima/Animus (inferior function)
Opposing persona (shadow of superior) (villain?)
Senex/Witch (shadow of auxiliary) (or critical parent)
Trickster (shadow of tertiary) (added deceiver for balance)
Demon/Daimon (shadow of inferior) (daemon?)

In my diagram, I’ve shown each function paired with its shadow function. And of course being Jungian, the terminology is full of gender distinctions, which could fairly easily be eliminated. There are also lots of arcane associations to MBTI and such, so this will take more than a quick look. Please just wait here, and I’ll be back with an update soon (or not)!

Further Reading:

The John Beebe 8 function model

Eight-Function Model

http://www.erictb.info/archetypes.html

https://www.careerplanner.com/8CognitiveFunctions/Cognitive-Functions-Simply-Explained.cfm

https://en.wikipedia.org/wiki/John_Beebe

Also see my posts:

Carl Jung’s Psychological Types

The Myers-Briggs Type Indicator

[*11.41, *11.43]

eightfolds, fourfolds, Metaphysics, Space and Time

An Eightfold Metaphysics

February 1, 2019 Martin K. Jones 5 Comments

If one juxtaposes the fourfold Space-Time-Matter-Energy with the fourfold Structure-Function-Part-Action, one sees that there are several relations between them. These are simple common-sense relations, not modern-physics types of relations. Below they are listed by Space-type relations and Time-type relations.

Space is required for the extension of Structures.
Matter constitutes Structures.
Parts in an arrangement make up Structures.

Structures extend and are organized in Space.
Matter is located in Space.
Parts occupy Space.

Time allows the expression of Functions.
Energy is required for the operation of Functions.
Actions in a sequence constitute Functions.

Functions have duration and reoccur in Time.
Energy requires and is dependent on Time.
Actions take place within Time.

Several analogies are also evident in this diagram.

Space : Structures :: Time : Functions
Space : Parts :: Time : Actions
Space : Matter :: Time : Energy
Structures : Parts :: Functions : Actions
Structures : Matter :: Functions : Energy
Parts : Matter :: Actions : Energy

Notice that analogies 1.-3. are “contains” relations, and 4.-6. are “part of” relations. One obtains the nesting of entities:

Space > Structures > Parts > Matter
Time > Functions > Actions > Energy

I suppose that any analogy could be shown in a figure like the one on the right (or even written as A / B // C / D) or with the two squares side by side, and that sets of four or six analogies that overlap could be nicely shown as above. If so, I wonder what can be gained by such representations? Probably individually not so much but with overlaps I think it would be interesting.

Further Reading:

The study of parts and wholes relations is called Mereology.

https://en.wikipedia.org/wiki/Mereology

https://plato.stanford.edu/entries/mereology/

[*10.157, *11.26, *11.34]

fourfolds, Philosophy, twelvefolds

The Rosetta Stone of Arthur M. Young

January 27, 2019 Martin K. Jones 2 Comments

In his book “The Geometry of Meaning”, Arthur M. Young developed a system which he called his “Rosetta Stone” because he claimed that it associated mental attributes to some physical measurements, giving a kind of map to mind-body dualism. In the above diagram the expressions and names of these twelve objective values or physical units are shown. Examine the list below to see the twelve subjective values or mental attributes (shown in parentheses) that Young ascribed to them.

I’ve rearranged the sequence of these somewhat because I don’t quite understand the reasons for the organization Young gave them. I’ve also relabeled the units using the more standard derivative notation L’ = dL/dT, L” = d^2 L/ dT^2, etc. His notation is given at the end of each line below in parentheses for comparison (essentially he removes the d’s from the derivatives, writing dL/dT as L/T, which seems confusing to me when then multiplied by L).

Given the measurement Position or Length (L), Moment is Mass (M) times Position giving ML, and Moment of Inertia is Moment (ML) times Position giving ML^2. Velocity (L’), Acceleration (L”), and Control (L”’) have similarly related pairs of values by multiplying each by M and by ML. This gives four sets of similar expressions differing by their common derivative. I should have probably named the 1st set to be 0th, 2nd set to be 1st, 3rd set to be 2nd, and 4th set to be 3rd after the order of the common derivative.

Position (Observation): L
Moment (Significance): ML
Moment of Inertia (Faith): ML^2
Velocity (Change): L′ (L/T)
Momentum (Transformation): ML′ (ML/T)
Action (Impulse): MLL′ (ML^2/T)
Acceleration (Spontaneous act): L” (L/T^2)
Force (Being): ML” (ML/T^2)
Work (Fact): MLL” (ML^2/T^2)
Control (Control): L′′′ (L/T^3)
Mass Control (Establishment): ML′′′ (ML/T^3)
Power (Knowledge): MLL′′′ (ML^2/T^3)

How does Young justify associating, for example, the mental attribute “Faith” with the physical unit “Moment of Inertia”? It seems to have something to do with his original cyclic order of these twelve measurements and the associations he gives them to the Astrological Signs of the Zodiac. This mapping is not shown but can easily be found in the links below.

Further Reading:

Arthur M. Young / The Geometry of Meaning

http://www.arthuryoung.com/

http://mindfire.ca/The%20Geometry%20of%20Meaning/Chapter%20Five%20-%20%20The%20Rosetta%20Stone%20of%20Meaning.htm

http://kairos.laetusinpraesens.org/meaning_1_h_15

The only early entry in my blog without a leading figure (and poor tables as well):

https://equivalentexchange.blog/2011/01/02/arthur-m-young%e2%80%99s-fourfold-theory-of-process/

[*7.78, *7.79, *8.2, *11.38]

fourfolds

The Rigor-Relevance Framework

January 18, 2019 Martin K. Jones Leave a comment

The Rigor-Relevance Framework is a fourfold developed by the International Center for Leadership in Education (www.leadered.com). It is an instrument to aid in planning and evaluation for teachers and their students. It is composed of

Acquisition: Low rigor and low relevance
Application: Low rigor and high relevance
Assimilation: High rigor and low relevance
Adaptation: High rigor and high relevance

Rigor (from low to high) is said to be based on the six levels of Bloom’s Taxonomy of Thinking (knowledge, comprehension, application, analysis, synthesis, evaluation; or revised as remember, understand, apply, analyze, evaluate, create).

Relevance (from low to high) is said to be based on Daggett’s Application Model (knowledge in one discipline, apply in discipline, apply across disciplines, apply to real-world predictable situations, apply to real-world predictable situations).

There seems to be quite a bit of information associated with this center so I’ll need to look at it some more!

Further Reading:

http://leadered.com/our-philosophy/rigor-relevance-framework.php

https://www.kaganonline.com/free_articles/dr_spencer_kagan/367/Kagan-Connections-The-Rigor-Relevance-Framework