From a post of January 3, 2024 —
"Hello darkness, my old friend.
I’ve come to talk with you again."
The above image was flipped to reverse left and right.
Related reading: Other posts tagged Darkness and …
Related material: Other posts tagged Star Brick and . . .
"And we may see the meadow in December,
icy white and crystalline"
— Song lyric, "Midnight Sun"
A recently coined phrase — "Negative Mathematics" — is related to the
better-known phrase "Negative Space."
The latter is closely related to the proof of the Cullinane diamond theorem.
For the former, see . . .
Related material: The proof symbol, i.e. the Halmos Tombstone.
From the Log24 search in the previous post for "Dimensions" —
"Hello darkness, my old friend.
I’ve come to talk with you again."
The above image was flipped to reverse left and right.
Related reading: Other posts tagged Darkness and …
"Weight limit 10 tons… Except local deliveries" —
Conclusion: the book is a mine of information, but
you sure have to dig for it. — Paul R. Halmos,
review of Topological Dynamics , November 1955
|
"Poincaré said that science is no more a collection of facts than a house is a collection of bricks. The facts have to be ordered or structured, they have to fit a theory, a construct (often mathematical) in the human mind. … Mathematics may be art, but to the general public it is a black art, more akin to magic and mystery. This presents a constant challenge to the mathematical community: to explain how art fits into our subject and what we mean by beauty. In attempting to bridge this divide I have always found that architecture is the best of the arts to compare with mathematics. The analogy between the two subjects is not hard to describe and enables abstract ideas to be exemplified by bricks and mortar, in the spirit of the Poincaré quotation I used earlier."
— Sir Michael Atiyah, "The Art of Mathematics" |
|
Gottschalk Review —
W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, The ending of the review — The most striking virtue of the book is its organization. The authors' effort to arrange the exposition in an efficient order, and to group the results together around a few central topics, was completely successful; they deserve to be congratulated on a spectacular piece of workmanship. The results are stated at the level of greatest available generality, and the proofs are short and neat; there is no unnecessary verbiage. The authors have, also, a real flair for the "right" generalization; their definitions of periodicity and almost periodicity, for instance, are very elegant and even shed some light on the classical concepts of the same name. The same is true of their definition of a syndetic set, which specializes, in case the group is the real line, to Bohr's concept of a relatively dense set. The chief fault of the book is its style. The presentation is in the brutal Landau manner, definition, theorem, proof, and remark following each other in relentless succession. The omission of unnecessary verbiage is carried to the extent that no motivation is given for the concepts and the theorems, and there is a paucity of illuminating examples. The striving for generality (which, for instance, has caused the authors to treat uniform spaces instead of metric spaces whenever possible) does not make for easy reading. The same is true of the striving for brevity; the shortest proof of a theorem is not always the most perspicuous one. There are too many definitions, especially in the first third of the book; the reader must at all times keep at his finger tips a disconcerting array of technical terminology. The learning of this terminology is made harder by the authors' frequent use of multiple statements, such as: "The term {asymptotic } {doubly asymptotic } means negatively {or} {and} positively asymptotic." Conclusion: the book is a mine of information, but you sure have to dig for it. — PAUL R. HALMOS |
Goes to Facebook !
https://www.nytimes.com/2021/10/04/
technology/facebook-down.html —
"Facebook’s global security operations center
determined the outage was 'a HIGH risk to
the People, MODERATE risk to Assets and
a HIGH risk to the Reputation of Facebook,'
the company memo said.
A small team of employees was soon dispatched
to Facebook’s Santa Clara, Calif., data center to
try a 'manual reset' of the company’s servers,
according to an internal memo."
On this date 15 years ago, other Santa Clara news . . .
* See today's previous post, Bookstores.
Douglas Rain, the voice of HAL in Kubrick's 2001 , reportedly
died at 90 on Sunday, Nov. 11, 2018. A piece from the Sunday,
April 1, 2018, print edition of The New York Times recalls that . . .
|
When HAL says, “I know I’ve made some very poor decisions recently, but I can give you my complete assurance that my work will be back to normal,” Mr. Rain somehow manages to sound both sincere and not reassuring. And his delivery of the line “I think you know what the problem is just as well as I do” has the sarcastic drip of a drawing-room melodrama and also carries the disinterested vibe of a polite sociopath. Kubrick had Mr. Rain sing the 1892 love song “Daisy Bell” (“I’m half crazy, all for the love of you”) almost 50 times, in uneven tempos, in monotone, at different pitches and even just by humming it. In the end, he used the very first take. Sung as HAL’s brain is being disconnected, it’s from his early programming days, his computer childhood. It brings to an end the most affecting scene in the entire film.
