Google Maps image
of the isle of Delos,
birthplace of Apollo:
“I faced myself that day with
the nonplused apprehension
of someone who has
come across a vampire
and has no crucifix in hand.”
— Joan Didion, “On Self-Respect,”
in Slouching Towards Bethlehem
“For every kind of vampire,
there is a kind of cross.”
— Thomas Pynchon,
Gravity’s Rainbow
Related material:
” ‘I know what it is you last saw,’ she said; ‘for that is also in my mind. Do not be afraid! But do not think that only by singing amid the trees, nor even by the slender arrows of elvenbows, is this land of Lothlórien maintained and defended against the Enemy. I say to you, Frodo, that even as I speak to you, I perceive the Dark Lord and know his mind, or all his mind that concerns the Elves. And he gropes ever to see me and my thought. But still the door is closed!’
She lifted up her white arms, and spread out her hands towards the East in a gesture of rejection and denial. Eärendil, the Evening Star, most beloved of the Elves, shone clear above. So bright was it that the figure of the Elven-lady cast a dim shadow on the ground. Its ray glanced upon a ring about her finger; it glittered like polished gold overlaid with silver light, and a white stone in it twinkled as if the Even-star had come to rest upon her hand. Frodo gazed at the ring with awe; for suddenly it seemed to him that he understood.
‘Yes,’ she said, divining his thought, ‘it is not permitted to speak of it, and Elrond could not do so. But it cannot be hidden from the Ring-Bearer, and one who has seen the Eye. Verily it is in the land of Lórien upon the finger of Galadriel that one of the Three remains. This is Nenya, the Ring of Adamant, and I am its keeper.’ ”
— J. R. R. Tolkien, The Lord of the Rings
The last 3 entries,
as well as
Mathematics and Narrative
“How much story
do you want?”
— George Balanchine
Related material:
Plato, Pegasus, and
the Evening Star,
Mathematics and Narrative,
“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
A Serious Position
“‘Teitelbaum,’ in German,
is ‘date palm.'”
— Generations, Jan. 2003
“In Hasidism, a mystical brand
of Orthodox Judaism, the grand rabbi
is revered as a kinglike link to God….”
— Today’s New York Times obituary
of Rabbi Moses Teitelbaum,
who died on April 24, 2006
(Easter Monday in the
Orthodox Church)
From Wikipedia, an unsigned story:
“In 1923 Alfred Teitelbaum and his brother Wacław changed their surnames to Tarski, a name they invented because it sounded very Polish, was simple to spell and pronounce, and was unused. (Years later, he met another Alfred Tarski in northern California.) The Tarski brothers also converted to Roman Catholicism, the national religion of the Poles. Alfred did so, even though he was an avowed atheist, because he was about to finish his Ph.D. and correctly anticipated that it would be difficult for a Jew to obtain a serious position in the new Polish university system.”
A Circle of Quiet
From the Harvard Math Table page:
“No Math table this week. We will reconvene next week on March 14 for a special Pi Day talk by Paul Bamberg.”
| Some friends of mine are in this band. They’re playing in a bar on Diversey, way down the bill, around… I said I’d be there. Great. Imaginary number? It’s a math joke. |
From the April 2006 Notices of the American Mathematical Society, a footnote in a review by Juliette Kennedy (pdf) of Rebecca Goldstein’s Incompleteness:
4 There is a growing literature in the area of postmodern commentaries of [sic] Gödel’s theorems. For example, Régis Debray has used Gödel’s theorems to demonstrate the logical inconsistency of self-government. For a critical view of this and related developments, see Bricmont and Sokal’s Fashionable Nonsense [13]. For a more positive view see Michael Harris’s review of the latter, “I know what you mean!” [9]….
[9] MICHAEL HARRIS, “I know what you mean!,” http://www.math.jussieu.fr/~harris/Iknow.pdf.
[13] ALAN SOKAL and JEAN BRICMONT, Fashionable Nonsense, Picador, 1999.
