A Game of Four-folds, Part 3

October 21, 2014 Martin K. Jones Leave a comment

Here’s another idea for the Game of Fourfolds, instead of using hexaflexagons. Since each fourfold can be permuted six ways, how about arranging those in six squares on the faces of a cube?

Here are the permutations of Heidegger’s “das Geviert” arranged on the faces of a cube, if one cuts it out and folds it up properly. Not only is the square the regular polygon of materiality, the cube is the regular polyhedron of Earth, the most material of the ancient four elements.

It might be awesome to have a cube of every fourfold I’ve talked about (at least the good ones), to combine and compare them. Even more awesome are the cubes for the REAL elements I’ve seen when I’ve searched images for “element cube”.

What I want to know is whether the six permutations can be arranged in a symmetric way on a cube?

[*8.81]

fourfolds, Mathematics

The Collatz Conjecture

October 17, 2014 Martin K. Jones 1 Comment

I thought I might be continuing my apparent effort to convert all triads into fourfolds, but I’ll take a little break to bring you this mathematical fourfold for the Collatz Conjecture. The Collatz algorithm starts with any specific nonzero integer, then iterates in the following way to turn the number into the next number to operate on, and so on. If X is even, then replace X with X/2. If X is odd, then replace X with 3*X + 1. If you get to 4, then you get 2, then 1, but you next go back to 4. In fact, if you have any power of 2, you quickly drop to 4 and so cycle. The Collatz Conjecture is that if you start with any number, then you will eventually reach 4. (Actually the conjecture is that you will reach 1, but since I’m partial to 4, I’ll state it in this way since it’s pretty much the same thing.)

The conjecture is not proven, but has been shown to be true for every number up to some very large numbers. Some numbers can jump around from larger to smaller to larger again, or up and down and back up, for quite a few steps before being reduced to the 4-2-1-4 sequence. That’s why these sequences are also called hailstone numbers.

One might think of the Collatz algorithm as a model of a toy universe in the following way. The procedure starts with a given number, which for the toy universe would be its initial state. As time moves from instant to instant, the procedure operates to halve the number, or multiply by three and add one. So instant after instant the procedure operates, making the number smaller, or larger as required. The conjecture, if true, would mean that no matter how it starts, our toy universe would always wind down and cycle endlessly through the sequence of 4-2-1-4-2-1-4-2-1-4…

Further Reading:

http://en.wikipedia.org/wiki/Collatz_conjecture

http://xkcd.com/710/

https://www.quantamagazine.org/mathematician-terence-tao-and-the-collatz-conjecture-20191211/

https://www.quantamagazine.org/why-mathematicians-still-cant-solve-the-collatz-conjecture-20200922/

Snakes and Ladders

[*8.88]

fourfolds, Philosophy

Plato’s Chariot

October 13, 2014 Martin K. Jones 2 Comments

At the risk of being labeled a neo-platonist, another triad of Plato’s that is often discussed is that of the Allegory of the Chariot. This analogy is supposed to bring insight into the workings of the human soul and consists of two horses, one good and one bad, and the charioteer whose duty is to control them.

You never hear about the chariot itself (but it is always pictured), but it is required to have a chariot, after all. The charioteer isn’t just standing on the backs of the horses, like Jean-Claude Van Damme doing his epic split, although that would be cool. (They do this at the circus, and I know I’ve seen it in old gladiator movies when the chariot loses a wheel and the charioteer has to cut away the chariot.)

Thus, unless you want to change the nature of the analogy, the chariot is required for everything to be connected together. This fourth material component completes the triad into a fourfold, and I place it at the lowest, fundamental position where I added The Real to The Beautiful, The True and The Good.

And of course everyone knows that Bad Horse is the Thoroughbred of Sin!

Further Reading:

http://en.wikipedia.org/wiki/Chariot_Allegory

https://drhorrible.fandom.com/wiki/Bad_Horse

[*8.86]

fourfolds, Philosophy

Plato Redux

October 9, 2014 Martin K. Jones Leave a comment

What is, is The True
What is, is The Good
What is, is The Beautiful
What is, is The Real

— Anonymous

I recently finished reading Rebecca Goldstein’s Plato at the Googleplex: Why Philosophy Won’t Go Away. In it there was much talk of The Beautiful, The True and The Good.

