Monday, December 11, 2017

A Diamond Metaphor

Filed under: Uncategorized — m759 @ 11:24 PM

For some remarks related to the title, see Black + Algebra + Metaphor.

Illustration of a 'diamond' in Scholze's 2014 lectures on p-adic geometry

There is apparently no relationship between Scholze's metaphor
and my own use of the word "diamond" in finite  geometry.

Sunday, December 10, 2017


Filed under: Uncategorized — Tags: — m759 @ 2:55 PM

Derrida quote from the previous post

See also Black + Algebra + Metaphor.

Friday, April 29, 2016

Blackboard Jungle…

Filed under: Uncategorized — Tags: — m759 @ 12:00 PM

Continues .

An older and wiser James Spader —

"Never underestimate the power of glitter."

Glitter by Josefine Lyche, as of diamond dust

Wednesday, April 13, 2016

Black List

Filed under: Uncategorized — Tags: — m759 @ 1:00 PM

A search for "Max Black" in this journal yields some images
from a post of August 30, 2006 . . .

A circular I Ching

The image “http://www.log24.com/log/pix06A/060830-SeventhSymbol.jpg” cannot be displayed, because it contains errors.

"Jackson has identified the seventh symbol."
— Stargate

The "Jackson" above is played by the young James Spader,
who in an older version currently stars in "The Blacklist."

"… the memorable models of science are 'speculative instruments,'
to borrow I. A. Richards' happy title. They, too, bring about a wedding
of disparate subjects, by a distinctive operation of transfer of the
implications  of relatively well-organized cognitive fields. And as with
other weddings, their outcomes are unpredictable."

Max Black in Models and Metaphors , Cornell U. Press, 1962

Thursday, December 5, 2013

Blackboard Jungle

Filed under: Uncategorized — Tags: , — m759 @ 11:07 AM

Continued from Field of Dreams, Jan. 20, 2013.

IMAGE- Richard Kiley in 'Blackboard Jungle,' with grids and broken records

That post mentioned the March 2011 AMS Notices ,
an issue on mathematics education.

In that issue was an interview with Abel Prize winner
John Tate done in Oslo on May 25, 2010, the day
he was awarded the prize. From the interview—

Research Contributions

Raussen and Skau: This brings us to the next
topic: Your Ph.D. thesis from 1950, when you were
twenty-five years old. It has been extensively cited
in the literature under the sobriquet “Tate’s thesis”.
Several mathematicians have described your thesis
as unsurpassable in conciseness and lucidity and as
representing a watershed in the study of number
fields. Could you tell us what was so novel and fruitful
in your thesis?

Tate: Well, first of all, it was not a new result, except
perhaps for some local aspects. The big global
theorem had been proved around 1920 by the
great German mathematician Erich Hecke, namely
​the fact that all L -functions of number fields,
abelian -functions, generalizations of Dirichlet’s
L -functions, have an analytic continuation
throughout the plane with a functional equation
of the expected type. In the course of proving
it Hecke saw that his proof even applied to a new
kind of L -function, the so-called L -functions with
Grössencharacter. Artin suggested to me that one
might prove Hecke’s theorem using abstract
harmonic analysis on what is now called the adele
ring, treating all places of the field equally, instead
of using classical Fourier analysis at the archimedian 
places and finite Fourier analysis with congruences 
at the p -adic places as Hecke had done. I think I did
a good job —it might even have been lucid and
concise!—but in a way it was just a wonderful 
exercise to carry out this idea. And it was also in the
air. So often there is a time in mathematics for 
something to be done. My thesis is an example. 
Iwasawa would have done it had I not.

[For a different perspective on the highlighted areas of
mathematics, see recent remarks by Edward Frenkel.]

"So often there is a time in mathematics for something to be done."

— John Tate in Oslo on May 25, 2010.

See also this journal on May 25, 2010, as well as
Galois Groups and Harmonic Analysis on Nov. 24, 2013.

Tuesday, June 12, 2012

Meet Max Black (continued)

Filed under: Uncategorized — m759 @ 11:59 PM

Background— August 30, 2006—

The Seventh Symbol:

The image “http://www.log24.com/log/pix06A/060830-Algebra.jpg” cannot be displayed, because it contains errors.

In the 2006 post, the above seventh symbol  110000 was
interpreted as the I Ching hexagram with topmost and
next-to-top lines solid, not broken— Hexagram 20, View .

In a different interpretation, 110000 is the binary for the decimal
number 48— representing the I Ching's Hexagram 48, The Well .

“… Max Black, the Cornell philosopher, and 
others have pointed out how ‘perhaps every science
must start with metaphor and end with algebra, and
perhaps without the metaphor there would never
have been any algebra’ ….”

– Max BlackModels and Metaphors,
Cornell U. Press, 1962, page 242, as quoted
in Dramas, Fields, and Metaphors,
by Victor Witter Turner, Cornell U. Press,
paperback, 1975, page 25

The algebra is certainly clearer than either I Ching
metaphor, but is in some respects less interesting.

For a post that combines both the above I Ching
metaphorsView  and Well  , see Dec. 14, 2007.

In memory of scholar Elinor Ostrom,
who died today—

"Time for you to see the field."
Bagger Vance

Thursday, May 31, 2012

Black Diamond

Filed under: Uncategorized — Tags: — m759 @ 12:26 PM

IMAGE- Four-elements-diamond test problem in the style of Raven's Progressive Matrices (answer: the black diamond)

“To say more is to say less.”
― Harlan Ellison, as quoted at goodreads.com

Saying less—

Saturday, April 7, 2018


Filed under: Uncategorized — Tags: — m759 @ 11:47 AM

The FBI holding cube in "The Blacklist" —

" 'The Front' is not the whole story . . . ."

— Vincent Canby, New York Times  film review, 1976,
     as quoted in Wikipedia.

See also Solomon's Cube in this  journal.

IMAGE- 'Solomon's Cube'

Webpage demonstrating symmetries of 'Solomon's Cube'

Some may view the above web page as illustrating the
Glasperlenspiel  passage quoted here in Summa Mythologica 

“"I suddenly realized that in the language, or at any rate
in the spirit of the Glass Bead Game, everything actually
was all-meaningful, that every symbol and combination of
symbols led not hither and yon, not to single examples,
experiments, and proofs, but into the center, the mystery
and innermost heart of the world, into primal knowledge.
Every transition from major to minor in a sonata, every
transformation of a myth or a religious cult, every classical
or artistic formulation was, I realized in that flashing moment,
if seen with a truly meditative mind, nothing but a direct route
into the interior of the cosmic mystery, where in the alternation
between inhaling and exhaling, between heaven and earth,
between Yin and Yang, holiness is forever being created.”

A less poetic meditation on the above 4x4x4 design cube —

"I saw that in the alternation between front and back,
between top and bottom, between left and right,
symmetry is forever being created."

See also a related remark by Lévi-Strauss in 1955

"…three different readings become possible:
left to right, top to bottom, front to back."

