Log24

Tuesday, April 26, 2016

A Sense of Identity

Filed under: Uncategorized — m759 @ 9:01 PM

Peter Schjeldahl on Wallace Stevens in the current New Yorker

"Stevens was born in 1879 in Reading, Pennsylvania,
the second of five children. His father, from humble
beginnings, was a successful lawyer, his mother a
former schoolteacher. Each night, she read a chapter
of the Bible to the children, who attended schools
attached to both Presbyterian and Lutheran churches,
where the music left an indelible impression on Stevens.
Both sides of the family were Pennsylvania Dutch,
an identity that meant little to him when he was young
but a great deal later on, perhaps to shore up a precarious
sense of identity."

See also this  journal on Christmas Day, 2010

http://www.log24.com/log/pix10B/101225-QuiltSymmetry.JPG

It's a start. For more advanced remarks from the same date, see Mere Geometry.

Interacting

Filed under: Uncategorized — m759 @ 8:31 PM

"… I would drop the keystone into my arch …."

— Charles Sanders Peirce, "On Phenomenology"

" 'But which is the stone that supports the bridge?' Kublai Khan asks."

— Italo Calvino, Invisible Cities, as quoted by B. Elan Dresher.

(B. Elan Dresher. Nordlyd  41.2 (2014): 165-181,
special issue on Features edited by Martin Krämer,
Sandra Ronai and Peter Svenonius. University of Tromsø –
The Arctic University of Norway.
http://septentrio.uit.no/index.php/nordlyd)

Peter Svenonius and Martin Krämer, introduction to the
Nordlyd  double issue on Features —

"Interacting with these questions about the 'geometric' 
relations among features is the algebraic structure
of the features."

For another such interaction, see the previous post.

This  post may be viewed as a commentary on a remark in Wikipedia

"All of these ideas speak to the crux of Plato's Problem…."

See also The Diamond Theorem at Tromsø and Mere Geometry.

Thursday, January 21, 2016

Dividing the Indivisible

Filed under: Uncategorized — m759 @ 11:00 AM

My statement yesterday morning that the 15 points
of the finite projective space PG(3,2) are indivisible 
was wrong.  I was misled by quoting the powerful
rhetoric of Lincoln Barnett (LIFE magazine, 1949).

Points of Euclidean  space are of course indivisible
"A point is that which has no parts" (in some translations).

And the 15 points of PG(3,2) may be pictured as 15
Euclidean  points in a square array (with one point removed)
or tetrahedral array (with 11 points added).

The geometry of  PG(3,2) becomes more interesting,
however, when the 15 points are each divided  into
several parts. For one approach to such a division,
see Mere Geometry. For another approach, click on the
image below.

IMAGE- 'Nocciolo': A 'kernel' for Pascal's Hexagrammum Mysticum: The 15 2-subsets of a 6-set as points in a Galois geometry.

Friday, August 14, 2015

Discrete Space

Filed under: Uncategorized — m759 @ 7:24 AM

(A review)

Galois space:

Image-- examples from Galois affine geometry

Counting symmetries of  Galois space:
IMAGE - The Diamond Theorem

The reason for these graphic symmetries in affine Galois space —

symmetries of the underlying projective Galois space:

Thursday, July 17, 2014

Paradigm Shift:

Filed under: Uncategorized — Tags: — m759 @ 11:01 AM
 

Continuous Euclidean space to discrete Galois space*

Euclidean space:

Point, line, square, cube, tesseract

From a page by Bryan Clair

Counting symmetries in Euclidean space:

Galois space:

Image-- examples from Galois affine geometry

Counting symmetries of  Galois space:
IMAGE - The Diamond Theorem

The reason for these graphic symmetries in affine Galois space —

symmetries of the underlying projective Galois space:

* For related remarks, see posts of May 26-28, 2012.

Monday, March 7, 2011

Point Taken

Filed under: Uncategorized — m759 @ 4:00 PM

Recommended— An essay (part 1 of 5 parts) in today's New York TImes—

THE ULTIMATUM

I don’t want to die in
a language I can’t understand.
— Jorge Luis Borges

Comment 71

"I agree with one of the earlier commenters that this is a piece of fine literary work. And in response to some of those who have wondered 'WHAT IS THE POINT?!' of this essay, I would like to say: Must literature always answer that question for us (and as quickly and efficiently as possible)?"

