Mathematics and Narrative
continued

"There is a pleasantly discursive treatment of Pontius Pilate's unanswered question 'What is truth?'"

--
H. S. M. Coxeter, 1987, introduction to Richard J. Trudeau's remarks on
the "Story Theory" of truth as opposed to  the "Diamond Theory" of
truth " in The Non-Euclidean Revolution

"I had an epiphany: I thought 'Oh my God, this is
it! People are talking about elliptic curves and of course they
think they are talking mathematics. But are they really? Or are
they talking about stories?'"

-- An organizer of last month's "Mathematics and Narrative" conference

"A new epistemology is
emerging to replace the Diamond Theory of truth. I will call it the
'Story Theory' of truth: There are no diamonds. People make up stories
about what they experience. Stories that catch on are called 'true.'
The Story Theory of truth is itself a story that is catching on. It is
being told and retold, with increasing frequency, by thinkers of many
stripes*...."

-- Richard J. Trudeau in The Non-Euclidean Revolution

"'Deniers' of truth...
insist that each of us is trapped in his own point of view; we make up
stories about the world and, in an exercise of power, try to impose
them on others."

-- Jim Holt in this week's New Yorker magazine.  Click on the box below.

* Many stripes --

   "What disciplines were represented at the meeting?"
   "Apart from historians, you mean? Oh, many: writers, artists,
philosophers, semioticians, cognitive psychologists – you
name it."

-- An organizer of last month's "Mathematics and Narrative" conference

    "The Game was at first nothing more than a witty
method for developing memory and ingenuity among students and musicians.
     The inventor, Bastian Perrot of Calw... found that the pupils at
the Cologne Seminary had a rather elaborate game they used to play. One
would call out, in the standardized abbreviations of their science,
motifs or initial bars of classical compositions, whereupon the other
had to respond with the continuation of the piece, or better still with
a higher or lower voice, a contrasting theme, and so forth. It was an
exercise in memory and improvisation quite similar to the sort of thing
probably in vogue among the ardent pupils of counterpoint in the days
of Schütz, Pachelbel, and Bach....
     Bastian Perrot... constructed a frame, modeled on a child's
abacus, a frame with several dozen wires on which could be strung glass
beads of various sizes, shapes, and colors...."

-- Hermann Hesse at The Glass Bead Game Defined

Narrative and Latin Squares

From The Independent, 15 August 2005:

"Millions of people now enjoy Sudoku puzzles. Forget the pseudo-Japanese
baloney: sudoku grids are a version of the Latin Square created by the
great Swiss mathematician Leonhard Euler in the late 18th century."

The Independent was discussing the conference on "Mathematics and Narrative" at Mykonos in July.

From the Wikipedia article on Latin squares:

"The popular Sudoku puzzles are a special case of Latin squares; any solution to a Sudoku puzzle is a Latin square. Sudoku imposes the additional restriction that 3×3 subgroups must also contain the digits 1–9 (in the standard version).

The Diamond 16 Puzzle illustrates a generalized concept of Latin-square orthogonality: that of "orthogonal squares" (Diamond Theory, 1976) or "orthogonal matrices"-- orthogonal, that is, in a combinatorial, not a linear-algebra sense (A. E. Brouwer, 1991)."

This last paragraph, added to Wikipedia on Aug. 14,  may or may not survive the critics there.

Kaleidoscope, continued:

Austere Geometry

From Noel Gray, The Kaleidoscope: Shake, Rattle, and Roll:

"... what we will be considering is how the ongoing production of meaning
can generate a tremor in the stability of the initial theoretical frame
of this instrument; a frame informed by geometry's long tradition of
privileging the conceptual ground over and above its visual
manifestation.  And to consider also how the possibility of a seemingly
unproblematic correspondence between the ground and its extrapolation,
between geometric theory and its applied images, is intimately
dependent upon the control of the truth status ascribed to the image by
the generative theory.  This status in traditional geometry has been
consistently understood as that of the graphic ancilla-- a maieutic
force, in the Socratic sense of that term-- an ancilla to lawful
principles; principles that have, traditionally speaking, their primary
expression in the purity of geometric idealities.*  It follows that the
possibility of installing a tremor in this tradition by understanding
the kaleidoscope's images as announcing more than the mere
subordination to geometry's theory-- yet an announcement that is still
in a sense able to leave in place this self-same tradition-- such a
possibility must duly excite our attention and interest.

* I refer here to Plato's utilisation in the Meno of graphic austerity as
the tool to bring to the surface, literally and figuratively, the
inherent presence of geometry in the mind of the slave."

See also

Noel Gray, Ph.D. thesis, U. of Sydney, Dept. of Art History and Theory, 1994:

"The Image
of Geometry: Persistence qua Austerity-- Cacography and The Truth to Space."

Kaleidoscope, continued:

In Derrida's Defense

The previous entry quoted an attack on Jacques Derrida for ignoring the
"kaleidoscope" metaphor of Claude Levi-Strauss.  Here is a quote
by Derrida himself:

-- Derrida,
J. (1983) ‘The Principle of Reason: The University
in the Eyes of its Pupils’, Diacritics 13.3:
3-20."

