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[ by Charles Cameron — divine geometry in Japan ]
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The sacred nature of number should not surprise us, though it often does. Here is the Neoplatonist Proclus, as quoted in Dantzig, Number: the Language of Science, p.78:

It is told that those who first brought out the irrationals from concealment into the open perished in shipwreck, to a man. For the unutterable and the formless must needs be concealed. And those who uncovered and touched this image of life were instantly destroyed and shall remain forever exposed to the play of the eternal waves.

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Here are two offerings from a Buddhist shrine in Japan:

and

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I found them in this delightful account of the Soroban or Japanese abacus:

The section on mathematical offerings begins around the 19.40 mark and goes to 20.18 — but the whole video is worth your attention.

[ by Charles Cameron — Aristotle, the Ismailis, CG Jung ]
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Two is the first number:

From the tenth-century Rasa’il Ikhwan al-Safa’ (Epistles or Treatises of the Brethren of Purity) as found in Seyyed Hossein Nasr & Mehdi Aminrazavi, eds., An Anthology of Philosophy in Persia, vol. 2, Ismaili Thought in the Classical Age:

We say that one is the source and generator of the numbers because when one is removed from existence all the numbers are removed with it, but when the numbers are removed from existence, one is not removed. We say that two is the first whole number because numbers are a plurality of ones, and the first plurality is two.

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I am delighted to find myself in the company of the Ismailis, having written in On two, one, seven plus or minus, and ten – towards infinity:

When I worked as senior analyst in a tiny think-shop, my boss would often ask me for an early indicator of some trend. My brain couldn’t handle that — I always needed two data points to see a pattern, and so I coined the mantra for myself, two is the first number.

One isn’t a number until there are two, because it’s limitless across all spectra and unique, and because it is its own, only context.

One isn’t a number unless there’s a mind to think of it — in which case it’s already an abstraction within that mind, and thus there are, minimally, two. At which point we are in the numbers game, and there may be many, many more than two — twenty, or plenty, or plenty-three, or the cube root of aleph null, or (ridiculous, I know) infinity-six…

Go, figure.

Two is the first number, because the two can mingle or separate, duel or duet: either way, there’s a connection, a link between them.

And in Of a non-comparative use of the DoubleQuotes method:

Putting it bluntly, one point is pointless — things could go anywhere from there. Two points suggest a line, a link, a connection — a possible, maybe even plausible, trend.

It’s that sense I have of two being the beginning of thought that makes me so fond of the DoubleQuotes format — and of Arthur Koestler‘s insight about creation occurring at the intersection of two spheres..

If I’m a fundamentalist about anything, it’s the notion that it takes two to tango!

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Furthermore..

From CG Jung, A Psychological Approach to the Dogma of the Trinity, in Psychology and Western Religion:

The number one claims an exceptional position, which we meet again in the natural philosophy of the Middle Ages. According to this, one is not a number at all; the first number is two. Two is the first number because, with it, separation and multiplication begin, which alone make counting possible. With the appearance of the number two, another appears alongside the one, a happening which is so striking that in many languages “the other” and “the second” are expressed by the same word.

And from Leonardo Tarán, Speusippus of Athens: A Critical Study With a Collection of the Related Texts and Commentary:

Now we know that Speusippus did consider one to be the first odd number, whereas Aristotle thinks that two is the first number and explicitly denies that one is a number. [ … ]

Concerning the One, then, the differences between Speusippus and Aristotle amount to this: For the latter the One is not a number, nor a separately existing entity, but the principle of number because, each number being a plurality of abstract monads, the One is its measure.

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As Emerson wrote:

The world is young; the former great men call to us affectionately.

[ by Charles Cameron — my own three-way play, and a little goose ethology from Konrad Lorenz ]
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Konrad Lorenz, Paul Leyhausen, Motivation of human and animal behavior: an ethological view, p 129

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Returning to the topic of ternary logics and scissors paper stone, here’s another old post of mine (Dec 2000):

I’ve always thought it was a training in the complete frustration of the assumption that conflict could be decided, most of all by means of skill of any kind. I mean, if you want to decide on a random basis, toss a coin, and if you want to decide on a basis of power or skill, set up a contest in which superior power or skill succeeds…

In scissors paper stone you have a setup where (as in heads and tails) no choice is superior to any other as far as chance is concerned, but there’s a troubling “threeness” about the thing which means it’s not aligned with an “either/or” logic but with a subtler logic altogether.

Two other places where this kind of “intransitivity” relation can be found would be (a) the geese studied by Konrad Lorenz which had a pecking order “a > b > c > a” (ie of the three particular geese, one would always defer to the second, which would always defer to the third, which in turn would always defer to the first) — and this is a perfectly “natural” effect which Lorenz encountered in “real life” — and (b) the impossible triangle devised by L S and R Penrose, which is clearly and unambiguously paradoxical (though logically equivalent to the Lorenz geese as far as I can tell).