— Gerry Flahive in the online New York Times , "A version of this article appears in print on , on Page AR13 of the New York edition with the headline: HAL 9000 Wasn’t Always So Eerily Calm." |
This journal on the above online date, March 30, 2018 —
"Program or be programmed."
— A saying by Douglas Rushkoff
See as well the following link from this journal
on Armistice Day, the reported date of Mr. Rain's death —
See also this journal on the above Catch-22 date — July 16, 2014 —
and a search in this journal for "Halmos + Tombstone."
Remarks related to a recent film and a not-so-recent film.
For some historical background, see Dirac and Geometry in this journal.
Also (as Thas mentions) after Saniga and Planat —
The Saniga-Planat paper was submitted on December 21, 2006.
Excerpts from this journal on that date —
"Open the pod bay doors, HAL."
The black rectangle at the end of Example 1.4
is known as the "end-of-proof symbol," "Halmos,"
or "tombstone."
For the late Paul Halmos —
Keith Devlin's column is a monthly feature of the Mathematical
Association of America (MAA). This month's column apparently
first appeared online on Tuesday, Feb. 17, 2015. The column
itself, however, is, as of today, dated 12:00 AM Feb. 1, 2015.
It is doubtful whether the professor honored in the column
would approve of this temporal subterfuge.
From Log24 on Feb. 26, 2008 —
|
“Many dreams have been The Return of the Author, On Sartre’s Les Mots – “Writing helps him find his own place within this vast comedy. He does not take to writing seriously yet, but he is eager to write books in order to escape the comedy he has been compelled to take part in.” |
"…the source of all great mathematics is the special case,
the concrete example. It is frequent in mathematics that
every instance of a concept of seemingly great generality
is in essence the same as a small and concrete special case."
— Paul Halmos in his autobiography
I Want to Be a Mathematician (1985).
For example:
For a memorable eccentric who reportedly died
on Wednesday, July 24, 2013—
Today's previous post recalled a post
from ten years before yesterday's date.
The subject of that post was the
Galois tesseract.
Here is a post from ten years before
today's date.
The subject of that post is the Halmos
tombstone:
"The symbol 
is used throughout the entire book
in place of such phrases as 'Q.E.D.' or 'This
completes the proof of the theorem' to signal
the end of a proof."
— Measure Theory (1950)
For exact proportions, click on the tombstone.
For some classic mathematics related
to the proportions, see September 2003.
Suggested by Peter J. Cameron's weblog post today on Halmos,
by a July 18 post in this journal on the Norwegian mathematician Abel,
by a link in the July 18 post to "Death Proof," and by today's
midday New York Lottery (459 and 7404)—
From July 4, 2004 (7404 interpreted as a date)—
"There are two unfortunate connotations of 'proof' that come from mathe-
matics and make the word inappropriate in discussions of the security of cryp-
tographic systems. The first is the notion of 100% certainty. Most people not
working in a given specialty regard a 'theorem' that is 'proved' as something
that they should accept without question. The second connotation is of an intri-
cate, highly technical sequence of steps. From a psychological and sociological
point of view, a 'proof of a theorem' is an intimidating notion: it is something
that no one outside an elite of narrow specialists is likely to understand in detail
or raise doubts about. That is, a 'proof' is something that a non-specialist does
not expect to really have to read and think about.
The word 'argument,' which we prefer here, has very different connotations."
— "Another Look at 'Provable Security',"
by Neal Koblitz and Alfred J. Menezes, July 4, 2004
(updated on July 16, 2004; October 25, 2004; March 31, 2005; and May 4, 2005)
As for 459, see Post 459 in this journal.
Related material: The Race, Crossing the Bridge, Aristophanic View, and Story Theory.
Continuing yesterday's lottery meditation…
The NY evening numbers yesterday were 244 and 2962.
The latter suggests Post 2962—
There is no Post 244 here, but a search within this journal for 244 yields…
See also Halmos Tombstone and Death Proof.
Part I: Literal
"Shinin' like a diamond,
she had tombstones in her eyes."
Part II: Figurative
See Halmos Tombstone in this journal.
“Logos and logic, crystal hypothesis,
Incipit and a form to speak the word
And every latent double in the word….”