Friday, April 6, 2018

Watching the Zero

Filed under: Uncategorized — Tags: — m759 @ 9:00 PM

From "The Blacklist" Season 5, Episode 11 —

– Remind me again what it is that we think we're doing here.
– The phone acts as a passive packet sniffer.
It's a trick Tom taught me.
– Packet sniffer? Ugh.
– The FBI uses them.
I'm sure your tech people know all about them.
It can intercept and log traffic that passes over a digital network.
– It is an absolute mystery to me how these gadgets work —
the Dick Tracy phones, these blueteeth connections.
Quite frankly, I miss the rotary phone.
Except for that zero.
Watching that zero crawl back.
Oh, my God.
It was painful.
– We have the code.
– Great.

Read more:  https://www.springfieldspringfield.co.uk/

And more:

Philip J. Davis reportedly turned 86 on January 2, 2009.
An image from this journal on that date

Rotary telephone dial

“You have the incorrect number.
I will tell you what you are doing:
you are turning the letter O
instead of the zero.”

— "Symbols and Signs,"
Vladimir Nabokov, 1948

Plan 9

Filed under: Uncategorized — Tags: — m759 @ 2:18 PM

Salinger's 'Nine Stories,' paperback with 3x3 array of titles on cover, adapted in a Jan. 2, 2009, Log24 post on Nabokov's 1948 'Signs and Symbols'

The Thread Phantom: A Death on Pi Day*

Filed under: Uncategorized — Tags: , — m759 @ 12:42 PM

The American Mathematical Society on April 4 posted a story
about a death that they said occurred on March 14  (Pi Day):

* Notes on the Title —

The Thread Part 

The Phantom Part 

"What a yarn!" — Raymond Reddington in "The Blacklist"

Fact check on the death date reported by the AMS —

But Davis's funeral-home obituary agrees with the Pi Day date.

Wednesday, April 4, 2018


Filed under: Uncategorized — Tags: , — m759 @ 2:08 PM

An image discussed in the previous post

From a search for "1943" in this journal — 

"Paradine found himself growing slightly confused . . . ."

Gold Bug Variations (Continued)

Filed under: Uncategorized — Tags: , — m759 @ 1:32 PM

See as well a search for "Gold Bug"  in this  journal.

From that search —

Richard Powers, The Gold Bug Variations , first published in 1991—

Botkin, whatever her gifts as a conversationist, is almost as old
as the rediscovery of Mendel. The other extreme in age, 
Joe Lovering, beat a time-honored path out of pure math 
into muddy population statistics. Ressler has seen the guy 
potting about in the lab, although exactly what the excitable kid 
does is anybody's guess. He looks decidedly gumfooted holding
any equipment more corporeal than a chi-square. Stuart takes
him to the Y for lunch, part of a court-your-resources campaign.
He has the sub, Levering the congealed mac and cheese. 
Hardly are they seated when Joe whips out a napkin and begins
sketching proofs. He argues that the genetic code, as an 
algorithmic formal system, is subject to Gödel's Incompleteness
Theorem. "That would mean the symbolic language of the code 
can't be both consistent and complete. Wouldn't that be a kick 
in the head?"

Kid talk, competitive showing off, intellectual fantasy. 
But Ressler knows what Joe is driving at. He's toyed with similar 
ideas, cast in less abstruse terms. We are the by-product of the 
mechanism in there. So it must be more ingenious than us. 
Anything complex enough to create consciousness may be too 
complex for consciousness to understand. Yet the ultimate paradox
is Lovering, crouched over his table napkin, using proofs to 
demonstrate proof's limits. Lovering laughs off recursion and takes
up another tack: the key is to find some formal symmetry folded
in this four-base chaos
. Stuart distrusts this approach even more.
He picks up the tab for their two untouched lunches, thanking 
Lovering politely for the insight.

"The key is to find some formal symmetry…."

IMAGE- Valéry on ornament in 'Method of Leonardo,' with Valéry's serpent-and-key emblem

Saturday, January 14, 2017

1984: A Space Odyssey

Filed under: Uncategorized — m759 @ 2:40 PM

See Eightfold 1984 in this journal.

Related material —

"… the object sets up a kind of
 frame or space or field
 within which there can be epiphany."

"… Instead of an epiphany of being,
we have something like
an epiphany of interspaces."

— Charles Taylor, "Epiphanies of Modernism,"
Chapter 24 of Sources of the Self ,
Cambridge University Press, 1989

"Perhaps every science must start with metaphor
and end with algebra; and perhaps without the metaphor
there would never have been any algebra."

— Max Black, Models and Metaphors ,
Cornell University Press, Ithaca, NY, 1962

Epiphany 2017 —

Click to enlarge:

Saturday, June 4, 2016


Filed under: Uncategorized — Tags: — m759 @ 12:00 PM

From this morning's news, a  cultural icon —

From November 18, 2015, four  icons —

— the three favicons above, and the following:

Jack in the Box, by Natasha Wescoat

Saturday, January 23, 2016


Filed under: Uncategorized — Tags: — m759 @ 9:00 PM

"Hard Science Fiction in the era of short attention spans,
crowd-sourcing, and rapid obsolescence"

— May 26, 2012, Dragon Press Bookstore symposium

Related material:  Posts now tagged Black Diamond.

IMAGE- 'The Stars My Destination' (with cover slightly changed)

Friday, January 22, 2016


Filed under: Uncategorized — Tags: — m759 @ 10:21 AM

The New Yorker , April 12, 2004 —

Friday, November 27, 2015

Einstein and Geometry

Filed under: Uncategorized — Tags: — m759 @ 2:01 PM

(A Prequel to Dirac and Geometry)

"So Einstein went back to the blackboard.
And on Nov. 25, 1915, he set down
the equation that rules the universe.
As compact and mysterious as a Viking rune,
it describes space-time as a kind of sagging mattress…."

— Dennis Overbye in The New York Times  online,
     November 24, 2015

Some pure  mathematics I prefer to the sagging Viking mattress —

Readings closely related to the above passage —

Thomas Hawkins, "From General Relativity to Group Representations:
the Background to Weyl's Papers of 1925-26
," in Matériaux pour
l'histoire des mathématiques au XXe siècle:
Actes du colloque
à la mémoire de Jean Dieudonné
, Nice, 1996  (Soc. Math.
de France, Paris, 1998), pp. 69-100.

The 19th-century algebraic theory of invariants is discussed
as what Weitzenböck called a guide "through the thicket
of formulas of general relativity."

Wallace Givens, "Tensor Coordinates of Linear Spaces," in
Annals of Mathematics  Second Series, Vol. 38, No. 2, April 1937, 
pp. 355-385.

Tensors (also used by Einstein in 1915) are related to 
the theory of line complexes in three-dimensional
projective space and to the matrices used by Dirac
in his 1928 work on quantum mechanics.

For those who prefer metaphors to mathematics —

"We acknowledge a theorem's beauty
when we see how the theorem 'fits' in its place,
how it sheds light around itself, like a Lichtung ,
a clearing in the woods." 
— Gian-Carlo Rota, Indiscrete Thoughts ,
Birkhäuser Boston, 1997, page 132

Rota fails to cite the source of his metaphor.
It is Heidegger's 1964 essay, "The End of Philosophy
and the Task of Thinking" —

"The forest clearing [ Lichtung ] is experienced
in contrast to dense forest, called Dickung  
in our older language." 
— Heidegger's Basic Writings 
edited by David Farrell Krell, 
Harper Collins paperback, 1993, page 441

Wednesday, November 18, 2015

Tuesday, November 17, 2015

Thursday, September 3, 2015

Rings of August

Filed under: Uncategorized — Tags: — m759 @ 7:20 AM

For the title, see posts from August 2007 tagged Gyges.

