Archive for the ‘fourfolds’ Category

The Tangram

June 1, 2019

Further Reading:

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

https://equivalentexchange.blog/2018/02/24/a-game-of-fourfolds-part-5/

[*11.100]

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The Chromatic Scale of Music

May 28, 2019

All art constantly aspires towards the condition of music.

— Walter Pater

I really don’t know much about music theory, but I know I like this chromatic “circle” with solmization center.

Also, if you count out seven triangles clockwise around the rim starting from C you get G, D, A, E, B, F#, etc., which is the Circle of Fifths for chords. (But you get that with any chromatic circle.)

And I imagine if you go counter-clockwise you get the Circle of Fourths, which is important for Jazz chords!

Further Reading:

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

https://www.skytopia.com/project/scale.html

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

http://mentalfloss.com/article/53280/why-are-notes-tonal-scale-called-do-re-mi

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

Music of the Squares: David Ramsay Hay and the Reinvention of Pythagorean Aesthetics

[*11.68, *11.69]

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Combogenesis: a constructive, emergent cosmos

May 22, 2019

I just finished reading Tyler Volk’s “Quarks to Culture: how we came to be”. In this book Volk outlines an interesting model for what he calls combo-genesis, a “great chain of being” leading from basic physical law up through the highest organizational structures that we know of, human societies. He traces a path through human knowledge: physics, chemistry, biology, zoology, sociology, etc., and I am reminded of E. O. Wilson’s “Consilience: the unity of knowledge” that argued for some of the same things.

But Volk’s work has some good new ideas. He details twelve hierarchical levels, where each level is constructed on a “lower” previous level, and the new “higher” level has new things and different abilities than its predecessor. These levels range from the level of fundamental quanta (the quarks of the title), to geo-political states (the culture of the title).

  • QUA: fundamental quanta
  • PRO: nucleons, which are protons and neutrons
  • NCL: atomic nuclei
  • ATM: atoms
  • MOL: molecules
  • PCL: prokaryotic cells
  • ECL: eukaryotic cells
  • ANI: multicellular organisms, including animals
  • ASG: animal social groups
  • HUM: human tribal meta-groups
  • AGR: agro-villages
  • STA: geo-political states

These levels are within three dynamical realms, the first realm being of physical laws and then those realms of biological and cultural evolution. Each of these realms has a base level that has a capability for great constructive and emergent potential via an “Alpha-kit”. An alpha-kit has two facets, an element set and a cornucopia set, that operate like an alphabet and the myriad combinations that that alphabet can produce.

Dynamical realms:

  • Realm of physical laws: QUA -> MOL
  • Realm of biological evolution PCL -> ASG
  • Realm of cultural evolution HUM -> STA

Base levels and their Alpha-kits:

  • QUA, means for chemistry and molecules (atomic)
  • PCL, mechanisms for biology and its evolution (genetic)
  • HUM, faculty for culture and its evolution (linguistic)

As Volk’s model has each higher level based on or constructed from the previous lower one, I make the following suggestion utilizing my four-fold Structure-Function. The structures of each lower level serve as the parts of the next higher level, and the functions of each lower level serve as the actions of the next higher level. In this way a chain of actions and parts, structures and functions are built giving different entities and capabilities to different operational domains.

In the diagram shown, the sets of structures (S) and functions (F) of level i are used for the sets of parts (P) and actions (A) of level i+1, so S(i)=>P(i+1) and F(i)=>A(i+1). Not all structural information or functional abilities are necessarily accessible in the higher level of parts and actions, similar to the information and method hiding in object-oriented programming, and so reducing overall complexity. And as I have argued before, parts are combined to create the structures and actions are combined to create the functions of each level, so P(i+1)=>S(i+1) and A(i+1)=>F(i+1). In this way we have a bottom-up combo-genesis leading from quarks to culture.

Are we now entering another dynamical realm, perhaps based on some technological or computational alpha-kit? But, unfortunately we have to ask, will it take us forwards or backwards?

