Log24

Thursday, September 15, 2016

Metaphysics at Scientific American

Filed under: General,Geometry — Tags: — m759 @ 9:36 PM

In 2011 Scientific American  magazine ran
the following promotional piece for one of their articles —

"Why 5, 8 and 24 Are the Strangest Numbers 
in the Universe
," by Michael Moyer, "the editor
in charge of physics and space coverage."

This is notably bad metaphysics. Numbers are, of course,
not  "in  the universe" — the universe, that is, of physics.

A passage from G. H. Hardy's Mathematician's Apology 
is relevant:

The contrast between pure and applied mathematics
stands out most clearly, perhaps, in geometry.
There is the science of pure geometry, in which there
are many geometries, projective geometry, Euclidean
geometry, non-Euclidean geometry, and so forth. Each
of these geometries is a model , a pattern of ideas, and
is to be judged by the interest and beauty of its particular
pattern. It is a map  or picture , the joint product of many
hands, a partial and imperfect copy (yet exact so far as
it extends) of a section of mathematical reality. But the
point which is important to us now is this, that there is
one thing at any rate of which pure geometries are not
pictures, and that is the spatio-temporal reality of the
physical world. It is obvious, surely, that they cannot be,
since earthquakes and eclipses are not mathematical
concepts.

By an abuse of language such as Burkard Polster's
quoted in the previous post, numbers may be said to be
in  the various "universes" of pure mathematics.

The Scientific American  article above is dated May 4, 2011.
See also Thomas Mann on metaphysics in this  journal
on that date.

Friday, October 11, 2019

Quest

Filed under: General — m759 @ 3:45 AM

John Horgan in Scientific American  magazine on October 8, 2019 —

"In the early 1990s, I came to suspect that the quest
for a unified theory is religious rather than scientific.
Physicists want to show that all things came from
one thing a force, or essence, or membrane
wriggling in eleven dimensions, or something that
manifests perfect mathematical symmetry. In their
search for this primordial symmetry, however,
physicists have gone off the deep end . . . ."

Other approaches —

See "Story Theory of Truth" in this  journal and, from the November 2019  
Notices of the American Mathematical Society . . .

Story Driven

More fundamental than the label of mathematician is that of human. And as humans, we’re hardwired to use stories to make sense of our world (story-receivers) and to share that understanding with others (storytellers) [2]. Thus, the framing of any communication answers the key question, what is the story we wish to share? Mathematics papers are not just collections of truths but narratives woven together, each participating in and adding to the great story of mathematics itself.

The first endeavor for constructing a good talk is recognizing and choosing just one storyline, tailoring it to the audience at hand. Should the focus be on a result about the underlying structures of group actions? . . . .

[2] Gottschall, J. , The Storytelling Animal ,
       Houghton Mifflin Harcourt, 2012.

— "Giving Good Talks,"  by Satyan L. Devadoss

"Before time began, there was the Cube." — Optimus Prime

Sunday, September 29, 2019

Spiritual Kin

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

"The 15 Puzzle and the Magic Cube
are spiritual kin …."

"Metamagical Themas"  column,
Douglas R. Hofstadter, Scientific American ,
Vol. 244, No. 3 (March 1981), pp. 20-39

As are the 15 Schoolgirls and the Eightfold Cube.

Saturday, August 4, 2018

Manifestations of Exquisite Geometry

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

An alleged manifestation in physics, from Scientific American  —

http://www.log24.com/log/pix18/180804-Exquisite_Geometry-subhead-Sciam-500w.jpg

Manifestations in pure mathematics, from Plato and R. T. Curtis  —

Counting symmetries with the orbit-stabilizer theorem

For some entertaining literary  manifestations, see Wrinkle.

Monday, June 11, 2018

Glitter

Filed under: General,Geometry — Tags: — m759 @ 8:32 PM

A Scientific American  headline today —

Glittering Diamond Dust in Space
Might Solve a 20-Year-Old Mystery

Related art —

"Never underestimate the power of glitter."

Glitter by Josefine Lyche, as of diamond dust

Background:  "Diamond Dust" + Glitter in this journal.

Tuesday, June 6, 2017

The Table

Filed under: General,Geometry — m759 @ 12:00 PM

John Horgan and James (Jim) McClellan, according to Horgan
in Scientific American  on June 1, 2017

Me: "Jim, you're a scholar! Professor! Esteemed historian of science! And yet you don’t really believe science is capable of producing truth."

Jim: "Science is stories we tell about nature. And some stories are better than other stories. And you can compare stories to each other on all kinds of grounds, but you have no access to"— he pauses for dramatic effect— "The Truth. Or any mode of knowing outside of your own story-telling capabilities, which include rationality, experiment, explanatory scope and the whole thing. I would love to have some means of making knowledge about the world that would allow us to say, 'This is really it. There really are goddamn electrons.'" He whacks the table.

See also posts tagged Dirac and Geometry and Glitch.

Monday, March 6, 2017

Bullshit Studies

Filed under: General — m759 @ 1:19 AM

From The Chronicle of Higher Education  on March 2, 2017 —

These days, in a world totally dependent on microprocessors, lasers, and nanotechnology, it has been estimated that 30 percent of the U.S. gross national product is based on inventions made possible by quantum mechanics. With the booming high-tech industry and the expected advent of quantum computers, this percentage will only grow. Within a hundred years, an esoteric theory of young physicists became a mainstay of the modern economy.

It took nearly as long for Einstein’s own theory of relativity, first published in 1905, to be used in everyday life in an entirely unexpected way. The accuracy of the global positioning system, the space-based navigation system that provides location and time information in today’s mobile society, depends on reading time signals of orbiting satellites. The presence of Earth’s gravitational field and the movement of these satellites cause clocks to speed up and slow down, shifting them by 38 milliseconds a day. In one day, without Einstein’s theory, our GPS tracking devices would be inaccurate by about seven miles.

Robbert Dijkgraaf, Director, Institute for Advanced Study, Princeton

The above paragraphs are clearly propaganda, not physics.

For "It has been estimated," see

The "without Einstein 's theory" statement may or may not be correct.
See the lengthy discussion at

http://physics.stackexchange.com/questions/1061/
why-does-gps-depend-on-relativity
.

See also Princeton's March of Mediocrity Continues.

Thursday, September 15, 2016

Metaphysics at Notre Dame

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

Recommended reading —

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

Related material —

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

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

Thursday, December 17, 2015

Hint of Reality

Filed under: General,Geometry — Tags: , , — m759 @ 12:45 PM

From an article* in Proceedings of Bridges 2014

As artists, we are particularly interested in the symmetries of real world physical objects.

Three natural questions arise:

1. Which groups can be represented as the group of symmetries of some real-world physical object?

2. Which groups have actually  been represented as the group of symmetries of some real-world physical object?

3. Are there any glaring gaps – small, beautiful groups that should have a physical representation in a symmetric object but up until now have not?

The article was cited by Evelyn Lamb in her Scientific American  
weblog on May 19, 2014.

The above three questions from the article are relevant to a more
recent (Oct. 24, 2015) remark by Lamb:

" finite projective planes [in particular, the 7-point Fano plane,
about which Lamb is writing] 
seem like a triumph of purely 
axiomatic thinking over any hint of reality…."

For related hints of reality, see Eightfold Cube  in this journal.

* "The Quaternion Group as a Symmetry Group," by Vi Hart and Henry Segerman

Thursday, November 5, 2015

The Monster

Filed under: General,Geometry — m759 @ 9:00 AM

In memory of Princeton mathematician John Nash

"For the past six years all over the world 
experts in the branch of abstract algebra
called group theory have been struggling
to capture a group known as the monster."

—Martin Gardner, Scientific American ,  June 1980

"When the Hawkline Monster moved to get a better view
of what was happening, the shadow, after having checked
all the possibilities of light, had discovered a way that it
could shift itself in front of the monster, so that the monster
at this crucial time would be blinded by darkness for a few
seconds, did so, causing confusion to befall the monster.

This was all that the shadow could do and it hoped that this
would give Greer and Cameron the edge they would need
to destroy the Hawkline Monster using whatever plan they
had come up with, for it seemed that they must have a plan
if they were to have any chance at all with the monster and
they did not seem like fools.

When Cameron yelled at Greer, the shadow interpreted this
as the time to move and did so. It obscured the vision of the
Hawkline Monster for a few seconds, knowing full well that if
the monster were destroyed it would be destroyed, too, but
death was better than going on living like this, being a part of
this evil."

