Wednesday, June 16, 2010

Geometry of Language

Filed under: General,Geometry — m759 @ 10:31 AM

(Continued from April 23, 2009, and February 13, 2010.)

Paul Valéry as quoted in yesterday’s post:

“The S[elf] is invariant, origin, locus or field, it’s a functional property of consciousness” (Cahiers, 15:170 [2: 315])

The geometric example discussed here yesterday as a Self symbol may seem too small to be really impressive. Here is a larger example from the Chinese, rather than European, tradition. It may be regarded as a way of representing the Galois field GF(64). (“Field” is a rather ambiguous term; here it does not, of course, mean what it did in the Valéry quotation.)

From Geometry of the I Ching

Image-- The 64 hexagrams of the I Ching

The above 64 hexagrams may also be regarded as
the finite affine space AG(6,2)— a larger version
of the finite affine space AG(4,2) in yesterday’s post.
That smaller space has a group of 322,560 symmetries.
The larger hexagram  space has a group of
1,290,157,424,640 affine symmetries.

From a paper on GL(6,2), the symmetry group
of the corresponding projective  space PG(5,2),*
which has 1/64 as many symmetries—

(Click to enlarge.)

Image-- Classes of the Group GL(6,)

For some narrative in the European  tradition
related to this geometry, see Solomon’s Cube.

* Update of July 29, 2011: The “PG(5,2)” above is a correction from an earlier error.

Saturday, February 13, 2010

Entertainment continued

Filed under: General,Geometry — m759 @ 9:28 AM

Logic is all about the entertaining of possibilities.”

– Colin McGinn, Mindsight: Image, Dream, Meaning,
   Harvard University Press, 2004

Geometry of Language,
continued from St. George's Day, 2009

Professor Arielle Saiber with chess set

Excerpt from Jasper Hopkins's 'Concise Introduction to the Philosophy of Nicholas of Cusa

Related material:

Prima Materia,
The Galois Quaternion,
and The Wake of Imagination.

See also the following from a physicist
(not of the most orthodox sort, but his remarks
  here on Heisenberg seem quite respectable)–

Ian J. Thompson, 7 Dec. 2009

Quantum mechanics describes the probabilities of actual outcomes in terms of a wave function, or at least of a quantum state of amplitudes that varies with time. The public always asks what the wave function is, or what the amplitudes are amplitudes of. Usually, we reply that the amplitudes are ‘probability amplitudes’, or that the wave function is a ‘probability wave function’, but neither answer is ontologically satisfying since probabilities are numbers, not stuff. We have already rehearsed the objections to the natural world being made out of numbers, as these are pure forms. In fact, ‘waves’, ‘amplitudes’ and ‘probabilities’ are all forms, and none of them can be substances. So, what are quantum objects made of: what stuff?

According to Heisenberg [6], the quantum probability waves are “a quantitative formulation of the concept of ‘dynamis’, possibility, or in the later Latin version, ‘potentia’, in Aristotle’s philosophy. The concept of events not determined in a peremptory manner, but that the possibility or ‘tendency’ for an event to take place has a kind of reality—a certain intermediate layer of reality, halfway between the massive reality of matter and the intellectual reality of the idea or the image—this concept plays a decisive role in Aristotle’s philosophy. In modern quantum theory this concept takes on a new form; it is formulated quantitatively as probability and subjected to mathematically expressible laws of nature.” Unfortunately Heisenberg does not develop this interpretation much beyond the sort of generality of the above statements, and the concept of ‘potentiality’ remains awkwardly isolated from much of his other thought on this subject [7]. It is unclear even what he means by ‘potentia’.


Heisenberg, W. 1961 On Modern Physics, London: Orion Press.


[6] W. Heisenberg, ‘Planck’s discovery and the philosophical problems of atomic physics’, pp. 3-20 in Heisenberg (1961).

[7] Heisenberg, for example, brings into his thought on quantum physics the Kantian phenomena/noumena distinction, as well as some of Bohr’s ideas on ‘complementarity’ in experimental arrangements.