– Notes Toward a Supreme Fiction
(Quoted here four years ago on October 2, 2006.)
(Background— Yesterday's Quarter to Three,
A Manifold Showing, Class of 64, and Child's Play.)
Hermeneutics
Fans of Gregory Chaitin and Harry Potter
may consult Writings for Yom Kippur
for the meaning of yesterday's evening 673.
(See also Lowry and Cabbala.)
Fans of Elizabeth Taylor, Ava Gardner,
and the Dark Lady may consult Prime Suspect
for the meaning of yesterday's midday 17.
For some more serious background, see Dante—
"….mirando il punto 
a cui tutti li tempi son presenti "
– Dante, Paradiso, XVII, 17-18
“The symbol 
is used throughout the entire book
in place of such phrases as ‘Q.E.D.’ or
‘This completes the proof of the theorem’
to signal the end of a proof.”
— Measure Theory, by Paul R. Halmos, Van Nostrand, 1950
Halmos died on the date of Yom Kippur —
October 2, 2006.
… on Friday, October 30, 2009 ………
(known to some as “Devil’s Night”)………
according to The New York Times
…
A search in this journal for “Levi-Strauss” yields various entries, the most recent being “Autistic Enchantment” (Sept. 3, 2009).
Related material:
Today’s New York Times on autism
(A Powerful Identity, a Vanishing Diagnosis)
and Log24 on enchantment.
An instance of the latter (from Feb. 15, 2008):
Door
Step:
“Many dreams have been
brought to your doorstep.
They just lie there
and they die there.”
— Lyricist Ray Evans,
who died at 92
one year ago today
Associated Press –
Feb. 15, 2008
Today in History –
Thought for Today:
Postscript of Nov. 3, 2009:
For more confusion, see
the works of Claude Levi-Strauss.
But according to The Telegraph, Levi-Strauss
died on Saturday, Oct. 31, All Hallows’ Eve.
According to Le Monde, he may have died
even later, on Sunday, Nov. 1, All Saints’ Day.
Tuesday, July 31, 2007Aesthetics for JesuitsJoke The Guardian, July 26, “… inspired satire, laced with Jewish and Christian polemics, sparkling wit and dazzlingly simple effects. For Golgotha a stagehand brings on three crosses. ‘Just two,’ says Jay. ‘The boy is bringing his own.’ Tabori often claimed that the joke was the most perfect literary form.” |
Update at noon,
Sept. 9, 2008:
Tabori, a Jew from Hungary
and former screenwriter
(“No Exit“), died at 93
on July 23, 2007.
For related material on
another Jew from Hungary
click on the black monolith
(also known as
the Halmos tombstone).
|
"Have liberty not as the air within a grave Or down a well. Breathe freedom, oh, my native, In the space of horizons that neither love nor hate."
— Wallace Stevens, |
A related visual
association of ideas —
("The association is the idea"
— Ian Lee, The Third Word War)
|
by John Braheny
"Hook" is the term you'll hear most often in the business and craft of commercial songwriting. (Well, maybe not as much as "Sorry, we can't use your song," but it's possible that the more you hear about hooks now, the less you'll hear "we can't use it" later.) |
See also UD's recent
A Must-Read and In My Day*
as well as the five
Log24 entries ending
Sept. 20, 2002.
* Hey Hey
|
"Credences of Summer," VII,
by Wallace Stevens, from
"Three times the concentred |
One possibility —
Bertram Kostant, Professor Emeritus of Mathematics at MIT, on an object discussed in a recent New Yorker:
"A word about E(8). In my opinion, and shared by others, E(8) is the most magnificent 'object' in all of mathematics. It is like a diamond with thousands of facets. Each facet offering a different view of its unbelievable intricate internal structure."
Another possibility —
A more modest object —
the 4×4 square.
Update of Aug. 20-21 —
Kostant's poetic comparison might be applied also to this object.
More precisely, there are 322,560 natural rearrangements– which a poet might call facets*— of the array, each offering a different view of the array's internal structure– encoded as a unique ordered pair of symmetric graphic designs. The symmetry of the array's internal structure is reflected in the symmetry of the graphic designs. For examples, see the Diamond 16 Puzzle.
For an instance of Stevens's "three times" process, see the three parts of the 2004 web page Ideas and Art.