— Richard Brautigan, The Hawkline Monster , 1974

From the post For Scientific Witch Hunters of October 30,
an illustration from The Boston Globe —

From the post Colorful Story (All Souls' Day),  
an Illustration from Google Book Search —

Earlier in Brautigan's tale

" Everybody started to leave the parlor to go downstairs
and pour out the Hawkline Monster but just as
they reached the door and one of the Hawkline women
had her hand on the knob, Cameron said, 'Hold it for a
second. I want to get myself a little whiskey.' "

Monday, November 2, 2015

Logic at Noon

Filed under: General — m759 @ 12:01 PM

Scientific American  photo caption —

"Monday, November 2, marks the 200-year anniversary
of the birth of the man who put True/False, 0/1, and
AND/OR and NOT on the map."

See Hardegree's Symbolic Logic  on "and/or" and its purported use
"to avoid ambiguity in legal contracts." His book is NOT recommended.

See also Bryan Garner in the ABA Journal  on "and/or" in legal usage
as well as a post in this  journal, The Witch of And/Or.

Wednesday, June 4, 2014

Monkey Business

Filed under: General,Geometry — Tags: — m759 @ 8:48 PM

The title refers to a Scientific American weblog item
discussed here on May 31, 2014:

Some closely related material appeared here on
Dec. 30, 2011:

IMAGE- Quaternion group acting on an eightfold cube

A version of the above quaternion actions appeared
at math.stackexchange.com on March 12, 2013:

"Is there a geometric realization of Quaternion group?" —

The above illustration, though neatly drawn, appeared under the
cloak of anonymity.  No source was given for the illustrated group actions.
Possibly they stem from my Log24 posts or notes such as the Jan. 4, 2012,
note on quaternion actions at finitegeometry.org/sc (hence ultimately
from my note "GL(2,3) actions on a cube" of April 5, 1985).

Saturday, May 31, 2014

Quaternion Group Models:

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

The ninefold square, the eightfold cube, and monkeys.

IMAGE- Actions of the unit quaternions in finite geometry, on a ninefold square and on an eightfold cube

For posts on the models above, see quaternion
in this journal. For the monkeys, see

"Nothing Is More Fun than a Hypercube of Monkeys,"
Evelyn Lamb's Scientific American  weblog, May 19, 2014:

The Scientific American  item is about the preprint
"The Quaternion Group as a Symmetry Group,"
by Vi Hart and Henry Segerman (April 26, 2014):

See also  Finite Geometry and Physical Space.

Friday, January 18, 2013

Solomon’s Rep-tiles

Filed under: General,Geometry — m759 @ 1:04 PM

"Rep-tiles Revisited," by Viorel Nitica, in MASS Selecta: Teaching and Learning Advanced Undergraduate Mathematics ,  American Mathematical Society, 2003—

"The goal of this note is to take a new look at some of the most amazing objects discovered in recreational mathematics. These objects, having the curious property of making larger copies of themselves, were introduced in 1962 by Solomon W. Golomb [2], and soon afterwards were popularized by Martin Gardner [3] in Scientific American…."

2.  S. W. Golomb: "Replicating Figures in the Plane," Mathematical Gazette  48, 1964, 403-412

3.  M. Gardner: "On 'Rep-tiles,' Polygons That Can Make Larger and Smaller Copies of Themselves," Scientific American  208, 1963, 154-157

Two such "amazing objects"—

Triangle

Square

For a different approach to the replicating properties of these objects, see the square-triangle theorem.

For related earlier material citing Golomb, see Not Quite Obvious (July 8, 2012; scroll down to see the update of July 15.).

Golomb's 1964 Gazette  article may now be purchased at JSTOR for $14.

Saturday, October 6, 2012

Black March

Filed under: General — m759 @ 5:18 PM
 

Log24, Dec. 18, 2006:

“I did a column in Scientific American on minimal art,
and I reproduced one of Ed Rinehart’s [sic ] black paintings.”
— Martin Gardner,
Notices of the American Mathematical Society ,  June/July 2005

“… the entire profession has received a very public
and very bad black mark.”
— Joan S. Birman,
Notices of the American Mathematical Society ,  January 2007

Related posts— See "Bad Black Mark" in this journal.

See also two items from St. Patrick's Day, March 17, 2005—

Midnight Drums for Larry  and…

IMAGE- St. Patrick's Day 2005 talk at Columbia by John D. McCarthy

IMAGE- John D. McCarthy at Birman conference, March 15-20, 2005

Click McCarthy photo for some more recent material.

Saturday, July 21, 2012

Saturday Morning Cartoon

Filed under: General — m759 @ 9:00 AM

"A Saturday morning cartoon is the colloquial term
for the animated television programming that has
typically been scheduled on Saturday mornings
on the major American television networks from
the 1960s to the present…." —Wikipedia

Martin Gardner in the Notices of the
American Mathematical Society 
,
June/July 2005:

“I did a column in Scientific American 
on minimal art, and I reproduced one of
Ed Rinehart’s [sic ] black paintings. 
Of course, it was just a solid square of
pure black.”

Black square 256x256

Click on picture for details.

For a cartoon graveyard

IMAGE- LA Times obits for two Saturday Night Live writers

Saturday, April 28, 2012

Sprechen Sie Deutsch?

Filed under: General,Geometry — m759 @ 10:48 AM

A Log24 post, "Bridal Birthday," one year ago today linked to
"The Discrete and the Continuous," a brief essay by David Deutsch.

From that essay—

"The idea of quantization—
the discreteness of physical quantities
turned out to be immensely fruitful."

Deutsch's "idea of quantization" also appears in
the April 12 Log24 post Mythopoetic

"Is Space Digital?" 

— Cover storyScientific American 
     magazine, February 2012

"The idea that space may be digital
  is a fringe idea of a fringe idea
  of a speculative subfield of a subfield."

— Physicist Sabine Hossenfelder 
     at her weblog on Feb. 5, 2012

"A quantization of space/time
 is a holy grail for many theorists…."

— Peter Woit in a comment 
      at his weblog on April 12, 2012

It seems some clarification is in order.

Hossenfelder's "The idea that space may be digital"
and Woit's "a quantization of space/time" may not
refer to the same thing.

Scientific American  on the concept of digital space—

"Space may not be smooth and continuous.
Instead it may be digital, composed of tiny bits."

Wikipedia on the concept of quantization—

Causal setsloop quantum gravitystring theory,
and 
black hole thermodynamics all predict
quantized spacetime….

For a purely mathematical  approach to the
continuous-vs.-discrete issue, see
Finite Geometry and Physical Space.

The physics there is somewhat tongue-in-cheek,
but the geometry is serious.The issue there is not
continuous-vs.-discrete physics , but rather
Euclidean-vs.-Galois geometry .

Both sorts of geometry are of course valid.
Euclidean geometry has long been applied to 
physics; Galois geometry has not. The cited
webpage describes the interplay of both  sorts
of geometry— Euclidean and Galois, continuous
and discrete— within physical space— if not
within the space of physics.

Thursday, April 12, 2012

Mythopoetic*

Filed under: General,Geometry — m759 @ 9:29 PM

"Is Space Digital?" 

Cover storyScientific American  magazine, February 2012

"The idea that space may be digital
  is a fringe idea of a fringe idea
  of a speculative subfield of a subfield."

— Physicist Sabine Hossenfelder
     at her weblog on Feb. 5, 2012

"A quantization of space/time
 is a holy grail for many theorists…."

— Peter Woit in a comment at his physics weblog today

See also 

* See yesterday's Steiner's Systems.

Friday, February 17, 2012

Pregeometry and Finite Geometry

Filed under: General,Geometry — Tags: — m759 @ 7:35 PM

Today's previous post, on the Feb. 2012 Scientific American
article "Is Space Digital?", suggested a review of a notion
that the theoretical physicist John Archibald Wheeler called
pregeometry .

From a paper on that topic—

"… the idea that geometry should constitute
'the magic building material of the universe'
had to collapse on behalf of what Wheeler
has called pregeometry  (see Misner et al. 1973,
pp. 1203-1212; Wheeler 1980), a somewhat
indefinite term which expresses “a combination
of hope and need, of philosophy and physics
and mathematics and logic” (Misner et al. 1973,
p. 1203)."

— Jacques Demaret, Michael Heller, and
Dominique Lambert, "Local and Global Properties
of the World," preprint of paper published in
Foundations of Science  2 (1): 137-176

Misner, C. W., Thorne, K. S. and Wheeler, J. A.
1973, Gravitation , W.H. Freeman and Company:
San Francisco.

Wheeler, J.A. 1980, "Pregeometry: Motivations
and Prospects," in: Quantum Theory and Gravitation ,
ed. A.R. Marlow, Academic Press: New York, pp. 1-11.

Some related material from pure mathematics—

http://www.log24.com/log/pix12/120217-Pregeometry_And_Geometry.jpg

Click image for further details.