For an excellent survey of the essay's historical context, see The Stanford Encyclopedia of Philosophy article

"The Incommensurability of Scientific Theories,"
First published Wed., Feb. 25, 2009,
by Eric Oberheim and Paul Hoyningen-Huene.

Related material from this journal—

Paradigms, Paradigms Lost, and a search for "mere geometry." This last includes remarks contrasting Euclid's definition of a point ("that which has no parts") with a later notion useful in finite geometry.

See also (in the spirit of The Abacus Conundrum )…

The Monolith Epiphany

http://www.log24.com/log/pix11/110307-Monolith.jpg

(Note the Borges epigraph above.)

Monday, December 27, 2010

Church Diamond

Filed under: Uncategorized — m759 @ 3:09 PM

IMAGE- The diamond property

Also known, roughly speaking, as confluence  or the Church-Rosser property.

From "NYU Lambda Seminar, Week 2" —

[See also the parent page Seminar in Semantics / Philosophy of Language or:
 What Philosophers and Linguists Can Learn From Theoretical Computer Science But Didn't Know To Ask)
]

A computational system is said to be confluent, or to have the Church-Rosser or diamond property, if, whenever there are multiple possible evaluation paths, those that terminate always terminate in the same value. In such a system, the choice of which sub-expressions to evaluate first will only matter if some of them but not others might lead down a non-terminating path.

The untyped lambda calculus is confluent. So long as a computation terminates, it always terminates in the same way. It doesn't matter which order the sub-expressions are evaluated in.

A computational system is said to be strongly normalizing if every permitted evaluation path is guaranteed to terminate. The untyped lambda calculus is not strongly normalizing: ω ω doesn't terminate by any evaluation path; and (\x. y) (ω ω) terminates only by some evaluation paths but not by others.

But the untyped lambda calculus enjoys some compensation for this weakness. It's Turing complete! It can represent any computation we know how to describe. (That's the cash value of being Turing complete, not the rigorous definition. There is a rigorous definition. However, we don't know how to rigorously define "any computation we know how to describe.") And in fact, it's been proven that you can't have both. If a computational system is Turing complete, it cannot be strongly normalizing.

There is no connection, apart from the common reference to an elementary geometric shape, between the use of "diamond" in the above Church-Rosser sense and the use of "diamond" in the mathematics of (Cullinane's) Diamond Theory.

Any attempt to establish such a connection would, it seems, lead quickly into logically dubious territory.

Nevertheless, in the synchronistic spirit of Carl Jung and Arthur Koestler, here are some links to such a territory —

 Link One — "Insane Symmetry"  (Click image for further details)—

http://www.log24.com/log/pix10B/101227-InsaneSymmetry.jpg

See also the quilt symmetry in this  journal on Christmas Day.

Link Two — Divine Symmetry

(George Steiner on the Name in this journal on Dec. 31 last year ("All about Eve")) —

"The links are direct between the tautology out of the Burning Bush, that 'I am' which accords to language the privilege of phrasing the identity of God, on the one hand, and the presumptions of concordance, of equivalence, of translatability, which, though imperfect, empower our dictionaries, our syntax, our rhetoric, on the other. That 'I am' has, as it were, at an overwhelming distance, informed all predication. It has spanned the arc between noun and verb, a leap primary to creation and the exercise of creative consciousness in metaphor. Where that fire in the branches has gone out or has been exposed as an optical illusion, the textuality of the world, the agency of the Logos in logic—be it Mosaic, Heraclitean, or Johannine—becomes 'a dead letter.'"

George Steiner, Grammars of Creation

(See also, from Hanukkah this year,  A Geometric Merkabah and The Dreidel is Cast.)

Link Three – Spanning the Arc —

Part A — Architect Louis Sullivan on "span" (see also Kindergarten at Stonehenge)

Part B — "Span" in category theory at nLab —

http://www.log24.com/log/pix10B/101227-nLabSpanImage.jpg

Also from nLab — Completing Spans to Diamonds

"It is often interesting whether a given span in some partial ordered set can be completed into a diamond. The property of a collection of spans to consist of spans which are expandable into diamonds is very useful in the theory of rewriting systems and producing normal forms in algebra. There are classical results e.g. Newman’s diamond lemma, Širšov-Bergman’s diamond lemma (Širšov is also sometimes spelled as Shirshov), and Church-Rosser theorem (and the corresponding Church-Rosser confluence property)."