* For the metaphor of rearrangements as facets, note that each symmetry (rearrangement) of a Platonic solid corresponds to a rotated facet: the number of symmetries equals the number of facets times the number of rotations (edges) of each facet–
If Greek geometers had started with a finite space (as in The Eightfold Cube), the history of mathematics might have dramatically illustrated Halmos's saying (Aug. 16) that
"The problem is– the genius is– given an infinite question, to think of the right finite question to ask. Once you thought of the finite answer, then you would know the right answer to the infinite question."
The Greeks, of course, answered the infinite questions first– at least for Euclidean space. Halmos was concerned with more general modern infinite spaces (such as Hilbert space) where the intuition to be gained from finite questions is still of value.
Seeing the Finite Structure
The following supplies some context for remarks of Halmos on combinatorics.
From Paul Halmos: Celebrating 50 years of Mathematics, by John H. Ewing, Paul Richard Halmos, Frederick W. Gehring, published by Springer, 1991–
Interviews with Halmos, “Paul Halmos by Parts,” by Donald J. Albers–
“Part II: In Touch with God*“– on pp. 27-28:
The Root of All Deep Mathematics
“Albers. In the conclusion of ‘Fifty Years of Linear Algebra,’ you wrote: ‘I am inclined to believe that at the root of all deep mathematics there is a combinatorial insight… I think that in this subject (in every subject?) the really original, really deep insights are always combinatorial, and I think for the new discoveries that we need– the pendulum needs– to swing back, and will swing back in the combinatorial direction.’ I always thought of you as an analyst.
Halmos: People call me an analyst, but I think I’m a born algebraist, and I mean the same thing, analytic versus combinatorial-algebraic. I think the finite case illustrates and guides and simplifies the infinite.
Some people called me full of baloney when I asserted that the deep problems of operator theory could all be solved if we knew the answer to every finite dimensional matrix question. I still have this religion that if you knew the answer to every matrix question, somehow you could answer every operator question. But the ‘somehow’ would require genius. The problem is not, given an operator question, to ask the same question in finite dimensions– that’s silly. The problem is– the genius is– given an infinite question, to think of the right finite question to ask. Once you thought of the finite answer, then you would know the right answer to the infinite question.
Combinatorics, the finite case, is where the genuine, deep insight is. Generalizing, making it infinite, is sometimes intricate and sometimes difficult, and I might even be willing to say that it’s sometimes deep, but it is nowhere near as fundamental as seeing the finite structure.”
Whether the above sketch of the passage from operator theory to harmonic analysis to Walsh functions to finite geometry can ever help find “the right finite question to ask,” I do not know. It at least suggests that finite geometry (and my own work on models in finite geometry) may not be completely irrelevant to mathematics generally regarded as more deep.
* See the Log24 entries following Halmos’s death.
Yesterday's entry discussed T.E. Hulme— a co-founder, with Ezra Pound, of the Imagist school of poetry. Recent entries on randomness, using the New York Lottery as a source of examples, together with Hulme's approach to poetry discussed yesterday, suggest the following meditation– what Charles Cameron might call a "bead game."
Part I:
Ezra Pound on Imagism (from Gaudier-Brzeska,* 1916):
Three years ago in Paris I got out of a "metro" train at La Concorde, and saw suddenly a beautiful face, and then another and another, and then a beautiful child’s face, and then another beautiful woman, and I tried all that day to find words for what this had meant to me, and I could not find any words that seemed to me worthy, or as lovely as that sudden emotion. [….]
The "one image poem" is a form of super-position, that is to say, it is one idea set on top of another. I found it useful in getting out of the impasse in which I had been left by my metro emotion. I wrote a thirty-line poem, and destroyed it because it was what we call work "of second intensity." Six months later I made a poem half that length; a year later I made the following hokku-like sentence: —
"The apparition of these
faces in the crowd:
Petals, on a
wet, black bough."
I dare say it is meaningless unless one has drifted into a certain vein of thought. In a poem of this sort one is trying to record the precise instant† when a thing outward and objective transforms itself, or darts into a thing inward and subjective.
Part II:
Eleanor Goodman on translation (in a July 7, 2008, weblog entry, "Pound and Process: An Introduction"):
"… all translations exist on an axis. Indeed, they exist in a manifold of many axes intersecting. One axis is that of foreignness and familiarity. One axis is that of structural mimicry, another of melodic mimicry. And one axis is that of semantic fidelity."
Goodman's use of the word "manifold" here is of course poetic, not mathematical.
