Log24

Wednesday, September 26, 2018

Analogies Between Analogies

Filed under: General — m759 @ 8:05 PM

On the new Netflix series "Maniac" —

(Spoiler alert)

"The treatment Owen and Annie sign up for promises to fix
its subjects’ brains with just three little pills—A, B, and C—
administered one after another over the span of three days.
The first forces you to relive your trauma;
the second exposes your blind spots; and
the third pill forces a confrontation."

— Kara Weisenstein at vice.com, Sept. 26, 2018, 12:19 PM

See also, from Log24 earlier 

A.  Monday — Mathematics as Art
B.  Tuesday — Trinity and Denkraum  Revisited
C.  Wednesday — Trinity Tale

'Out of nothing' opening of 'Maniac' at Netflix

Friday, June 29, 2018

Analogies Between Analogies:

Filed under: General — m759 @ 3:33 AM

Literary Meditation for the Feast of  SS Peter and Paul

Background McLuhan on analogy.

See a publication offering facsimiles of the original 4×6 cards
of John Shade's "Pale Fire," as Nabokov described them.

Regarding these card proportions, note that 4/6 = 333/500, approximately —
the proportions of the text box in a post from yesterday.

"Continue a search for thirty-three and three" — Katherine Neville.

These rather pointless, but vaguely poetic, analogies were suggested by

  • Yesterday morning's "The Corrections," a post
    featuring spider ballooning and a dead poet, and
     
  • "Blue Dream," a post of Feb. 11, 2006.

Tuesday, January 10, 2017

Analogies Between Analogies

Filed under: General — m759 @ 9:27 PM

"Do you like puzzles?" — J. K. Simmons

See also Sunday's post "A Theory of Everything"
and an obituary in this evening's New York Times .

Wednesday, August 10, 2016

Analogies Test

Filed under: General — Tags: — m759 @ 12:00 AM

Obituary for Wilford Stanton Miller, author in 1926
of the Miller Analogies Test  —  
         

Marshall McLuhan writing to Ezra Pound on Dec. 21, 1948—

"The American mind is not even close to being amenable
to the ideogram principle as yet.  The reason is simply this.
America is 100% 18th Century. The 18th century had
chucked out the principle of metaphor and analogy—
the basic fact that as A is to B so is C to D.  AB:CD.   
It can see AB relations.  But relations in four terms are still
verboten.  This amounts to deep occultation of nearly all
human thought for the U.S.A.

I am trying to devise a way of stating this difficulty as it exists.  
Until stated and publicly recognized for what it is, poetry and
the arts can’t exist in America."

A line for W. S. Miller, taken from "Annie Hall" —

"You know nothing of my work."

Tuesday, August 9, 2016

Analogies Between Analogies (continued)

Filed under: General — m759 @ 11:30 PM

2:3 :: 4:6

Midrash —

Piano keys with C, E, G as 4, 5, 6

Notes and frequency ratios

Friday, December 21, 2012

Analogies*

Filed under: General,Geometry — Tags: — m759 @ 4:30 PM

The Moore correspondence may be regarded
as an analogy between the 35 partitions of an
8-set into two 4-sets and the 35 lines in the
finite projective space PG(3,2).

Closely related to the Moore correspondence
is a correspondence (or analogy) between the
15 2-subsets of a 6-set and the 15 points of PG(3,2).

An analogy between  the two above analogies
is supplied by the exceptional outer automorphism of S6.
See…

The 2-subsets of a 6-set are the points of a PG(3,2),
Picturing outer automorphisms of  S6, and
A linear complex related to M24.

(Background: InscapesInscapes III: PG(2,4) from PG(3,2),
and Picturing the smallest projective 3-space.)

* For some context, see Analogies and
  "Smallest Perfect Universe" in this journal.

Monday, May 27, 2019

But Seriously . . .

Filed under: General — Tags: — m759 @ 9:32 PM

McLuhan on Analogy.

I prefer the simple "four dots" figure
of the double colon:

For those who prefer stranger analogies . . .

Actors from "The Eiger Sanction" —

Doctor Strange on Mount Everest —

Dr. Strange at beyondtheopposites.com on 2016/12/02

See as well this  journal on the above Strange date, 2016/12/02,
in posts tagged Lumber Room.

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.

