Archive for the ‘fourfolds’ Category

The Four Binary Operators of Linear Logic, Part 2

April 6, 2019

Ordinarily, inference rules in natural deduction are written using a horizontal line, with the known, true, assumed or proven things written above the line and the inferred things written below the line. Here I’ve taken the artistic liberty to use diagonal lines instead of horizontal ones, and so tried to represent the introduction rules for the four binary operators of Linear Logic. In order to fit additive disjunction “plus” into this schema, I’ve broken the inference rule diagonal and written the duplicate inferred introduction below only once. I’m sure no self-respecting logician would do such a thing.

Further Reading:

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

https://plato.stanford.edu/entries/logic-linear/

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

https://equivalentexchange.blog/2012/04/17/the-four-binary-operators-of-linear-logic/

https://equivalentexchange.blog/2011/03/09/j-y-girards-linear-logic/

[*11.60]

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Recipe for Mathematics

April 3, 2019

Guided only by their feelings for symmetry, simplicity, and generality, and an indefinable sense of the fitness of things, creative mathematicians now, as in the past, are inspired by the art of mathematics rather than by any prospect of ultimate usefullness.

— E. T. Bell

A smile fell on the grass.
Irretrievable!

And how will your night dances
Lose themselves. In mathematics?

— Sylvia Plath, from The Night Dances

Further Reading:

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

https://hellopoetry.com/poem/710/the-night-dances/

https://poetrywithmathematics.blogspot.com/

[*11.14]

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Colors of Light and Matter

March 26, 2019

Colors of light and colors of matter operate somewhat differently from each other. Colors of lights are additive, but colors of material pigments are subtractive. Of course, they both are conditioned by the biology of the human eye and cultural preferences. Above I have tried to create a synthesis of sorts for both types, and whereas it is pleasing in some ways, it is lacking in others.

White light is combined red, green, and blue light, and combinations of green and red light yield yellow light, combinations of red and blue light yield magenta light, and combinations of green and blue light yield cyan light. Similarly, black is mixed yellow, magenta, and cyan pigments, mixing cyan and magenta pigments produce blue pigment, mixing cyan and yellow pigments produce green pigment, and mixing yellow and magenta pigments produce red pigment.

Further Reading:

http://learn.leighcotnoir.com/artspeak/elements-color/primary-colors/

[*11.63]

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Degen’s Eight-square Identity

March 25, 2019

Here is another identity but this time corresponding to an eight-fold: the Eight-square identity of Ferdinand Degen found about 1818. You know the drill: it states that a product of two numbers that are each the sum of eight squares is itself the sum of eight squares!

(a12 + a22 + a32 + a42 + a52 + a62 + a72 + a82)(b12 + b22 + b32 + b42 + b52 + b62 + b72 + b82) =

…The sum of the expressions in the eight triangles written in the diagram above. (Please consult the Wikipedia entry below for the textual formulas, as it’s too hard to write in HTML.)

Note that the expressions above have an interesting symmetry, aside from the one on the upper left. Indeed, Euler’s Four-square Identity has a similar simpler symmetry. There is also a connection with Octonions if you are interested in digging for it. If you are anticipating that there is such a formula for sums of sixteen squares, there is, but not a bilinear one, and it is much more complicated!

Further Reading:

https://en.wikipedia.org/wiki/Degen%27s_eight-square_identity

https://en.wikipedia.org/wiki/Pfister%27s_sixteen-square_identity

[*11.64]

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Euler’s Four-square Identity

March 24, 2019

Here’s a nice little math identity that mathematician Leonard Euler wrote down in a letter dated 1748. It states that the product of two numbers that are sums of four squares is itself the sum of four squares.

(a12+a22+a32+a42)(b12+b22+b32+b42) =

(a1b1-a2b2-a3b3-a4b4)2 + (a1b2+a2b1+a3b4-a4b3)2 + (a1b3+a3b1+a4b2-a2b4)2 + (a1b4+a4b1+a2b3-a3b2)2

It can be proved with elementary algebra or even by quaternions!

