Monday, February 27, 2017

Logic for Jews

Filed under: General — Tags: — m759 @ 9:07 PM


Adam Gopnik in The New Yorker  today reacts to the startling
outcomes of three recent contests: the presidential election,
the Super Bowl, and the Oscar for Best Picture —

"The implicit dread logic is plain."

Related material —

Transformers in this journal and

“Lord Arglay had a suspicion that the Stone would be
purely logical.  Yes, he thought, but what, in that sense,
were the rules of its pure logic?”

Many Dimensions  (1931), by Charles Williams

See also

The above figure is from Ian Stewart's 1996 revision of a 1941 classic, 
What Is Mathematics? , by Richard Courant and Herbert Robbins.

One wonders how the confused slave boy of Plato's Meno  would react
to Stewart's remark that

"The number of copies required to double an
 object's size depends on its dimension."

Saturday, February 28, 2015

Logical Loop

Filed under: General — Tags: — m759 @ 10:45 PM

In memory of Theodore SturgeonLeonard Nimoy,
and William Thomas McKinley —

From the Boston Modern Orchestra Project today :

"In a good way"

Or not.

Sunday, November 24, 2013

Logic for Jews*

Filed under: General,Geometry — 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.

Thursday, December 15, 2011

Logic for Jews

Filed under: General — Tags: — m759 @ 11:02 PM

New York Daily News , 2:55 PM EST today—

Joe Simon, who dreamed up the star-spangled super hero Captain America while riding on a Manhattan bus during the early days of World War II, died Thursday [Dec. 15] after an undisclosed illness. He was 98.

New York Times , about 10 PM EST today—

Joe Simon, a writer, editor and illustrator of comic books who was a co-creator of the superhero Captain America, conceived out of a patriotic impulse as war was roiling Europe, died on Wednesday [Dec. 14] at his home in Manhattan. He was 98.

The discrepancy is perhaps due to initial reports that quoted Simon's family as saying he died "Wednesday night."

Simon was a co-creator of Captain America. For some background on Simon and a photo with his fellow comic artist Jerry Robinson, co-creator of The Joker, see a Washington Post article from this afternoon. Robinson died on either Wednesday, Dec. 7, or Thursday, Dec. 8, 2011.

Los Angeles Times

Jerry Robinson, a pioneer in the early days of Batman comics and a key force in the creation of Robin the Boy Wonder; the Joker;  Bruce Wayne’s butler, Alfred; and Two-Face, died Wednesday afternoon [Dec. 7] in New York City. He was 89.


Cartoonist Jerry Robinson, who worked on the earliest Batman comics and claimed credit for creating the super-villain The Joker, died Thursday [Dec. 8] at the age of 89, his family confirmed.

A picture by Robinson—

IMAGE- The Joker with calendar page for November 27

The Joker in January 1943
with a Nov. 27 calendar page

A non-joke from a more recent November 27—

Simplex Sigillum Veri

"Oh, what a tangled web we weave…"

Sunday, October 20, 2019

MSRI (Pronounced “Misery”)

Filed under: General — Tags: — m759 @ 10:29 AM

Friday, October 11, 2019

Dick Date (YouTube, August 7, 2013)

Filed under: General — Tags: — m759 @ 3:47 PM

Down the Rabbit Hole  with Stephen King

Wednesday, October 9, 2019

Philosophical Infanticide

Filed under: General — Tags: — m759 @ 5:51 AM

From Wallace Stevens —

"Reality is the beginning not the end,
Naked Alpha, not the hierophant Omega,
Of dense investiture, with luminous vassals."

— “An Ordinary Evening in New Haven” VI

From The Point  magazine yesterday, October 8, 2019
Parricide:  On Irad Kimhi's Thinking and Being .
Book review by Steven Methven.

The conclusion:

"Parricide is nothing that the philosopher need fear . . . .
What sustains can be no threat. Perhaps what the
unique genesis of this extraordinary work suggests is that
the true threat to philosophy is infanticide."

