Log24

Wednesday, October 23, 2024

The Delta Transform

Filed under: General — Tags: , , , — m759 @ 7:04 am

Rothko — "… the elimination of all obstacles between the painter and
the idea, and between the idea and the observer."

Walker Percy has similarly discussed elimination of obstacles between
the speaker and the word, and between the word and the hearer.

Walker Percy's chapter on 'The Delta Factor' from 'Message in the Bottle'

Click images to enlarge.

Related mathematics —

The source: http://finitegeometry.org/sc/gen/typednotes.html.

A document from the above image —

AN INVARIANCE OF SYMMETRY

BY STEVEN H. CULLINANE

We present a simple, surprising, and beautiful combinatorial
invariance of geometric symmetry, in an algebraic setting.

DEFINITION. A delta transform of a square array over a 4-set is
any pattern obtained from the array by a 1-to-1 substitution of the
four diagonally-divided two-color unit squares for the 4-set elements.

THEOREM. Every delta transform of the Klein group table has
ordinary or color-interchange symmetry, and remains symmetric under
the group G of 322,560 transformations generated by combining
permutations of rows and colums with permutations of quadrants.

PROOF (Sketch). The Klein group is the additive group of GF (4);
this suggests we regard the group's table  T as a matrix over that
field. So regarded, T is a linear combination of three (0,1)-matrices
that indicate the locations, in  T, of the 2-subsets of field elements.
The structural symmetry of these matrices accounts for the symmetry
of the delta transforms of  T, and is invariant under G.

All delta transforms of the 45 matrices in the algebra generated by
the images of  T under G are symmetric; there are many such algebras. 

THEOREM. If 1 m ≤ n2+2, there is an algebra of 4m
2n x 2n matrices over GF(4) with all delta transforms symmetric.

An induction proof constructs sets of basis matrices that yield
the desired symmetry and ensure closure under multiplication.

REFERENCE

S. H. Cullinane, Diamond theory (preprint).

Update of 1:12 AM ET on Friday, Oct. 25, 2024 —

The above "invariance of symmetry" document was written in 1978
for submission to the "Research Announcements" section of the
Bulletin of the American Mathematical Society .  This pro forma 
submission was, of course, rejected.  Though written before
I learned of similar underlying structures in the 1974 work of
R. T. Curtis on his "Miracle Octad Generator," it is not without
relevance to his work.

Sunday, September 22, 2024

Seattle Jam

Filed under: General — Tags: — m759 @ 9:11 am
 

CBS Sunday Morning today suggests a review of an old post featuring Pearl Jam. From that post . . .

Mathematics and Narrative, continued…

Out of What Chaos, a novel by Lee Oser

"This book is more or less what one would expect if Walker Percy wrote about a cynical rock musician who converts to Catholicism, and then Nabokov added some of his verbal pyrotechnics, and then Buster Keaton and the Marquis de Sade and Lionel Trilling inserted a few extra passages. It is a loving and yet appalled description of the underground music scene in the Pacific Northwest. And it is a convincing representation of someone very, very smart."

Matt Greenfield in The Valve

"If Evelyn Waugh had lived amid the American Northwest rock music scene, he might have written a book like this."

–Anonymous Amazon.com reviewer

A possible source for Oser's title–

"…Lytton Strachey described Pope's theme as 'civilization illumined by animosity; such was the passionate and complicated material from which he wove his patterns of balanced precision and polished clarity.' But out of what chaos did that clarity and precision come!"

Authors at Work, by  Herman W. Liebert and Robert H. Taylor, New York, Grolier Club, 1957, p. 16

Tuesday, February 20, 2024

Backlight

Filed under: General — m759 @ 12:09 am

The epigraph of the previous post

"To Phaedrus, this backlight from the conflict between
the Sophists and the Cosmologists adds an entirely
new dimension to the Dialogues of Plato." — Robert M. Pirsig

Related reading and art for academic nihilists — See . . .

Reading and art I prefer —

Love in the Ruins , by Walker Percy, and . . .

Van Gogh  (by Ed Arno) and an image and
a passage from The Paradise of Childhood
(by Edward Wiebé):

'Dear Theo' cartoon of van Gogh by Ed Arno, adapted to illustrate the eightfold cube

Monday, February 19, 2024

Theology for Sophists

Filed under: General — m759 @ 10:15 pm

"To Phaedrus, this backlight from the conflict between
the Sophists and the Cosmologists adds an entirely
new dimension to the Dialogues of Plato." — Robert M. Pirsig

"It’s all in Plato, all in Plato;
bless me, what do  they
teach them at these schools?”

— C. S. Lewis in
The Narnia Chronicles

Compare and Contrast — Plato's Diamond.

Plato's diamond in Jowett's version of the Meno dialogue

Tuesday, April 21, 2020

Living Water

Filed under: General — m759 @ 6:47 pm

The Tesseract Timeline:
Where The Cube Has Been In The …

www.cinemablend.com › news › the-tesseract-timeline-…
Mar 13, 2019 – With HYDRA. In 1942, Johann Schmidt, a.k.a.
the Red Skull, arrived in Tønsberg to procure the Tesseract
from an ancient church. While he …

Related material from posts tagged Aqua
(suggested by a name in the previous post) —

Lene from Tønsberg

 

Tuesday, March 29, 2016

The Sudden Inkling

Filed under: General — m759 @ 2:01 pm

In memory of "a cultural icon" —

Walker Percy's chapter on 'The Delta Factor' from 'Message in the Bottle'

Click image to enlarge.

Tuesday, July 1, 2014

Latin Word:

Filed under: General — Tags: , — m759 @ 4:36 am

Aqua

Version 1:

(See the June 30 posts Toward Evening,
Joke, and High Concept.)