Related theological remarks:

Boolean  spaces (old) vs. Galois  spaces  (new) in 
"The Quality Without a Name"
(a post from August 26, 2015) and the

Related literature:  A search for Borogoves in this journal will yield
remarks on the 1943 tale underlying the above film.

Saturday, June 13, 2015

Egg Tales

Filed under: Uncategorized — Tags: , — m759 @ 12:00 PM

"And not all the king's men nor his horses
 Will resurrect his corpus."

Finnegans Wake

See as well Andy Weir's "The Egg" and Working Backward.

Friday, April 24, 2015

Speak, Memory

Filed under: Uncategorized — Tags: — m759 @ 10:30 AM

For "Blacklist" fans —

See also Mimsy.

Tuesday, April 7, 2015

For Times Square Church

Filed under: Uncategorized — Tags: — m759 @ 9:55 AM

The Times  version —

Wednesday, April 1, 2015

Math’s Big Lies

Filed under: Uncategorized — Tags: , — m759 @ 11:00 AM

Two mathematicians, Barry Mazur and Edward Frenkel,
have, for rhetorical effect, badly misrepresented the
history of some basic fields of mathematics. Mazur and
Frenkel like to emphasize the importance of new 
research by claiming that it connects fields that previously
had no known connection— when, in fact, the fields were
known to be connected since at least the nineteenth century.

For Mazur, see The Proof and the Lie; for Frenkel, see posts
tagged Frenkel-Metaphors.

See also a story and video on Robert Langlands from the
Toronto Star  on March 27, 2015:

"His conjectures are called functoriality and
reciprocity. They made it possible to link up
three branches of math: harmonic analysis,
number theory, and geometry. 

To mathematicians, this is mind-blowing stuff
because these branches have nothing to do
with each other."

For a much earlier link between these three fields, see the essay
"Why Pi Matters" published in The New Yorker  last month.

Sunday, March 29, 2015

Mathematics for Jews*

Filed under: Uncategorized — Tags: , — m759 @ 11:00 PM

Headline at the Toronto Star  on Friday, March 27, 2015:

Robert Langlands: The Canadian
who reinvented mathematics

“He’s like a modern-day Einstein.”

Apparently, unlike God, Langlands würfelt .

* See also Blockheads  in this journal.

Wednesday, March 4, 2015

Frenkel’s Rashomon

Filed under: Uncategorized — Tags: — m759 @ 2:02 PM

Saturday, February 21, 2015

Putting the Con in Conceptual

Filed under: Uncategorized — Tags: — m759 @ 6:00 AM

Edward Frenkel in The New York Times ,
in an op-ed piece dated Feb. 20, 2015 —

"… I suggest that we regard the paradoxes
of quantum physics as a metaphor for
the unknown infinite possibilities
of our own existence. This is poignantly
and elegantly expressed in the Vedas:
'As is the atom, so is the universe;
as is the microcosm, so is the macrocosm;
as is the human body, so is the cosmic body;
as is the human mind, so is the cosmic mind.'"

The Times : "Edward Frenkel, a professor of mathematics
at the University of California, Berkeley, is the author of
Love and Math: The Heart of Hidden Reality. "

See also Con Vocation (Sept. 2, 2014).

Saturday, February 7, 2015

Word and Object

Filed under: Uncategorized — Tags: , — m759 @ 7:00 PM

From actor James Spader, whose birthday is today —

"… my father taught English. My mother taught art…."

— Spader in a 2014 interview

See as well the 2013 film "Words and Pictures"
and Log24 posts on a 2007 film, "The Last Mimzy."

Above: A scene from Spader's TV series "The Blacklist"
that was aired on Thursday, February 5, 2015.

Thursday, May 15, 2014

Mathematics as Religion

Filed under: Uncategorized — Tags: — m759 @ 11:00 PM

On Edward Frenkel:

"Math is, for him, 'a narrative' of human endeavor
that shares much with art, music and religion.

For instance, he describes new mathematical insights
as 'revelations,' and the utterly unchanging truths of
mathematical ideas are 'nothing short of a miracle.'"


Saturday, March 15, 2014

Greene on Mathematics

Filed under: Uncategorized — Tags: , — m759 @ 1:20 AM

A reply in the March 8 LA Times  to the opinion piece by Edward Frenkel discussed here yesterday

"It is completely wrong to imply that Euclidean geometry is somehow not interesting because it is old. Actually, Euclidean geometry appeals not only in its intrinsic mathematical nature but also in its power to explain what one sees around one spatially.

If this subject is taught badly, like any other subject it can seem tedious. If it is taught well, it arouses the sense of the intellectual power and attractiveness of mathematical thought as well as or better than anything else that can be presented to a beginner.

One of the points of mathematics educationally is to introduce students to a subject in which precise thought exists. They are surrounded by a world of baloney versions of science. Mathematics is where they find out that really precise thought exists. The last thing they need is to be given the impression that mathematics is another subject in which learning a few buzzwords is the whole show.

Modern mathematics can exist only because older mathematics has existed.

Robert E. Greene

Pacific Palisades

The writer is a professor of mathematics at UCLA."

And several other things too.

Friday, March 14, 2014

Whitewashing Picasso

Filed under: Uncategorized — Tags: , — m759 @ 4:30 PM

A search today for Edward Frenkel's phrase
"portals into the magic world of modern math"
leads to a reprint of his March 2 LA Times  opinion piece
in The Salem News —

IMAGE- Edward Frenkel in The Salem News

To hell with Picasso, I'll take Tom Sawyer.


Filed under: Uncategorized — 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 —


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. ….


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 14, 2013

Sacred and Profane

Filed under: Uncategorized — Tags: , — m759 @ 10:00 AM

(Continued from yesterday afternoon)

This journal on December 12th, 2009

Rothstein's 'Emblems of Mind,' 1995, cover illustrations by Pinturicchio from Vatican

Cover illustration— Arithmetic and Music,
Borgia Apartments, The Vatican

Compare and contrast with Frenkel at the Fields Institute

Thursday, December 5, 2013


Filed under: Uncategorized — Tags: , , — m759 @ 1:20 AM

Edward Frenkel recently claimed for Robert Langlands
the discovery of a link between two "totally different"
fields of mathematics— number theory and harmonic analysis.
He implied that before Langlands, no relationship between
these fields was known.

See his recent book, and his lecture at the Fields Institute
in Toronto on October 24, 2013.

Meanwhile, in this journal on that date, two math-related
quotations for Stephen King, author of Doctor Sleep

"Danvers is a town in Essex County, Massachusetts, 
United States, located on the Danvers River near the
northeastern coast of Massachusetts. Originally known
as Salem Village, the town is most widely known for its
association with the 1692 Salem witch trials. It is also
known for the Danvers State Hospital, one of the state's
19th-century psychiatric hospitals, which was located here." 

"The summer's gone and all the roses fallin' "

For those who prefer their mathematics presented as fact, not fiction—

(Click for a larger image.)