Following the trail marked by Ms. Kennedy, we find the following in Harris’s paper:
“Their [Sokal’s and Bricmont’s] philosophy of mathematics, for instance, is summarized in the sentence ‘A mathematical constant like
doesn’t change, even if the idea one has about it may change.’ ( p. 263). This claim, referring to a ‘crescendo of absurdity’ in Sokal’s original hoax in Social Text, is criticized by anthropologist Joan Fujimura, in an article translated for IS*. Most of Fujimura’s article consists of an astonishingly bland account of the history of non-euclidean geometry, in which she points out that the ratio of the circumference to the diameter depends on the metric. Sokal and Bricmont know this, and Fujimura’s remarks are about as helpful as FN’s** referral of Quine’s readers to Hume (p. 70). Anyway, Sokal explicitly referred to “Euclid’s pi”, presumably to avoid trivial objections like Fujimura’s — wasted effort on both sides.32 If one insists on making trivial objections, one might recall that the theorem
that p is transcendental can be stated as follows: the homomorphism Q[X] –> R taking X to
More interestingly, one can ask what kind of object
as a creation rather than a discovery. But Platonism does correspond to the familiar experience that there is something about mathematics, and not just about other mathematicians, that precisely doesn’t let us get away with saying ‘évidemment’!38
32 There are many circles in Euclid, but no pi, so I can’t think of any other reason for Sokal to have written ‘Euclid’s pi,’ unless this anachronism was an intentional part of the hoax. Sokal’s full quotation was ‘the
33 This is not mere sophistry: the construction of models over number fields actually uses arguments of this kind. A careless construction of the equations defining modular curves may make it appear that pi is included in their field of scalars.
34 Unless you claim, like the present French Minister of Education [at the time of writing, i.e. 1999], that real numbers exist in nature, while imaginary numbers were invented by mathematicians. Thus
35 Cf. M. Kline, Mathematics The Loss of Certainty, p. 324.
36 Compare Morris Hirsch’s remarks in BAMS April 94.
37 IS*, p. 38, footnote 26. Weinberg’s remarks are contained in his article “Sokal’s Hoax,” in the New York Review of Books, August 8, 1996.
38 Metaphors from virtual reality may help here.”
* Earlier defined by Harris as “Impostures Scientifiques (IS), a collection of articles compiled or commissioned by Baudouin Jurdant and published simultaneously as an issue of the journal Alliage and as a book by La Découverte press.”
** Earlier defined by Harris as “Fashionable Nonsense (FN), the North American translation of Impostures Intellectuelles.”
What is the moral of all this French noise?
Perhaps that, in spite of the contemptible nonsense at last summer’s Mykonos conference on mathematics and narrative, stories do have an important role to play in mathematics — specifically, in the history of mathematics.
Despite his disdain for Platonism, exemplified in his remarks on the noteworthy connection of pi with the zeta function in the formula given above, Harris has performed a valuable service to mathematics by pointing out the excellent historical work of Catherine Goldstein. Ms. Goldstein has demonstrated that even a French nominalist can be a first-rate scholar. Her essay on circles that Harris cites in a French version is also available in English, and will repay the study of those who, like Barry Mazur and other Harvard savants, are much too careless with the facts of history. They should consult her “Stories of the Circle,” pp. 160-190 in A History of Scientific Thought, edited by Michel Serres, Blackwell Publishers (December 1995).
For the historically-challenged mathematicians of Harvard, this essay would provide a valuable supplement to the upcoming “Pi Day” talk by Bamberg.
For those who insist on limiting their attention to mathematics proper, and ignoring its history, a suitable Pi Day observance might include becoming familiar with various proofs of the formula, pictured above, that connects pi with the zeta function of 2. For a survey, see Robin Chapman, Evaluating Zeta(2) (pdf). Zeta functions in a much wider context will be discussed at next May’s politically correct “Women in Mathematics” program at Princeton, “Zeta Functions All the Way” (pdf).