The concepts in this last paragraph may or may not have influenced the diamond theory of Rudolf Kaehr (apparently dating from 2007).

They certainly have nothing to do with the Diamond Theory of Steven H. Cullinane (dating from 1976).

For more on what the above San Francisco art curator is pleased to call "insane symmetry," see this journal on Christmas Day.

For related philosophical lucubrations (more in the spirit of Kaehr than of Steiner), see the New York Times  "The Stone" essay "Span: A Remembrance," from December 22—

“To understand ourselves well,” [architect Louis] Sullivan writes, “we must arrive first at a simple basis: then build up from it.”

Around 300 BC, Euclid arrived at this: “A point is that which has no part. A line is breadthless length.”

See also the link from Christmas Day to remarks on Euclid and "architectonic" in Mere Geometry.

Saturday, December 25, 2010

Not-So-Fearful Symmetry

Filed under: Uncategorized — m759 @ 12:25 AM

From a Mennonite homeschooling family

http://www.log24.com/log/pix10B/101225-QuiltSymmetry.JPG

It's a start. For more advanced remarks from the same date, see Mere Geometry.

Tuesday, May 4, 2010

Mathematics and Narrative, continued

Filed under: Uncategorized — Tags: — m759 @ 8:28 PM

Romancing the
Non-Euclidean Hyperspace

Backstory
Mere Geometry, Types of Ambiguity,
Dream Time, and Diamond Theory, 1937

The cast of 1937's 'King Solomon's Mines' goes back to the future

For the 1937 grid, see Diamond Theory, 1937.

The grid is, as Mere Geometry points out, a non-Euclidean hyperspace.

For the diamonds of 2010, see Galois Geometry and Solomon’s Cube.

Monday, April 26, 2010

Types of Ambiguity

Filed under: Uncategorized — Tags: — m759 @ 10:31 AM

From Ursula K. Le Guin’s novel
The Dispossessed: An Ambiguous Utopia
(1974)—

Chapter One

“There was a wall. It did not look important. It was built of uncut rocks roughly mortared. An adult could look right over it, and even a child could climb it. Where it crossed the roadway, instead of having a gate it degenerated into mere geometry, a line, an idea of boundary. But the idea was real. It was important. For seven generations there had been nothing in the world more important than that wall.

Like all walls it was ambiguous, two-faced. What was inside it and what was outside it depended upon which side of it you were on.”

Note—

“We note that the phrase ‘instead of having a gate it degenerated into mere geometry’ is mere fatuousness. If there is an idea here, degenerate, mere, and geometry  in concert do not fix it. They bat at it like a kitten at a piece of loose thread.”

— Samuel R. Delany, The Jewel-Hinged Jaw: Notes on the Language of Science Fiction  (Dragon Press, 1977), page 110 of revised edition, Wesleyan University Press, 2009

(For the phrase mere geometry  elsewhere, see a note of April 22. The apparently flat figures in that note’s illustration “Galois Affine Geometry” may be regarded as degenerate  views of cubes.)

Later in the Le Guin novel—

“… The Terrans had been intellectual imperialists, jealous wall builders. Even Ainsetain, the originator of the theory, had felt compelled to give warning that his physics embraced no mode but the physical and should not be taken as implying the metaphysical, the philosophical, or the ethical. Which, of course, was superficially true; and yet he had used number, the bridge between the rational and the perceived, between psyche and matter, ‘Number the Indisputable,’ as the ancient founders of the Noble Science had called it. To employ mathematics in this sense was to employ the mode that preceded and led to all other modes. Ainsetain had known that; with endearing caution he had admitted that he believed his physics did, indeed, describe reality.