Thursday, September 15, 2016

Metaphysics at Notre Dame

Filed under: General,Geometry — Tags: , — m759 @ 11:07 PM

Recommended reading —

"When Analogies Fail," by Alexander Stern,
a doctoral candidate in philosophy at Notre Dame, in
The Chronicle of Higher Education  online September 11, 2016.

Related material —

That same Alexander Stern in this  journal on April 17, 2016:

See also the eightfold cube in the previous post,
Metaphysics at Scientific American:

Friday, August 26, 2016

Wolfe vs. Chomsky

Filed under: General — Tags: , — m759 @ 12:00 AM

1.  Tom Wolfe has a new book on Chomsky, "The Kingdom of Speech."

2.  This suggests a review of a post of Aug. 11, 2014, Syntactic/Symplectic.

To paraphrase Wittgenstein, sentence 1 above is about "correlating in real life"
(cf. Crooked House and Wolfe's From Bauhaus to Our House ), and may be 
compared to sentence 2 above, which links to a sort of "correlating in
mathematics" that is a particular example of the more general sort of
mathematical correlating mentioned by Wittgenstein in 1939.

Sunday, May 29, 2016

The Ideogram Principle …

Filed under: General,Geometry — Tags: — m759 @ 4:23 PM

According to McLuhan

Marshall McLuhan writing to Ezra Pound on Dec. 21, 1948—

"The American mind is not even close to being amenable
to the ideogram principle as yet.  The reason is simply this.
America is 100% 18th Century. The 18th century had
chucked out the principle of metaphor and analogy—
the basic fact that as A is to B so is C to D.  AB:CD.   
It can see AB relations.  But relations in four terms are still
verboten.  This amounts to deep occultation of nearly all
human thought for the U.S.A.

I am trying to devise a way of stating this difficulty as it exists.  
Until stated and publicly recognized for what it is, poetry and
the arts can’t exist in America."

For context, see Cameron McEwen,
"Marshall McLuhan, John Pick, and Gerard Manley Hopkins."
(Renascence , Fall 2011, Vol. 64 Issue 1, 55-76)

A relation in four terms

A : B  ::  C : D   as   Model : Crutch  ::  Metaphor : Ornament —

See also Dueling Formulas and Symmetry.

Thursday, April 14, 2016

Banach Revisited

Filed under: General — m759 @ 12:00 PM

A  1960  analogy by Max Black

"Those who see a model as a mere crutch
are like those who consider metaphor
a mere decoration or ornament."

This suggests a search for "Analogies between Analogies" —

“A mathematician is a person who can find analogies
between theorems; a better mathematician is one who
can see analogies between proofs and the best
mathematician can notice analogies between theories.
One can imagine that the ultimate mathematician is one
who can see analogies between analogies.”

— Stefan Banach, according to MacTutor.

Wednesday, December 23, 2015

Analogy

Filed under: General — m759 @ 2:12 AM

"An analogy between mathematics and religion is apposite."

Harvard Magazine  review by Avner Ash of
     Mathematics without Apologies
    
(Princeton University Press, January 18, 2015)

See as well Analogies in this  journal.

Friday, March 14, 2014

Quotation

Filed under: General — Tags: , — m759 @ 1:09 PM

Edward Frenkel in a vulgar and stupid
LA Times  opinion piece, March 2, 2014 —

"In the words of the great mathematician Henri Poincare, mathematics is valuable because 'in binding together elements long-known but heretofore scattered and appearing unrelated to one another, it suddenly brings order where there reigned apparent chaos.' "

My attempts to find the source of these alleged words of Poincaré were fruitless.* Others may have better luck.

The search for Poincaré's words did, however, yield the following passage —

HENRI POINCARÉ
THE FUTURE OF MATHEMATICS

If a new result is to have any value, it must unite elements long since known, but till then scattered and seemingly foreign to each other, and suddenly introduce order where the appearance of disorder reigned. Then it enables us to see at a glance each of these elements in the place it occupies in the whole. Not only is the new fact valuable on its own account, but it alone gives a value to the old facts it unites. Our mind is frail as our senses are; it would lose itself in the complexity of the world if that complexity were not harmonious; like the short-sighted, it would only see the details, and would be obliged to forget each of these details before examining the next, because it would- be incapable of taking in the whole. The only facts worthy of our attention are those which introduce order into this complexity and so make it accessible to us.