Further Reading:

Tyler Volk / Quarks to Culture: how we came to be

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

https://blogs.scientificamerican.com/cross-check/how-quarks-turned-into-cultures/

http://www.integralworld.net/smith56.html

http://www.integralworld.net/visser114.html

https://inquisitivebiologist.wordpress.com/2019/03/06/book-review-quarks-to-culture-how-we-came-to-be/

Tyler Volk and Robert Wright discuss:

mol-2017-09-28-wright-volk

Edward O. Wilson / Consilience: the unity of knowledge

https://en.wikipedia.org/wiki/Consilience_(book)

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

[*11.98, *11.99]

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Fire and Ice

May 16, 2019

Apologies to Robert Frost.

Further Reading:

https://www.poetryfoundation.org/poems/44263/fire-and-ice

https://en.wikipedia.org/wiki/Fire_and_Ice_(poem)

[*8.128, *11.96]

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You Are What

May 14, 2019

Identity, simplified? Pop psychology? Truth by Google search suggestion? Go to http://www.google.com and type in “you are what you”.

Further Reading:

https://www.blog.google/products/search/how-google-autocomplete-works-search/

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

[*11.82]

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The Hero’s Journey

May 11, 2019

Professor of Literature Joseph Campbell [1] popularized the monomyth of the “hero’s journey” [2], a recurring template for the plot arc or cycle of many heroic characters found in mythologies and even popular modern stories. From his most famous book, “The Hero with a Thousand Faces”:

A hero ventures forth from the world of common day into a region of supernatural wonder: fabulous forces are there encountered and a decisive victory is won: the hero comes back from this mysterious adventure with the power to bestow boons on his fellow man. [3]

Quite a few diagrams are available showing the generic journey of the hero or heroine; some are more complete than others, and the one I present here is no exception. Campbell’s schema is said to have seventeen basic parts to it, but not all elements need to be present in any specific narrative. I list four basic “acts” (separation, descent, ascent, unification) and twelve basic events, including four thresholds: 1st) crossing into the special world, 2nd) surmounting the supreme ordeal, 3rd) returning to the normal world, and 4th) journey’s end and / or beginning.

  • Call and Refusal
  • ——–< Separation >
  • Meet the Mentor
  • Crossing Over (1st Threshold)
  • Tests, Allies, Enemies
  • ——–< Into the Abyss / Descent >
  • Inmost Cave / Whale Belly
  • Supreme Ordeal (2nd Threshold)
  • Ultimate Boon / Reward
  • ——–< Magic Flight / Ascent >
  • Road Back / Refusal
  • Crossing Back (3rd Threshold)
  • Return with Elixir / Resurrection
  • ——–< Unification >
  • Master of Two Worlds
  • Freedom to Live (or Die) (4th Threshold)

Any hero’s journey must be transformational, and so has associations with alchemical change. It must be of such difficulty that it utterly changes the nature and the mindset of the traveler. Even though many supporting characters may influence, help, and even hinder our hero, it is ultimately about their journey, their sacrifice, and their reward. But what the hero brings back with them is also key to the story, because it is for the benefit of others as well as for themselves.

An interesting dissertation by Richard Warm that I ran across recently proposes that leaders, like heroes, need ordeals and trials in order to bring back the inspiration and wisdom to motivate others. From the abstract:

This dissertation will explore leadership as a mytho-poetic transformational journey toward self-knowledge, authenticity, and ultimately wisdom; the power to make meaning and give something back to the world in which we live; and the necessity of transformation. I view leadership as a transformative process and a transformational responsibility. As leaders we must undergo our own transformation in order to lead change on a larger scale. The dissertation will be both philosophical and theoretical, exploring how the threads of the hero’s journey, transformation, wisdom, and leadership intertwine. …[4]

These ordeals may also be of an intellectual or moral kind, and not just feats of strength or stamina, although those are important too. Warm also has a company [5] for coaching you on your leadership, life, and legacy.