Physics vs. Geometry

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

Physics

The February 2012 issue of Scientific American 
has a cover article titled "Is Space Digital?".

http://www.log24.com/log/pix12/120217-IsSpaceDigital.jpg

The article discusses whether physical space
"is made of chunks. Blocks. Bits."

Maybe it is, maybe it isn't.

Geometry

The word "space" in pure mathematics
(as opposed to physics) applies to
a great variety of structures.

Some are continuous, some are not.

For some purely mathematical structures
that are not  continuous, (i.e., are made of
"chunks, blocks, bits") see finitegeometry.org/sc
in particular, the pages on Finite Geometry and Physical Space
and on Noncontinuous Groups.

The geometry of these structures may or may not eventually
be relevant to the "21st-century physics" discussed
in the February Scientific American.

Tuesday, January 17, 2012

Khora as Synchronicity

Filed under: General,Geometry — m759 @ 11:01 AM

A search for khora  + tao  yields a paper on Derrida—

http://www.log24.com/log/pix12/120117-IanEdwards-OnKhora.gif

http://www.log24.com/log/pix12/120117-IanEdwards.jpg

A check of the above date— Nov. 18, 2010— yields…

Thursday, November 18, 2010

Frontiers of Speculation

 m759 @ 8:02 AM

Peter Woit has a post on Scientific American 's new Garrett Lisi article, "A Geometric Theory of Everything."

The Scientific American  subtitle is "Deep down, the particles and forces of the universe are a manifestation of exquisite geometry."

See also Rhetoric (Nov. 4, 2010) and Exquisite Geometries (May 19, 2009).

Related material on the temptation of physics
for a pure mathematician—

This morning's post on khora  and Cardinal Manning, and,
from Hawking's birthday this year, Big Apple.

Within this  post, by leading us to the apple,
Derrida as usual plays the role of Serpent.

Thursday, May 5, 2011

On Art and Magic

Filed under: General,Geometry — Tags: — m759 @ 10:30 PM

Two Blocks Short of a Design:

A sequel to this morning's post on Douglas Hofstadter

http://www.log24.com/log/pix11A/110505-ThemeAndVariations-Hofstadter.jpg

Photo of Hofstadter by Mike McGrath taken May 13, 2006

Related material — See Lyche's  "Theme and Variations" in this journal
and Hofstadter's "Variations on a Theme as the Essence of Imagination"
Scientific American  October 1982

A quotation from a 1985 book by Hofstadter—

"… we need to entice people with the beauties of clarity, simplicity, precision,
elegance, balance, symmetry, and so on.

Those artistic qualities… are the things that I have tried to explore and even
to celebrate in Metamagical Themas .  (It is not for nothing that the word
'magic' appears inside the title!)"

The artistic qualities Hofstadter lists are best sought in mathematics, not in magic.

An example from Wikipedia —

http://www.log24.com/log/pix11A/110505-BlockDesignTheory.jpg

Mathematics

http://www.log24.com/log/pix11A/110505-WikipediaFanoPlane.jpg

The Fano plane block design

Magic

http://www.log24.com/log/pix11A/110505-DeathlyHallows.jpg

The Deathly Hallows  symbol—
Two blocks short of  a design.

Friday, April 15, 2011

Exercise

Filed under: General — Tags: — m759 @ 2:00 AM

The April Scientific American  on the partition function p (n )

"… in January, Ono and another collaborator [Bruinier] described the first formula that directly calculates p (n ) for any n, a feat that had eluded number theorists for centuries."

Exercise: Is this remarkable claim true or false?

For commentary here, see Jan. 27, "Indiana Jones and the Magical Oracle."

For further comments (the most recent from March 11), see mathoverflow.net, "Exact formulas for the partition function?"

Thursday, November 18, 2010

Frontiers of Speculation

Filed under: General,Geometry — m759 @ 8:02 AM

Peter Woit has a post on Scientific American 's new Garrett Lisi article, "A Geometric Theory of Everything."

The Scientific American  subtitle is "Deep down, the particles and forces of the universe are a manifestation of exquisite geometry."

See also Rhetoric (Nov. 4, 2010) and Exquisite Geometries (May 19, 2009).

Friday, September 17, 2010

Fade to Blacker

Filed under: General,Geometry — m759 @ 1:22 PM

From Peter J. Cameron's web journal today—

Eliot’s Four Quartets  has been one of my favourite works of poetry since I was a student…. 

Of course, a poem doesn’t have a single meaning, especially one as long and complex as Four Quartets.  But to me the primary meaning of the poem is about the relationship between time and eternity, which is something maybe of interest to mathematicians as well as to mystics.

Curiously, the clearest explanation of what Eliot is saying that I have found is in a completely different work, Pilgrimage of Dreams  by the artist Thetis Blacker, in which she describes a series of dreams she had which stood out as being completely different from the confusion of normal dreaming. In one of these dreams, “Mr Goad and the Cathedral”, we find the statements

“Eternity isn’t a long time

and

“Eternity is always now, but …”
“Now isn’t always eternity”.

In other words, eternity is not the same as infinity; it is not the time line stretched out to infinity. Rather, it is an intimation of a different dimension, which we obtain only because we are aware of the point at which that dimension intersects the familiar dimension of time. In a recurring motif in the second Quartet, “East Coker”, Eliot says,

Time future and time past
Are both somehow contained in time present

and, in “Little Gidding”,

   … to apprehend
The point of intersection of the timeless
With time, is an occupation for the saint

From this  journal on the date of Blacker's death
what would, if she were a Catholic saint, be called her dies natalis

Monday December 18, 2006

m759 @ 7:20 AM
 
Fade to Black:

Martin Gardner in the Notices of the American Mathematical SocietyJune/July 2005 (pdf):

“I did a column in Scientific American  on minimal art, and I reproduced one of Ed Rinehart’s [sic ] black paintings.  Of course, it was just a solid square of pure black.”

Black square 256x256

Click on picture for details.

The Notices of the American Mathematical SocietyJanuary 2007 (pdf):

“This was just one of the many moments in this sad tale when there were no whistle-blowers. As a result the entire profession has received a very public and very bad black mark.”

– Joan S. Birman
Professor Emeritus of Mathematics
Barnard College and
Columbia University

Thursday, July 1, 2010

Omega at Eight

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

The "compact key to universal wisdom" passage in the previous post seemed
too well written to be the work of an anonymous webforum author.

Here is a slightly expanded version—

Throughout history mystics and philosophers have sought
a compact key to universal wisdom, a finite formula or text
that would provide the answer to every question. The use of
the Bible, the Koran and the I Ching for divination and the
tradition of the secret books of Hermes Trismegistus and the
medieval Jewish Cabala exemplify this belief or hope.  Such
sources of universal wisdom are traditionally protected from
casual use by being difficult to find as well as difficult to un-
derstand and dangerous to use, tending to answer more quest-
ions and deeper ones than the searcher wishes to ask. The
esoteric book is, like God, simple yet undescribable. It is om-
niscient, and it transforms all who know it. The use of clas-
sical texts to foretell mundane events is considered supersti-
tious nowadays, yet in another sense science is in quest of its
own Cabala, a concise set of natural laws that would explain
all phenomena. In mathematics, where no set of axioms can
hope to prove all true statements, the goal might be a concise
axiomatization of all "interesting" true statements.
      Ω is in many senses a Cabalistic number. It can be known
of through human reason, but not known. To know it in detail
one must accept its uncomputable sequence of digits on faith,
like words of a sacred text.   

This is Martin Gardner's* and Charles H. Bennett's
revised version of a passage from Bennett's  paper
"On Random and Hard-to-Describe Numbers," 1979.

The original passage from Bennett's paper—

Throughout history mystics and philosophers have sought a compact key to
universal wisdom, a finite formula or text which, when known and understood,
would provide the answer to every question. The Bible, the Koran, the mythical
secret books of Hermes Trismegistus, and the medieval Jewish Cabala have
been so regarded. Sources of universal wisdom are traditionally protected from
casual use by being hard to find, hard to understand when found, and dangerous
to use, tending to answer more and deeper questions than the user wishes to
ask. Like God the esoteric book is simple yet undescribable, omniscient, and
transforms all who know It. The use of classical texts to fortell [sic] mundane events
is considered superstitious nowadays, yet, in another sense, science is in quest of
its own Cabala, a concise set of natural laws which would explain all phenomena.
In mathematics, where no set of axioms can hope to prove all true statements,
the goal might be a concise axiomatization of all "interesting" true statements.
      Ω is in many senses a Cabalistic number. It can be known of, but not known,
through human reason. To know it in detail, one would have to accept its un-
computable digit sequence on faith, like words of a sacred text.

The Bennett paper deals with Gregory Chaitin's concept of an "Omega Number."

I prefer the Omega of Josefine Lyche—

Image-- Uncertified copy of 1986 figures by Cullinane in a 2009 art exhibit in Oslo

Click for further details.