Further Reading:

https://en.wikipedia.org/wiki/Euler%27s_four-square_identity

[*11.62]

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Seven Sermons to the Dead, V2

March 19, 2019

We must, therefore, distinguish the qualities of the pleroma. The qualities are pairs of opposites, such as—

The Effective and the Ineffective.
Fullness and Emptiness.
Living and Dead.
Difference and Sameness.
Light and Darkness.
The Hot and the Cold.
Force and Matter.
Time and Space.
Good and Evil.
Beauty and Ugliness.
The One and the Many. etc.

— Carl Jung, from Seven Sermons to the Dead

I previously mentioned two fourfolds which are perhaps better combined as an eightfold. These concepts or entities are from Carl Jung’s “Seven Sermons to the Dead”.

To the left are the “principal gods”, who are said to be in one-to-one correspondence with the “world’s measurements”. What then are these measurements? Are they the coordinates of relativistic Space-Time or something else?

  • The God-Sun: The beginning
  • Eros: Binding of two together, outspreading and brightening
  • The Tree of Life: Filling space with bodily forms
  • The Devil: The Void, opening all that is closed, dissolving all formed bodies, and destroyer of everything

And to the right we have:

  • The Pleroma: The spiritual universe as the abode of gods and of the totality of the divine powers and emanations.
  • The Creatura: The living world, subject to perceptual difference, distinction, and information
  • Abraxas: The supreme power of being transcending all divinities and demons and uniting all opposites into one
  • Philemon: Jung’s spiritual guide and the narrator, also said to be Simon Magus or Basilides

Further Reading:

http://gnosis.org/library/7Sermons.htm

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

https://jungiangenealogy.weebly.com/seven-sermons.html

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

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

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

http://philemonfoundation.org/about-philemon/who-is-philemon/

https://equivalentexchange.blog/2018/02/11/seven-sermons-to-the-dead/

https://equivalentexchange.blog/2018/02/14/seven-sermons-to-the-dead-p2/

[*10.76, *11.49]

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Four Forms Make a Universe, Part 2

March 17, 2019

This is a continuation of my last entry. Above is a different representation of the LICO alphabet, with the letters turned 45 degrees counter-clockwise, and rearranged into a symmetric pattern. The letters seem to arise more naturally in this orientation, but then Schmeikal rotates them into his normal schema.

And to the right is a diagram of the logical expressions that correspond to the letters above.

After making these new diagrams, I became inspired and made a few other figures to share with you.

These two versions, with triangles instead of line segments, and also with borders between adjacent triangles removed:

 

 

 

 

 

And these two versions, with quarter circles, and also with edges between adjacent quarter circles removed:

 

 

 

 

 

 

Further Reading:

https://equivalentexchange.blog/2019/03/12/four-forms-make-a-universe/

[* 11.50, *11.58]

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Four Forms Make a Universe

March 12, 2019

How could I not love a paper with this title? I’ve struggled with it for a bit, and I’ve only managed a couple of diagrams relating the author’s LICO (Linear Iconic) alphabet made up of 16 letters. However, I see that there are a few other papers by Schmeikal available on ResearchGate that look easier to understand. But also however, the first one says to read the “Four Forms” paper first!

At any rate, I present a sixteen-fold of the LICO alphabet, and another of the binary Boolean operators that are in a one-to-one mapping with LICO. There is much to understand from these papers, including much syncretism between various mathematical sixteen-folds, so please forgive me if I don’t explain it all with immediate ease. However, I believe it is well worth the effort to understand.

(Please note that the characters of the LICO alphabet are oriented so that the bottoms of the letters are downward, but the Boolean operators are oriented so that the bottoms of the equations are towards the right angles of the triangles.)

The title comes from the result that four elements of LICO can reproduce the other twelve via linear combinations. These four forms are 1) Boolean True (A or ~A), 2) A, 3) B, and 4) A=B. These are within the interior right-hand triangles in the LICO diagram. Of course, it is well known from Computer Science that the NAND operator (~A or ~B) can also generate all other fifteen operators, but this is by multiple nested operations instead of simple Boolean arithmetic. There are several other “universal” binary gates that can do this as well.