One of my favorite books that I think of when the hero’s journey comes to mind is “Figures of Earth: a comedy of appearances” by James Branch Cabell. It is a bit of a spoof on fantasy tropes, well before its time. However, I’m not quite sure what the “magic elixir” is that the hero returns with except for his story: that life is what you make of it, or what it appears to be to others, or what you can convince yourself that it is, or that it is what it is and must be sufficient to you in the end.

References:

[1] https://en.wikipedia.org/wiki/Joseph_Campbell

[2] https://en.wikipedia.org/wiki/Hero%27s_journey

[3] Joseph Campbell / The Hero with a Thousand Faces

[4] https://aura.antioch.edu/etds/113/

[5] https://mythoscoaching.com/

Further Reading:

https://tvtropes.org/pmwiki/pmwiki.php/Main/TheHerosJourney

http://www.movieoutline.com/articles/the-hero-journey-mythic-structure-of-joseph-campbell-monomyth.html

http://www.sfcenter.ku.edu/Workshop-stuff/Joseph-Campbell-Hero-Journey.htm

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

“There are one or two things that I do not fancy the looks of in this torture-chamber.”

[*6.38, *6.133, *6.149, *9.200, *11.70]

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Patrick Geddes and the Notation of Life

May 8, 2019

Over the last few decades, there has been renewed interest in the work of biologist, sociologist, and city planner Patrick Geddes [1]. This is due to his efforts for holistic considerations for the entirety of the modes of human life and the facilities appropriate for their function. That is, he asked what makes a city or a town ideal for life, and how can we plan to bring this ideality into being? To this day, cities fail in many important ways.

Geddes embraced the new (at the time) Victorian notion of evolution in his work and thought of how cities could and should evolve to meet their shortcomings as well as provide environments for future developments. For example, common institutions such as schools, churches, and governments (polity) need to cooperate with family dwellings to provide for synergy and functional enrichment.

Geddes often used grids of words to explore relations between concepts, such as place, work, and folk. Placing these words along the diagonal of a square allowed one to consider the paired concepts of place-work, work-place, place-folk, etc. For example, how does the place-work compare with the work-place? His “notation of life” was a complicated schematic for exploring relations between a city’s facilities and the activities that they should promote.

  • Town / Acts : place, work, folk
  • School / Facts : sense, experience, feeling (alt. lore, learn, love)
  • City / Deeds : ethno-polity, synergy, achievement (alt. polity, culture art)
  • Cloister / Dreams (Thoughts) : emotion, ideation, imagery (alt. ideals, ideas, imagery)

Two locales are objective, two are subjective, two are passive, and two are active:

  • In-World (Subjective) : School and Cloister
  • Out-World (Objective) : Town and City
  • Passive : Town and School
  • Active : City and Cloister

And so:

  • Passive & Subjective : School
  • Active & Subjective : Cloister
  • Passive & Objective : Town
  • Active & Objective : City

Any important thinker is inspired and influenced by those that were previous or are contemporary to them and in turn is inspiration to those that follow. James H. Cousins was an important syncronic influence on Geddes, and please see [2] and [3] for information about him. The integral theory of Ken Wilber [4] is also compared to Geddes in [5]. I understand architect Lewis Mumford was a disciple of Geddes and I hope to find out more at a future time, perhaps by reading my copy of [6].

References:

[1] https://en.wikipedia.org/wiki/Patrick_Geddes

[2] http://hodgers.com/mike/patrickgeddes/feature.html

[3] https://equivalentexchange.blog/2017/07/13/a-study-in-synthesis/

[4] https://equivalentexchange.blog/2010/06/10/ken-wilbers-aqal/

[5] Theodore S. Eisenman, Tom Murray / An Integral Lens on Patrick Geddes, Landscape and Urban Planning,
http://dx.doi.org/10.1016/j.landurbplan.2017.05.011

[6] Donald T. Miller / Lewis Mumford, a Life

Further Reading:

http://architectureandurbanism.blogspot.com/2010/10/volker-m-welter-biopolis-patrick-geddes.html

https://www.dundee.ac.uk/geddesinstitute/projects/citythink/

http ://medium.com/@designforsustainability/design-and-planning-for-people-in-place-sir-patrick-geddes-1854-1932-and-the-emergence-of-2efa4886317e