Part III:
New York Lottery, mid-day on July 7, 2008: 771.
A Google search on manifold 771 reveals that 771 is, according to Google's scanners, an alternate form (a "translation," via structural mimicry) of a script version of the letter M. (See Part V below.)
Part V:
Long version of a
one-image poem —
"Random apparition:
manifold translated."
This poem summarizes the
relationship (See Part IV above) of
the (apparently) random number 771
to the rather non-random concept of
a linear manifold:
[Such lines and planes have not
been, in mathematical language,
"translated."]
— Paul R. Halmos,
Finite Dimensional Vector Spaces,
Princeton University Press, 1948
Short version of the
above one-image poem —
771:
* Gaudier-Brzeska created the artifact shown on the cover of Solid Objects, a work of literary theory by Douglas Mao. For more on that artifact and on the New York Lottery, see Sermon for St. Peter's Day. "It is not in the premise that reality/ Is a solid…." –Wallace Stevens
† "I was like, Oh My God." —Poet Billy Collins at Chautauqua Institution, morning of July 7, 2008
Some thoughts from
the preceding Friday,
the birthday of actor
Kevin “You’re Next“
McCarthy:
|
“Many dreams have been “Writing helps him find his own place within this vast comedy. He does not take to writing seriously yet, but he is eager to write books in order to escape the comedy he has been compelled to take part in.”
|
The obituary of Burghardt
and The Four Last Things.
“Hell is other people.”
— Jean-Paul Sartre,
No Exit
With a laugh track.
Step:
“Many dreams have been
brought to your doorstep.
They just lie there
and they die there.”
— Lyricist Ray Evans,
who died at 92
one year ago today
Associated Press –
Today in History –
Thought for Today:
The Return of the Author, by Eugen Simion:
On Sartre’s Les Mots —
The craft of writing appeared to me as an adult activity, so ponderously serious, so trifling, and, at bottom, so lacking in interest that I didn’t doubt for a moment that it was in store for me. I said to myself both ‘that’s all it is’ and ‘I am gifted.’ Like all dreamers, I confused disenchantment with truth.”
This is given in The Oxford Dictionary of Quotations (1999) as
Like all dreamers, I mistook disenchantment for truth.
Also from the AP’s
Today in History —
Today’s Birthdays:
Actor Kevin McCarthy is 94.
Hopkins at Heaven’s Gate
(In context: October 2007)–
“Dolly’s Little Diner–
Home from Home”
The three main characters “spend much of the play discussing quantum mechanics, string theory and Schrödinger’s Cat experiment….
Ms. Buggé’s frequently clever script makes the audience feel smart by offering up fairly recognizable literary references (from, among other things, T. S. Eliot’s ‘Love Song of J. Alfred Prufrock’ and William Wordsworth’s ‘Tintern Abbey’). But the play suffers from abrupt, sometimes motivation-free exits and entrances.”
As does life itself.
Yau rated the conjecture as one of the major mathematical puzzles of the 20th Century. |
| Five years have passed; five summers, with the length Of five long winters! |
There is one story and one story only
That will prove worth your telling….
— Robert Graves,
“To Juan at the Winter Solstice”
Exits and Entrances:
Halmos exited on Yom Kippur.
He may or may not achieve
re-entry. For details, see
Log24 entries of Oct. 1-15:
Ticket Home
Related material:
The Unity of Mathematics,
Heisenberg on Beauty, and
Theme and Variations.
Geometry
from Point
to Hyperspace
by Steven H. Cullinane
Euclid is “the most famous
geometer ever known
and for good reason:
for millennia it has been
his window
that people first look through
when they view geometry.”
— Euclid’s Window:
The Story of Geometry
from Parallel Lines
to Hyperspace,
by Leonard Mlodinow
“…the source of
all great mathematics
is the special case,
the concrete example.
It is frequent in mathematics
that every instance of a
concept of seemingly
great generality is
in essence the same as
a small and concrete
special case.”
— Paul Halmos in
I Want To Be a Mathematician
Euclid’s geometry deals with affine
spaces of 1, 2, and 3 dimensions
definable over the field
of real numbers.
Each of these spaces
has infinitely many points.
Some simpler spaces are those
defined over a finite field–
i.e., a “Galois” field–
for instance, the field
which has only two
elements, 0 and 1, with
addition and multiplication
as follows:
|
|
From these five finite spaces,
we may, in accordance with
Halmos’s advice,
select as “a small and
concrete special case”
the 4-point affine plane,
which we may call
Galois’s Window.