The arrows in the figure at the right are an attempt to say visually that 
the diamond theorem is related to various fields of mathematics.
There is no claim that prior to the theorem, these fields were not  related.

See also Scott Carnahan on arrow diagrams, and Mathematical Imagery.

Tuesday, November 26, 2013

Edward Frenkel, Your Order Is Ready.

Filed under: Uncategorized — Tags: — m759 @ 11:00 AM

Backstory: Frenkel's Metaphors and Waitressing for Godot.

In a recent vulgarized presentation of the Langlands program,
Edward Frenkel implied that number theory and harmonic
analysis were, before Langlands came along, quite unrelated.

This is false.

"If we think of different fields of mathematics as continents,
then number theory would be like North America and
harmonic analysis like Europe." 

Edward Frenkel, Love and Math , 2013

For a discussion of pre-Langlands connections between 
these "continents," see


"Fourier Analysis in Number Theory, my senior thesis, under the advisory of Patrick Gallagher.

This thesis contains no original research, but is instead a compilation of results from analytic
number theory that involve Fourier analysis. These include quadratic reciprocity (one of 200+
published proofs), Dirichlet's theorem on primes in arithmetic progression, and Weyl's criterion.
There is also a function field analogue of Fermat's Last Theorem. The presentation of the
material is completely self-contained."

Shanshan Ding, University of Pennsylvania graduate student

Monday, November 25, 2013

Pythagoras Wannabe*

Filed under: Uncategorized — Tags: — m759 @ 10:10 AM

A scholium on the link to Pythagoras
in this morning's previous post Figurate Numbers:

For related number mysticism, see Chapter 8, "Magic Numbers,"
in Love and Math: The Heart of Hidden Reality
by Edward Frenkel (Basic Books, Oct. 1, 2013).

(Click for clearer image.)

See also Frenkel's Metaphors in this journal. 

* The wannabe of the title is of course not Langlands, but Frenkel.

Sunday, November 24, 2013

Galois Groups and Harmonic Analysis

Filed under: Uncategorized — Tags: — m759 @ 9:29 AM

"In 1967, he [Langlands] came up with revolutionary
insights tying together the theory of Galois groups 
and another area of mathematics called harmonic
analysis. These two areas, which seem light years
, turned out to be closely related."

— Edward Frenkel, Love and Math, 2013

"Class field theory expresses Galois groups of
abelian extensions of a number field F
in terms of harmonic analysis on the
multiplicative group of [a] locally compact
topological ring, the adèle ring, attached to F."

— Michael Harris in a description of a Princeton
    mathematics department talk of October 2012

Related material: a Saturday evening post.

See also Wikipedia on the history of class field theory.
For greater depth, see Tate's [1950] thesis and the book
Fourier Analysis on Number Fields .

Logic for Jews*

Filed under: Uncategorized — Tags: , — m759 @ 7:20 AM

The search for 1984 at the end of last evening's post
suggests the following Sunday meditation.

My own contribution to this genre—

A triangle-decomposition result from 1984:

American Mathematical Monthly ,  June-July 1984, p. 382


Triangles are square

"Every triangle consists of n  congruent copies of itself"
is true if and only if  is a square. (The proof is trivial.) 
— Steven H. Cullinane

The Orwell slogans are false. My own is not.

* The "for Jews" of the title applies to some readers of Edward Frenkel.

Saturday, November 23, 2013

Light Years Apart?

Filed under: Uncategorized — Tags: — m759 @ 9:00 PM

From a recent attempt to vulgarize the Langlands program:

"Galois’ work is a great example of the power of a mathematical insight…. 

And then, 150 years later, Langlands took these ideas much farther. In 1967, he came up with revolutionary insights tying together the theory of Galois groups and another area of mathematics called harmonic analysis. These two areas, which seem light years apart, turned out to be closely related."

— Frenkel, Edward (2013-10-01).
     Love and Math: The Heart of Hidden Reality
     (p. 78, Basic Books, Kindle Edition) 

(Links to related Wikipedia articles have been added.)


Wikipedia on the Langlands program

The starting point of the program may be seen as Emil Artin's reciprocity law [1924-1930], which generalizes quadratic reciprocity. The Artin reciprocity law applies to a Galois extension of algebraic number fields whose Galois group is abelian, assigns L-functions to the one-dimensional representations of this Galois group; and states that these L-functions are identical to certain Dirichlet L-series or more general series (that is, certain analogues of the Riemann zeta function) constructed from Hecke characters. The precise correspondence between these different kinds of L-functions constitutes Artin's reciprocity law.

For non-abelian Galois groups and higher-dimensional representations of them, one can still define L-functions in a natural way: Artin L-functions.

The insight of Langlands was to find the proper generalization of Dirichlet L-functions, which would allow the formulation of Artin's statement in this more general setting.


From "An Elementary Introduction to the Langlands Program," by Stephen Gelbart (Bulletin of the American Mathematical Society, New Series , Vol. 10, No. 2, April 1984, pp. 177-219)

On page 194:

"The use of group representations in systematizing and resolving diverse mathematical problems is certainly not new, and the subject has been ably surveyed in several recent articles, notably [ Gross and Mackey ]. The reader is strongly urged to consult these articles, especially for their reformulation of harmonic analysis as a chapter in the theory of group representations.

In harmonic analysis, as well as in the theory of automorphic forms, the fundamental example of a (unitary) representation is the so-called 'right regular' representation of G….

Our interest here is in the role representation theory has played in the theory of automorphic forms.* We focus on two separate developments, both of which are eventually synthesized in the Langlands program, and both of which derive from the original contributions of Hecke already described."

Gross ]  K. I. Gross, On the evolution of non-commutative harmonic analysis . Amer. Math. Monthly 85 (1978), 525-548.

Mackey ]  G. Mackey, Harmonic analysis as the exploitation of symmetry—a historical survey . Bull. Amer. Math. Soc. (N.S.) 3 (1980), 543-698.

* A link to a related Math Overflow article has been added.

In 2011, Frenkel published a commentary in the A.M.S. Bulletin  
on Gelbart's Langlands article. The commentary, written for
a mathematically sophisticated audience, lacks the bold
(and misleading) "light years apart" rhetoric from his new book 
quoted above.

In the year the Gelbart article was published, Frenkel was
a senior in high school. The year was 1984.

For some remarks of my own that mention
that year, see a search for 1984 in this journal.

Thursday, November 21, 2013

Twelfth Step

Filed under: Uncategorized — Tags: , , — m759 @ 7:59 AM

Continued from 24 hours ago.

From this morning's 6 AM (ET) post

"… you never made a Twelfth Step
call on an active alcoholic by yourself,
unless the alkie in question was safely
incarcerated in a hospital, detox, or the
local bughouse."

— Stephen King, Doctor Sleep

Related material from a math addict, a likely victim
of a professor's misleading rhetoric —

"Frenkel is the real deal, a professor at Berkeley…."

— "Math Porn Update" by David Justice,
       Nov. 20, 2013

The rhetoric link above leads to remarks by Frenkel.
For a similar professor's earlier misleading remarks,
see Barry Mazur in this journal.