Mathematicians attach a great importance to the elegance of their methods and of their results, and this is not mere dilettantism. What is it that gives us the feeling of elegance in a solution or a demonstration? It is the harmony of the different parts, their symmetry, and their happy adjustment; it is, in a word, all that introduces order, all that gives them unity, that enables us to obtain a clear comprehension of the whole as well as of the parts. But that is also precisely what causes it to give a large return; and in fact the more we see this whole clearly and at a single glance, the better we shall perceive the analogies with other neighbouring objects, and consequently the better chance we shall have of guessing the possible generalizations. Elegance may result from the feeling of surprise caused by the unlooked-for occurrence together of objects not habitually associated. In this, again, it is fruitful, since it thus discloses relations till then unrecognized. It is also fruitful even when it only results from the contrast between the simplicity of the means and the complexity of the problem presented, for it then causes us to reflect on the reason for this contrast, and generally shows us that this reason is not chance, but is to be found in some unsuspected law. ….

HENRI POINCARÉ
L'AVENIR DES MATHÉMATIQUES

Si un résultat nouveau a du prix, c'est quand en reliant des éléments connus depuis longtemps, mais jusque-là épars et paraissant étrangers les uns aux autres, il introduit subitement l'ordre là où régnait l'apparence du désordre. Il nous permet alors de voir d'un coup d'œil chacun de ces éléments et la place qu'il occupe dans l'ensemble. Ce fait nouveau non-seulement est précieux par lui-même, mais lui seul donne leur valeur à tous les faits anciens qu'il relie. Notre esprit est infirme comme le sont nos sens; il se perdrait dans la complexité du monde si cette complexité n'était harmonieuse, il n'en verrait que les détails à la façon d'un myope et il serait forcé d'oublier chacun de ces détails avant d'examiner le suivant, parce qu'il serait incapable de tout embrasser. Les seuls faits dignes de notre attention sont ceux qui introduisent de l'ordre dans cette complexité et la rendent ainsi accessible.

Les mathématiciens attachent une grande importance à l'élégance de leurs mé-thodes et de leurs résultats; ce n'est pas là du pur dilettantisme. Qu'est ce qui nous donne en effet dans une solution, dans une démonstration, le sentiment de l'élégance? C'est l'harmonie des diverses parties, leur symétrie, leur heureux balancement; c'est en un mot tout ce qui y met de l'ordre, tout ce qui leur donne de l'unité, ce qui nous permet par conséquent d'y voir clair et d'en comprendre l'ensemble en même temps que les détails. Mais précisément, c'est là en même temps ce qui lui donne un grand rendement ; en effet, plus nous verrons cet ensemble clairement et d'un seul coup d'œil, mieux nous apercevrons ses analogies avec d'autres objets voisins, plus par conséquent nous aurons de chances de deviner les généralisations possibles. L'élé-gance peut provenir du sentiment de l'imprévu par la rencontre inattendue d'objets qu'on n'est pas accoutumé à rapprocher; là encore elle est féconde, puisqu'elle nous dévoile ainsi des parentés jusque-là méconnues; elle est féconde même quand elle ne résulte que du contraste entre la simplicité des moyens et la complexité du problème posé ; elle nous fait alors réfléchir à la raison de ce contraste et le plus souvent elle nous fait voir que cette raison n'est pas le hasard et qu'elle se trouve dans quelque loi insoupçonnée. ….

* Update of 1:44 PM ET March 14 — A further search, for "it suddenly brings order," brought order. Words very close to Frenkel's quotation appear in a version of Poincaré's "Future of Mathematics" from a 1909 Smithsonian report

"If a new result has value it is when, by binding together long-known elements, until now scattered and appearing unrelated to each other, it suddenly brings order where there reigned apparent disorder."

Saturday, December 22, 2012

Web Links:

Filed under: General,Geometry — m759 @ 5:01 PM

Spidey Goes to Church

More realistically

  1. "Nick Bostrom is a Swedish philosopher at 
    St. Cross College, University of Oxford…."
  2. "The early location of St Cross was on a site in
    St Cross Road, immediately south of St Cross Church."
  3. "The church building is located on St Cross Road
    just south of Holywell Manor."
  4. "Balliol College has had a presence in the area since
    the purchase by Benjamin Jowett, the Master, in the 1870s
    of the open area which is the Balliol sports ground
    'The Master's Field.' "
  5. Leaving Wikipedia, we find a Balliol field at Log24:
  6. A different view of the same field, from 1950—
    .
  7. A view from 1974, thanks to J. J. Seidel — 
  8. Yesterday's Analogies.