This remark suggests revisiting a post from Monday

Monday, October 7, 2019

Berlekamp Garden vs. Kinder Garten

Filed under: General — m759 @ 11:00 PM

Stevens's Omega and Alpha (see previous post)
suggest a review.

Omega — The Berlekamp Garden. 
                  See Misère Play (April 8, 2019).
Alpha  —  The Kinder Garten. 
                  See Eighfold Cube.

. . . .

Monday, October 7, 2019


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

(A sequel to Simplex Sigillum Veri and 
Rabbit Hole Meets Memory Hole)

" Wittgenstein does not, however, relegate all that is not inside the bounds
of sense to oblivion. He makes a distinction between saying  and showing  
which is made to do additional crucial work. 'What can be shown cannot
be said,' that is, what cannot be formulated in sayable (sensical)
propositions can only be shown. This applies, for example, to the logical
form of the world, the pictorial form, etc., which show themselves in the
form of (contingent) propositions, in the symbolism, and in logical
propositions. Even the unsayable (metaphysical, ethical, aesthetic)
propositions of philosophy belong in this group — which Wittgenstein
finally describes as 'things that cannot be put into words. They make
themselves manifest. They are what is mystical' " (Tractatus  6.522).

Stanford Encyclopedia of Philosophy , "Ludwig Wittgenstein"

From Tractatus Logico-Philosophicus  by Ludwig Wittgenstein.

(First published in Annalen der Naturphilosophie ,1921.
English edition first published 1922 by Kegan Paul, Trench and Trübner. This translation first published 1961 by Routledge & Kegan Paul. Revised edition 1974.)


The solutions of the problems of logic must be simple, since they set the standard of simplicity.

Men have always had a presentiment that there must be a realm in which the answers to questions are symmetrically combined — a priori — to form a self-contained system.

A realm subject to the law: Simplex sigillum veri.

Somehow, the old Harvard seal, with its motto "Christo et Ecclesiae ,"
was deleted from a bookplate in an archived Harvard copy of Whitehead's 
The Axioms of Projective Geometry  (Cambridge U. Press, 1906).

In accordance with Wittgenstein's remarks above, here is a new
bookplate seal for Whitehead, based on a simplex

Saturday, October 5, 2019

Midnight Landmarks

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

Monday, September 23, 2019

Rabbit Hole Meets Memory Hole:

Filed under: General — Tags: , , — m759 @ 9:11 AM

The disappearance of "Christo et Ecclesiae" at Harvard

Rabbit Hole 

Memory Hole

The above Harvard seal in a PDF —

The same page, minus the seal, today at the Internet Archive — 

For a larger image of the seal-less page, click here.

Happy Fall 2019!

Click to enlarge.

Sunday, September 22, 2019

Colorful Tale

Filed under: General — Tags: , — m759 @ 7:59 PM

“Perhaps the philosophically most relevant feature of modern science
is the emergence of abstract symbolic structures as the hard core
of objectivity behind— as Eddington puts it— the colorful tale of
the subjective storyteller mind.”

— Hermann Weyl, Philosophy of  Mathematics and
    Natural Science 
, Princeton, 1949, p. 237

"The bond with reality is cut."

— Hans Freudenthal, 1962

Indeed it is.

From page 180, Logicomix — It was a dark and stormy night


Saturday, September 15, 2018


Filed under: G-Notes,General,Geometry — Tags: , — m759 @ 9:50 AM

Tieszen— 'Kurt Godel and Phenomenology' — 1992

Update of 10:18 AM the same day —

See also Logicomix  in this  journal and, at Harvard,


  • September 6, 2018:  Eric Maskin, Amartya Sen and I
    are giving a course this semester: 'Axiomatic Reasoning'
    (PHIL 273B). Introduction to Axiomatic Reasoning gives a
    general sense of what we intend to cover.

Update of 10:48 AM the same day —


See Log24 on the date of Tieszen's death.