See also All Hallows' Eve, 2002.

* Martin Gardner's Mathematical Games  column
"The Random Number Omega Bids Fair to Hold the Mysteries of the Universe,"
Scientific American, November 1979, 241(5), pp. 20–34.
The column is reprinted as "Chaitin's Omega," Ch. 21, pp. 307-319 in the
collection of Gardner's columns titled Fractal Music, Hypercards and More,
W.H. Freeman & Co., 1991

Sunday, December 13, 2009

Ein Kampf

Filed under: General — Tags: — m759 @ 11:30 AM

YouTube: Hitler Plans Burning Man

(Click on image for video.)

See also Tyger! Tyger! and
The Stars My Destination.

Hitler's Peer Review–

YouTube: Hitler's Peer Review-- The Abstract

YouTube: Hitler's Peer Review-- Scientific American


See also Abstract 79T-A37
and Scientific American
.

Sunday, November 16, 2008

Sunday November 16, 2008

Filed under: General,Geometry — m759 @ 8:00 PM
Art and Lies

Observations suggested by an article on author Lewis Hyde– “What is Art For?“–  in today’s New York Times Magazine:

Margaret Atwood (pdf) on Lewis Hyde’s
Trickster Makes This World: Mischief, Myth, and Art

“Trickster,” says Hyde, “feels no anxiety when he deceives…. He… can tell his lies with creative abandon, charm, playfulness, and by that affirm the pleasures of fabulation.” (71) As Hyde says, “…  almost everything that can be said about psychopaths can also be said about tricksters,” (158), although the reverse is not the case. “Trickster is among other things the gatekeeper who opens the door into the next world; those who mistake him for a psychopath never even know such a door exists.” (159)

What is “the next world”? It might be the Underworld….

The pleasures of fabulation, the charming and playful lie– this line of thought leads Hyde to the last link in his subtitle, the connection of the trickster to art. Hyde reminds us that the wall between the artist and that American favourite son, the con-artist, can be a thin one indeed; that craft and crafty rub shoulders; and that the words artifice, artifact, articulation and art all come from the same ancient root, a word meaning to join, to fit, and to make. (254) If it’s a seamless whole you want, pray to Apollo, who sets the limits within which such a work can exist. Tricksters, however, stand where the door swings open on its hinges and the horizon expands: they operate where things are joined together, and thus can also come apart.

For more about
“where things are
joined together,” see
 Eight is a Gate and
The Eightfold Cube.
Related material:

The Trickster
and the Paranormal

and
Martin Gardner on
   a disappearing cube —

“What happened to that… cube?”

Apollinax laughed until his eyes teared. “I’ll give you a hint, my dear. Perhaps it slid off into a higher dimension.”

“Are you pulling my leg?”

“I wish I were,” he sighed. “The fourth dimension, as you know, is an extension along a fourth coordinate perpendicular to the three coordinates of three-dimensional space. Now consider a cube. It has four main diagonals, each running from one corner through the cube’s center to the opposite corner. Because of the cube’s symmetry, each diagonal is clearly at right angles to the other three. So why shouldn’t a cube, if it feels like it, slide along a fourth coordinate?”

— “Mr. Apollinax Visits New York,” by Martin Gardner, Scientific American, May 1961, reprinted in The Night is Large

For such a cube, see

Cube with its four internal diagonals

ashevillecreative.com

this illustration in

The Religion of Cubism
(and the four entries
preceding it —
 Log24, May 9, 2003).

Beware of Gardner’s
“clearly” and other lies.

Monday, April 9, 2007

Monday April 9, 2007

Filed under: General — m759 @ 7:20 PM
Symmetry
for Beavis and Butt-Head

7:20 in the Book
(An illustration from
Mathematics and Narrative;
the “Book” is The Gospel
According to St. Matthew
.)

From Ian Stewart’s new book,
Why Beauty is Truth:
A History of Symmetry

Beauty, Truth, Symmetry

Is Beauty Truth and Truth Beauty?,”
a review by famed vulgarizer
Martin Gardner of the new book
by his fellow vulgarizer Ian Stewart
in the April 2007 Scientific American:

“Associated with every kind of symmetry is a ‘group.’ Stewart explains the group concept in a simple way by considering operations on an equilateral triangle. Rotate it 60 degrees in either direction, and it looks the same. Every operation has an ‘inverse,’ that cancels the operation. Imagine the corners of the triangle labeled A, B and C. A 60-degree clockwise rotation alters the corners’ positions. If this is followed by a similar rotation the other way, the original positions are restored. If you do nothing to the triangle, this is called the ‘identity’ operation. The set of all symmetry transformations of the triangle constitutes its group.”

“Is Beauty Truth?”
asked jesting Gardner…

The reasoned reply of
Beavis and Butt-Head:

“Sixty degrees, a hundred
and twenty degrees, who
gives a rat’s ass?”

Saturday, December 23, 2006

Saturday December 23, 2006

Filed under: General,Geometry — m759 @ 9:00 AM
Black Mark

Bernard Holland in The New York Times on Monday, May 20, 1996:

“Philosophers ponder the idea of identity: what it is to give something a name on Monday and have it respond to that name on Friday….”

Log24 on Monday,
Dec. 18, 2006:

“I did a column in
Scientific American
on minimal art, and
I reproduced one of
Ed Rinehart’s [sic]
black paintings.”

Martin Gardner (pdf)

“… the entire profession
has received a very public
and very bad black mark.”

Joan S. Birman (pdf)

Lottery on Friday,
Dec. 22, 2006:

The image “http://www.log24.com/log/pix06B/061222-PAlottery.jpg” cannot be displayed, because it contains errors.

5/04
, 2005:

Analysis of the structure
of a 2x2x2 cube

The Eightfold Cube

via trinities of
projective points
in a Fano plane.

7/15, 2005:

“Art history was very personal
through the eyes of Ad Reinhardt.”

  — Robert Morris,
Smithsonian Archives
of American Art

Also on 7/15, 2005,
a quotation on Usenet:

“A set having three members is a
single thing wholly constituted by
its members but distinct from them.
After this, the theological doctrine
of the Trinity as ‘three in one’
should be child’s play.”

— Max Black,
Caveats and Critiques:
Philosophical Essays in
Language, Logic, and Art

Monday, December 18, 2006

Monday December 18, 2006

Filed under: General — m759 @ 7:20 AM
Fade to Black:
Mathematics and Narrative
continued

Martin Gardner in the Notices of the American Mathematical Society, June/July 2005 (pdf):

“I did a column in Scientific American on minimal art, and I reproduced one of Ed Rinehart’s [sic] black paintings.  Of course, it was just a solid square of pure black.”

Black square 256x256

Click on picture
for details.

The Notices of the American Mathematical Society, January 2007 (pdf):

“This was just one of the many moments in this sad tale when there were no whistle-blowers. As a result the entire profession has received a very public and very bad black mark.”

— Joan S. Birman
Professor Emeritus of Mathematics
Barnard College and
Columbia University

Saturday, August 19, 2006

Saturday August 19, 2006

Filed under: General — Tags: — m759 @ 4:28 PM
Metaphysical
Wonderlands

"With no means to verify its truth, superstring theory, in the words of Burton Richter, director emeritus of the Stanford Linear Accelerator Center, may turn out to be 'a kind of metaphysical wonderland.' Yet it is being pursued as vigorously as ever, its critics complain, treated as the only game in town."

— "The Inelegant Universe," by George Johnson, in the Sept. 2006 Scientific American

Some may prefer metaphysics of a different sort:

"To enter Cervantes’s world, we cross a threshold that is Shakespearean and quixotic into a metaphysical wonderland where time expands to become space and vast vaulted distances bend back on themselves, where the threads of fiction and the strands of history shuttle back and forth in the great loom of the artist’s imagination."

As wonderlands go, I personally prefer Clive Barker's Weaveworld.
 

Thursday, November 10, 2005

Thursday November 10, 2005

Filed under: General — m759 @ 4:00 PM
The Rhetoric of
Scientism

Kansas, Where “Ignorant”
is the New “Educated”:

“… the Board of Education went as far as to redefine what science is: it’s no longer just a search for natural explanations for natural phenomena. Now it’s a search for… well, that’s a bit hard to say. Any sort of explanation, apparently. Pixies, ghosts, telekinesis, auras, ancient astronauts, excesses of choleric humor, they all seem to be fair game in the interest of ‘academic freedom.'”