The interior lines of the picture
are by no means irrelevant to
the space’s structure, as may be
seen by examining the cases of
the above Galois affine 3-space
and Galois affine hyperplane
in greater detail.
For more on these cases, see
The Eightfold Cube,
Finite Relativity,
The Smallest Projective Space,
Latin-Square Geometry, and
Geometry of the 4×4 Square.
(These documents assume that
the reader is familar with the
distinction between affine and
projective geometry.)
These 8- and 16-point spaces
may be used to
illustrate the action of Klein’s
simple group of order 168
and the action of
a subgroup of 322,560 elements
within the large Mathieu group.
The view from Galois’s window
also includes aspects of
quantum information theory.
For links to some papers
in this area, see
Elements of Finite Geometry.
"Every great magic trick consists of three acts. The first act is called 'The Pledge.' The magician shows you something ordinary, but of course… it probably isn't. The second act is called 'The Turn.' The magician makes his ordinary 'some thing' do something extraordinary. Now if you're looking for the secret… you won't find it. That's why there's a third act, called 'The Prestige.' This is the part with the twists and turns, where lives hang in the balance, and you see something shocking you've never seen before."
"You can fool
too many of the people
too much of the time."
— James Thurber,
American humorist
(1894-1961)
For more detail,
— Gian-Carlo Rota in
Indiscrete Thoughts, p. 152
The Devil’s Bible and
Nothing Nothings (Again).
The Context
One context for the Rota quote
is Paul Halmos’s remark, quoted
in today’s New York Times,
that mathematics is
“almost like being
in touch with God.”
Another context is
Log24 on Aug. 29, 2005.
Click on the halmos
for further details from
today’s New York Times.
|
A Poem for Pinter
Oct. 13, 2005 The Guardian on Harold Pinter, winner of this year's Nobel Prize for Literature: "Earlier this year, he announced his decision to retire from playwriting in favour of poetry," Michael Muskal in today's Los Angeles Times: "Pinter, 75, is known for his sparse and thin style as well as his etched characters whose crystal patter cuts through the mood like diamond drill bits." Robert Stone, A Flag for Sunrise (See Jan. 25): "'That old Jew gave me this here.' Egan looked at the diamond…. 'It's worth a whole lot of money– you can tell that just by looking– but it means something, I think. It's got a meaning, like.'
'Let's see,' Egan said, 'what would it mean?' He took hold of Pablo's hand cupping the stone and held his own hand under it. '"The jewel is in the lotus," perhaps that's what it means. The eternal in the temporal….'"
"Modal logic was originally developed to investigate logic under the modes of necessary and possible truth. The words 'necessary' and 'possible' are called modal connectives, or modalities. A modality is a word that when applied to a statement indicates when, where, how, or under what circumstances the statement may be true. In terms of notation, it is common to use a box [] for the modality 'necessary' and a diamond <> for the modality 'possible.'"
Commentary:
"Waka" also means Japanese poem or Maori canoe. (For instance, this Japanese poem and this Maori canoe.)
For a meditation on "bang splat," see Sept. 25-29. For the meaning of "tick tick," see Emily Dickinson on "degreeless noon." "Hash," of course, signifies "checkmate." (See previous three entries.) |
For language more suited to
the year's most holy day, see
this year's Yom Kippur entry,
from October 2.
That was also the day of the
Amish school killings in
Pennsylvania and the day that
mathematician Paul Halmos died.
For more on the former, see
Death in Two Seconds.
For more on the latter, see
The Halmos Tombstone.
(Title of an interview with
the late Paul Halmos, mathematician)
From a 1990 interview:
“What’s the best part of being a mathematician? I’m not a religious man, but it’s almost like being in touch with God when you’re thinking about mathematics. God is keeping secrets from us, and it’s fun to try to learn some of the secrets.”
I personally prefer Annie Dillard on God:
“… if Holy the Firm is matter at its dullest, Aristotle’s materia prima, absolute zero, and since Holy the Firm is in touch with the Absolute at base, then the circle is unbroken. And it is…. Holy the Firm is in short the philosopher’s stone.”
Some other versions of
the philosopher’s stone:
This last has the virtue of
being connected with Halmos
via his remarks during the
“In Touch with God” interview:
See also the remark of Halmos that serves as an epigraph to Theme and Variations.
has also served
at least one interpreter
as a philosopher’s stone,
and is also the original
“Halmos tombstone.”
