Mathematics and Narrative
by Steven H. Cullinane
Notes from Log24.net suggested by
a
conference on mathematics and narrative
scheduled for July 2005 in Mykonos
Sunday, June 19, 2005
4:00 AM
ART WARS:
Darkness Visible
"Ed Rinehart [sic]
made a fortune painting canvases that were just one solid color.
He had his black period in which the canvas was totally black.
And then he had a blue period in which he was painting the canvas
blue. He was exhibited in top shows in New York, and his pictures
wound up in museums. I did a column in Scientific American
on minimal art, and I reproduced one of Ed Rinehart's black
paintings. Of course, it was just a solid square of pure
black. The publisher insisted on getting permission from the
gallery to reproduce it."
Fade to Black
"...that
ineffable constellation of talents that makes the player of rank: a
gift for conceiving abstract schematic possibilities; a sense of
mathematical poetry in the light of which the infinite chaos of
probability and permutation is crystallized under the pressure of
intense concentration into geometric blossoms; the ruthless focus of
force on the subtlest weakness of an opponent."
-- Trevanian, Shibumi
"'Haven't there been splendidly elegant colors in Japan since ancient times?'
'Even black has various subtle shades,' Sosuke nodded."
-- Yasunari Kawabata, The Old Capital
An Ad Reinhardt painting
described in the entry of
noon, November 9, 2004
is illustrated below.
Ad Reinhardt,
Abstract Painting, 1960-66.
Oil on canvas, 60 x 60 inches.
Solomon R. Guggenheim Museum
The viewer may need to tilt
the screen to see that this
painting is not uniformly black,
but is instead a picture of a
Greek cross, as described below.
"The
grid is a staircase to the Universal.... We could think about Ad
Reinhardt, who, despite his repeated insistence that 'Art is art,'
ended up by painting a series of... nine-square grids in which the
motif that inescapably emerges is a Greek cross. Greek Cross
There
is no painter in the West who can be unaware of the symbolic power of
the cruciform shape and the Pandora's box of spiritual reference that
is opened once one uses it." -- Rosalind Krauss,
Meyer Schapiro Professor
of Modern Art and Theory
at Columbia University
(Ph.D., Harvard U., 1969),
in "Grids"
Krauss |
Sunday, July 10, 2005
6:00 PM
| From Artemiadis's website: |
| 1986: | Elected Regular Member
of the Academy of Athens |
| 1999: | Vice President
of the Academy of Athens |
| 2000: | President
of the Academy of Athens |
"First of all, I'd like to
thank the Academy..."
(Remark attributed to Plato)
Thursday, June 23, 2005
3:00 PM
Mathematics and Metaphor
The current (June/July) issue of the Notices of the American Mathematical Society has two feature articles. The first, on the vulgarizer Martin Gardner, was dealt with here in a June 19 entry, Darkness Visible. The second is related to a letter of André Weil (pdf) that is in turn related to mathematician Barry Mazur's attempt to rewrite mathematical history and to vulgarize other people's research by using metaphors drawn, it would seem, from the Weil letter.
A Mathematical Lie
conjectures that Mazur's revising of history was motivated by a desire
to dramatize some arcane mathematics, the Taniyama conjecture, that
deals with elliptic curves and modular forms, two areas of mathematics
that have been known since the nineteenth century to be closely
related.
Mazur led author Simon Singh to believe that these
two areas of mathematics were, before Taniyama's conjecture of 1955,
completely unrelated --
"Modular forms and
elliptic equations live in completely different regions of the
mathematical cosmos, and nobody would ever have believed that there was
the remotest link between the two subjects." -- Simon Singh, Fermat's Enigma, 1998 paperback, p. 182
This is false. See Robert P. Langlands, review of Elliptic Curves, by Anthony W. Knapp, Bulletin of the American Mathematical Society, January 1994.
It now appears that Mazur's claim was in part motivated by a desire to
emulate the great mathematician André Weil's manner of speaking;
Mazur parrots Weil's "bridge" and "Rosetta stone" metaphors --
From Peter Woit's weblog, Feb. 10, 2005:
"The focus of Weil's letter is the analogy between number fields and
the field of algebraic functions of a complex variable. He describes
his ideas about studying this analogy using a third, intermediate
subject, that of function fields over a finite field, which he thinks
of as a 'bridge' or 'Rosetta stone.'"
In "A 1940 Letter of André Weil on Analogy in Mathematics," (pdf), translated by Martin H. Krieger, Notices of the A.M.S., March 2005, Weil writes that
"The purely algebraic theory of algebraic functions in any arbitrary
field of constants is not rich enough so that one might draw useful
lessons from it. The 'classical' theory (that is, Riemannian) of
algebraic functions over the field of constants of the complex numbers
is infinitely richer; but on the one hand it is too much so, and in the
mass of facts some real analogies become lost; and above all, it is too
far from the theory of numbers. One would be totally obstructed if
there were not a bridge between the two. And just as God defeats the devil: this bridge exists; it is the theory of the field of algebraic functions over a finite field of constants....