View at Medium.com

https://quadralectics.wordpress.com/4-representation/4-1-form/4-1-4-cities-in-the-mind/4-1-4-1-the-ideal-city/

Patrick Geddes / Cities in Evolution

https://archive.org/details/citiesinevolutio00gedduoft/page/n10

[*9.12, *9.14, *11.92]

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Every Fourth Thing

April 30, 2019

What do I mean by the subtitle of this blog? If you search on Google for “every fourth thing”, after you see a few references to this blog you see mentions like: being out of something at the druggist, or saying a particular actor is appearing in a random show, or showing extra interest to a comment from a spouse, or things being requested that are forgotten by waiters, or subtexts of statements issuing from celebrities or politicians, or noting that every such organism ever discovered since the beginning of time was in fact a beetle.

So for the most part, a frequency of occurrence, like you have everything of a certain type that exists or can be named or thought about or the facts of the world in a line, and you are counting down and picking out every fourth one of them. For the usual sense of the phrase “every fourth thing”, the three things not picked are ignored because they don’t meet the condition desired, but for the sense I want, they are still “fourths” of things. They are as important as the one picked, either because they implicitly complete the wholeness in some way, or complete some other wholeness.

Or instead of the other three things passed over completing the whole, you have to count out your fourths four times, so you are looking at sixteen things but skipping twelve things to end up with four fourths. Or here is another idea, you have everything (of a certain type) in the world in a bin, and the condition of selection is that it is a fourth by its nature rather than counting it out. This is probably closer to my intention, but the ambiguity between these alternate senses is why I continue to enjoy the subtitle “every fourth thing”.

Once you’ve figured out your selection process, do any four things you select comprise a four-fold, or do only things that go together in a “special way” deserve to be called that? Do you pick with replacement, or once picked do things become unavailable for further selection? You have quite a lot of picking to do, so I would say no to replacement at first consideration; but since I do reuse fourths ad nauseam, replacement must be ok.

And one of the other amusing things that I like about “every fourth thing” is that I can pretty much talk about anything I want to with the minor condition that it has four parts to it (or some multiple of four), or that it’s four things together, or whatever. I used to be far more rigorous and selective, but now I don’t really care so much, and I’ll burden my readers with all sorts of nonsense to while away the time until the sun explodes. There’s really a never ending supply that goes on and on and on (and on).

Putting aside those questions for now, since they are evidently more complicated than my initial expectations, for this analysis I am trying to enumerate the different ways that four fourths could combine to form a unity. This list depends heavily on notions of sameness and difference. I have also discussed notions of equivalence before, but I have not returned to those considerations for a while and they may be erroneous or incomplete. But let’s soldier on and at least give it an honest try before we are exhausted by all of our doubts or side issues.

The first way that comes to mind is if each of the fourths is either identically the same thing or merely indistinguishable from each other (via replacement or being indiscernible). I’m not sure at the moment how to differentiate the two concepts so I’ll lump them together for now. Mathematically, we could represent this by the simple sum 1 + 1 + 1 + 1, or maybe the lambda expression λa.4a. For an example we could use the protons in a beryllium atom, or the sides of a square.

The second way is if each of the fourths is definitely a different thing. Then you might well ask why they are a unity and I’ll just say because I said so, that’s why! Or if I’m feeling magnanimous, I’ll try to at least give some reasons. Mathematically, we could represent this type of four-fold by a set, e.g. {a, b, c, d}, or by λa λb λc λd.a + b + c + d. As an example, a piano quartet would be nice consisting of piano, violin, viola, and cello. Also, the Four Causes, at first blush.

For the third way, we could consider the cross product of two sets, each having two elements, say {A, B} x {C, D}. Again, the linear expression λa λb.2a + 2b is different. And most real world examples have opposite relations like {A, ~A} x {B, ~B} (what I’ve called double duals before), so I’m having difficulty here. Maybe there are really 3 ways here that should be differentiated. Examples: the nucleus of a helium atom, Heidegger’s Four-fold, the Four Qualities, etc.