Wednesday, November 20, 2013

Quad Rants

Filed under: Uncategorized — Tags: — m759 @ 6:00 AM

IMAGE- 'Development of Mathematics in the Nineteenth Century,' by Felix Klein

"… the message is clear on what is the main
accomplishment of 19th [century] mathematics:
complex function theory, comprising almost half
the book. The heart and soul of this theory is the
theory of elliptic functions and its generalisations
(abelian functions, elliptic modular functions,
automorphic functions)."

Viktor Blasjo, Feb. 27, 2006:

Tune for an entertainer —

Tuesday, November 19, 2013


Filed under: Uncategorized — Tags: — m759 @ 6:29 AM

IMAGE- The Klein Four-Group, 'Vierergruppe': the group's four elements in four colors. Blue, red, green arrows represent pairs of transpositions, and the four black points, viewed as stationary, represent the identity.

* Update of 8 PM Nov. 19:
   The title refers to a work by Beckett.
  "There is nothing outside itself that Quad
   might be about." — Sue Wilson.
   The Klein group is not so limited.

Monday, November 18, 2013

Teleportation Web?

Filed under: Uncategorized — Tags: — m759 @ 8:45 PM

"In this book, I will describe one of the biggest ideas
to come out of mathematics in the last fifty years:
the Langlands Program, considered by many as
the Grand Unified Theory of mathematics. It’s a
fascinating theory that weaves a web of tantalizing
connections between mathematical fields that
at first glance seem to be light years apart:
algebra, geometry, number theory, analysis
and quantum physics. If we think of those fields as
continents in the hidden world of mathematics, then
the Langlands Program is the ultimate teleportation
device, capable of getting us instantly from one of
them to another, and back."

— Edward Frenkel, excerpt from his new book
     in today's online New York Times  

The four areas of pure mathematics that Frenkel
names do not, of course, seem to be "light years
apart" to those familiar with the development of
mathematics in the nineteenth century.

Related material:  Sunday morning's post.

Sunday, November 17, 2013

The X-Men Tree

Filed under: Uncategorized — Tags: — m759 @ 7:59 AM

Continued from November 12, 2013. A post on that date
showed the tree from Waiting for Godot  along with the two
X-Men patriarchs. See also last night's Chapel post,
which shows a more interesting tree—

A recent book on the Langlands program by Edward Frenkel
repeats a metaphor about building a bridge  between unrelated
worlds within mathematics. A review of the Frenkel book by
Marcus du Sautoy replaces the bridge  metaphor with a wormhole .
Some users of such metaphors seem to feel they are justified, 
for maximum rhetorical effect, in lying about the unrelatedness of
the worlds being connected. The connections they discuss are
surprising (see the Eichler function discussed by Frenkel and
du Sautoy), but the connections occur, at least in the case of
elliptic curves and modular forms, between areas of mathematics
long known to be, in less subtle ways, related. See remarks
from 2005 by Diamond and Shurman below.

Related material:

Saturday, November 16, 2013

Mathematics and Rhetoric

Filed under: Uncategorized — Tags: — m759 @ 12:00 PM

Jim Holt in the current (Dec. 5) New York Review of Books

One form of Eros is the sexual desire aroused by the physical beauty of a particular beloved person. That, according to Diotima, is the lowest form. With philosophical refinement, however, Eros can be made to ascend toward loftier and loftier objects. The penultimate of these—just short of the Platonic idea of beauty itself—is the perfect and timeless beauty discovered by the mathematical sciences. Such beauty evokes in those able to grasp it a desire to reproduce—not biologically, but intellectually, by begetting additional “gloriously beautiful ideas and theories.” For Diotima, and presumably for Plato as well, the fitting response to mathematical beauty is the form of Eros we call love.

Consider (for example) the beauty of the rolling donut

            (Animation source: MIQEL.com)

Saturday, June 29, 2013

Thursday, June 27, 2013

Tuesday, January 22, 2013

Raven Light

Filed under: Uncategorized — Tags: , — m759 @ 11:40 AM

"…a fundamental cognitive ability known as 'fluid' intelligence: the capacity to solve novel problems, to learn, to reason, to see connections and to get to the bottom of things. …

…matrices are considered the gold standard of fluid-intelligence tests. Anyone who has taken an intelligence test has seen matrices like those used in the Raven’s: three rows, with three graphic items in each row, made up of squares, circles, dots or the like. Do the squares get larger as they move from left to right? Do the circles inside the squares fill in, changing from white to gray to black, as they go downward? One of the nine items is missing from the matrix, and the challenge is to find the underlying patterns— up, down and across— from six possible choices. Initially the solutions are readily apparent to most people, but they get progressively harder to discern. By the end of the test, most test takers are baffled."

— Dan Hurley, "Can You Make Yourself Smarter?," NY Times , April 18, 2012

See also "Raven Steals the Light" in this  journal.

Related material:

Plan 9 from MIT and, perhaps exemplifying crystallized  rather than fluid  intelligence, Black Diamond.

Thursday, November 1, 2012

For All Saints’ Day

Filed under: Uncategorized — Tags: , — m759 @ 5:31 AM

Conclusion of "The Storyteller," a story 
by Cynthia Zarin about author Madeleine L'Engle—

The New Yorker , April 12, 2004 —

Note the black diamond at the story's end.

Monday, September 17, 2012

The Count

Filed under: Uncategorized — Tags: , — m759 @ 11:01 PM

… I saw a shadow
sliding around the ropes
to get at me. The referee
moved it back, and then
went over and picked up the count.
"One!" The fog was clearing.

I rose to a knee,
and at "nine" to my feet.

— Louis Simpson, "The Appointment"

Simpson reportedly died on Holy Cross Day.

That day in this journal—

IMAGE- Log24 posts 'Please Mister Please' and 'Plan 9'

Saturday, May 26, 2012

Talk Amongst Yourselves

Filed under: Uncategorized — Tags: , — m759 @ 3:33 PM

Hard Science Fiction weekend at Dragon Press Bookstore

Saturday May 26:
11am-noon Playing with the net up:
Hard Science Fiction in the era of
short attention spans, crowd-sourcing,
and rapid obsolescence
( Greg Benford, James Cambias, Kathryn Cramer)
3pm-4:30 Technological optimism and pessimism;
utopia and dystopia; happy endings & sad endings:
what do these oppositions have to do with one another?
Are they all the same thing? How are they different
from one another? Group discussion.

My own interests in this area include…

(Click image for some context)

IMAGE- 'The Stars My Destination' (with cover slightly changed)

    The above was adapted from a 1996 cover

IMAGE- PyrE on the 1996 Vintage Books cover of 'The Stars My Destination'

 Vintage Books, July 1996. Cover: Evan Gaffney.

For the significance of the flames, 
see PyrE in the book. For the significance
of the cube in the altered cover, see
The 2×2×2 Cube and The Diamond Archetype.

Friday, December 10, 2010

Cruel Star, Part II

Filed under: Uncategorized — Tags: — m759 @ 2:00 PM

Symmetry, Duality, and Cinema

— Title of a Paris conference held June 17, 2010

From that conference, Edward Frenkel on symmetry and duality

"Symmetry plays an important role in geometry, number theory, and quantum physics. I will discuss the links between these areas from the vantage point of the Langlands Program. In this context 'duality' means that the same theory, or category, may be described in two radically different ways. This leads to many surprising consequences."