Monday, May 21, 2012

Wittgenstein’s Kindergarten

Filed under: General,Geometry — m759 @ 12:25 PM

A web search for the author Cameron McEwen  mentioned
in today's noon post was unsuccessful, but it did yield an
essay, quite possibly by a different  Cameron McEwen, on

The Digital Wittgenstein:

"The fundamental difference between analog
and digital systems may be understood as
underlying philosophical discourse since the Greeks."

The University of Bergen identifies the Wittgenstein 
McEwen as associated with InteLex  of Charlottesville.

The title of this post may serve to point out an analogy*
between the InteLex McEwen's analog-digital contrast
and the Euclidean-Galois contrast discussed previously
in this journal.

The latter contrast is exemplified in Pilate Goes to Kindergarten.

* An analogy, as it were, between  analogies.

Tuesday, April 3, 2012

Catholic View

Filed under: General — m759 @ 9:48 AM

"When shall we three meet again?"

Left to right— John von Neumann, Richard Feynman, Stanislaw Ulam

The source of the above book's title, "Analogies between Analogies,"
was misattributed in a weblog post linked to here on March 4th, 2012.
It occurs in a quote due not to Stanislaw Ulam but to Stefan Banach

IMAGE- 'Catholic view' quote in foreword of book 'Analogies between Analogies'

Ulam was Jewish. Banach was not.

From a webpage on Banach

"On 3 April 1892, he was baptized in the Roman Catholic
 Parish of St. Nicholas in Krakow."

See also…

  1. a post of Sunday, April 2, 2006,
  2. yesterday's Pennsylvania lottery, and
  3. post 585 in this journal. 

(At Los Alamos, Ulam developed the Monte Carlo method.)

Sunday, March 4, 2012

Look, Buster…

Filed under: General — m759 @ 11:00 AM

(Continued from previous posts)


  Detail from Washington Post  page today (below)…

  Click to enlarge

In related news…

  The Hallowed Crucible

"After all the pretty contrast of life and death
 Proves that these opposite things partake of one,
 At least that was the theory…."

— Wallace Stevens, "Connoisseur of Chaos"

Friday, June 11, 2010

Toward the Light

Filed under: General — m759 @ 11:01 AM

The title is a reference to yesterday's noon post.

For the late Vladimir Igorevich Arnold

"All things began in order, so shall they end, and so shall they begin again; according to the ordainer of order and mystical Mathematicks of the City of Heaven."

— Sir Thomas Browne, The Garden of Cyrus, Chapter V

Arnold's own mystical mathematics may be found in his paper

"Polymathematics: Is Mathematics a Single Science or a Set of Arts?"

Page 13–
"In mathematics we always encounter mysterious analogies, and our trinities [page 8] represent only a small part of these miracles."

Also from that paper—

Page 5, footnote 2–
"The Russian way to formulate problems is to mention the first nontrivial case (in a way that no one would be able to simplify it). The French way is to formulate it in the most general form making impossible any further generalization."

Arnold died in Paris on June 3. A farewell gathering was held there on June 8—

"Celles et ceux qui le souhaitent pourront donner un dernier adieu à Vladimir Igorevitch
mardi 8 juin, de 14h a 16h, chambre mortuaire de l'hopital Saint Antoine…."

An International Blue Diamond

In Arnold's memory—  Here, in the Russian style, is a link to a "first nontrivial case" of a blue diamond— from this journal on June 8 (feast of St. Gerard Manley Hopkins). For those who prefer French style, here is a link to a blue diamond from May 18

From French cinema—

Duelle

"a 'non-existent myth' of a battle between
goddesses of the sun and the moon
for a mysterious blue diamond
that has the power to make
mortals immortal and vice versa"

Wednesday, June 25, 2008

Wednesday June 25, 2008

Filed under: General,Geometry — Tags: — m759 @ 7:20 PM
The Cycle of
the Elements

John Baez, Week 266
(June 20, 2008):

“The Renaissance thinkers liked to
organize the four elements using
a chain of analogies running
from light to heavy:

fire : air :: air : water :: water : earth

They also organized them
in a diamond, like this:”

Diamond of the four ancient elements, figure by John Baez

This figure of Baez
is related to a saying
attributed to Heraclitus:

Diamond  showing transformation of the four ancient elements

For related thoughts by Jung,
see Aion, which contains the
following diagram:

Jung's four-diamond figure showing transformations of the self as Imago Dei

“The formula reproduces exactly the essential features of the symbolic process of transformation. It shows the rotation of the mandala, the antithetical play of complementary (or compensatory) processes, then the apocatastasis, i.e., the restoration of an original state of wholeness, which the alchemists expressed through the symbol of the uroboros, and finally the formula repeats the ancient alchemical tetrameria, which is implicit in the fourfold structure of unity.”