Besides Plato’s Divided Line, which was mentioned in The Republic and consists of four parts, the threesome of The Beautiful, The True and The Good is mentioned in various dialogues.

Being the quadraphile I am, I thought adding The Real to the threesome makes the now foursome nicely balanced. Usually one hears of just the three, without the fourth, but why is that?

Some argue loud and long that The Real has no part in this threesome of Universals, that the three are sufficient among themselves. Others disagree. Which side would you say I’d be on?

Further Reading:

http://en.wikipedia.org/wiki/Transcendentals

http://en.wikipedia.org/wiki/Analogy_of_the_Divided_Line

Henry Rutgers Marshall / The True, The Good and the Beautiful (The Philosophical Review, Vol. 31, No. 5, Sep. 1922 pp. 449-470)

Michael Boylan / The Good, the True and the Beautiful

[*4.82, *8.72, *8.82]

fourfolds

Hayden White’s Metahistory

October 3, 2014 Martin K. Jones 1 Comment

In 1973, historian Hayden White published Metahistory: The Historical Imagination in Nineteenth-century Europe. His key fourfold was that of literary emplotments due to Northrop Frye: Romance, Comedy, Satire, and Tragedy. White also derived a synoptic table that associated other fourfolds by Stephen Pepper (Organicist, Mechanicist, Formist, and Contextualist), tropes (Metaphor, Metonymy, Synecdoche, and Irony), plus various modes, ideologies, representational historians, and key philosophers.

I’ve had this fourfold sitting around for a while, and I haven’t written anything about it, because I don’t really agree with the synopticisms as given. For instance, I would pair Romance with Organicism, Comedy with Formism, Satire with Contextualism, and Tragedy with Mechanism. Why? Romance is the “drama of self-identification”, as the organism is self-identified, being that the “individual part of the whole is more than the sum of its parts”. Comedy is “harmony between the natural and the social”, as Formism is created by social “classifying, labelling, and categorizing” of natural objects. Tragedy is about the “limitations of the world”, as Mechanism is “finding laws the govern the operations of human activities”. Finally, Satire is the “opposite of romance — people are captives in the world until they die”, whereas Contextualism is “events explained by their relationships to similar events”. That last one isn’t very convincing, but I’ve only switched Romance and Comedy, and left Satire alone.

Again, as with my problems with the synoptic table of Arthur M. Young, it would be nice to play the Game of Fourfolds to see if I can find better arguments for my synoptic claims, or to convince myself that the claims of others are better. Of course, it would probably help for me to read the original works, instead of reading summaries. One might care to look elsewhere for better exposition.

Perhaps I could elaborate further at a later time:

Romance: The individual succeeds.
Comedy: Most succeed. Society wins.
Satire: Most fail. Society comes up short.
Tragedy: The individual fails.

Further Reading:

http://en.wikipedia.org/wiki/Metahistory

[*3.172, *3.173]

fourfolds

Northrop Frye’s Anatomy of Criticism

September 17, 2014 Martin K. Jones 1 Comment

In the late 50’s literary critic Northrop Frye published Anatomy of Criticism: Four Essays, where his analysis of the criticism of literature followed literature itself in being systemically separated into Rhetorical, Historical, Archetypal, and Ethical criticism. Each of these criticisms in turn corresponded to theories of Genres, Modes, Myths, and Symbols.

References:

http://en.wikipedia.org/wiki/Northrop_Frye

http://en.wikipedia.org/wiki/Anatomy_of_Criticism

[*7.7]

fourfolds, linear logic, Mathematics, Science, Space and Time, spacetime, technology

Four Dimensional Space-time

September 1, 2014 Martin K. Jones 2 Comments

Here’s a simple fourfold I’ve been ignoring just because it’s so trivial, but that triviality can be deceiving. Space-time as formulated in special relativity has four dimensions: three of space and one of time. Our everyday experience shows us the three dimensions of space: length, width (or breadth), and depth (or height), but time is a different kind of thing because we cannot see or move forward and backward through time with our eyes or body, like we can along the axes of space.