Related material —


See also  "Black Swan" in this journal, Ingmar Bergman's production of Yukio Mishima's "Madame de Sade," and Duality and Symmetry, 2001.

This journal on the date of the Paris conference
had a post, "Nighttown," with some remarks about
the duality of darkness and light. Its conclusion—

"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

Thursday, December 9, 2010

Cruel Star

Filed under: Uncategorized — Tags: — m759 @ 4:23 AM

This morning's New York Times  obituaries describe a memoir titled "Under a Cruel Star."

This is not  the story of Kayshonne Insixieng May, who appears with mathematics professor Edward Frenkel in his recent homage to Yukio Mishima, "Rites of Love and Math." (See press kit pdf.)


Mathematics Professor Edward Frenkel

For further details, see yesterday's East Bay Express

Erotica, Intrigue, and Arithmetic in 'Rites of Love and Math'
Berkeley professor Edward Frenkel brings his passion for math
to the masses — by starring in an erotic film.

Professor Frenkel also appears in last Saturday's post "Forgive Us Our Transgressions."

Related material —

“I carry the past inside me like an accordion, like a book of picture postcards that people bring home as souvenirs from foreign cities, small and neat,” she wrote in her memoir. “But all it takes is to lift one corner of the top card for an endless snake to escape, zigzag joined to zigzag, the sign of the viper, and instantly all the pictures line up before my eyes.”

Today's New York Times  on Heda Kovaly, author of Under a Cruel Star

See also the endless snake in a post from last Sunday, the day of Kovaly's death.

Tuesday, February 24, 2009

Tuesday February 24, 2009

Filed under: Uncategorized — Tags: — m759 @ 1:00 PM
Hollywood Nihilism
Pantheistic Solipsism

Tina Fey to Steve Martin
at the Oscars:
"Oh, Steve, no one wants
 to hear about our religion
… that we made up."

Tina Fey and Steve Martin at the 2009 Oscars

From Wallace Stevens: A World of Transforming Shapes, by Alan D. Perlis, Bucknell University Press, 1976, p. 117:

… in 'The Pediment of Appearance,' a slight narrative poem in Transport to Summer

 A group of young men enter some woods 'Hunting for the great ornament, The pediment of appearance.' Though moving through the natural world, the young men seek the artificial, or pure form, believing that in discovering this pediment, this distillation of the real, they will also discover the 'savage transparence,' the rude source of human life. In Stevens's world, such a search is futile, since it is only through observing nature that one reaches beyond it to pure form. As if to demonstrate the degree to which the young men's search is misaligned, Stevens says of them that 'they go crying/The world is myself, life is myself,' believing that what surrounds them is immaterial. Such a proclamation is a cardinal violation of Stevens's principles of the imagination.

Superficially the young men's philosophy seems to resemble what Wikipedia calls "pantheistic solipsism"– noting, however, that "This article has multiple issues."

As, indeed, does pantheistic solipsism– a philosophy (properly called "eschatological pantheistic multiple-ego solipsism") devised, with tongue in cheek, by science-fiction writer Robert A. Heinlein.

Despite their preoccupation with solipsism, Heinlein and Stevens point, each in his own poetic way, to a highly non-solipsistic topic from pure mathematics that is, unlike the religion of Martin and Fey, not made up– namely, the properties of space.


"Sharpie, we have condensed six dimensions into four, then we either work by analogy into six, or we have to use math that apparently nobody but Jake and my cousin Ed understands. Unless you can think of some way to project six dimensions into three– you seem to be smart at such projections."
    I closed my eyes and thought hard. "Zebbie, I don't think it can be done. Maybe Escher could have done it."


A discussion of Stevens's late poem "The Rock" (1954) in Wallace Stevens: A World of Transforming Shapes, by Alan D. Perlis, Bucknell University Press, 1976, p. 120:

For Stevens, the poem "makes meanings of the rock." In the mind, "its barrenness becomes a thousand things/And so exists no more." In fact, in a peculiar irony that only a poet with Stevens's particular notion of the imagination's function could develop, the rock becomes the mind itself, shattered into such diamond-faceted brilliance that it encompasses all possibilities for human thought:

The rock is the gray particular of man's life,
The stone from which he rises, up—and—ho,
The step to the bleaker depths of his descents ...

The rock is the stern particular of the air,
The mirror of the planets, one by one,
But through man's eye, their silent rhapsodist,

Turquoise the rock, at odious evening bright
With redness that sticks fast to evil dreams;
The difficult rightness of half-risen day.

The rock is the habitation of the whole,
Its strength and measure, that which is near,
     point A
In a perspective that begins again

At B: the origin of the mango's rind.

                    (Collected Poems, 528)

Stevens's rock is associated with empty space, a concept that suggests "nothingness" to one literary critic:

B. J. Leggett, "Stevens's Late Poetry" in The Cambridge Companion to Wallace Stevens— On the poem "The Rock":

"… the barren rock of the title is Stevens's symbol for the nothingness that underlies all existence, 'That in which space itself is contained'….  Its subject is its speaker's sense of nothingness and his need to be cured of it."

This interpretation might appeal to Joan Didion, who, as author of the classic novel Play It As It Lays, is perhaps the world's leading expert on Hollywood nihilism.

More positively…

Space is, of course, also a topic
in pure mathematics…
For instance, the 6-dimensional
affine space
(or the corresponding
5-dimensional projective space)

The 4x4x4 cube

over the two-element Galois field
can be viewed as an illustration of
Stevens's metaphor in "The Rock."

Heinlein should perhaps have had in mind the Klein correspondence when he discussed "some way to project six dimensions into three." While such a projection is of course trivial for anyone who has taken an undergraduate course in linear algebra, the following remarks by Philippe Cara present a much more meaningful mapping, using the Klein correspondence, of structures in six (affine) dimensions to structures in three.


Philippe Cara on the Klein correspondence
Here the 6-dimensional affine
space contains the 63 points
of PG(5, 2), plus the origin, and
the 3-dimensional affine
space contains as its 8 points
Conwell's eight "heptads," as in
Generating the Octad Generator.

Saturday, May 12, 2007

Saturday May 12, 2007

Filed under: Uncategorized — Tags: — m759 @ 11:07 AM
Artistic Vision

Last night's entry "A Midrash for Hollywood" discussed a possible interpretation of yesterday's Pennsylvania Lottery numbers– mid-day 384, evening 952.

In memory of a blacklisted Hollywood screenwriter who died yesterday, here is another interpretation of those numbers.

First, though, it seems appropriate to quote again the anonymous source from "Heaven, Hell, and Hollywood" on screenwriters– "You can be replaced by some Ping Pong balls and a dictionary."  An example was given illustrating this saying.  Here is another example:

Yesterday's PA lottery numbers in the dictionary–

Webster's New World Dictionary,
College Edition, 1960–

Page 384: "Defender of the Faith"
Related Log24 entries:
"To Announce a Faith," Halloween 2006,
and earlier Log24 entries from
that year's Halloween season

Page 952: "monolith"
Related Log24 entries:
"Shema, Israel," and "Punch Line"
(with the four entries that preceded it).