Wednesday, September 20, 2017

The Imaginary Professor

Filed under: General — Tags: — m759 @ 8:35 PM
- I was a teacher.
- You're being modest, aren't you?
  You were a professor at Boston University...
  Isn't that right?
- Yes, well, assistant professor.
- And what'd you teach?
- Philosophy. Truth and logic. 
  That sort of thing.

Read more: 

Compare and contrast with a real  Boston University professor,
John Stachel, quoted here on Sept. 5, 2017.

Tuesday, September 5, 2017

Annals of Critical Epistemology

Filed under: General,Geometry — Tags: — m759 @ 5:36 PM

"But unlike many who left the Communist Party, I turned left
rather than right, and returned—or rather turned for the first time—
to a critical examination of Marx's work. I found—and still find—
that his analysis of capitalism, which for me is the heart of his work,
provides the best starting point, the best critical tools, with which—
suitably developed—to understand contemporary capitalism.
I remind you that this year is also the sesquicentennial of the
Communist Manifesto , a document that still haunts the capitalist world."

— From "Autobiographical Reflections," a talk given on June 5, 1998, by
John Stachel at the Max Planck Institute for the History of Science in Berlin
on the occasion of a workshop honoring his 70th birthday, 
"Space-Time, Quantum Entanglement and Critical Epistemology."

From a passage by Stachel quoted in the previous post

From the source for Stachel's remarks on Weyl and coordinatization —

Note that Stachel distorted Weyl's text by replacing Weyl's word 
"symbols" with the word "quantities." —

This replacement makes no sense if the coordinates in question
are drawn from a Galois field — a field not of quantities , but rather
of algebraic symbols .

"You've got to pick up every stitch… Must be the season of the witch."
— Donovan song at the end of Nicole Kidman's "To Die For"

Florence 2001

Filed under: General,Geometry — Tags: — m759 @ 4:44 AM

Or:  Coordinatization for Physicists

This post was suggested by the link on the word "coordinatized"
in the previous post.

I regret that Weyl's term "coordinatization" perhaps has
too many syllables for the readers of recreational mathematics —
for example, of an article on 4×4 magic squares by Conway, Norton,
and Ryba to be published today by Princeton University Press.

Insight into the deeper properties of such squares unfortunately
requires both the ability to learn what a "Galois field" is and the
ability to comprehend seven-syllable words.

Monday, September 4, 2017

Identity Revisited

Filed under: General — Tags: — m759 @ 8:00 PM

From the Log24 post "A Point of Identity" (August 8, 2016) —

A logo that may be interpreted as one-eighth of a 2x2x2 array
of cubes —

The figure in white above may be viewed as a subcube representing,
when the eight-cube array is coordinatized, the identity (i.e., (0, 0, 0)).

Tuesday, December 20, 2016

The Project’s Central Problem

Filed under: General — Tags: — m759 @ 10:00 AM

From page 180, Logicomix —


Alfred North Whitehead in the first of
the above-named years, 1906 —

"But the project's central problem was always there."

"The deeper we got into our Quest…
The more I doubted its premises."

— Attributed to Bertrand Russell
by Apostolos Doxiadis and Christos
Papadimitriou in Logicomix  (2008-9)

Wednesday, April 8, 2015

The Crosswicks Curse

Filed under: General — Tags: , — m759 @ 8:08 PM


It was a dark and stormy night 


— Page 180, Logicomix

Saturday, February 28, 2015

Elegy with Stars

Filed under: General — Tags: , — m759 @ 10:00 PM

This evening's New York Times —

"William Thomas McKinley, a prolific American composer
whose music was infused with the jazz he had performed
since childhood, died on Feb. 3 at his home in Reading,
Mass. He was 76.

He died in his sleep, his son Elliott said."

"William Thomas McKinley: Elegy for Strings (2006)

[Elliott McKinley]  

137 views as of 9:45 PM ET Feb. 28, 2015

Published on Feb 11, 2015

Composed as an elegy and tribute for friends and family
that have passed, spurred by the passing of McKinley's
long time friend, drummer Roger Ryan. The performance
heard here is by the Seattle Symphony under the direction
of Gerard Schwarz. 