“Prof. Paul Halmos died of pneumonia early in the morning of October 2, 2006. He was 90 years old. He is survived by his wife, Virginia Halmos. An obituary may be found at the website of the Mathematical Association of America….”
— Halmos’s home page
at Santa Clara University
For a memorial of sorts, see
Lovely, Dark and Deep.
Update of 8 PM Oct. 4 —
From Google Book Search:
This is the source of the
“Halmos tombstone” symbol,
which has been described in a
different form at Wikipedia:
“The tombstone, or halmos–
symbol ∎ (Unicode U+220E)–
is used in mathematics to denote
the end of a proof.”
This Unicode character is rendered
as an empty square in Explorer
and as a black square in Firefox.
Related material:
Tombstone
From today's New York Times:
"Jiri Frel, a mercurial and eccentric curator who helped build the J. Paul Getty Museum into a major center for Greek and Roman art but resigned after revelations about unscrupulous acquisition practices, died on April 29. He was 82…. a well-regarded expert in Greek tombstones…."
"ATHENS, May 16 — After four hours of talks here with the Greek culture minister, the director of the J. Paul Getty Museum in Los Angeles said Tuesday that he would press for the return of some of the Getty's most prized ancient artifacts to Greece…. Greece is seeking the repatriation of a… tombstone…."
From a photo accompanying the obituary:
To Aster, from Plato
Asteras eisathreis, Aster emos.
Eithe genoimen ouranos,
'os pollois ommasin eis se blepo.
You gaze at stars, my Star.
Would that I were born the starry sky,
that I with many eyes might gaze at you.
Related material:
Example
“…the source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly great generality is in essence the same as a small and concrete special case.”
— Paul Halmos in
I Want To Be a Mathematician
ART WARS for
Trotsky’s Birthday
Part I:
Symbols
From my entry of July 26, 2003, in memory
of Marathon Man director John Schlesinger:
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Bright Star and Dark Lady “Mexico is a solar country — but it is also a black country, a dark country. This duality of Mexico has preoccupied me since I was a child.” — Octavio Paz, |
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Bright Star
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Amen. |
Dark Lady
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For the meaning of the above symbols, see
Kubrick’s 1x4x9 monolith in 2001,
the Halmos tombstone in Measure Theory,
and the Fritz Leiber Changewar stories.
Part II:
Sunday in the Park with Death
To Leon from Diego —
Details of a mural,
A Dream of a Sunday Afternoon
in Alameda Park,
Fresco, 1947-48,
Alameda Hotel, Mexico City:
Three’s a Crowd:
Symbol:
Funeral March
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John Schlesinger dead at 77; LOS ANGELES – Oscar-winning director John Schlesinger, who daringly brought gay characters into mainstream cinema with Midnight Cowboy and tapped into nightmares with the teeth-drilling torture of Marathon Man, died Friday at 77. The British-born filmmaker…. died about 5:30 a.m…. |
Schlesinger also directed The Day of the Locust, based on a novel by Nathanael West.
See Heaven, Hell, and Hollywood and
From the latter:
“Then you know your body’s sent,
Don’t care if you don’t pay rent,
Sky is high and so am I,
If you’re a viper — a vi-paah.”
— The Day of the Locust,
by Nathanael West (1939),
New Directions paperback,
1969, page 162
This song may be downloaded at
That same site begins with a traditional Mexican song…
“La cucaracha, la cucaracha,
ya no puede caminar,
porque no quiere,
porque le falta
marihuana que fumar.”
(“The cockroach, the cockroach,
can’t walk anymore,
because he doesn’t want to,
because he has no
marihuana to smoke.”)
This suggests an appropriate funeral march for John Schlesinger:
“Ya murió la cucaracha, ya la llevan a enterrar…” – La Cucaracha
Those attending Schlesinger’s wake, as opposed to his funeral, may wish to perform other numbers from the Pot Culture page, which offers a variety of “viper” songs.
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Bright Star and Dark Lady “Mexico is a solar country — but it is also a black country, a dark country. This duality of Mexico has preoccupied me since I was a child.” — Octavio Paz, |
||
|
Bright Star
|
Amen.
|
Dark Lady
|
For the meaning of the above symbols, see
Kubrick’s 1x4x9 monolith in 2001,
the Halmos tombstone in Measure Theory,
and the Fritz Leiber Changewar stories.