John Rennie, editor in chief of
  Scientific American, Nov. 8, 2005

The shocking redefinition
(with changes highlighted):
Kansas Definition of Science
Adopted Feb. 14, 2001

Science is the human activity of seeking natural explanations for what we observe in the world around us.  Science does so through the use of observation, experimentation, and logical argument while maintaining strict empirical standards and healthy skepticism. Scientific explanations are built on observations, hypotheses, and theories. A hypothesis is a testable statement about the natural world that can be used to build more complex inferences and explanations. A theory is a well-substantiated explanation of some aspect of the natural world that can incorporate observations, inferences, and tested hypotheses

Kansas Definition of Science
Approved Nov. 8, 2005

Science is a systematic method of continuing investigation that uses observations, hypothesis testing, measurement, experimentation, logical argument and theory building to lead to more adequate explanations of natural phenomena. Science does so while maintaining strict empirical standards and healthy skepticism. Scientific explanations are built on observations, hypotheses, and theories. A hypothesis is a testable statement about the natural world that can be used to build more complex inferences and explanations. A theory is a well-substantiated explanation of some aspect of the natural world that can incorporate observations, inferences, and tested hypotheses.

Scientific explanations must meet certain criteria. Scientific explanations are consistent with experimental and/or observational data and testable by scientists through additional experimentation and/or observation. Scientific explanation must meet criteria that govern the repeatability of observations and experiments. The effect of these criteria is to insure that scientific explanations about the world are open to criticism and that they will be modified or abandoned in favor of new explanations if empirical evidence so warrants. Because all scientific explanations depend on observational and experimental confirmation, all scientific knowledge is, in principle, subject to change as new evidence becomes available. The core theories of science have been subjected to a wide variety of confirmations and have a high degree of reliability within the limits to which they have been tested. In areas where data or understanding are incomplete, new data may lead to changes in current theories or resolve current conflicts. In situations where information is still fragmentary, it is normal for scientific ideas to be incomplete, but this is also where the opportunity for making advances may be greatest. Science has flourished in different regions during different time periods, and in history, diverse cultures have contributed scientific knowledge and technological inventions. Changes in scientific knowledge usually occur as gradual modifications, but the scientific enterprise also experiences periods of rapid advancement. The daily work of science and technology results in incremental advances in our understanding of the world about us.” Scientific explanations must meet certain criteria. Scientific explanations are consistent with experimental and/or observational data and testable by scientists through additional experimentation and/or observation. Scientific explanation must meet criteria that govern the repeatability of observations and experiments. The effect of these criteria is to insure that scientific explanations about the world are open to criticism and that they will be modified or abandoned in favor of new explanations if empirical evidence so warrants. Because all scientific explanations depend on observational and experimental confirmation, all scientific knowledge is, in principle, subject to change as new evidence becomes available. The core theories of science have been subjected to a wide variety of confirmations and have a high degree of reliability within the limits to which they have been tested. In areas where data or understanding is incomplete, new data may lead to changes in current theories or resolve current conflicts. In situations where information is still fragmentary, it is normal for scientific ideas to be incomplete, but this is also where the opportunity for making advances may be greatest. Science has flourished in different regions during different time periods, and in history, diverse cultures have contributed scientific knowledge and technological inventions. Changes in scientific knowledge usually occur as gradual modifications, but the scientific enterprise also experiences periods of rapid advancement. The daily work of science and technology results in incremental advances in understanding the world.”

From both old (2001) and
new (2005) Kansas standards:


Teaching With Tolerance and Respect

“A teacher is an important role model for  demonstrating respect, sensitivity, and civility. Teachers should not ridicule, belittle or embarrass a student for expressing an alternative view or belief.”

It’s a very ancient saying,
But a true and honest thought,
That if you become a teacher,
By your pupils you’ll be taught.

— Oscar Hammerstein,
“Getting to Know You”

Scientism and Civility:

A Google blog search for
fucking kansas evolution standards -fuck
yields “about 47” entries.

A search for
fuck kansas evolution standards -fucking
yields  “about 34” entries.

A search for
fuck fucking kansas evolution standards
yields “about 42” entries.

Thursday, August 25, 2005

Thursday August 25, 2005

Filed under: General,Geometry — m759 @ 3:09 PM
Analogical
Train of Thought

Part I: The 24-Cell

From S. H. Cullinane,
 Visualizing GL(2,p),
 March 26, 1985–

Visualizing the
binary tetrahedral group
(the 24-cell):

The image “http://www.log24.com/theory/images/VisuBinaryTetGrp.jpg” cannot be displayed, because it contains errors.

Another representation of
the 24-cell
:

The image “http://www.log24.com/theory/images/24-cell.jpg” cannot be displayed, because it contains errors.

 From John Baez,
This Week’s Finds in
Mathematical Physics (Week 198)
,”
September 6, 2003: 

Noam Elkies writes to John Baez:

Hello again,

You write:

[…]

“I’d like to wrap up with a few small comments about last Week.  There I said a bit about a 24-element group called the ‘binary tetrahedral group’, a 24-element group called SL(2,Z/3), and the vertices of a regular polytope in 4 dimensions called the ’24-cell’.  The most important fact is that these are all the same thing! And I’ve learned a bit more about this thing from here:”

[…]

Here’s yet another way to see this: the 24-cell is the subgroup of the unit quaternions (a.k.a. SU(2)) consisting of the elements of norm 1 in the Hurwitz quaternions – the ring of quaternions obtained from the Z-span of {1,i,j,k} by plugging up the holes at (1+i+j+k)/2 and its <1,i,j,k> translates. Call this ring A. Then this group maps injectively to A/3A, because for any g,g’ in the group |g-g’| is at most 2 so g-g’ is not in 3A unless g=g’. But for any odd prime p the (Z/pZ)-algebra A/pA is isomorphic with the algebra of 2*2 matrices with entries in Z/pZ, with the quaternion norm identified with the determinant. So our 24-element group injects into SL2(Z/3Z) – which is barely large enough to accommodate it. So the injection must be an isomorphism.

Continuing a bit longer in this vein: this 24-element group then injects into SL2(Z/pZ) for any odd prime p, but this injection is not an isomorphism once p>3. For instance, when p=5 the image has index 5 – which, however, does give us a map from SL2(Z/5Z) to the symmetric group of order 5, using the action of SL2(Z/5Z) by conjugation on the 5 conjugates of the 24-element group. This turns out to be one way to see the isomorphism of PSL2(Z/5Z) with the alternating group A5.

Likewise the octahedral and icosahedral groups S4 and A5 can be found in PSL2(Z/7Z) and PSL2(Z/11Z), which gives the permutation representations of those two groups on 7 and 11 letters respectively; and A5 is also an index-6 subgroup of PSL2(F9), which yields the identification of that group with A6.

NDE


The enrapturing discoveries of our field systematically conceal, like footprints erased in the sand, the analogical train of thought that is the authentic life of mathematics – Gian-Carlo Rota

Like footprints erased in the sand….

Part II: Discrete Space

The James Joyce School
 of Theoretical Physics
:


Log24, May 27, 2004

  “Hello! Kinch here. Put me on to Edenville. Aleph, alpha: nought, nought, one.” 

  “A very short space of time through very short times of space….
   Am I walking into eternity along Sandymount strand?”

   — James Joyce, Ulysses, Proteus chapter

A very short space of time through very short times of space….

   “It is demonstrated that space-time should possess a discrete structure on Planck scales.”

   — Peter Szekeres, abstract of Discrete Space-Time

   “A theory…. predicts that space and time are indeed made of discrete pieces.”

   — Lee Smolin in Atoms of Space and Time (pdf), Scientific American, Jan. 2004

   “… a fundamental discreteness of spacetime seems to be a prediction of the theory….”

   — Thomas Thiemann, abstract of Introduction to Modern Canonical Quantum General Relativity

   “Theories of discrete space-time structure are being studied from a variety of perspectives.”

   — Quantum Gravity and the Foundations of Quantum Mechanics at Imperial College, London

Disclaimer:

The above speculations by physicists
are offered as curiosities.
I have no idea whether
 any of them are correct.

Related material:

Stephen Wolfram offers a brief
History of Discrete Space.

For a discussion of space as discrete
by a non-physicist, see John Bigelow‘s
Space and Timaeus.

Part III: Quaternions
in a Discrete Space

Apart from any considerations of
physics, there are of course many
purely mathematical discrete spaces.
See Visible Mathematics, continued
 (Aug. 4, 2005):

The image “http://www.log24.com/theory/images/Quaternions2.jpg” cannot be displayed, because it contains errors.