“In The Painted Word, a rumination on the state of American painting in the 1970s, Tom Wolfe described an epiphany….”
— Peter Berkowitz, “Literature in Theory”
“I had an epiphany.”
— Apostolos Doxiadis, organizer of last summer’s conference on mathematics and narrative. See the Log24 entry of 1:06 PM last August 23 and the four entries that preceded it.
“… das Durchleuchten des ewigen Glanzes des ‘Einen’ durch die materielle Erscheinung“
— A definition of beauty from Plotinus, via Werner Heisenberg
“By groping toward the light we are made to realize how deep the darkness is around us.”
— Arthur Koestler, The Call Girls: A Tragi-Comedy, Random House, 1973, page 118, quoted in The Shining of May 29
“Perhaps we are meant to see the story as a cubist retelling of the crucifixion….”
— Adam White Scoville, quoted in Cubist Crucifixion, on Iain Pears’s novel, An Instance of the Fingerpost
From James A. Michener‘s The Source:
“Trouble started in a quarter that neither Uriel nor Zadok could have foreseen. For many generations the wiser men of Zadok’s clan had worshipped El-Shaddai with the understanding that whereas Canaanites and Egyptians could see their gods directly, El-Shaddai was invisible and inhabited no specific place. Unequivocally the Hebrew patriarchs had preached this concept and the sager men of the clans accepted it, but to the average Hebrew who was not a philosopher the theory of a god who lived nowhere, who did not even exist in corporeal form, was not easy to comprehend. Such people were willing to agree with Zadok that their god did not live on this mountain– the one directly ahead– but they suspected that he did live on some mountain nearby, and when they said this they pictured an elderly man with a white beard who lived in a proper tent and whom they might one day see and touch. If questioned, they would have said that they expected El-Shaddai to look much like their father Zadok, but with a longer beard, a stronger voice, and more penetrating eyes.
Now, as these simpler-minded Hebrews settled down outside the walls of Makor, they began to see Canaanite processions leave the main gate and climb the mountain to the north, seeking the high place where Baal lived, and they witnessed the joy which men experienced when visiting their god, and the Hebrews began in subtle ways and easy steps to evolve the idea that Baal, who obviously lived in a mountain, and El-Shaddai, who was reported to do so, must have much in common. Furtively at first, and then openly, they began to climb the footpath to the place of Baal, where they found a monolith rising from the highest point of rock. Here was a tangible thing they could comprehend, and after much searching along the face of the mountain, a group of Hebrew men found a straight rock of size equal to the one accorded Baal, and with much effort they dragged it one starless night to the mountain top, where they installed it not far from the home of Baal….”
The above monolith is perhaps more
closely related to El-Shaddai than to
Madonna, Grammy Night, and Baal.
It reflects my own interests
(Mathematics and Narrative)
and those of Martin Buber
(Jews on Fiction):
|
— Harrison Ford in
“Indiana Jones and the
Temple of Doom”
In today’s online New York Times:
(1) A review of pop-archaeology TV,
“Digging for the Truth,”
(2) a Sunday news story,
“Looking for the Lie,”
(3) and a profile,
“Storyteller in the Family.”
From (1):
“The season premiere ‘Digging for the Truth: The Real Temple of Doom,’ showed Mr. Bernstein in South America, exploring tunnels….”
From (2):
“… scientists are building a cognitive theory of deception to show what lying looks like….”
From (3):
“I did feel one had to get not just the facts, but the emotional underpinnings.”
— and Mathematics and Narrative.
See also Saturday’s entry,
Raiders of the Lost Matrix,
for logic as an aid in
detecting lies.
Jews on Fiction
See Tony Kushner and E.L. Doctorow in today’s New York Times, Rebecca Goldstein’s talk from last summer’s Mykonos conference on mathematics and narrative, and Martin Buber on the Bible.
Mathematics and Narrative
Rebecca Goldstein, Mathematics and the Character of Tragedy:
“It was Plato who best expressed– who veritably embodied– the tension between the narrative arts and mathematics.”