Personally, only our memory and imagination can let us range through time. Of course, after the invention of language and more recent technologies, the spoken word, writings, photographs, audio recordings, and videos can also be used. But it’s not the same as shifting one’s gaze along the length of something or moving one’s body across a width.

So, we can move semi-freely through the three spatial dimensions but our movement in time seems to be fixed into a relentless forward motion that we have no control over. And because gravity pulls us down onto the surface of the world, one of the spatial dimensions (depth or height) is more limiting than the other two.

Thus another interesting comparison to this fourfold is to that of linear logic. One observation is that length and width can be considered reversible but depth and time can be considered somewhat irreversible. That’s not true of course, but because of gravity it is easier to descend than to ascend, and it’s far easier to move into the future than into the past. But we can see into the distant past, just not our own, as we turn our telescopes to the heavens.

Space without time could have four or even higher dimensions, but we have no empirical evidence that it is so. Mathematically, however, we can easily construct multidimensional spaces. One representation of four dimensional space is by using quaternions, which have four dimensions to the complex numbers’ two. Tuples of real numbers or even vector spaces can also be used. However, the geometry of space-time is not Euclidean; it is described by the Minkowski metric.

Novels about characters living in different numbers of spatial dimensions are an interesting way to learn and think about them. The very first was Flatland by Edwin Abbott Abbott, about a being limited to two dimensions that learns about a third outside his experience when a three dimensional being comes to visit. Just recently I’ve finished reading Spaceland by Rudy Rucker, about an ordinary human person limited to the three dimensions of space that learns about the fourth dimension by similar reasons.

Links:

http://en.wikipedia.org/wiki/Special_relativity

http://en.wikipedia.org/wiki/Minkowski_space

http://en.wikipedia.org/wiki/Four-dimensional_space

http://en.wikipedia.org/wiki/Flatland

http://en.wikipedia.org/wiki/Flatland_%282007_film%29

http://en.wikipedia.org/wiki/Spaceland_%28novel%29

[*8.72]

fourfolds, Physics, Science

Maxwell’s Equations

August 17, 2014 Martin K. Jones Leave a comment

Actually, just the left hand side of the equations, because it looks better that way. The right hand sides are just a click away.

E and B are the electric and magnetic fields, respectively. $\nabla$ is the “del” operator.

$The divergences \nabla \cdot E$ and $\nabla \cdot B are the “fields emanating from the sources.”$

$The curls \nabla x E$ and $\nabla x B are the “circulation of the fields.”$

Further Reading:

http://en.wikipedia.org/wiki/Maxwell%27s_equations

fourfolds

All Watched Over

August 15, 2014 Martin K. Jones Leave a comment

I like to think
(it has to be!)
of a cybernetic ecology
where we are free of our labors
and joined back to nature,
returned to our mammal
brothers and sisters,
and all watched over
by machines of loving grace.

— Richard Brautigan from All Watched Over by Machines of Loving Grace

References:

http://en.wikipedia.org/wiki/All_Watched_Over_by_Machines_of_Loving_Grace

http://en.wikipedia.org/wiki/All_Watched_Over_by_Machines_of_Loving_Grace_(TV_series)

[*6.150, *8.70]

fourfolds

There is Nothing Wrong

August 13, 2014 Martin K. Jones Leave a comment

There is nothing wrong with your television set. Do not attempt to adjust the picture. We are controlling transmission. If we wish to make it louder, we will bring up the volume. If we wish to make it softer, we will tune it to a whisper. We will control the horizontal. We will control the vertical. We can roll the image, make it flutter. We can change the focus to a soft blur or sharpen it to crystal clarity. For the next hour, sit quietly and we will control all that you see and hear. We repeat: there is nothing wrong with your television set. You are about to participate in a great adventure. You are about to experience the awe and mystery which reaches from the inner mind to – The Outer Limits.

— The Control Voice from The Outer Limits

Links:

http://en.wikipedia.org/wiki/The_Outer_Limits_%281963_TV_series%29

http://tvtropes.org/pmwiki/pmwiki.php/Series/TheOuterLimits

[*8.70]