— Carl Gustav Jung

That the words Maximus of Tyre (second century A.D.) attributed to Heraclitus imply a cycle of the elements (analogous to the rotation in Jung’s diagram) is not a new concept. For further details, see “The Rotation of the Elements,” a 1995 webpage by one  “John Opsopaus.”

Related material:

Log24 entries of June 9, 2008, and

Quintessence: A Glass Bead Game,”
by Charles Cameron.

Thursday, June 7, 2007

Thursday June 7, 2007

Filed under: General — m759 @ 4:15 PM
Framing
truth

On “framing” and “spin”
in journalism:

“… Packaging is unavoidable.
Facts rarely, if ever, 
  speak for themselves.”

Matthew C. Nisbet,  
Assistant Professor
  of “Communication,”
June 6, 2007

If they could, they might
say “We was framed!”

Facts cannot, of course,
speak for themselves
to those who do not
understand their language.

Example:

A picture that appeared in
Log24 on June 7, 2005:

Natural Transformation

Click for details.

Attempt to
frame the picture:

Analogies

“A functor is an analogy.”
— Anonymous

  The best mathematicians “see
analogies between analogies.”
Banach, according to Ulam 

For further details,
click on the link
Analogies” above.

See also the analogies in
the previous entry.

Saturday, August 6, 2005

Saturday August 6, 2005

Filed under: General,Geometry — Tags: — m759 @ 9:00 AM
For André Weil on
the seventh anniversary
of his death:

 A Miniature
Rosetta Stone

The image “http://www.log24.com/log/pix05B/grid3x3med.bmp” cannot be displayed, because it contains errors.

In a 1940 letter to his sister Simone,  André Weil discussed a sort of “Rosetta stone,” or trilingual text of three analogous parts: classical analysis on the complex field, algebraic geometry over finite fields, and the theory of number fields.  

John Baez discussed (Sept. 6, 2003) the analogies of Weil, and he himself furnished another such Rosetta stone on a much smaller scale:

“… a 24-element group called the ‘binary tetrahedral group,’ a 24-element group called ‘SL(2,Z/3),’ and the vertices of a regular polytope in 4 dimensions called the ’24-cell.’ The most important fact is that these are all the same thing!”

For further details, see Wikipedia on the 24-cell, on special linear groups, and on Hurwitz quaternions,

The group SL(2,Z/3), also known as “SL(2,3),” is of course derived from the general linear group GL(2,3).  For the relationship of this group to the quaternions, see the Log24 entry for August 4 (the birthdate of the discoverer of quaternions, Sir William Rowan Hamilton).

The 3×3 square shown above may, as my August 4 entry indicates, be used to picture the quaternions and, more generally, the 48-element group GL(2,3).  It may therefore be regarded as the structure underlying the miniature Rosetta stone described by Baez.

“The typical example of a finite group is GL(n,q), the general linear group of n dimensions over the field with q elements. The student who is introduced to the subject with other examples is being completely misled.”

 — J. L. Alperin, book review,
    Bulletin (New Series) of the American
    Mathematical Society 10 (1984), 121

Thursday, June 23, 2005

Thursday June 23, 2005

Filed under: General,Geometry — m759 @ 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.

Tuesday, March 30, 2004

Tuesday March 30, 2004

Filed under: General — Tags: — m759 @ 12:00 AM

Banach’s Birthday

“A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies.”

— Stefan Banach, according to MacTutor.

The quotation is perhaps taken from Through a Reporter’s Eyes: The Life of Stefan Banach, by Roman Kauza (a.k.a. Roman Kaluza).

“What we today call ‘Banach spaces’
are called
‘spaces of type (B)’
in Banach’s book.”
Sheldon Axler

Sunday, July 13, 2003

Sunday July 13, 2003

Filed under: General,Geometry — Tags: , — m759 @ 5:09 PM

ART WARS, 5:09

The Word in the Desert

For Harrison Ford in the desert.
(See previous entry.)