On the other hand, between the function fields and the 'Riemannian'
fields, the distance is not so large that a patient study would not
teach us the art of passing from one to the other, and to profit in the
study of the first from knowledge acquired about the second, and of the
extremely powerful means offered to us, in the study of the latter,
from the integral calculus and the theory of analytic functions. That
is not to say that at best all will be easy; but one ends up by
learning to see something there, although it is still somewhat
confused. Intuition makes much of it; I mean by this the faculty of
seeing a connection between things that in appearance are completely
different; it does not fail to lead us astray quite often. Be that as
it may, my work consists in deciphering a trilingual text {[cf. the Rosetta Stone]};
of each of the three columns I have only disparate fragments; I have
some ideas about each of the three languages: but I know as well there
are great differences in meaning from one column to another, for which
nothing has prepared me in advance. In the several years I have worked
at it, I have found little pieces of the dictionary. Sometimes I worked
on one column, sometimes under another."
Here is another statement of the Rosetta-stone metaphor, from Weil's translator, Martin H. Krieger, in the A.M.S. Notices of November 2004, "Some of What Mathematicians Do" (pdf):
"Weil refers to three columns, in analogy with the Rosetta Stone’s
three languages and their arrangement, and the task is to 'learn to
read Riemannian.' Given an ability to read one column, can you
find its translation in the other columns? In the first column
are Riemann’s transcendental results and, more generally, work in
analysis and geometry. In the second column is algebra, say
polynomials with coefficients in the complex numbers or in a finite
field. And in the third column is arithmetic or number theory and
combinatorial properties."
For greater clarity, see Armand Borel (pdf) on Weil's Rosetta stone,
where the three columns are referred to as Riemannian (transcendental),
Italian ("algebraico-geometric," over finite fields), and arithmetic
(i.e., number-theoretic).
From Fermat's Enigma, by Simon Singh, Anchor paperback, Sept. 1998, pp. 190-191:
Barry
Mazur: "On the one hand you have the elliptic world, and on the other
you have the modular world. Both these branches of mathematics
had been studied intensively but separately.... Than along comes the
Taniyama-Shimura conjecture, which is the grand surmise that there's a bridge between these two completely different worlds. Mathematicians love to build bridges."
Simon Singh: "The value of mathematical bridges
is enormous. They enable communities of mathematicians who have
been living on separate islands to exchange ideas and explore each
other's creations.... The great potential of the Taniyama-Shimura
conjecture was that it would connect two islands and allow them to
speak to each other for the first time. Barry Mazur thinks of the
Taniyama-Shimura conjecture as a translating device similar to the Rosetta stone.... 'It's as if you know one language and this Rosetta stone
is going to give you an intense understanding of the other language,'
says Mazur. 'But the Taniyama-Shimura conjecture is a Rosetta stone with a certain magical power.'"
If Mazur, who is scheduled to speak at a conference on Mathematics and Narrative this July, wants more material on stones with magical powers, he might consult The Blue Matrix and The Diamond Archetype.
Monday, July 25, 2005
web page
Ashay Dharwadker,
The Four-Color Theorem,
and Usenet Postings
by Steven H. Cullinane
July 25, 2005
In 2000,
Ashay Dharwadker claimed
to have proved the four-color theorem by an argument that involved the
Steiner system S(5,8,24). Since my own work involved a
connection, via
the MOG of R. T. Curtis, to this Steiner system, I wrote
a rough critique of Dharwadker's claim. That critique has led* to my being called
a pathological liar,
a sociopath,
a crank,
a nut,
someone who should be "locked up in gitmo,"
a moron,
a lunatic,
a fraudster,
paranoid,
an idiot,
stupid,
a pompous fool,
and
evil.
The above may be of some use to students of crankery.
* For the trail that leads from my critique of Dharwadker to the above list of epithets, see
Non-computer proof of 4 color Theorem,
2000 Oct. 13-Nov. 30,
sci.math, 23 posts
Open Directory Abuse,
2002 Oct. 2-Oct. 14,
sci.math, 8 posts
Open Directory Abuse,
2002 Oct. 2-Oct. 15,
comp.misc, 2 posts
Steven Cullinane is a Liar,
2002 Nov. 1-Nov.16,
geometry.research, 2 posts
Four-colour proof claim,
2003 Aug. 10-Sept.1,
sci.math, 9 posts
Proof of 4 colour theorem No computer!!!,
2003 Aug. 10-Aug. 20,
alt.sci.math.combinatorics, 8 posts
Steven Cullinane is a Crank,
2005 July 5-July 21
sci.math, 70 posts
Friday, July 22, 2005
5:55 AM
By Their Fruits Today's birthdays:
Don Henley and Willem Dafoe
Related material:
Mathematics and Narrative,
Crankbuster.