In 1946, Robert Graves published “King Jesus, an historical novel based on the theory and Graves’ own historical conjecture that Jesus was, in fact, the rightful heir to the Israelite throne… written while he was researching and developing his ideas for The White Goddess.”
In 1948, C. S. Lewis finished the first draft of The Lion, The Witch, and The Wardrobe, a novel in which one of the main characters is “the White Witch.”
In 1948, Robert Graves published The White Goddess.
In 1949, Robert Graves published Seven Days in New Crete [also titled Watch the North Wind Rise], “a novel about a social distopia in which Goddess worship is (once again?) the dominant religion.”
Lewis died on November 22, 1963, the day John F. Kennedy was killed.
Related material:
Log24, December 10, 2005
Graves died on December 7 (Pearl Harbor Day), 1985.
Related material:
Log24, December 7, 2005, and
Log24, December 11, 2005
Jesus died, some say, on April 7 in the year 30 A.D.
Related material:
Art Wars, April 7, 2003:
Geometry and Conceptual Art,Eight is a Gate, and
Plato’s Diamond
— Motto of
Plato’s Academy
“How much story do you want?”
— George Balanchine
Intelligence: A file on James Jesus Angleton at namebase.org, a site run by Daniel Brandt.
Counterintelligence: Hollywood on James Jesus Angleton–
"From a screenplay by 'Forrest Gump' screenwriter Eric Roth, 'The Good Shepherd' tells the mostly true story of James Wilson (a character reported to be based on legendary CIA spymaster James Jesus Angleton, and played in the film by Matt Damon), one of the founding members of the Central Intelligence Agency. Beginning as an scholar at Yale, the film follows Wilson as he is recruited to join the secret Skull and Bones fraternity, a brotherhood and breeding ground for future world leaders, where his acute mind, spotless reputation and sincere belief in the American way of life render him a prime candidate for a career in intelligence."
— Edward Havens, FilmJerk.com, 8/30/2005
The Forrest Gump Award goes to Good Will Hunting* for this choice of roles.
Counterintelligence
illustrated:
Forrest Gump (l.)
and JFK (r.)
* See Log24, April 4, 2003, Mathematics Awareness Month. For some related material, see Mathematics and Narrative.
For St. Andrew’s Day
“The miraculous enters…. When we investigate these problems, some fantastic things happen….”
— John H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, preface to first edition (1988)
The remarkable Mathieu group M24, a group of permutations on 24 elements, may be studied by picturing its action on three interchangeable 8-element “octads,” as in the “Miracle Octad Generator” of R. T. Curtis.
A picture of the Miracle Octad Generator, with my comments, is available online.
Related material:
Mathematics and Narrative.
Wikipedia on the tesseract:
Robert A. Heinlein in Glory Road:
Johnny Cash: “And behold, a white horse.”
On The Last Battle, a book in the Narnia series by C. S. Lewis:
Lewis said in “The Weight of Glory”—
On enchantments that need to be broken:
See the description of the Eater of Souls in Glory Road and of Scientism in
"…his eyes ranged the Consul's books disposed quite neatly… on high shelves around the walls: Dogme et Ritual de la Haute Magie, Serpent and Siva Worship in Central America, there were two long shelves of this, together with the rusty leather bindings and frayed edges of the numerous cabbalistic and alchemical books, though some of them looked fairly new, like the Goetia of the Lemegaton of Solomon the King, probably they were treasures, but the rest were a heterogeneous collection…."
— Malcolm Lowry, Under the Volcano, Chapter VI
"… when Saul does reach for a slim leather-bound volume Eliza cannot help but feel that something momentous is about to happen. There is care in the way he carries the book on the short journey from its shelf, as if it were constructed not of leather and parchment but of flesh and blood….
"Otzar Eden HaGanuz," Saul says. "The Hidden Eden. In this book, Abulafia describes the process of permutation…. Once you have mastered it, you will have mastered words, and once you have mastered words, you will be ready to receive shefa."