Sunday, June 19, 2005

Sunday June 19, 2005

Filed under: General,Geometry — Tags: — m759 @ 4:00 AM
ART WARS:
Darkness Visible
“No light, but rather darkness visible
 Serv’d only to discover sights of woe”
John Milton, Paradise Lost,
Book I,  lines 63-64

From the cover article (pdf) in the
June/July 2005 Notices of the
American Mathematical Society–

Martin Gardner


A famed vulgarizer, Martin Gardner,
summarizes the art of Ad Reinhardt
(Adolph Dietrich Friedrich Reinhardt,
  Dec. 24, 1913 – Aug. 30, 1967):

“Ed Rinehart [sic] made a fortune painting canvases that were just one solid color.  He had his black period in which the canvas was totally black.  And then he had a blue period in which he was painting the canvas blue.  He was exhibited in top shows in New York, and his pictures wound up in museums.  I did a column in Scientific American on minimal art, and I reproduced one of Ed Rinehart’s black paintings.  Of course, it was just a solid square of pure black.  The publisher insisted on getting permission from the gallery to reproduce it.”

Related material
from Log24.net,
Nov. 9-12, 2004:

Fade to Black

“…that ineffable constellation of talents that makes the player of rank: a gift for conceiving abstract schematic possibilities; a sense of mathematical poetry in the light of which the infinite chaos of probability and permutation is crystallized under the pressure of intense concentration into geometric blossoms; the ruthless focus of force on the subtlest weakness of an opponent.”

— Trevanian, Shibumi

“‘Haven’t there been splendidly elegant colors in Japan since ancient times?’

‘Even black has various subtle shades,’ Sosuke nodded.”

— Yasunari Kawabata, The Old Capital

An Ad Reinhardt painting
described in the entry of
noon, November 9, 2004
is illustrated below.

Ad Reinhardt,  Greek Cross

Ad Reinhardt,
Abstract Painting,
1960-66.
Oil on canvas, 60 x 60 inches.
Solomon R. Guggenheim Museum

The viewer may need to tilt
the screen to see that this
painting is not uniformly black,
but is instead a picture of a
Greek cross, as described below.

“The grid is a staircase to the Universal…. We could think about Ad Reinhardt, who, despite his repeated insistence that ‘Art is art,’ ended up by painting a series of… nine-square grids in which the motif that inescapably emerges is a Greek cross.

Greek Cross

There is no painter in the West who can be unaware of the symbolic power of the cruciform shape and the Pandora’s box of spiritual reference that is opened once one uses it.”

— Rosalind Krauss,
Meyer Schapiro Professor
of Modern Art and Theory
at Columbia University

(Ph.D., Harvard U., 1969),
in “Grids”

The image “http://www.log24.com/log/pix04B/041109-Krauss.jpg” cannot be displayed, because it contains errors.

Krauss

 
In memory of
St. William Golding
(Sept. 19, 1911 – June 19, 1993)

Tuesday, April 26, 2005

Tuesday April 26, 2005

Filed under: General — m759 @ 6:29 AM
The Ring of Falsehood

In memory of Philip Morrison, bombmaker,

The image “http://www.log24.com/log/pix05/050426-Morrison.jpg” cannot be displayed, because it contains errors.
Morrison

Scientific American columnist,
  pioneer of the
Search for Extraterrestrial Intelligence (SETI)
and author of
The Ring of Truth


Morrison died
in Cambridge, Massachusetts,
on Friday, April 22, 2005.

From The Measure of a Life:

Does religion play a role in attitudes toward ETIs? Philip Morrison gave his considered opinion… “Well, it might, but I think that it’s just one of the permissive routes; it isn’t an essential factor. My parents were Jewish. Their beliefs were conventional but not very deep. They belonged to the Jewish community; they went to services infrequently, on special occasions—funerals and high holidays”….

Although Sagan did not believe in God, he nevertheless said this about SETI’s importance… “It touches deeply into myth, folklore, religion, mythology; and every human culture in some way or another has wondered about that type of question. It’s one of the most basic questions there is.” In fact, in Sagan’s novel/film Contact, described by Keay Davidson as “one of the most religious science-fiction tales ever written”… Ellie discovers that pi—the ratio of the circumference of a circle to its diameter—is numerically encoded in the cosmos and this is proof that a super-intelligence designed the universe…

The universe was made on purpose, the circle said. In whatever galaxy you happen to find yourself, you take the circumference of a circle, divide it by its diameter, measure closely enough, and uncover a miracle—another circle, drawn kilometers downstream of the decimal point. In the fabric of space and in the nature of matter, as in a great work of art, there is, written small, the artist’s signature. Standing over humans, gods, and demons, subsuming Caretakers and Tunnel builders, there is an intelligence that antedates the universe.



Nell

See also yesterday’s entry Mathematical Style.

Extra credit:
Discuss the difference betweeen physical constants and mathematical constants. Use the results of your discussion to show that the above discussion of pi is nonsense.

Monday, April 4, 2005

Monday April 4, 2005

Filed under: General — m759 @ 4:04 AM
Fourth Day of the Fourth Month,
4:04:04

“My wife took, unnoticed, this picture, unposed, of me in the act of writing a novel…. The date (discernible in the captured calendar) is February 27, 1929. The novel, Zashchita Luzhina (The Defense), deals with the defense invented by an insane chess player….”
— Vladimir Nabokov, note to photograph following page 256 in Speak, Memory: An Autobiography Revisited, Vintage International paperback, August 1989

— Quoted in The Matthias Defense

From a site titled Meaning of the Twentieth Century —

“Freeman Dyson has expressed some thoughts on craziness. In a Scientific American article called ‘Innovation in Physics,’ he began by quoting Niels Bohr. Bohr had been in attendance at a lecture in which Wolfgang Pauli proposed a new theory of elementary particles. Pauli came under heavy criticism, which Bohr summed up for him: ‘We are all agreed that your theory is crazy. The question which divides us is whether it is crazy enough to have a chance of being correct. My own feeling is that is not crazy enough.’ To that Freeman added: ‘When a great innovation appears, it will almost certainly be in a muddled, incomplete and confusing form. To the discoverer, himself, it will be only half understood; to everyone else, it will be a mystery. For any speculation which does not at first glance look crazy, there is no hope!’ “

Kenneth Brower, The Starship and the Canoe, 1979, pp. 146, 147

It is my hope that the speculation, implied in The Matthias Defense, that the number 162 has astonishing mystical properties (as a page number, article number, etc.) is sufficiently crazy to satisfy Pauli and his friend Jung as well as the more conventional thinkers Bohr and Dyson.

— Log24.net, Feast of St. Mark, 2003

See also The Black Queen and The Eight.
 
In accordance with the theology of the previous entry, based on Zein’s list of the most common Chinese characters, here are some meanings of

character 162:

[si4] {sì} /to watch/to wait/to examine/to spy/
[si4] {sì} /to seem/to appear/similar/like/to resemble/
[si4] {sì} /until/wait for/
[si4] {sì} /rhinoceros indicus/
[si4] {sì} /four/
[si4] {sì} /(surname)/wife of older brother/
[si4] {sì} /Buddhist temple/
[si4] {sì} /6th earthly branch/9-11 a.m./
[si4] {sì} /stream which returns after branching/
[si4] {sì} /place name/snivel/
[si4] {sì} /offer sacrifice to/
[si4] {sì} /hamper/trunk/
[si4] {sì} /plough/ploughshare/
[si4] {sì} /four (fraud-proof)/market/
[si4] {sì} /to feed/
[si4] {sì} /to raise/to rear/to feed/
[si4] {sì} /team of 4 horses/
[si4 bai3 wan4] {sì bǎi wàn} /four million/
[si4 bai3 yi4] {sì bǎi yì} /40 billion/
[si4 cao2] {sì cáo} /feeding trough/
[si4 cao3] {sì cǎo} /forage grass/
[si4 chu4] {sì chù} /all over the place/everywhere and all directions/
[si4 chuan1] {sì chuān} /Sichuan province, China/
[si4 chuan1 sheng3] {sì chuān shěng} /(N) Sichuan, a south west China province/
[si4 de5] {sì de} /seem as if/rather like/
[si4 fang1] {sì fāng} /four-way/four-sided/
[si4 fen1 zhi1 yi1] {sì fēn zhī yī} /one-quarter/
[si4 fu2] {sì fú} /servo/
[si4 fu2 qi4] {sì fú qì} /server (computer)/
[si4 ge4 xiao3 shi2] {sì gè xiǎo shí} /four hours/
[si4 hu5] {sì hu} /apparently/to seem/to appear/as if/seemingly/
[si4 hu5 hen3 an1 quan2] {sì hu hěn ān quán} /to appear (to be) very safe/
[si4 ji1] {sì jī} /to watch for one's chance/
[si4 ji4] {sì jì} /(n) the four seasons/
[si4 liao4] {sì liào} /feed/fodder/
[si4 lun2 ma3 che1] {sì lún mǎ chē} /chariot/
[si4 men2 jiao4 che1] {sì mén jiào chē} /sedan (motor car)/
[si4 mian4 ba1 fang1] {sì miàn bā fāng} /in all directions/all around/far and near/
[si4 mian4 ti3] {sì miàn tǐ} /tetrahedron/
[si4 miao4] {sì miào} /temple/monastery/shrine/
[si4 nian2] {sì nián} /four years/
[si4 nian2 qian2] {sì nián qián} /four years previously/
[si4 nian2 zhi4 de5 da4 xue2] {sì nián zhì de dà xué} /four-year university/
[si4 qian1] {sì qiān} /four thousand/4 000/
[si4 shi2] {sì shí} /forty/40/
[si4 shi2 duo1] {sì shí duō} /more than 40/
[si4 shi2 liu4] {sì shí liù} /forty six/46/
[si4 shi2 san1] {sì shí sān} /43/forty three/
[si4 shi4 er2 fei1] {sì shì ér fēi} /(saying) appeared right but actually was wrong/
[si4 tian1] {sì tiān} /four days/
[si4 xiao4 fei1 xiao4] {sì xiào fēi xiào} /(saying) resemble a smile yet not smile/
[si4 xue3] {sì xuě} /snowy/
[si4 yang3] {sì yǎng} /to raise/to rear/
[si4 yang3 zhe3] {sì yǎng zhě} /feeder/
[si4 yuan4] {sì yuàn} /cloister/
[si4 yue4] {sì yuè} /April/fourth month/
[si4 yue4 shi2 qi1 hao4] {sì yuè shí qī hào} /April 17/
[si4 zhi1] {sì zhī} /(n) the four limbs of the body/
[si4 zhou1] {sì zhōu} /all around/