Photos by Elliott McKinley (Rho Ophiuchi nebula complex…
and the Pleiades…) shot at Cherry Springs State Park."

Related material from the date of McKinley's death —
Expanding the Spielraum.

Recycled Religion

Filed under: General — Tags: , — m759 @ 3:16 PM

The previous post's Kirkridge link leads to
a mention of religious philosopher Parker J. Palmer.

From an Utne Reader  page on Palmer:

See also Theodore Sturgeon's 1949 story "What Dead Men Tell"—

"… He’d read about it in a magazine or somewhere.
He took a strip of scrap film about eighteen
inches long and put the ends together. He turned
one end over and spliced ’em. Now, if you trace
that strip, or mark it with a grease pencil, right up
the center, you find that the doggone thing only
has one side!”
The doctor nodded, and the girl said:
“A Möbius strip.”
“That what they call it?” said Hulon. “Well, I figured
this corridor must be something like that. On that
strip, a single continuous line touched both sides.
All I had to do was figure out an object built so that
a continuous line would cover all three of three sides,
and I’d have it. So I sat down and thought it out…."

— and the following mathematical illustration —

A Kirk for Spock

Filed under: General — Tags: , — m759 @ 2:09 PM

Related material:

What Dead Men Tell*

Filed under: General — Tags: , — m759 @ 1:23 PM

Theodore Sturgeon, 1949 :

"I thought I had an important idea.
It's part of a  call it a philosophy,
if that doesn't sound too high-
falutin'," he said.

"It's a philosophy," she said.
"We can call things by their names."

Leonard Nimoy,  2015 :

"A life is like a garden. 
Perfect moments can be had, 
but not preserved, except in memory."  

* A tale from Astounding Science Fiction
   Vol. 44, No. 3, November 1949

Friday, February 27, 2015

Final Tweet

Filed under: General — Tags: , — m759 @ 1:44 PM

"Final Tweet Will Make You Cry"

Or not.

Wednesday, May 21, 2014

Through the Vanishing Point*

Filed under: General,Geometry — Tags: , , — m759 @ 9:48 AM

Marshall McLuhan in "Annie Hall" —

"You know nothing of my work."

Related material — 

"I need a photo opportunity
I want a shot at redemption
Don't want to end up a cartoon
In a cartoon graveyard"

— Paul Simon

It was a dark and stormy night…


— Page 180, Logicomix

A photo opportunity for Whitehead
(from Romancing the Cube, April 20, 2011)—

IMAGE- Whitehead on Fano's construction of the 15-point projective Galois space over GF(2)

See also Absolute Ambition (Nov. 19, 2010).

* For the title, see Vanishing Point in this journal.

Friday, June 8, 2012

Cartoon Graveyard

Filed under: General — Tags: — m759 @ 2:01 AM

Whitehead and Russell, 'Logicomix' page 181

For some background, see "Cartoon Graveyard" and "Many Dimensions."

Wednesday, April 20, 2011

Romancing the Cube

Filed under: General,Geometry — Tags: — m759 @ 1:00 AM

It was a dark and stormy night…


— Page 180, Logicomix

“… the class of reflections is larger in some sense over an arbitrary field than over a characteristic zero field.”

– Julia Hartmann and Anne V. Shepler, “Jacobians of Reflection Groups

For some context, see the small cube in “A Simple Reflection Group of Order 168.”

See also the larger cube in “Many Dimensions” + Whitehead in this journal (scroll down to get past the current post).

That search refers to a work by Whitehead published in 1906, the year at the top of the Logicomix  page above—


A related remark on axiomatics that has metaphysical overtones suitable for a dark and stormy night

“An adequate understanding of mathematical identity requires a missing theory that will account for the relationships between formal systems that describe the same items. At present, such relationships can at best be heuristically described in terms that invoke some notion of an ‘intelligent user standing outside the system.'”