It may not be entirely irrelevant that a headline in last night's entry– "Lonesome No More!"– was linked to a discussion of Kurt Vonnegut's Slapstick, that a film version of that novel starred Jerry Lewis, and that yesterday afternoon's entry quoted a vision of "an Ingmar Bergman script as directed by Jerry Lewis."


See also April 7, 2003:


April is Math Awareness Month.
This year's theme is "mathematics and art."

"Art isn't easy."
— Stephen Sondheim    

Wednesday, August 30, 2006

Wednesday August 30, 2006

Filed under: Uncategorized — m759 @ 10:07 AM
The Seventh Symbol:

A Multicultural Farewell

to a winner of the
Nobel Prize for Literature,
the Egyptian author of
The Seventh Heaven:
Supernatural Stories

The image “http://www.log24.com/theory/images/GF64-63cycleA495.gif” cannot be displayed, because it contains errors.

The image “http://www.log24.com/log/pix06A/060830-SeventhSymbol.jpg” cannot be displayed, because it contains errors.

"Jackson has identified
the seventh symbol."

Other versions of
the seventh symbol —

Chinese version:

The image “http://www.log24.com/log/pix06A/060830-hexagram20.gif” cannot be displayed, because it contains errors.

pictorial version:

The image “http://www.log24.com/log/pix06A/060830-Box.jpg” cannot be displayed, because it contains errors.

algebraic version:

The image “http://www.log24.com/log/pix06A/060830-Algebra.jpg” cannot be displayed, because it contains errors.

"… Max Black, the Cornell philosopher, and others have pointed out how 'perhaps every science must start with metaphor and end with algebra, and perhaps without the metaphor there would never have been any algebra' …."

— Max Black, Models and Metaphors, Cornell U. Press, 1962, page 242, as quoted in Dramas, Fields, and Metaphors, by Victor Witter Turner, Cornell U. Press, paperback, 1975, page 25

Sunday, August 27, 2006

Sunday August 27, 2006

Filed under: Uncategorized — m759 @ 4:00 PM
Today’s Saint:

Philosopher Max Black,
who died on this date
in 1988

“… Max Black, the Cornell philosopher, and others have pointed out how ‘perhaps every science must start with metaphor and end with algebra, and perhaps without the metaphor there would never have been any algebra’ ….”

— Max Black, Models and Metaphors, Cornell U. Press, 1962, page 242, as quoted in Dramas, Fields, and Metaphors, by Victor Witter Turner, Cornell U. Press, paperback, 1975, page 25

Sunday, March 12, 2006

Sunday March 12, 2006

Filed under: Uncategorized — m759 @ 1:00 PM

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.”

The image “http://www.log24.com/log/pix06/060312-PaulBamberg21.jpg” cannot be displayed, because it contains errors.

Paul Bamberg

Transcript of the movie “Proof”–

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.

They’re all in the math department.
They’re good.
They have this song called “i.”
You’d like it. Lowercase i.
They just stand there.
They don’t play anything for three minutes.

Imaginary number?

It’s a math joke.
You see why they’re way down the bill.

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 The image “http://www.log24.com/log/pix06/060312-Char-pi.jpg” cannot be displayed, because it contains errors. 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 The image “http://www.log24.com/log/pix06/060312-Char-pi.jpg” cannot be displayed, because it contains errors. is injective.  In other words, The image “http://www.log24.com/log/pix06/060312-Char-pi.jpg” cannot be displayed, because it contains errors. can be identified algebraically with X, the variable par excellence.33

The image “http://www.log24.com/log/pix06/060312-X.jpg” cannot be displayed, because it contains errors.

More interestingly, one can ask what kind of object The image “http://www.log24.com/log/pix06/060312-Char-pi.jpg” cannot be displayed, because it contains errors. was before the formal definition of real numbers. To assume the real numbers were there all along, waiting to be defined, is to adhere to a form of Platonism.34  Dedekind wouldn’t have agreed.35  In a debate marked by the accusation that postmodern writers deny the reality of the external world, it is a peculiar move, to say the least, to make mathematical Platonism a litmus test for rationality.36 Not that it makes any more sense simply to declare Platonism out of bounds, like Lévy-Leblond, who calls Stephen Weinberg’s gloss on Sokal’s comment ‘une absurdité, tant il est clair que la signification d’un concept quelconque est évidemment affectée par sa mise en oeuvre dans un contexte nouveau!’37 Now I find it hard to defend Platonism with a straight face, and I prefer to regard the formula

The image “http://www.log24.com/log/pix06/060312-pi.jpg” cannot be displayed, because it contains errors.

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 The image “http://www.log24.com/log/pix06/060312-Char-pi.jpg” cannot be displayed, because it contains errors. of Euclid and the G of Newton, formerly thought to be constant and universal, are now perceived in their ineluctable historicity.’  But there is no need to invoke non-Euclidean geometry to perceive the historicity of the circle, or of pi: see Catherine Goldstein’s ‘L’un est l’autre: pour une histoire du cercle,’ in M. Serres, Elements d’histoire des sciences, Bordas, 1989, pp. 129-149.
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 The image “http://www.log24.com/log/pix06/060312-Char-pi.jpg” cannot be displayed, because it contains errors. would be a physical constant, like the mass of the electron, that can be determined experimentally with increasing accuracy, say by measuring physical circles with ever more sensitive rulers. This sort of position has not been welcomed by most French mathematicians.
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).

Saturday, June 4, 2005

Saturday June 4, 2005

Filed under: Uncategorized — m759 @ 7:00 PM
  Drama of the Diagonal
   The 4×4 Square:
  French Perspectives

The image “http://www.log24.com/log/pix05A/050604-Fuite1.jpg” cannot be displayed, because it contains errors.
   Les Anamorphoses:
   The image “http://www.log24.com/log/pix05A/050604-DesertSquare.jpg” cannot be displayed, because it contains errors.
  “Pour construire un dessin en perspective,
   le peintre trace sur sa toile des repères:
   la ligne d’horizon (1),
   le point de fuite principal (2)
   où se rencontre les lignes de fuite (3)
   et le point de fuite des diagonales (4).”
  Serge Mehl,
   Perspective &
  Géométrie Projective:
   “… la géométrie projective était souvent
   synonyme de géométrie supérieure.
   Elle s’opposait à la géométrie
   euclidienne: élémentaire
  La géométrie projective, certes supérieure
   car assez ardue, permet d’établir
   de façon élégante des résultats de
   la géométrie élémentaire.”
  Finite projective geometry
  (in particular, Galois geometry)
   is certainly superior to
   the elementary geometry of
  quilt-pattern symmetry
  and allows us to establish
   de façon élégante
   some results of that
   elementary geometry.
  Other Related Material…
   from algebra rather than
   geometry, and from a German
   rather than from the French:  

This is the relativity problem:
to fix objectively a class of
equivalent coordinatizations
and to ascertain
the group of transformations S
mediating between them.”
— Hermann Weyl,
The Classical Groups,
Princeton U. Press, 1946

The image “http://www.log24.com/log/pix05/050124-galois12s.jpg” cannot be displayed, because it contains errors.