ktmatu.com Chinese-English dictionary

Wednesday, August 18, 2004

Wednesday August 18, 2004

Filed under: General — m759 @ 3:00 AM

Drunk Bird


T. Charles Erickson
Shizuo Kakutani
in the 1980’s

Kakutani died yesterday.

“A drunk man will find his way home, but a drunk bird may get lost forever.”

— Shizuo Kakutani, quoted by J. Chang in Stochastic Processes (ps), p. 1-19.  Chang says the quote is from an R. Durrett book on probability.

Meaning:

A random walk in d dimensions is recurrent if d = 1 or d = 2, but transient if d is greater than or equal to 3.


From a web page on Kylie Minogue:

Turns out she’s a party girl
who loves Tequila:
“Time disappears with Tequila.  
  It goes elastic, then vanishes.”



Kylie sings
“Locomotion”

From a web page on Malcolm Lowry’s classic novel Under the Volcano

The day begins with Yvonne’s arrival at the Bella Vista bar in Quauhnahuac. From outside she hears Geoffrey’s familiar voice shouting a drunken lecture this time on the topic of the rule of the Mexican railway that requires that  “A corpse will be transported by express!” (Lowry, Volcano, p. 43).

For further literary details in memory of Shizuo Kakutani, Yale mathematician and father of book reviewer Michiko Kakutani, see

Santa Versus the Volcano.

Of course, Kakutani himself would probably prefer the anti-Santa, Michael Shermer.  For a refutation of Santa by this high priest of Scientism, see

Miracle on Probability Street

(Scientific American, July 26, 2004). 

Sunday, November 30, 2003

Sunday November 30, 2003

Filed under: General — m759 @ 3:27 PM

The Proof and the Lie

A mathematical lie has been circulating on the Internet.

It concerns the background of Wiles’s recent work on mathematics related to Fermat’s last theorem, which involves the earlier work of a mathematician named Taniyama.

This lie states that at the time of a conjecture by Taniyama in 1955, there was no known relationship between the two areas of mathematics known as “elliptic curves” and “modular forms.”

The lie, due to Harvard mathematician Barry Mazur, was broadcast in a TV program, “The Proof,” in October 1997 and repeated in a book based on the program and in a Scientific American article, “Fermat’s Last Stand,” by Simon Singh and Kenneth Ribet, in November 1997.

“… elliptic curves and modular forms… are from opposite ends of the mathematical spectrum, and had previously been studied in isolation.”

Site on Simon Singh’s 1997 book Fermat’s Last Theorem

“JOHN CONWAY: What the Taniyama-Shimura conjecture says, it says that every rational elliptic curve is modular, and that’s so hard to explain.

BARRY MAZUR: So, let me explain.  Over here, you have the elliptic world, the elliptic curves, these doughnuts.  And over here, you have the modular world, modular forms with their many, many symmetries.  The Shimura-Taniyama conjecture makes a bridge between these two worlds.  These worlds live on different planets.  It’s a bridge.  It’s more than a bridge; it’s really a dictionary, a dictionary where questions, intuitions, insights, theorems in the one world get translated to questions, intuitions in the other world.

KEN RIBET: I think that when Shimura and Taniyama first started talking about the relationship between elliptic curves and modular forms, people were very incredulous….”

Transcript of NOVA program, “The Proof,” October 1997

The lie spread to other popular accounts, such as the column of Ivars Peterson published by the Mathematical Association of America:

“Elliptic curves and modular forms are mathematically so different that mathematicians initially couldn’t believe that the two are related.”

Ivars Peterson, “Curving Beyond Fermat,” November 1999 

The lie has now contaminated university mathematics courses, as well as popular accounts:

“Elliptic curves and modular forms are completely separate topics in mathematics, and they had never before been studied together.”

Site on Fermat’s last theorem by undergraduate K. V. Binns

Authors like Singh who wrote about Wiles’s work despite their ignorance of higher mathematics should have consulted the excellent website of Charles Daney on Fermat’s last theorem.

A 1996 page in Daney’s site shows that Mazur, Ribet, Singh, and Peterson were wrong about the history of the known relationships between elliptic curves and modular forms.  Singh and Peterson knew no better, but there is no excuse for Mazur and Ribet.

Here is what Daney says:

“Returning to the j-invariant, it is the 1:1 map betweem isomorphism classes of elliptic curves and C*. But by the above it can also be viewed as a 1:1 map j:H/r -> C.  j is therefore an example of what is called a modular function. We’ll see a lot more of modular functions and the modular group. These facts, which have been known for a long time, are the first hints of the deep relationship between elliptic curves and modular functions.”

“Copyright © 1996 by Charles Daney,
All Rights Reserved.
Last updated: March 28, 1996″

Update of Dec. 2, 2003

For the relationship between modular functions and modular forms, see (for instance) Modular Form in Wikipedia.

Some other relevant quotations:

From J. S. Milne, Modular Functions and Modular Forms:

“The definition of modular form may seem strange, but we have seen that such functions arise naturally in the [nineteenth-century] theory of elliptic functions.”

The next quote, also in a nineteenth-century context, relates elliptic functions to elliptic curves.

From Elliptic Functions, a course syllabus:

“Elliptic functions parametrize elliptic curves.”

Putting the quotes together, we have yet another description of the close relationship, well known in the nineteenth century (long before Taniyama’s 1955 conjecture), between elliptic curves and modular forms.

Another quote from Milne, to summarize:

“From this [a discussion of nineteenth-century mathematics], one sees that arithmetic facts about elliptic curves correspond to arithmetic facts about special values of modular functions and modular forms.”

Serge Lang apparently agrees:

Elliptic functions parametrize elliptic curves, and the intermingling of the analytic and algebraic-arithmetic theory has been at the center of mathematics since the early part of the nineteenth century.”

Editorial description of Lang’s Elliptic Functions (second edition, 1987)

Update of Dec. 3, 2003

The theory of modular functions and modular forms, defined on the upper half-plane H and subject to appropriate tranformation laws with respect to the group Gamma = SL(2, Z) of fractional linear transformations, is closely related to the theory of elliptic curves, because the family of all isomorphism classes of elliptic curves over C can be parametrized by the quotient Gamma\H. This is an important, although formal, relation that assures that this and related quotients have a natural structure as algebraic curves X over Q. The relation between these curves and elliptic curves predicted by the Taniyama-Weil conjecture is, on the other hand, far from formal.”

Robert P. Langlands, review of Elliptic Curves, by Anthony W. Knapp.  (The review  appeared in Bulletin of the American Mathematical Society, January 1994.)

Thursday, May 22, 2003

Thursday May 22, 2003

Filed under: General — m759 @ 7:29 PM

Seek and Ye Shall Find:

On the Mystical Properties
of the Number 162

On this date in history:

May 22, 1942:  Unabomber Theodore John Kaczynski is born in the Chicago suburb of Evergreen Park, Ill., to Wanda Kaczynski and her husband Theodore R. Kaczynski, a sausage maker. His mother brings him up reading Scientific American.

From the June 2003 Scientific American:

“Seek and ye shall find.” – Michael Shermer

From my note Mark of April 25, 2003:

“Tell me of runes to grave
 That hold the bursting wave,
 Or bastions to design
 For longer date than mine.”