For the fourth way, let’s consider the fourths to be distinguishable, but there also is an order to them. Mathematically, how about the ordinal 4 = {0, 1, 2, 3} (but not the cardinal!). As an example, consider common time in music (4 4 time), having four quarter notes to a measure. The notes are different because of their order in the sequence, even if they are the same tone. Also the Four Seasons, except that it’s a double dual plus it has a sequence.

Now the fifth way. Mathematically: the Cartesian Plane, R x R, or the Complex Plane, C. Latitude and longitude, or maybe even North, South, East, West are examples. For every ordered pair (x,y) there is (-x, -y), (x, -y), and (-x, y). Or complex x + iy, -x – iy, x – iy, -x + iy.

Sixth. Three and one. Not the song “Three and One” by Thad Jones, even though it’s a great tune. Not the object (or is it the subject) of Trinitarianism, because that’s three IN one, and also the three are different. Ditto for three ingredient recipes. No on the yet to be released 4th Neon Genesis Evangelion movie 3.0 + 1.0, since that’s also three different things as far as I know (the first three movies). Perhaps ammonia (NH3) is an appropriate example with three hydrogens and one nitrogen . But I can’t say I’m very excited about this one, but it may be important due to my selection questions above. Now do the math: λa λb.3a + b.

Seventh. Two and one and one. λa λb λc.2a + b + c. I like this one even less, but at least it lets you see that the lambda expressions can be put into hierarchical order based on partitions of four. Perhaps the nuances of further structure can be teased out by different expressions, such as set theoretical or logical (instead of arithmetical).

Further Reading:

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

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

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

https://en.wikipedia.org/wiki/Time_signature#common_time

https://en.wikipedia.org/wiki/Category:Trinitarianism

[10.156, *10.190, *11.80, *11.81]

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Particle Man

April 26, 2019

Particle man, particle man
Doing the things a particle can
What’s he like? It’s not important
Particle man

— From “Particle Man”, by They Might Be Giants

…nothing is missing from your understanding of “Particle Man”…

— John Linnell of They Might Be Giants

In the song “Particle Man”, four “men” that are maybe super heroes are discussed:

  • Particle Man: even if something was known about this entity, except for being minute, it would seem to be of no consequence
  • Person Man: whether this individual just has low self esteem or actually deserves their belittled state is indeterminate
  • Triangle Man: hates Particle Man and Person Man for some reason, and bests each one individually in fights; a bully perhaps
  • Universe Man: might be identified with the whole of space-time itself and so may be too large or important for petty conflicts

Further Reading:

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

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

https://en.wikipedia.org/wiki/Flood_(They_Might_Be_Giants_album)

[*11. 88]

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The Lambda Cube

April 25, 2019

More or less from Wikipedia:

In mathematical logic and type theory, the λ-cube is a framework introduced by Henk Barendregt to investigate the different dimensions in which the calculus of constructions is a generalization of the simply typed λ-calculus. Each dimension of the cube corresponds to a new way of making objects depend on other objects, namely

    1. terms allowed to depend on types, corresponding to polymorphism.
    2. types depending on terms, corresponding to dependent types.
    3. types depending on types, corresponding to type operators.

The different ways to combine these three dimensions yield the 8 vertices of the cube, each corresponding to a different kind of typed system.

So in the diagram above, we have emblazoned the names of these type systems ordered from lower left to upper right:

  • λ→: the simply typed lambda calculus, our base system
  • λ2: add 1. above to λ→, giving what is also known as System F or the Girard–Reynolds polymorphic lambda calculus
  • λP: add 2. above to λ→
  • λ_ω_: add 3. above to λ→
  • λP2: combine 1. and 2., λ2 and λP
  • λω: combine 1. and 3., λ2 and λ_ω_
  • λP_ω_: combine 2. and 3., λP and λ_ω_
  • λC: combine 1., 2., and 3., giving the calculus of constructions

Further Reading:

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

http://www.rbjones.com/rbjpub/logic/cl/tlc001.htm

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

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

[* 11.86, *11.87]

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