The above quotation comes from Simon Wortham,  who thinks the "optical device" of Derrida is a mirror.  The same quotation appears in Desiring Dualisms at thispublicaddress.com, where the "optical device" is interpreted as a kaleidoscope.

Derrida's "optical device" may (for university pupils desperately seeking an essay topic) be compared with Joyce's "collideorscape."  For a different connection with Derrida, see The 'Collideorscape' as Différance.

Kaleidoscope, continued

From Clifford Geertz, The Cerebral Savage:

"Savage
logic works like a kaleidoscope whose chips can fall into a variety of
patterns while remaining unchanged in quantity, form, or color. The
number of patterns producible in this way may be large if the chips are
numerous and varied enough, but it is not infinite. The patterns
consist in the disposition of the chips vis-a-vis one another (that is,
they are a function of the relationships among the chips rather than
their individual properties considered separately).  And their
range of possible transformations is strictly determined by the
construction of the kaleidoscope, the inner law which governs its
operation. And so it is too with savage thought.  Both anecdotal
and geometric, it builds coherent structures out of 'the odds and ends
left over from psychological or historical process.'

These
odds and ends, the chips of the kaleidoscope, are images drawn from
myth, ritual, magic, and empirical lore....  as in a kaleidoscope,
one always sees the chips distributed in some pattern, however
ill-formed or irregular.   But, as in a kaleidoscope, they are
detachable from these structures and arrangeable into different ones of
a similar sort....  Levi-Strauss generalizes this permutational
view of thinking to savage thought in general.  It is all a matter
of shuffling discrete (and concrete) images--totem animals, sacred
colors, wind directions, sun deities, or whatever--so as to produce
symbolic structures capable of formulating and communicating objective
(which is not to say accurate) analyses of the social and physical
worlds.

.... And the point is general.  The relationship between a symbolic structure and its referent, the basis of its meaning, 
is fundamentally 'logical,' a coincidence of form-- not affective, not
historical, not functional.  Savage thought is frozen reason and
anthropology is, like music and mathematics, 'one of the few true
vocations.'

Or like linguistics."

Edward Sapir on Linguistics, Mathematics, and Music:

"...
linguistics has also that profoundly serene and satisfying quality
which inheres in mathematics and in music and which may be described as
the creation out of simple elements of a self-contained universe of
forms.  Linguistics has neither the sweep nor the instrumental
power of mathematics, nor has it the universal aesthetic appeal of
music.  But under its crabbed, technical, appearance there lies
hidden the same classical spirit, the same freedom in restraint, which
animates mathematics and music at their purest."

-Edward Sapir, "The Grammarian and his Language,"
  American Mercury 1:149-155,1924

From Robert de Marrais, Canonical Collage-oscopes:

"...underwriting
the form languages of ever more domains of mathematics is a set of deep
patterns which not only offer access to a kind of ideality that Plato
claimed to see the universe as created with in the Timaeus;
more than this, the realm of Platonic forms is itself subsumed in this
new set of design elements-- and their most general instances are not
the regular solids, but crystallographic reflection groups.  You
know, those things the non-professionals call . . . kaleidoscopes! *  (In the next exciting episode, we'll see how Derrida claims mathematics is the key to freeing us from 'logocentrism' **-- then ask him why, then, he jettisoned the deepest structures of mathematical patterning just to make his name...)

**
... contemporary with the Johns Hopkins hatchet job that won him
American marketshare, Derrida was also being subjected to a series of
probing interviews in Paris by the hometown crowd.  He first
gained academic notoriety in France for his book-length reading of
Husserl's two-dozen-page essay on 'The Origin of Geometry.'  The
interviews were collected under the rubric of Positions
(Chicago: U. of Chicago Press, 1981...).  On pp.
34-5 he says the following: 'the resistance to logico-mathematical
notation has always been the signature of logocentrism and phonologism
in the event to which they have dominated metaphysics and the classical
semiological and linguistic projects.... A grammatology that would
break with this system of presuppositions, then, must in effect
liberate the mathematization of language.... The effective progress of
mathematical notation thus goes along with the deconstruction of
metaphysics, with the profound renewal of mathematics itself, and the
concept of science for which mathematics has always been the
model.'  Nice campaign speech, Jacques; but as we'll see, you
reneged on your promise not just with the kaleidoscope (and we'll investigate, in
depth, the many layers of contradiction and cluelessness you put on
display in that disingenuous 'playing to the house'); no, we'll see
how, at numerous other critical junctures, you instinctively took the
wrong fork in the road whenever mathematical issues arose...
henceforth, monsieur, as Joe Louis once said, 'You can run, but you
just can't hide.'...."

Kaleidoscope

A new web page simplifies the Diamond 16 Puzzle and relates the resulting "kaleidoscope" to Hesse's Bead Game.