Two other representations that have four elements that can generate the other twelve via linear combinations come from CL(3,1), the Minkowski algebra. These representations are called “Idempotents” and  “Colorspace vectors”. Because of this algebra’s association with space and time in relativity, Schmeikal claims that LICO has ramifications in many far-ranging conceptualizations.

Further Reading:

Bernd Schmeikal / Four Forms Make a Universe, in Advances in Applied Clifford Algebras (2015), Springer Basel (DOI 10.1007/s00006-015-0551-z)

https://link.springer.com/article/10.1007/s00006-015-0551-z

At http://www.researchgate.net:

Bernd Schmeikal / Free Linear Iconic Calculus – AlgLog Part 1: Adjunction, Disconfirmation and Multiplication Tables

Bernd Schmeikal / LICO a Reflexive Domain in Space-time (AlgLog Part 3)

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

https://www.allaboutcircuits.com/technical-articles/universal-logic-gates/

[*9.145, *11.50]

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The Visions of W. B. Yeats

March 8, 2019

William Butler Yeats published two editions of his book, “A Vision”, the first in 1925 and the second in 1937. They are interesting amalgamations of metaphysics, poetry and esoteric speculations. After years of neglect I return to mention these works briefly.

Previously I showed the fourfold of the Faculties (Creative Mind, Body of Fate, Mask, and Will), but ignored the complementary fourfold of the Principles (Spirit, Celestial Body, Passionate Body, and the Husk). Here I present an eightfold of both Faculties and Principles, and it makes a bit more sense with all its accompanying structure.

Each of these eight elements has certain dual properties which I have indicated by the second diagram: Primary (+) vs. Antithetical (/), Solar (Sun) vs. Lunar (Moon), Static (=) vs. Active (~). Primary is also called Objective, Antithetical is also called Subjective, Solar is also called Core, and Lunar is also called Changing.

In my diagram I have the Faculties as down-pointing triangles (+ and /) and the Principles as up-pointing triangles (Sun and Moon). Elements that are Active (~) are in the upper-left and lower-right squares and those that are Static (=) are in upper-right and lower-left. Since explanations of these elements usually involve interpenetrating cones and gyres and such, whether this diagram has any value is left for a later time.

I also just found out that Neil Mann, the author of the excellent and enduring web site about “A Vision”, has a new book out. It’s called “A Reader’s Guide to Yeats’s ‘A Vision'” and is published by Clemson University Press. It’s a bit pricey but I’m sure it’s full of valuable insights into these works. Perhaps I’ll just have to raid my piggy bank.

Further Reading:

http://www.yeatsvision.com/

http://www.yeatsvision.com/Yeats.html

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

https://blogs.clemson.edu/press/2018/02/13/forthcoming-a-readers-guide-to-yeatss-a-vision/

https://global.oup.com/academic/product/a-readers-guide-to-yeatss-a-vision-9781942954620?cc=us&lang=en&#038;

https://equivalentexchange.blog/2013/10/23/a-vision-by-w-b-yeats-faculties/

[*7.199, *7.200, *8.1, *11.56]

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Education is Not a Product

March 1, 2019

A recent meme popular in education:

  • Education is not a product.
  • Students are not customers.
  • Professors are not tools.
  • The University is not a factory.

I’m not sure who originally came up with this interesting saying.

Examining the anti-analogy, you have “customers buy products made in a factory with tools”. Perhaps a better fourfold would be to change this to “workers use tools to make products in a factory”, so you could say “Students are not workers.” But then you lose the capitalistic “consumer culture” geist.

Or you could say the Professors are (not) factory workers and the Students are (not) the raw material. As a homework assignment, write an essay or develop a mathematical theory comparing this anti-analogy fourfold (or a close derivative thereof) to Aristotle’s Four Causes. Please show your work.

Further Reading:

Note the similarity with some of the elements of the following (which also reminds me for the Four Causes):

https://equivalentexchange.blog/2015/03/11/j-y-girards-transcendental-syntax-v2/

[*11.18]

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