Anyhow, what I’m trying to suggest is that “scissors paper stone” can work a little like a koan, it can somehow challenge our usual binary asusmptions and train us in a more flexible and “lateral” way of thinking..

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Some thoughts about a game (real or agent-based) in which three players (or sides) compete, two teaming up against one in constantly different configurations of challenge and victory (June 2005):

The idea is simplicity itself, putting it into words unambiguously is difficult.

Yesterday I was at a hotel swimming pool where two sons of another guest and my own son Emlyn were playing around together, all aged around 10-12, with nobody else there, and I asked them to play a game where one of the three would be “top guy” and the other two would try to dunk and generally overthrow him, and as soon as it became clear that the top guy had been dunked and dethroned, there would be a new “top guy” (one of the two who had been doing the overthrowing and come out the obvious winner) and the other two would pounce on him…

They were not evenly matched, but any one of them was easily outmatched by the other two working together, and being “top guy” meant your head was above water and you were, so to speak, “comfortable” — so there was a real premium on being “top guy”, and the other two were (at any time) eager to collaborate to get the position.

And there was no tendency for the two brothers to gang up against my son, because the immediacy of two defeating one was more urgent and compelling than any ideas of teaming by kinship interests, minor differences in strength and skill, etc…

More:

Prisoners’ Dilemma deals with the essence of twoness, as do many win / lose games such as tennis or chess, and this game deals with the essence of threeness, as (in a very different mode) does paper scissors rock.

My sense from PSR, and reason for being particularly interested in threeness in games, is that PSR involves thinking in flagrant defiance of our usual binary logic as humans.

I suspect my threeness game could be paradigmatic for cooperation and competition, in that each player at any given moment is either “top guy” or “member of the opposition” — and thus plays either “sole defender” (competing without a trace of collaboration) or “team attacker” (cooperating to compete) at any give moment, but with a very rapid turnover between the two.

The kids seemed to enjoy the in-the-pool version yesterday, until it the water-play got a bit too rough after about forty minutes or an hour of enthusiastic splashing and dunking.

And finally:

As a game designer, I want to think through this mode of play, two against one with switching, in a number of media — water play, three-person tag, possible card-game instantiations, board games, a chess variant perhaps, a variant on my own HipBone Games, etc — and as a conceptual matrix for game-theorizing, modeling, understanding conflict and conflict resolution, etc.

I wonder what ideas and possible uses the basic idea might trigger in others…

[ by Charles Cameron — in which 359 turns out to be just a degree closer to zero than 358 is ]
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For those Zenpundit readers no more fluent in Turkish than I am, Acting Prime Minister Ahmet Davutoglu‘s interview above was reported by Today’s Zaman:

During the interview on Wednesday night Davutoglu also commented on the Ankara bombings on Saturday which left nearly 100 people dead in a suspected Islamic State of Iraq and the Levant (ISIL) attack. “There is a 360-degree, not 180 degrees, difference between the Islam we defend and daesh has in its mind,” the acting prime minister said in an apparent gaffe that sparked a series of tweets mocking his statements which meant the government and ISIL are on the same page on their interpretation of Islam. Daesh is the Arabic acronym for ISIL.

“Davutoglu said there is a 360-degree difference between ISIL and them. They would sue others who would say that,” academic Koray Çaliskan said.

“Professor Prime Minister Ahmet Davutoglu, a closed disc has 360 degrees,” Liberal Democratic Party (LDP) leader Cem Toker tweeted, recalling that Davutoglu is a professor.

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Today’s Zaman also quoted the “famous Turkish geometry teacher” whose classes on YouTube have garnered him an impressive following, Mustafa Mete Ekolan aka Ekol Hoca. Ekol tweeted the following graphic illustration today:

360 derece pic.twitter.com/uBBawFWHeb

— Ekol Hoca (@ekolhoca) October 14, 2015

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It’s worth noting that Today’s Zaman Editor-in-Chief Bulent Kenes was released pending trial today. A fine legal point made by his defense lawyer:

Gunaydin said in the petition to the court there is no crime in the Turkish Penal Code (TCK) for “sending tweets.”

[ by Charles Cameron — when not dancing innumerably on the heads of pins ]
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I responded to a question from David Ronfeldt today by saying I don’t know of a book that offers a “a sustained, point-for-point, systematic, thorough comparison” between earlier Christian and contemporary Islamist religious violence.

I do share his concern for point-for-point comparisons, however, and this one popped up as I was working on my response to David:

These two quotes are abstract, indeed metaphysical, I know, and don’t deal with human-on-human violence as such — but the DoubleQuote they form is nevertheless a point-for-point comparison, and the very exactness of its counterpoint gives it the sort of power the best haiku have, I believe, offering us wit in brevity, multum in parvo, small is beautiful.