"In the Inner Game, we call the Game Dhum Welur, the Mind of God."
— The Gameplayers of Zan, a novel featuring games based on cellular automata
"Regarding cellular automata, I'm trying to think in what SF books I've seen them mentioned. Off the top of my head, only three come to mind:
The Gameplayers of Zan M.A. Foster
Permutation City Greg Egan
Glory Season David Brin"
— Jonathan L. Cunningham, Usenet
"If all that 'matters' are fundamentally mathematical relationships, then there ceases to be any important difference between the actual and the possible. (Even if you aren't a mathematical Platonist, you can always find some collection of particles of dust to fit any required pattern. In Permutation City this is called the 'logic of the dust' theory.)….
… Paul Durham is convinced by the 'logic of the dust' theory mentioned above, and plans to run, just for a few minutes, a complex cellular automaton (Permutation City) started in a 'Garden of Eden' configuration — one which isn't reachable from any other, and which therefore must have been the starting point of a simulation…. I didn't understand the need for this elaborate set-up, but I guess it makes for a better story than 'well, all possible worlds exist, and I'm going to tell you about one of them.'"
— Danny Yee, review of Permutation City
"Y'know, I never imagined the competition version involved so many tricky permutations."
— David Brin, Glory Season, 1994 Spectra paperback, p. 408
|
Figure 2
"… matter is consciousness expressed in the intermixing of force and form, but so heavily structured and constrained by form that its behaviour becomes describable using the regular and simple laws of physics. This is shown in Figure 2. |
Figure 3
"This quaternary is a Kabbalistic representation of God-the-Knowable, in the sense that it the most abstract representation of God we are capable of comprehending…. — A Depth of Beginning: Notes on Kabbalah by Colin Low (pdf) |
See also
Cognitive Blending and the Two Cultures,
Mathematics and Narrative,
Deep Game,
and the previous entry.
From Google News this afternoon–
See also the previous entry.
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."
"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 a 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)
— Many Dimensions, by Charles Williams, 1931 (Eerdmans paperback, April 1979, pp. 43-44)
— 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  ).
"The possibilia that exist,
— Michael Sudduth, |
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
Part I: The Light
The Shining of May 29
and
Diamond Theory
Part II: The Darkness
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.
From today's New York Times:
From the Associated Press,
filed at 4:34 PM ET July 27, 2005:
"Held once described his work this way: 'Historically, the priests and wise men believed that it was the artist's job to make images of heaven and hell believable, even though nobody had experienced these places.'
'Today,' he went on, 'scientists talk about vast worlds and universes that the senses cannot experience. The purpose of the nonobjective artist is to create these images.'"
"Most modern men do not believe in hell because they have not been there."
— Review of Malcolm Lowry's novel Under the Volcano (1947)
Related material:
Hollywood images:
And from Mathematics and Narrative:
By Their Fruits
Today's (July 22) birthdays:
Don Henley and Willem Dafoe
Related material:
"And the fruit is rotten;
the serpent's eyes shine
as he wraps around the vine
in the Garden of Allah."
Today's birthdays:
Don Henley and Willem Dafoe
Related material:
"And the fruit is rotten;
the serpent's eyes shine
as he wraps around the vine
in the Garden of Allah."
“Do not underestimate Evil Cullinane’s plan for World Domination! http://www.log24.com now shows that he has crossed over to the dark side, making sacrifices to the Ancient Hindu Goddess ‘Kalli’ to ward off our attacks! ‘Kalli’-nane will soon appear as the top result on every Google search.
Soon, all young mathematicians will be hypnotised by his dark diamonds of falsehood. At least, that’s his plan. But wait, who’s that brilliant mathematician who shines the light right through Cullinane’s fraud and exposes him to the whole world?! Crankbuster saves the day! (applause)”
Related material:
“And if the band you’re in
starts playing different tunes
I’ll see you on
the dark side of the moon.”