— Gian-Carlo Rota, “Syntax, Semantics, and…” in Indiscrete Thoughts . See also the original 1988 article.

Tuesday, January 25, 2011

Another Reappearing Number

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

(A sequel to yesterday's reappearing number)

25 —

5x5 ultra super magic square

See "Quine, Newton, logic" in this journal.

Tuesday, November 23, 2010


Filed under: General — Tags: , , — m759 @ 8:00 PM

A reviewer says Steve Martin finds in his new novel An Object of Beauty  "a sardonic morality tale."

From this journal on the day The Cube  was published (see today's Art Object ) —

Monday February 20, 2006

m759 @ 12:00 AM

The Past Revisited

From Log24 a year ago on this date, a quote from Many Dimensions  (1931), by Charles Williams:

“Lord Arglay had a suspicion that the Stone would be purely logical.  Yes, he thought, but what, in that sense, were the rules of its pure logic?”

For the rest of the story, see the downloadable version at Project Gutenberg of Australia.

See also a post on Mathematics and Narrative from Nov. 14, 2009.

That post compares characters in Many Dimensions  to those in Logicomix

Whitehead and Russell, 'Logicomix' page 181

Saturday, November 20, 2010


Filed under: General,Geometry — Tags: — m759 @ 4:00 AM

An Epic Search for Truth

— Subtitle of Logicomix , a work reviewed in the December 2010 Notices of the American Mathematical Society  (see previous post).

Some future historian of mathematics may contrast the lurid cover of the December 2010 Notices


Excerpts from Logicomix

with the 1979 cover found in a somewhat less epic search —


Larger view of Google snippet —


For some purely mathematical background, see Finite Geometry of the Square and Cube.

For some background related to searches for truth, see "Coxeter + Trudeau" in this journal.

Friday, November 19, 2010

Absolute Ambition

Filed under: General — Tags: , — m759 @ 10:30 AM

"It's my absolute ambition that you are touched to the core of your being with the content…."

— Julie Taymor on Spider-Man: Turn Off the Dark  (Playbill video, undated)

Another ambitious comic-book promotion —

"What Logicomix  does that few works in any medium do is to make intellectual passion palpable. That is its greatest strength. And it’s here that its form becomes its substance."

— Judith Roitman, review (pdf, 3.7 MB) of Logicomix: An Epic Search for Truth , in …


 The December 2010 AMS Notices  cover has excerpts from Logicomix.

Related material:

"In the classical grammarians’ sense of the power of form over 'content' and style over 'substance,' he originated the phrase, 'the medium is the message.'"

— Joseph P. Duggan on Marshall McLuhan at The University Bookman

See also, in this  journal, The Medium is the Message, Wechsler, and Blockheads .

Sunday, February 7, 2010

Today’s Sermon

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

Mathematics and Religion, continued–

Calvin Jongsma, review of an anthology titled Mathematics and the Divine

"Believers of many faiths have found significant points of contact between their religious outlooks and mathematics. Not all of these claims were made in the distant past or by certified crackpots…."

Edward Nelson in "Warning Signs of a Possible Collapse of Contemporary Mathematics"–

"The most impressive feature of Cantor’s theory is that he showed that there are different sizes of infinity, by his famous diagonal argument. But Russell applied this argument to establish his paradox: the set of all sets that are not elements of themselves both is and is not an element of itself."

Jongsma's assertion appears to be true. Nelson's appears to be false. Discuss.


Saying that someone applied some argument– any argument will do here– to establish a paradox– any paradox will do here– casts into doubt the validity of either the argument, the application of the argument, or both. In the Cantor-Russell case, such doubt is unnecessary, since the paradox is clearly independent of the diagonal argument. There is certainly an historical connection between Cantor's argument and Russell's paradox– see, for instance, Wikipedia on the latter. The historical connection is, however, not a logical connection.

For Russell discovering his paradox without the use of Cantor's diagonal argument, see Logicomix

Russell discovers his paradox

Click for some context.