— A. E. Housman, quoted by G. H. Hardy in A Mathematician’s Apology

“Here, as examples, are one rune and one bastion…. (illustrations: the Dagaz rune and the Nike bastion of the Acropolis)…. Neither the rune nor the bastion discussed has any apparent connection with the number 162… But seek and ye shall find.”

Here is a connection to runes:

Mayer, R.M., “Runenstudien,” Beiträge zur Geschichte der deutschen Sprache und Literatur 21 (1896): pp. 162 – 184.

Here is a connection to Athenian bastions from a UN article on Communist educational theorist Dimitri Glinos:

“Educational problems cannot be scientifically solved by theory and reason alone….” (D. Glinos (1882-1943), Dead but not Buried, Athens, Athina, 1925, p. 162)

“Schools are…. not the first but the last bastion to be taken by… reform….”

“…the University of Athens, a bastion of conservatism and counter-reform….”

I offer the above with tongue in cheek as a demonstration that mystical numerology may have a certain heuristic value overlooked by fanatics of the religion of Scientism such as Shermer.

For a more serious discussion of runes at the Acropolis, see the photo on page 16 of the May 15, 2003, New York Review of Books, illustrating the article “Athens in Wartime,” by Brady Kiesling.

Friday, April 25, 2003

Friday April 25, 2003

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

Mark

Today is the feast of Saint Mark.  It seems an appropriate day to thank Dr. Gerald McDaniel for his online cultural calendar, which is invaluable for suggesting blog topics.

Yesterday's entry "Cross-Referenced" referred to a bizarre meditation of mine titled "The Matthias Defense," which combines some thoughts of Nabokov on lunacy with some of my own thoughts on the Judeo-Christian tradition (i.e., also on lunacy).  In this connection, the following is of interest:

From a site titled Meaning of the Twentieth Century —

"Freeman Dyson has expressed some thoughts on craziness. In a Scientific American article called 'Innovation in Physics,' he began by quoting Niels Bohr. Bohr had been in attendance at a lecture in which Wolfgang Pauli proposed a new theory of elementary particles. Pauli came under heavy criticism, which Bohr summed up for him: 'We are all agreed that your theory is crazy. The question which divides us is whether it is crazy enough to have a chance of being correct. My own feeling is that is not crazy enough.' To that Freeman added: 'When a great innovation appears, it will almost certainly be in a muddled, incomplete and confusing form. To the discoverer, himself, it will be only half understood; to everyone else, it will be a mystery. For any speculation which does not at first glance look crazy, there is no hope!' "

Kenneth Brower, The Starship and the Canoe, 1979, pp. 146, 147

It is my hope that the speculation, implied in The Matthias Defense, that the number 162 has astonishing mystical properties (as a page number, article number, etc.) is sufficiently crazy to satisfy Pauli and his friend Jung as well as the more conventional thinkers Bohr and Dyson.  It is no less crazy than Christianity, and has a certain mad simplicity that perhaps improves on some of that religion's lunatic doctrines. 

Some fruits of the "162 theory" —

Searching on Google for muses 162, we find the following Orphic Hymn to Apollo and a footnote of interest:

27 Tis thine all Nature's music to inspire,
28 With various-sounding, harmonising lyre;
29 Now the last string thou tun'ft to sweet accord,
30 Divinely warbling now the highest chord….

"Page 162 Verse 29…. Now the last string…. Gesner well observes, in his notes to this Hymn, that the comparison and conjunction of the musical and astronomical elements are most ancient; being derived from Orpheus and Pythagoras, to Plato. Now, according to the Orphic and Pythagoric doctrine, the lyre of Apollo is an image of the celestial harmony…."

For the "highest chord" in a metaphorical sense, see selection 162 of the 1919 edition of The Oxford Book of English Verse (whose editor apparently had a strong religious belief in the Muses (led by Apollo)).  This selection contains the phrase "an ever-fixèd mark" — appropriately enough for this saint's day.  The word "mark," in turn, suggests a Google search for the phrase "runes to grave" Hardy, after a poem quoted in G. H. Hardy's A Mathematician's Apology.

Such a search yields a website that quotes Housman as the source of the "runes" phrase, and a further search yields what is apparently the entire poem:

Smooth Between Sea and Land

by A. E. Housman

Smooth between sea and land
Is laid the yellow sand,
And here through summer days
The seed of Adam plays.

Here the child comes to found
His unremaining mound,
And the grown lad to score
Two names upon the shore.

Here, on the level sand,
Between the sea and land,
What shall I build or write
Against the fall of night?

Tell me of runes to grave
That hold the bursting wave,
Or bastions to design
For longer date than mine.

Shall it be Troy or Rome
I fence against the foam
Or my own name, to stay
When I depart for aye?

Nothing: too near at hand
Planing the figured sand,
Effacing clean and fast
Cities not built to last
And charms devised in vain,
Pours the confounding main.

(Said to be from More Poems (Knopf, 1936), p. 64)

Housman asks the reader to tell him of runes to grave or bastions to design.  Here, as examples, are one rune and one bastion.

 


The rune known as
"Dagaz"

Represents
the balance point or "still point."


The Nike Bastion

 Dagaz: (Pronounced thaw-gauze, but with the "th" voiced as in "the," not unvoiced as in "thick") (Day or dawn.)

From Rune Meanings:

 Dagaz means "breakthrough, awakening, awareness. Daylight clarity as opposed to nighttime uncertainty. A time to plan or embark upon an enterprise. The power of change directed by your own will, transformation. Hope/happiness, the ideal. Security and certainty. Growth and release. Balance point, the place where opposites meet."

Also known as "the rune of transformation."

For the Dagaz rune in another context, see Geometry of the I Ching.  The geometry discussed there does, in a sense, "hold the bursting wave," through its connection with Walsh functions, hence with harmonic analysis.

 Temple of Athena Nike on the Nike Bastion, the Acropolis, Athens.  Here is a relevant passage from Paul Valéry's Eupalinos ou L'Architecte about another temple of four columns:

Et puis… Écoute, Phèdre (me disait-il encore), ce petit temple que j'ai bâti pour Hermès, à quelques pas d'ici, si tu savais ce qu'il est pour moi ! — Où le passant ne voit qu'une élégante chapelle, — c'est peu de chose: quatre colonnes, un style très simple, — j'ai mis le souvenir d'un clair jour de ma vie. Ô douce métamorphose ! Ce temple délicat, nul ne le sait, est l'image mathématique d'une fille de Corinthe que j'ai heureusement aimée. Il en reproduit fidèlement les proportions particulières. Il vit pour moi !

Four columns, in a sense more suited to Hardy's interests, are also a recurrent theme in The Diamond 16 Puzzle and Diamond Theory.

Apart from the word "mark" in The Oxford Book of English Verse, as noted above, neither the rune nor the bastion discussed has any apparent connection with the number 162… but seek and ye shall find.
 

Monday, October 21, 2002

Monday October 21, 2002

Filed under: General — Tags: — m759 @ 12:01 AM

Birthdays for a Small Planet

Today's birthdays:

The entry below, "Theology for a Small Planet," sketches an issue that society has failed to address since the fall of 1989, when it was first raised by the Harvard Divinity Bulletin.

In honor mainly of Ursula K. Le Guin, but also of her fellow authors above, I offer Le Guin's solution. It is not new. It has been ignored mainly because of the sort of hateful and contemptible arrogance shown by

  • executives in the tradition of Henry Ford and later Ford Foundation and Ford Motors employees McGeorge Bundy and Robert McNamara (see yesterday's entry below for Ford himself), by
  • theologians in the tradition of the Semitic religions — Judaism, Christianity, and Islam — and by
  • self-proclaimed "shamans of scientism" like Michael Shermer in the tradition of Scientific American magazine.

Here is an introduction to the theology that should replace the ridiculous and outdated Semitic religions.

According to Le Guin,

"Scholarly translators of the Tao Te Ching, as a manual for rulers, use a vocabulary that emphasizes the uniqueness of the Taoist 'sage,' his masculinity, his authority. This language is perpetuated, and degraded, in most popular versions. I wanted a Book of the Way accessible to a present-day, unwise, unpowerful, and perhaps unmale reader, not seeking esoteric secrets, but listening for a voice that speaks to the soul. I would like that reader to see why people have loved the book for 2500 years.

It is the most lovable of all the great religious texts, funny, keen, kind, modest, indestructibly outrageous and inexhaustibly refreshing. Of all the deep springs, this is the purest water. To me it is also the deepest spring."

Tao Te Ching: Chapter 6
translated by Ursula K. Le Guin

The valley spirit never dies
Call it the mystery, the woman.

The mystery,
the Door of the Woman,
is the root
of earth and heaven.

Forever this endures, forever.
And all its uses are easy.

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