— Billy Graham Evangelistic Association,
according to messiahpage.com
"… 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
(that is to say, a finite number
of elements: also said to be a Galois
field, or earlier 'Galois imaginaries'
because Galois first defined them
and studied them….)"
— André Weil, 1940 letter to his sister,
Simone Weil, alias Simone Galois
(see previous entry)
Related material:
Billy Graham and the City:
A Later Look at His Words
— New York Times, June 24, 2005
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.
Recommended geometry:
Click on picture to enlarge.
Related material:
Birthday Links
Today’s birthdays:
Gene Wilder and Adrienne Barbeau.
For Gene:
A discussion of Frankenstein as
The Modern Prometheus at
Mathematics and Narrative.
For Adrienne:
Chinese Arithmetic.
Logos Alogos
by S. H. Cullinane
"To a mathematician, mathematical entities have their own existence, they habitate spaces created by their intention. They do things, things happen to them, they relate to one another. We can imagine on their behalf all sorts of stories, providing they don't contradict what we know of them. The drama of the diagonal, of the square…"
— Dennis Guedj, abstract of "The Drama of Mathematics," a talk to be given this July at the Mykonos conference on mathematics and narrative.
For the drama of the diagonal of the square, see
The Turning
Readers who have an Amazon.com account may view book pages relevant to the previous entry. See page 77 of The Way We Think, by Fauconnier and Turner (Amazon search term = Meno). This page discusses both the Pythagorean theorem and Plato's diamond figure in the Meno, but fails to "blend" these two topics. See also page 53 of The History of Mathematics, by Roger Cooke (first edition), where these two topics are in fact blended (Amazon search term = Pythagorean). The illustration below is drawn from the Cooke book.
Cooke demonstrates how the Pythagorean theorem might have been derived by "blending" Plato's diamond (left) with the idea of moving the diamond's corners (right).
The previous entry dealt with a conference on mathematics and narrative. Above is an example I like of mathematics…. Here is an example I like of narrative:
Kate felt quite dizzy. She didn't know exactly what it was
that had just happened, but she felt pretty damn certain that
it was the sort of experience that her mother would not have
approved of on a first date.
"Is this all part of what we have to do to go to Asgard?"
she said. "Or are you just fooling around?"
"We will go to Asgard...now," he said.
At that moment he raised his hand as if to pluck an apple,
but instead of plucking he made a tiny, sharp turning movement.
The effect was as if he had twisted the entire world through a
billionth part of a billionth part of a degree. Everything
shifted, was for a moment minutely out of focus, and then
snapped back again as a suddenly different world.
— Douglas Adams, The Long Dark Tea-Time of the Soul
And here is a blend of the concepts "Asgard" and "conference":
"Asgard
During the Interuniverse Society conference,
a bridge was opened to Valhalla…."
Bifrost
In Norse myth, the rainbow bridge
that connected Earth to Asgard,
home of the gods. It was extended
to Tellus Tertius during the
Interuniverse Society conference"
— From A Heinlein Concordance
— Front page picture from a
local morning newspaper published
today, Wednesday, May 25, 2005
“Poetry is a satisfying of
the desire for resemblance….
If resemblance is described as
a partial similarity between
two dissimilar things,
it complements and reinforces
that which the two dissimilar things
have in common.
It makes it brilliant.”
— Wallace Stevens,
“Three Academic Pieces” in
The Necessary Angel (1951)
Two dissimilar things:
1. A talk to be given at a conference on “Mathematics and Narrative” in Mykonos in July:
Mark Turner,
“The Role of Narrative Imagining in Blended Mathematical Concepts” —
Abstract:
“The Way We Think (Gilles Fauconnier and Mark Turner; Basic Books, 2002) presents a theory of conceptual integration, or “blending,” as a basic mental operation. See http://blending.stanford.edu. This talk will explore some ways in which narrative imagining plays a role in blended mathematical concepts.”
2. An application of the “conceptual blending” of Fauconnier and Turner to some journal entries of 2004: Cognitive Blending and the Two Cultures.
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