Log24

Monday, June 26, 2017

Four Dots

Analogies — "A : B  ::  C : D"  may be read  "A is to B  as  C is to D."

Gian-Carlo Rota on Heidegger…

"… The universal as  is given various names in Heidegger's writings….

The discovery of the universal as  is Heidegger's contribution to philosophy….

The universal 'as' is the surgence of sense in Man, the shepherd of Being.

The disclosure of the primordial as  is the end of a search that began with Plato….
This search comes to its conclusion with Heidegger."

— "Three Senses of 'A is B' in Heideggger," Ch. 17 in Indiscrete Thoughts
 

See also Four Dots in this journal. 

Some context:  McLuhan + Analogy.

Tuesday, July 6, 2010

What “As” Is

Filed under: General,Geometry — Tags: , , , — m759 @ 8:00 PM

or:  Combinatorics (Rota) as Philosophy (Heidegger) as Geometry (Me)

"Dasein’s full existential structure is constituted by
the 'as-structure' or 'well-joined structure' of the rift-design*…"

— Gary Williams, post of January 22, 2010

Background—

Gian-Carlo Rota on Heidegger…

"… The universal as  is given various names in Heidegger's writings….

The discovery of the universal as  is Heidegger's contribution to philosophy….

The universal 'as' is the surgence of sense in Man, the shepherd of Being.

The disclosure of the primordial as  is the end of a search that began with Plato….
This search comes to its conclusion with Heidegger."

— "Three Senses of 'A is B' in Heideggger," Ch. 17 in Indiscrete Thoughts

… and projective points as separating rifts

Image-- The Three-Point Line: A
 Finite Projective Space

    Click image for details.

* rift-design— Definition by Deborah Levitt

"Rift.  The stroke or rending by which a world worlds, opening both the 'old' world and the self-concealing earth to the possibility of a new world. As well as being this stroke, the rift is the site— the furrow or crack— created by the stroke. As the 'rift design' it is the particular characteristics or traits of this furrow."

— "Heidegger and the Theater of Truth," in Tympanum: A Journal of Comparative Literary Studies, Vol. 1, 1998

Window, continued

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

"Simplicity, simplicity, simplicity!  I say, let your affairs
be as two or three,
and not a hundred or a thousand;
instead of a million count half a dozen,
and keep your accounts on your thumb-nail."
— Henry David Thoreau, Walden

This quotation is the epigraph to Section 1.1 of
Alexandre V. Borovik's
Mathematics Under the Microscope:

Notes on Cognitive Aspects of Mathematical Practice
(American Mathematical Society, Jan. 15, 2010, 317 pages).

From Peter J. Cameron's review notes for
his new course in group theory

http://www.log24.com/log/pix10A/100705-CameronExample.jpg

From Log24 on June 24

Geometry Simplified

Image-- The Four-Point Plane: A Finite Affine Space
(an affine  space with subsquares as points
and sets  of subsquares as hyperplanes)

Image-- The Three-Point Line: A Finite Projective Space
(a projective  space with, as points, sets
  of line segments that separate subsquares)

Exercise

Show that the above geometry is a model
for the algebra discussed by Cameron.

Monday, July 5, 2010

Window

Filed under: General — Tags: , — m759 @ 9:00 AM

"Examples are the stained-glass
  windows of knowledge." — Nabokov

Image-- Example of group actions on the set Omega of three partitions of a 4-set into two 2-sets

Related material:

Thomas Wolfe and the
Kernel of Eternity

Thursday, June 24, 2010

Midsummer Noon

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

Geometry Simplified

Image-- The Three-Point Line: A Finite Projective Space
(a projective space)

The above finite projective space
is the simplest nontrivial example
of a Galois geometry (i.e., a finite
geometry with coordinates in a
finite (that is, Galois) field.)

The vertical (Euclidean) line represents a
 (Galois) point, as does the horizontal line
and also the vertical-and-horizontal
cross that represents the first two points'
binary sum (i.e., symmetric difference,
if the lines are regarded as sets).

Homogeneous coordinates for the
points of this line —

(1,0), (0,1), (1,1).

Here 0 and 1 stand for the elements
of the two-element Galois field GF(2).

The 3-point line is the projective space
corresponding to the affine space
(a plane, not a line) with four points —

http://www.log24.com/log/pix10A/100624-The4PointPlane.bmp
(an affine space)

The (Galois) points of this affine plane are
  not the single and combined (Euclidean)
line segments that play the role of
  points in the 3-point projective line,
but rather the four subsquares
that the line segments separate.

For further details, see Galois Geometry.

There are, of course, also the trivial
two-point affine space and the corresponding
trivial one-point projective space —

http://www.log24.com/log/pix10A/100624-TrivialSpaces.bmp

Here again, the points of the affine space are
represented by squares, and the point of the
projective space is represented by a line segment
separating the affine-space squares.

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