Concluding Unscientific Postscript:
Oh, yes… the question of
Heaven or Hell for John Schlesinger…
Recall that he also directed the delightful
Cold Comfort Farm and see
last year’s entry for this date.
Symbols —
Broadway:
The Sound of Silence
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Hello darkness, my old friend. I’ve come to talk with you again. (See previous entry, Mar. 7, “Lovely, Dark and Deep.) |
And the people bowed and prayed to the neon god they made. (See CNN.com Broadway City Arcade club story of Mar. 9) |
The words of the prophets are written on the subway walls. (See picture in NY Times Book Review, Mar. 9, page 31.) |
See also the footnote on the Halmos “tombstone” symbol in the previous entry, the entry “Dustin in Wonderland” of Feb. 24, the film “Marathon Man,” and the entry “Geometry for Jews” of March 6.
Lovely, Dark and Deep
On this date in 1923, "Stopping by Woods on a Snowy Evening," by Robert Frost, was published. On this date in 1999, director Stanley Kubrick died. On this date in 1872, Piet Mondrian was born.
"….mirando il punto
a cui tutti li tempi son presenti"
— Dante, Paradiso, XVII, 17-18
Chez Mondrian
Kertész, Paris, 1926
6:23 PM Friday, March 7:
From Measure Theory, by Paul R. Halmos, Van Nostrand, 1950:
"The symbol is used throughout the entire book in place of such phrases as 'Q.E.D.' or 'This completes the proof of the theorem' to signal the end of a proof."
The Boys from Uruguay
If one were to write a “secret history” of the twentieth century, one possible organizing theme might be the religious struggle between worshippers of the Semitic deity (variously known as Yahweh, God, and Allah) and worshippers of the Aryan deities… notably, the Aryan god of music, light, and reason, Apollo.
(See my jounal notes of Monday, Sept. 2, 2002, below.)
In perhaps the best academic website I have ever seen, Karey L. Perkins quotes Walker Percy:
“The truth is that man’s capacity for symbol-mongering in general and language in particular is…intimately part and parcel of his being human, of his perceiving and knowing, of his very consciousness…”
The greatest symbol-monger of the twentieth century was, of course, Adolf Hitler. His use of the Aryan sun-wheel symbol rose to the level of genius. Of course, it ultimately failed to win the approval of the sun god himself, Apollo, who is also the god of reason.
Since symbol-mongering cannot be avoided, let us hope that it can be done in a somewhat more reasonable way than that of the National Socialist movement. Two examples suggest themselves.
From Karey Perkins’s website:
On this Rosh Hashanah, the cross as a symbol of intelligence may be offensive to some worshippers of Yahweh. Let them read The Archivist, a novel by Martha Cooley, and then my journal note The Matthias Defense.
They might also contemplate the biblical quotation in the musical “Contact” broadcast from Lincoln Center on September 1, 2002: “Let there be light!”
Three Jews named Paul have been associated with light…
Saul of Tarsus, who later assumed an alias.
Paul Newman, whose performance in “The Verdict” continues, indirectly, to trouble Cardinal Law of Boston.
Paul R. Halmos, a personal hero of mine ever since I saw his Finite Dimensional Vector Spaces and Measure Theory as an ignorant young undergraduate browsing the bookstores of Harvard Square.
In accordance with the “secret history” theme mentioned above, the struggle between Aryan and Semitic religions may also be viewed in the light of the struggle between Christianity and Communism. Hitler exploited this viewpoint very successfully, pretending to be the champion of the Christians against the godless Reds. Peggy Noonan also successfully uses this strategy. Both Hitler and Noonan manage to ignore the fact that Christianity is itself one of the Semitic religions, and that at least two of its three deities are Jewish.
As for me, I rather identify with the young Hitler clone at the end of the film “The Boys from Brazil.” Forced to decide between Gregory Peck and Sir Laurence Olivier, he sides with Olivier. His reason? Peck lied.
In a similar situation, forced to decide between Peggy Noonan and the Jew Halmos, I would probably side with Halmos. Halmos, who should, if not a saint, be at least dubbed a knight, does not, unlike the great majority of the damned human race, lie.
See Halmos’s memoir, I Want to Be a Mathematician. In particular, see the single index entry “communist by allegation” and the 29 entries under “Uruguay.”
Happy birthday to Elia Kazan and Peggy Noonan, and a happy and prosperous New Year to should-be-Sir Paul R. Halmos.
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