Category Archives: eightfolds

The Eight Auspicious Symbols

Pause only for a moment to contemplate Ashtamangala, or the Eight Auspicious Symbols of Buddhism:

  • The Endless Knot
  • The Treasure Vase
  • The Lotus Flower
  • Two Golden Fish
  • The Fancy Parasol
  • The Conch Shell
  • The Victory Banner
  • The Dharma Wheel

Further Reading:

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

http://www.buddhanet.net/e-learning/history/b8symbol.htm

The Whitewater Rafter’s Guide to the 8 Auspicious Symbols of Buddhism

[*12.14]

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The Devolution of Trust

The Prisoner’s Dilemma is a simple game designed to show how the success or failure of cooperation between individuals can be contingent on various factors, primarily some sort of reward. Shown above is a representative payoff matrix between two players; each square shows the two choices and the two winnings for each. Each player cooperates (A or B) or cheats (A’ or B’) with the other player, so for example if A and B’ obtains (A cooperates but B cheats) then A loses 1 and B wins 3.

Each player knows all the values of the payoff matrix so it is said they have perfect information, except they don’t know what their opponent will do. If they are rational and believe their opponent to be as well, the wisest thing to do is for both to cooperate to maximize their winnings, knowing that their opponent knows that they could also cheat. If the game is played only once, however, that is clearly not the case.

If the game is iterated, things change. If each player remembers what their opponent did previously, and it is considered to be informative for what they might do next, then the player could use it to condition their decision to cooperate or cheat. Different algorithms or personalities can be considered for the players, with more or less thinking about what to do and more or less willingness to cooperate, and it is interesting to try different strategies, all the while seeing what adjustments of the payoff matrix might do to the results.

This Evolution of Trust site is a very nice lesson in some of the complications that can result for such algorithms and adjustments. On the whole, this site indicates that rationality and consideration for others can thrive, if conditions are right. In the traditional Prisoner’s Dilemma, the reward values in the payoff matrix are usually considered to be jail sentence time (so less is better), or for the site mentioned above where I’ve taken the representative matrix, monetary value (so more is better).

One thing of note in these examples is that each player doesn’t distinguish their opponent by anything other than their posteriori plays, because these players are supposed to be all part of the same group or society. But what if there is an a priori distinction that conditions their decision? So, if your opponent is a known Y, and you are a X, then you might want to raise your social credit with your other Xs by punishing a Y, even if it punishes you or even other Xs in the long run.

For example if you are a member of gang X, you wouldn’t want to cheat against another X. But cheating against a member of gang Y might raise your in-group social capital and be as important as the value of the reward. Or you might want to punish your opponent in group Y by not granting them any benefits even at the cost of your own benefit. Such distinctions are not usually part and parcel of the Prisoner’s Dilemma game, but they would add an interesting and realistic dimension to the game.

And thus lend insight into the woes of our modern political scene and culturally diverse society.

Further Reading:

https://ncase.me/trust/

https://en.wikipedia.org/wiki/Prisoner%27s_dilemma

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

https://equivalentexchange.blog/2017/08/03/the-prisoners-dilemma/

https://en.wikipedia.org/wiki/Devolution_(biology)

[*11.24, *11.172]

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Tractatus Logico-Philosophicus

Just the facts, ma’am.

— Detective Joe Friday

  1. The World (Die Welt): The world is everything that is the case.
  2. The Case (Der Fall): What is the case, the fact, is the existence of atomic facts.
  3. The Picture (Das Bild): The logical picture of the facts is the thought.
  4. Thought (Gedanke): The thought is the significant proposition.
  5. Propositions (Der Satz): Propositions are truth-functions of elementary propositions.
  6. The Form (Die Form): The general form of truth-function is: [p-bar, xi-bar, N(xi-bar)].
  7. Silence (Schweigen): Whereof one cannot speak, thereof one must be silent.
  8. ( ):

The Tractatus has seven propositions, most with sub-propositions and sub-sub-propositions, etc. I have added the eighth, which is the actual silence of all one cannot speak of. Quite a large section, for all its emptiness.

I also thought it would be nice to have an internet version where you could click and expand down through the sub-sentences. There are already many such versions available for your enjoyment.

I can’t decide whether I like the English or the German version better, so here are both.

Further Reading:

https://en.wikipedia.org/wiki/Tractatus_Logico-Philosophicus

http://www.tractatuslogico-philosophicus.com/

http://www.bazzocchi.com/wittgenstein/tractatus/eng/index.htm

http://www.kfs.org/jonathan/witt/ten.html

https://pbellon.github.io/tractatus-tree/#/

https://tractatus-online.appspot.com/Tractatus/jonathan/index.html

http://daxoliver.com/tractatus/

http://people.umass.edu/klement/tlp/

The Tricky Truth about Tractatus Trees (updated)

[*11.170]

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Tanabata, V2

Soon it will be Tanabata (七夕) in Japan on July 7th.

Make a wish!

  • Orihime : Vega
  • Hikoboshi : Altair
  • Bridge of Birds : Deneb
  • Silver River : Milky Way

Further Reading:

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

http://www.iromegane.com/japan/make-a-wish-for-tanabata/

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

[*10.48, *11.114]

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Patrick Geddes and the Notation of Life

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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The Lambda Cube

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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Noether-Pauli-Jung, V2

What happens when the fourfold of Noether’s Theorem is spliced together with the fourfold of Pauli-Jung? Both have Space-Time and Matter-Energy. The former has Conservation and Symmetry, and the latter has Causality and Synchronicity. And if Space-Time and Matter-Energy are both divided into Space and Time and Matter and Energy, one obtains this eight-fold.

Causality means that some action or cause in time (say a process) of things in space can have an effect (another process, say) on different things in space, and Synchronicity means that different events (say processes) separated in space can have non-causal relationships between them.  Conservation means the consistency of a quantity of matter or energy or matter-energy through time, and Symmetry means the consistency of a measure of a structure or form through space.

I am reminded of my fourfold Four Bindings, consisting of Chains, Grids, Blocks, and Cycles. Causality and Synchronicity are Chains (or non-chains for the latter) Space and Time are Grids (or flexible meshes), Matter and Energy are Blocks (or chunks of stuff), and Symmetry or Conservation are Cycles (of the group-theoretic kind or the equivalence class kind or just loops).

Further Reading:

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

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

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

https://equivalentexchange.wordpress.com/2012/05/04/noethers-theorem/

https://equivalentexchange.wordpress.com/2018/01/23/atom-and-archetype/

https://equivalentexchange.blog/2016/04/06/four-bindings/

This is a reworking of a previous six-fold diagram that I believe is served better as an eight-fold.

https://equivalentexchange.blog/2018/03/29/noether-pauli-jung/

[*10.68, *10.155, *11.55]

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