    Words strain,
Crack and sometimes break,
    under the burden,
Under the tension, slip, slide, perish,
Will not stay still. Shrieking voices
Scolding, mocking, or merely chattering,
Always assail them.
    The Word in the desert
Is most attacked by voices of temptation,
The crying shadow in the funeral dance,
The loud lament of
    the disconsolate chimera.

— T. S. Eliot, Four Quartets

The link to the word "devilish" in the last entry leads to one of my previous journal entries, "A Mass for Lucero," that deals with the devilishness of postmodern philosophy.  To hammer this point home, here is an attack on college English departments that begins as follows:

"William Faulkner's Snopes trilogy, which recounts the generation-long rise of the drily loathsome Flem Snopes from clerk in a country store to bank president in Jefferson, Mississippi, teems with analogies to what has happened to English departments over the past thirty years."

For more, see

The Word in the Desert,
by Glenn C. Arbery
.

See also the link on the word "contemptible," applied to Jacques Derrida, in my Logos and Logic page.

This leads to an National Review essay on Derrida,

The Philosopher as King,
by Mark Goldblatt

A reader's comment on my previous entry suggests the film "Scotland, PA" as viewing related to the Derrida/Macbeth link there.

I prefer the following notice of a 7-11 death, that of a powerful art museum curator who would have been well cast as Lady Macbeth:

Die Fahne Hoch,
Frank Stella,
1959


Dorothy Miller,
MOMA curator,

died at 99 on
July 11, 2003
.

From the Whitney Museum site:

"Max Anderson: When artist Frank Stella first showed this painting at The Museum of Modern Art in 1959, people were baffled by its austerity. Stella responded, 'What you see is what you see. Painting to me is a brush in a bucket and you put it on a surface. There is no other reality for me than that.' He wanted to create work that was methodical, intellectual, and passionless. To some, it seemed to be nothing more than a repudiation of everything that had come before—a rational system devoid of pleasure and personality. But other viewers saw that the black paintings generated an aura of mystery and solemnity.

The title of this work, Die Fahne Hoch, literally means 'The banner raised.'  It comes from the marching anthem of the Nazi youth organization. Stella pointed out that the proportions of this canvas are much the same as the large flags displayed by the Nazis.

But the content of the work makes no reference to anything outside of the painting itself. The pattern was deduced from the shape of the canvas—the width of the black bands is determined by the width of the stretcher bars. The white lines that separate the broad bands of black are created by the narrow areas of unpainted canvas. Stella's black paintings greatly influenced the development of Minimalism in the 1960s."

From Play It As It Lays:

   She took his hand and held it.  "Why are you here."
   "Because you and I, we know something.  Because we've been out there where nothing is.  Because I wanted—you know why."
   "Lie down here," she said after a while.  "Just go to sleep."
   When he lay down beside her the Seconal capsules rolled on the sheet.  In the bar across the road somebody punched King of the Road on the jukebox again, and there was an argument outside, and the sound of a bottle breaking.  Maria held onto BZ's hand.
   "Listen to that," he said.  "Try to think about having enough left to break a bottle over it."
   "It would be very pretty," Maria said.  "Go to sleep."

I smoke old stogies I have found…    

Cigar Aficionado on artist Frank Stella:

" 'Frank actually makes the moment. He captures it and helps to define it.'

This was certainly true of Stella's 1958 New York debut. Fresh out of Princeton, he came to New York and rented a former jeweler's shop on Eldridge Street on the Lower East Side. He began using ordinary house paint to paint symmetrical black stripes on canvas. Called the Black Paintings, they are credited with paving the way for the minimal art movement of the 1960s. By the fall of 1959, Dorothy Miller of The Museum of Modern Art had chosen four of the austere pictures for inclusion in a show called Sixteen Americans."

For an even more austere picture, see

Geometry for Jews:

For more on art, Derrida, and devilishness, see Deborah Solomon's essay in the New York Times Magazine of Sunday, June 27, 1999:

 How to Succeed in Art.

"Blame Derrida and
his fellow French theorists…."

See, too, my site

Art Wars: Geometry as Conceptual Art

For those who prefer a more traditional meditation, I recommend

Ecce Lignum Crucis

("Behold the Wood of the Cross")

THE WORD IN THE DESERT

For more on the word "road" in the desert, see my "Dead Poet" entry of Epiphany 2003 (Tao means road) as well as the following scholarly bibliography of road-related cultural artifacts (a surprising number of which involve Harrison Ford):

A Bibliography of Road Materials

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