Saturday, November 14, 2009

Mathematics and Narrative, continued:

Filed under: General,Geometry — Tags: , — m759 @ 10:10 PM

A graphic novel reviewed in the current Washington Post  features Alfred North Whitehead and Bertrand Russell–

Whitehead and Russell, 'Logicomix' page 181

Related material:

Whitehead on Fano’s finite projective three-space:

“This is proved by the consideration of a three dimensional geometry in which there are only fifteen points.”

The Axioms of Projective Geometry , Cambridge University Press, 1906

A related affine six-space:

Grey cube, 4x4x4

Further reading:

See Solomon’s Cube and the link at the end of today’s previous entry, then compare and contrast the above portraits of Whitehead and Russell with Charles Williams’s portraits of Sir Giles Tumulty and Lord Arglay in the novel Many Dimensions .

It was a dark and stormy night….

Friday, June 6, 2008

Friday June 6, 2008

Filed under: General,Geometry — Tags: — m759 @ 2:45 AM
The Dance of Chance

"Harvard seniors have
every right to demand a
    Harvard-calibre speaker."

— Adam Goldenberg in
The Harvard Crimson

"Look down now, Cotton Mather"

— Wallace Stevens,
Harvard College
Class of 1901

For Thursday, June 5, 2008,
commencement day for Harvard's
Class of 2008, here are the
Pennsylvania Lottery numbers:

Mid-day 025
Evening 761

Thanks to the late
Harvard professor
Willard Van Orman Quine,
the mid-day number 025
suggests the name
"Isaac Newton."

(For the logic of this suggestion,
see On Linguistic Creation
and Raiders of the Lost Matrix.)

Thanks to Google search, the
  name of Newton, combined with
  Thursday's evening number 761,
suggests the following essay:

Science 10 August 2007:
Vol. 317. no. 5839, pp. 761-762

The Cha-Cha-Cha Theory
of Scientific Discovery

Daniel E. Koshland Jr.*

* D. E. Koshland Jr. passed away on 23 July 2007. He was a professor of biochemistry and molecular and cell biology at the University of California, Berkeley, since 1965. He served as Science's editor-in-chief from 1985 to 1995.

What can a non-scientist add?

Perhaps the Log24 entries for
the date of Koshland's death:

The Philosopher's Stone
and The Rock.

Or perhaps the following

On the figure of 25 parts
discussed in
"On Linguistic Creation"–

5x5 ultra super magic square

"The Moslems thought of the
central 1 as being symbolic
of the unity of Allah.

— Clifford Pickover  

"At the still point,
there the dance is.

— T. S. Eliot,
Harvard College
Class of 1910

Thursday, June 21, 2007

Thursday June 21, 2007

Filed under: General,Geometry — Tags: — m759 @ 12:07 PM
Let No Man
Write My Epigraph

(See entries of June 19th.)

"His graceful accounts of the Bach Suites for Unaccompanied Cello illuminated the works’ structural logic as well as their inner spirituality."

Allan Kozinn on Mstislav Rostropovich in The New York Times, quoted in Log24 on April 29, 2007

"At that instant he saw, in one blaze of light, an image of unutterable conviction…. the core of life, the essential pattern whence all other things proceed, the kernel of eternity."

— Thomas Wolfe, Of Time and the River, quoted in Log24 on June 9, 2005

"… the stabiliser of an octad preserves the affine space structure on its complement, and (from the construction) induces AGL(4,2) on it. (It induces A8 on the octad, the kernel of this action being the translation group of the affine space.)"

— Peter J. Cameron, "The Geometry of the Mathieu Groups" (pdf)

"… donc Dieu existe, réponse!"

— Attributed, some say falsely,
to Leonhard Euler
"Only gradually did I discover
what the mandala really is:
'Formation, Transformation,
Eternal Mind's eternal recreation'"

(Faust, Part Two, as
quoted by Jung in
Memories, Dreams, Reflections)


Wolfgang Pauli as Mephistopheles

"Pauli as Mephistopheles
in a 1932 parody of
Goethe's Faust at Niels Bohr's
institute in Copenhagen.
The drawing is one of
many by George Gamow
illustrating the script."
Physics Today


"Borja dropped the mutilated book on the floor with the others. He was looking at the nine engravings and at the circle, checking strange correspondences between them.