Thursday, April 23, 2009

Thursday April 23, 2009

Filed under: General,Geometry — Tags: , — m759 @ 10:00 AM


The Geometry
of Language

(continued from April 16)


Professor Arielle Saiber with chess set

Click on the image for an
interview with the author of
Giordano Bruno and
the Geometry of Language

Related material:

Joyce on language —

The sigla of 'Finnegans Wake'

Bruno, Joyce, and coincidentia oppositorum

Cullinane on geometry —

Geometry of the I Ching (for comparison to Joyce's 'sigla')

Click on images for details.

Friday, April 17, 2009

Friday April 17, 2009

Filed under: General,Geometry — Tags: — m759 @ 10:31 AM

Begettings of
the Broken Bold

Thanks for the following
quotation (“Non deve…
nella testa“) go to the
weblog writer who signs
himself “Conrad H. Roth.”

of Goethe

(Vol. II, London, Bell & Daldy,
1868, at Google Books):

… Yesterday I took leave of my Captain, with a promise of visiting him at Bologna on my return. He is a true


representative of the majority of his countrymen. Here, however, I would record a peculiarity which personally distinguished him. As I often sat quiet and lost in thought he once exclaimed “Che pensa? non deve mai pensar l’uomo, pensando s’invecchia;” which being interpreted is as much as to say, “What are you thinking about: a man ought never to think; thinking makes one old.” And now for another apophthegm of his; “Non deve fermarsi l’uomo in una sola cosa, perche allora divien matto; bisogna aver mille cose, una confusione nella testa;” in plain English, “A man ought not to rivet his thoughts exclusively on any one thing, otherwise he is sure to go mad; he ought to have in his head a thousand things, a regular medley.”

Certainly the good man could not know that the very thing that made me so thoughtful was my having my head mazed by a regular confusion of things, old and new. The following anecdote will serve to elucidate still more clearly the mental character of an Italian of this class. Having soon discovered that I was a Protestant, he observed after some circumlocution, that he hoped I would allow him to ask me a few questions, for he had heard such strange things about us Protestants that he wished to know for a certainty what to think of us.

Notes for Roth:

Roth and Corleone in Havana

The title of this entry,
“Begettings of the Broken Bold,”
is from Wallace Stevens’s
“The Owl in the Sarcophagus”–

This was peace after death, the brother of sleep,
The inhuman brother so much like, so near,
Yet vested in a foreign absolute,

Adorned with cryptic stones and sliding shines,
An immaculate personage in nothingness,
With the whole spirit sparkling in its cloth,

Generations of the imagination piled
In the manner of its stitchings, of its thread,
In the weaving round the wonder of its need,

And the first flowers upon it, an alphabet
By which to spell out holy doom and end,
A bee for the remembering of happiness.

Peace stood with our last blood adorned, last mind,
Damasked in the originals of green,
A thousand begettings of the broken bold.

This is that figure stationed at our end,
Always, in brilliance, fatal, final, formed
Out of our lives to keep us in our death....

Related material:

  • Yesterday’s entry on Giordano Bruno and the Geometry of Language
  • James Joyce and Heraldry
  • “One might say that he [Joyce] invented a non-Euclidean geometry of language; and that he worked over it with doggedness and devotion….” —Unsigned notice in The New Republic, 20 January 1941
  • Joyce’s “collideorscape” (scroll down for a citation)
  • “A Hanukkah Tale” (Log24, Dec. 22, 2008)
  • Stevens’s phrase from “An Ordinary Evening in New Haven” (Canto XXV)

Some further context:

Roth’s entry of Nov. 3, 2006–
Why blog, sinners?“–
and Log24 on that date:
First to Illuminate.”

Thursday, April 16, 2009

Thursday April 16, 2009

Filed under: General,Geometry — m759 @ 1:00 PM
Happy Birthday,
Benedict XVI:

A Game for Bishops
continued from April 3

Professor Arielle Saiber with chess set

Click on the image for an
interview with the author of
Giordano Bruno and
the Geometry of Language

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