Evariste Galois

 Weyl also says that the profound branch
of mathematics known as Galois theory

   “… is nothing else but the
   relativity theory for the set Sigma,
   a set which, by its discrete and
    finite character, is conceptually
   so much simpler than the
   infinite set of points in space
   or space-time dealt with
   by ordinary relativity theory.”
  — Weyl, Symmetry,
   Princeton U. Press, 1952
   Metaphor and Algebra  

“Perhaps every science must
start with metaphor
and end with algebra;
and perhaps without metaphor
there would never have been
any algebra.” 

   — attributed, in varying forms, to
   Max Black, Models and Metaphors, 1962

For metaphor and
algebra combined, see  

  “Symmetry invariance
  in a diamond ring,”

  A.M.S. abstract 79T-A37,
Notices of the
American Mathematical Society,
February 1979, pages A-193, 194 —
the original version of the 4×4 case
of the diamond theorem.

More on Max Black

“When approaching unfamiliar territory, we often, as observed earlier, try to describe or frame the novel situation using metaphors based on relations perceived in a familiar domain, and by using our powers of association, and our ability to exploit the structural similarity, we go on to conjecture new features for consideration, often not noticed at the outset. The metaphor works, according to Max Black, by transferring the associated ideas and implications of the secondary to the primary system, and by selecting, emphasising and suppressing features of the primary in such a way that new slants on it are illuminated.”

— Paul Thompson, University College, Oxford,
    The Nature and Role of Intuition
     in Mathematical Epistemology

  A New Slant…  

That intuition, metaphor (i.e., analogy), and association may lead us astray is well known.  The examples of French perspective above show what might happen if someone ignorant of finite geometry were to associate the phrase “4×4 square” with the phrase “projective geometry.”  The results are ridiculously inappropriate, but at least the second example does, literally, illuminate “new slants”– i.e., diagonals– within the perspective drawing of the 4×4 square.

Similarly, analogy led the ancient Greeks to believe that the diagonal of a square is commensurate with the side… until someone gave them a new slant on the subject.

Wednesday, March 9, 2005

Wednesday March 9, 2005

Filed under: Uncategorized — Tags: — m759 @ 4:02 AM
Women's History Month,

American Activities

Col. Mary A. Hallaren,
a much-decorated WW II veteran and
head of the Women's Army Corps,
died on Feb. 13, 2005.


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             U.S. Army Photo
Col. Mary A. Hallaren in 1950.

Happy Year of the Rooster.

"The entertaining script was adapted from the novel by Charles Portis, by well-known, long time writer, Marguerite Roberts who liked to write scripts for tough men. She wrote scripts for MGM in the '30's, '40's, until she was blacklisted in 1952, for not revealing names to The Committee on Un-American Activities."

Monday, January 24, 2005

Monday January 24, 2005

Filed under: Uncategorized — Tags: — m759 @ 2:45 PM

Old School Tie

From a review of A Beautiful Mind:

"We are introduced to John Nash, fuddling flat-footed about the Princeton courtyard, uninterested in his classmates' yammering about their various accolades. One chap has a rather unfortunate sense of style, but rather than tritely insult him, Nash holds a patterned glass to the sun, [director Ron] Howard shows us refracted patterns of light that take shape in a punch bowl, which Nash then displaces onto the neckwear, replying, 'There must be a formula for how ugly your tie is.' "

The image “http://www.log24.com/log/pix05/050124-Tie.gif” cannot be displayed, because it contains errors.

"Three readings of diamond and box
have been extremely influential."

Draft of
Computing with Modal Logics
(pdf), by Carlos Areces
and Maarten de Rijke

"Algebra in general is particularly suited for structuring and abstracting. Here, structure is imposed via symmetries and dualities, for instance in terms of Galois connections….

… diamonds and boxes are upper and lower adjoints of Galois connections…."

— "Modal Kleene Algebra
and Applications: A Survey"
(pdf), by Jules Desharnais,
Bernhard Möller, and
Georg Struth, March 2004
See also
Galois Correspondence

The image “http://www.log24.com/log/pix05/050124-galois12s.jpg” cannot be displayed, because it contains errors.

Evariste Galois

and Log24.net, May 20, 2004:

"Perhaps every science must
start with metaphor
and end with algebra;
and perhaps without metaphor
there would never have been
any algebra."

— attributed, in varying forms
(1, 2, 3), to Max Black,
Models and Metaphors, 1962

For metaphor and
algebra combined, see

"Symmetry invariance
in a diamond ring,"

A.M.S. abstract 79T-A37,
Notices of the Amer. Math. Soc.,
February 1979, pages A-193, 194 —
the original version of the 4×4 case
of the diamond theorem.

Thursday, May 20, 2004

Thursday May 20, 2004

Filed under: Uncategorized — m759 @ 7:00 AM


“A comparison or analogy. The word is simply a transliteration of the Greek word: parabolé (literally: ‘what is thrown beside’ or ‘juxtaposed’), a term used to designate the geometric application we call a ‘parabola.’….  The basic parables are extended similes or metaphors.”


“If one style of thought stands out as the most potent explanation of genius, it is the ability to make juxtapositions that elude mere mortals.  Call it a facility with metaphor, the ability to connect the unconnected, to see relationships to which others are blind.”

Sharon Begley, “The Puzzle of Genius,” Newsweek magazine, June 28, 1993, p. 50

“The poet sets one metaphor against another and hopes that the sparks set off by the juxtaposition will ignite something in the mind as well. Hopkins’ poem ‘Pied Beauty’ has to do with ‘creation.’ “

Speaking in Parables, Ch. 2, by Sallie McFague

“The Act of Creation is, I believe, a more truly creative work than any of Koestler’s novels….  According to him, the creative faculty in whatever form is owing to a circumstance which he calls ‘bisociation.’ And we recognize this intuitively whenever we laugh at a joke, are dazzled by a fine metaphor, are astonished and excited by a unification of styles, or ‘see,’ for the first time, the possibility of a significant theoretical breakthrough in a scientific inquiry. In short, one touch of genius—or bisociation—makes the whole world kin. Or so Koestler believes.”

— Henry David Aiken, The Metaphysics of Arthur Koestler, New York Review of Books, Dec. 17, 1964

For further details, see

Speaking in Parables:
A Study in Metaphor and Theology

by Sallie McFague

Fortress Press, Philadelphia, 1975

Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7

“Perhaps every science must start with metaphor and end with algebra; and perhaps without metaphor there would never have been any algebra.”

— attributed, in varying forms (1, 2, 3), to Max Black, Models and Metaphors, 1962

For metaphor and algebra combined, see

“Symmetry invariance in a diamond ring,” A.M.S. abstract 79T-A37, Notices of the Amer. Math. Soc., February 1979, pages A-193, 194 — the original version of the 4×4 case of the diamond theorem.

Thursday, March 13, 2003

Thursday March 13, 2003

Filed under: Uncategorized — Tags: — m759 @ 5:24 AM

Death Knell

In memory of Howard Fast, novelist and Jewish former Communist,
who died yesterday, a quotation:

"For many of us, the geometry course sounded the death knell
for our progress — and interest — in mathematics."

— "Shape and Space in Geometry"

© 1997-2003 Annenberg/CPB. All rights reserved.
Legal Policy

See also
Geometry for Jews.

Added March 16, 2003: See, too, the life of
John Sanford, blacklisted Jewish writer,
who died on March 6, 2003 —
Michelangelo's birthday and the date of
Geometry for Jews."

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