'To meet someone' was his enigmatic answer. 'To search for the stone that the Great Architect rejected, the philosopher's stone, the basis of the philosophical work. The stone of power. The devil likes metamorphoses, Corso.'"

The Club Dumas, basis for the Roman Polanski film "The Ninth Gate" (See 12/24/05.)

"Pauli linked this symbolism
with the concept of automorphism."

The Innermost Kernel
 (previous entry)

And from
"Symmetry in Mathematics
and Mathematics of Symmetry
(pdf), by Peter J. Cameron,
a paper presented at the
International Symmetry Conference,
Edinburgh, Jan. 14-17, 2007,
we have

The Epigraph–

Weyl on automorphisms
(Here "whatever" should
of course be "whenever.")

Also from the
Cameron paper:

Local or global?

Among other (mostly more vague) definitions of symmetry, the dictionary will typically list two, something like this:

• exact correspondence of parts;
• remaining unchanged by transformation.

Mathematicians typically consider the second, global, notion, but what about the first, local, notion, and what is the relationship between them?  A structure M is homogeneous if every isomorphism between finite substructures of M can be extended to an automorphism of M; in other words, "any local symmetry is global."

Some Log24 entries
related to the above politically
(women in mathematics)–

Global and Local:
One Small Step

and mathematically–

Structural Logic continued:
Structure and Logic

This entry cites
Alice Devillers of Brussels–

Alice Devillers

"The aim of this thesis
is to classify certain structures
which are, from a certain
point of view, as homogeneous
as possible, that is which have
  as many symmetries as possible."

"There is such a thing
as a tesseract."

Madeleine L'Engle 

Monday, April 30, 2007

Monday April 30, 2007

Filed under: General,Geometry — Tags: — m759 @ 6:24 PM
Structure and Logic

The phrase “structural logic” in yesterday’s entry was applied to Bach’s cello suites.  It may equally well be applied to geometry.  In particular:

“The aim of this thesis is to classify certain structures which are, from a certain point of view, as homogeneous as possible, that is which have as many symmetries as possible.”

Alice Devillers, “Classification of Some Homogeneous and Ultrahomogeneous Structures,” Ph.D. thesis, Université Libre de Bruxelles, academic year 2001-2002

Related material:

New models of some small finite spaces

In Devillers’s words, the above spaces with 8 and 16 points are among those structures that have “as many symmetries as possible.” For more details on what this means, see Devillers’s thesis and Finite Geometry of the Square and Cube.

The above models for the corresponding projective spaces may be regarded as illustrating the phrase “structural logic.”

For a possible application of the 16-point space’s “many symmetries” to logic proper, see The Geometry of Logic.

Sunday, April 29, 2007

Sunday April 29, 2007

Filed under: General — Tags: — m759 @ 7:00 AM

Rostropovich at Christ the Savior Cathedral

“The coffin of the cellist and conductor Mstislav Rostropovich seen inside Christ the Savior Cathedral during a farewell ceremony in Moscow, Sunday, April 29, 2007. Hundreds of Russians on Sunday came to bid final farewell to the great cellist and conductor Mstislav Rostropovich who won world fame for his masterly play and his courage in defending human rights. Rostropovich, who fought for the rights of Soviet-era dissidents and later triumphantly played Bach suites below the crumbling Berlin Wall, died Friday at age 80. (AP Photo/Mikhail Metzel)” —AP News

“His graceful accounts of the Bach Suites for Unaccompanied Cello illuminated the works’ structural logic as well as their inner spirituality.” —Allan Kozinn in Friday’s New York Times

Sunday, March 12, 2006

Sunday March 12, 2006

Filed under: General,Geometry — 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).

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