"And the fruit is rotten;
the serpent's eyes shine
as he wraps around the vine
in the Garden of Allah.
"
Friday, August 19, 2005
2:00 PM
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
Saturday, August 20, 2005
2:07 PM
Truth vs. Bullshit
Background:
For an essay on the above topic
from this week's New Yorker,
click on the box below.
| Representing truth: 
Rebecca Goldstein | Representing bullshit: 
Apostolos Doxiadis |
Goldstein's truth:
Gödel was a Platonist who believed in objective truth.
See Rothstein's review of Goldstein's new book Incompleteness.
| Doxiadis's bullshit:
Gödel, along with Darwin, Marx, Nietzsche, Freud, Einstein, and Heisenberg, destroyed a tradition of certainty that began with Plato and Euclid. |
"Examples are the stained-glass
windows of knowledge." -- Nabokov
Monday, August 22, 2005
4:07 PM
The Hole
Part I: Mathematics and Narrative
Apostolos Doxiadis on last month's conference on "mathematics and narrative"--
Doxiadis is describing how talks by two noted mathematicians were related to
"... a sense of a 'general theory bubbling up' at the meeting... a general theory of the deeper relationship of mathematics to narrative.... "
Doxiadis says both talks had "a big hole in the middle."
"Both began by saying something like: 'I believe there is an important connection between story and mathematical thinking. So, my talk has two parts. [In one part] I’ll tell you a few things about proofs. [And in the other part] I’ll tell you about stories.' .... And in both talks it was in fact implied by a variation of the post hoc propter hoc, the principle of consecutiveness implying causality, that the two parts of the lectures were intimately related, the one somehow led directly to the other."
"And the hole?"
"This was exactly at the point of the link... [connecting math and narrative]... There is this very well-known Sidney Harris cartoon... where two huge arrays of formulas on a blackboard are connected by the sentence 'THEN A MIRACLE OCCURS.' And one of the two mathematicians standing before it points at this and tells the other: 'I think you should be more explicit here at step two.' Both... talks were one half fascinating expositions of lay narratology-- in fact, I was exhilarated to hear the two most purely narratological talks at the meeting coming from number theorists!-- and one half a discussion of a purely mathematical kind, the two parts separated by a conjunction roughly synonymous to 'this is very similar to this.' But the similarity was not clearly explained: the hole, you see, the 'miracle.' Of course, both [speakers]... are brilliant men, and honest too, and so they were very clear about the location of the hole, they did not try to fool us by saying that there was no hole where there was one."
Part II: Possible Worlds
"At times, bullshit can only be countered with superior bullshit."
-- Norman Mailer
Many Worlds and Possible Worlds in Literature and Art, in Wikipedia:
"The concept of possible worlds dates back to at least Leibniz who in his Théodicée tries to justify the apparent imperfections of the world by claiming that it is optimal among all possible worlds. Voltaire satirized this view in his picaresque novel Candide....
Borges' seminal short story El jardín de senderos que se bifurcan ("The Garden of Forking Paths") is an early example of many worlds in fiction."
Background:
Modal Logic in Wikipedia
Possible Worlds in Wikipedia
Possible-Worlds Theory, by Marie-Laure Ryan
(entry for The Routledge Encyclopedia of Narrative Theory)
The God-Shaped Hole
Part III: Modal Theology "'What is this Stone?' Chloe asked....
'...It is told that, when the Merciful One made the worlds, first of all He created that Stone and gave it to the Divine One whom the Jews call Shekinah, and as she gazed upon it the universes arose and had being.'"
-- Many Dimensions, by Charles Williams, 1931 (Eerdmans paperback, April 1979, pp. 43-44)
"The lapis was thought of as a unity and therefore often stands for the prima materia in general."
--
Aion, by C. G. Jung, 1951 (Princeton paperback, 1979, p. 236)
"Its discoverer was of the opinion that he had produced the equivalent of the primordial protomatter which exploded into the Universe."
"We symbolize logical necessity with the box ( ) and logical possibility with the diamond ( )." -- Keith Allen Korcz 
"The possibilia that exist, and out of which the Universe arose, are located in a necessary being...." -- Michael Sudduth, Notes on
God, Chance, and Necessity
by Keith Ward, Regius Professor of Divinity, Christ Church College, Oxford (the home of Lewis Carroll) |
Page created May 27, 2005;
last modified Dec. 29, 2006.