Strangeness and familiarity: in every movement of the Terran’s thought Shevek caught this combination, was constantly intrigued. And sympathetic: for Ainsetain, too, had been after a unifying field theory. Having explained the force of gravity as a function of the geometry of spacetime, he had sought to extend the synthesis to include electromagnetic forces. He had not succeeded. Even during his lifetime, and for many decades after his death, the physicists of his own world had turned away from his effort and its failure, pursuing the magnificent incoherences of quantum theory with its high technological yields, at last concentrating on the technological mode so exclusively as to arrive at a dead end, a catastrophic failure of imagination. Yet their original intuition had been sound: at the point where they had been, progress had lain in the indeterminacy which old Ainsetain had refused to accept. And his refusal had been equally correct– in the long run. Only he had lacked the tools to prove it– the Saeba variables and the theories of infinite velocity and complex cause. His unified field existed, in Cetian physics, but it existed on terms which he might not have been willing to accept; for the velocity of light as a limiting factor had been essential to his great theories. Both his Theories of Relativity were as beautiful, as valid, and as useful as ever after these centuries, and yet both depended upon a hypothesis that could not be proved true and that could be and had been proved, in certain circumstances, false.

But was not a theory of which all the elements were provably true a simple tautology? In the region of the unprovable, or even the disprovable, lay the only chance for breaking out of the circle and going ahead.

In which case, did the unprovability of the hypothesis of real coexistence– the problem which Shevek had been pounding his head against desperately for these last three days. and indeed these last ten years– really matter?

He had been groping and grabbing after certainty, as if it were something he could possess. He had been demanding a security, a guarantee, which is not granted, and which, if granted, would become a prison. By simply assuming the validity of real coexistence he was left free to use the lovely geometries of relativity; and then it would be possible to go ahead. The next step was perfectly clear. The coexistence of succession could be handled by a Saeban transformation series; thus approached, successivity and presence offered no antithesis at all. The fundamental unity of the Sequency and Simultaneity points of view became plain; the concept of interval served to connect the static and the dynamic aspect of the universe. How could he have stared at reality for ten years and not seen it? There would be no trouble at all in going on. Indeed he had already gone on. He was there. He saw all that was to come in this first, seemingly casual glimpse of the method, given him by his understanding of a failure in the distant past. The wall was down. The vision was both clear and whole. What he saw was simple, simpler than anything else. It was simplicity: and contained in it all complexity, all promise. It was revelation. It was the way clear, the way home, the light.”

Related material—

Time Fold, Halloween 2005, and May and Zan.

See also The Devil and Wallace Stevens

“In a letter to Harriet Monroe, written December 23, 1926, Stevens refers to the Sapphic fragment that invokes the genius of evening: ‘Evening star that bringest back all that lightsome Dawn hath scattered afar, thou bringest the sheep, thou bringest the goat, thou bringest the child home to the mother.’ Christmas, writes Stevens, ‘is like Sappho’s evening: it brings us all home to the fold’ (Letters of Wallace Stevens, 248).”

— “The Archangel of Evening,” Chapter 5 of Wallace Stevens: The Intensest Rendezvous, by Barbara M. Fisher, The University Press of Virginia, 1990

Thursday, April 22, 2010

Mere Geometry

Filed under: Uncategorized — Tags: — m759 @ 1:00 PM

Image-- semeion estin ou meros outhen

Image-- Euclid's definition of 'point'

Stanford Encyclopedia of Philosophy

Mereology (from the Greek μερος, ‘part’) is the theory of parthood relations: of the relations of part to whole and the relations of part to part within a whole. Its roots can be traced back to the early days of philosophy, beginning with the Presocratics….”

A non-Euclidean* approach to parts–

Image-- examples from Galois affine geometry

Corresponding non-Euclidean*
projective points —

Image-- The smallest Galois geometries

Richard J. Trudeau in The Non-Euclidean Revolution, chapter on “Geometry and the Diamond Theory of Truth”–

“… Plato and Kant, and most of the philosophers and scientists in the 2200-year interval between them, did share the following general presumptions:

(1) Diamonds– informative, certain truths about the world– exist.
(2) The theorems of Euclidean geometry are diamonds.

Presumption (1) is what I referred to earlier as the ‘Diamond Theory’ of truth. It is far, far older than deductive geometry.”

Trudeau’s book was published in 1987. The non-Euclidean* figures above illustrate concepts from a 1976 monograph, also called “Diamond Theory.”

Although non-Euclidean,* the theorems of the 1976 “Diamond Theory” are also, in Trudeau’s terminology, diamonds.

* “Non-Euclidean” here means merely “other than  Euclidean.” No violation of Euclid’s parallel postulate is implied.

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