Version 2:

Version 3:

Thursday, December 3, 2009

Out of What Chaos

Filed under: General — Tags: — m759 @ 2:28 pm

Mathematics and Narrative, continued…

Out of What Chaos, a novel by Lee Oser

"This book is more or less what one would expect if Walker Percy wrote about a cynical rock musician who converts to Catholicism, and then Nabokov added some of his verbal pyrotechnics, and then Buster Keaton and the Marquis de Sade and Lionel Trilling inserted a few extra passages. It is a loving and yet appalled description of the underground music scene in the Pacific Northwest. And it is a convincing representation of someone very, very smart."

Matt Greenfield in The Valve

"If Evelyn Waugh had lived amid the American Northwest rock music scene, he might have written a book like this."

–Anonymous Amazon.com reviewer

A possible source for Oser's title–

"…Lytton Strachey described Pope's theme as 'civilization illumined by animosity; such was the passionate and complicated material from which he wove his patterns of balanced precision and polished clarity.' But out of what chaos did that clarity and precision come!"

Authors at Work, by  Herman W. Liebert and Robert H. Taylor, New York, Grolier Club, 1957, p. 16

Related material:

Unthought Known

Pearl Jam 'Backspacer' album released Sept. 20, 2009

and the

Catholic Analyst's Couch, White Cube Gallery, 2002

White Cube Gallery, 2002

Friday, September 5, 2008

Friday September 5, 2008

Filed under: General — m759 @ 6:23 pm
Adult Books
 

On author Madeleine L’Engle:

“Madeleine’s adult books– including the autobiographical titles that eventually would be grouped together as the Crosswicks Journals– A Circle of Quiet (1971), The Summer of the Great-Grandmother (1974), The Irrational Season (1976), and Two-Part Invention (1988)– were edited by Robert Giroux. If Roger Straus was FSG’s [Farrar, Straus & Giroux’s] worldly sophisticate presiding over editorial meetings, Bob Giroux was the white-haired, rosy-cheeked favorite uncle (if you happened to have an erudite uncle who had edited T. S. Eliot, Robert Lowell, Isaac Bashevitz Singer, Elizabeth Bishop, Flannery O’Connor, and Walker Percy).”

Sandra Jordan, School Library Journal, November 1, 2007

On Robert Giroux, who died early this morning:

the gold standard of literary taste.”

For a less demanding standard, see today’s previous entry.

Tuesday, June 6, 2006

Tuesday June 6, 2006

Filed under: General — m759 @ 4:29 am
Pentecost
and
Queer Theory

“Stuff comes up,
weird doors open,
people fall into things.”
— David Sedaris,
baccalaureate address
at Princeton on Sunday,
June 4, 2006,
the Feast of Pentecost

“The truth is that man’s capacity for symbol-mongering in general and language in particular is so intimately part and parcel of his being human, of his perceiving and knowing, of his very consciousness itself, that it is all but impossible for him to focus on the magic prism through which he sees everything else.”

Walker Percy, The Message in the Bottle: How Queer Man Is, How Queer Language Is, and What One Has to Do with the Other. New York: Farrar, Straus, and Girox, 1975, page 29.

Monday, June 5, 2006

Monday June 5, 2006

Filed under: General — Tags: — m759 @ 12:06 am

Sermon

Baccalaureate:
A farewell address
in the form of a sermon
delivered to a graduating class.

"Stuff comes up,
weird doors open,
people fall into things."
— David Sedaris,
baccalaureate address
at Princeton yesterday

"The truth is that man's capacity for symbol-mongering in general and language in particular is so intimately part and parcel of his being human, of his perceiving and knowing, of his very consciousness itself, that it is all but impossible for him to focus on the magic prism through which he sees everything else."

Walker Percy, The Message in the Bottle: How Queer Man Is, How Queer Language Is, and What One Has to Do with the Other. New York: Farrar, Straus, and Giroux, 1975, page 29.

Review: ART WARS
on Sept. 12, 2002:

Und was für ein Bild des Christentums 
ist dabei herausgekommen?

Voilà:

The image �http://www.log24.com/log/pix06A/060604-Primitive.gif� cannot be displayed, because it contains errors.

Related material:
Bright Star.

Wednesday, September 15, 2004

Wednesday September 15, 2004

Filed under: General — Tags: — m759 @ 7:11 pm

High Holy
Hexagram

7:11:20 PM

For a poetic interpretation
of this symbol, see
Hexagram 20,
Contemplation (View).

For a religious interpretation
suited to the High Holy Days,
see the film

.

 “The truth is that man’s capacity for symbol-mongering in general and language in particular is so intimately part and parcel of his being human, of his perceiving and knowing, of his very consciousness itself, that it is all but impossible for him to focus on the magic prism through which he sees everything else.”

— Walker Percy, The Message in the Bottle

Sunday, August 29, 2004

Sunday August 29, 2004

Filed under: General — m759 @ 11:07 am

Olympic Arc

Thomas Becker, president of Chautauqua Institution, on Friday, Aug. 27, 2004:

“I’m really proud of this lecture platform this year.  We started with Phil Wilcox on the first day of the season and finished with Sandra Day O’Connor.  The arc of participation between them was really amazing.”

Phil Wilcox: See 

Israel and Palestine:
Let’s Separate Myth from Reality
,

by Philip C. Wilcox, Jr., President,
Foundation for Middle East Peace,
Chautauquan Daily, June 28, 2004

Sandra Day O’Connor: See

The Majesty of the Law:
Reflections of a
Supreme Court Justice
,
by Sandra Day O’Connor

The O’Connor link above is to a page at the Chautauqua Bookstore.

For Justice O’Connor:

Reflections on Themis
(Log24, Aug. 17, 2004) 

For Wilcox:

The Zen of Abraham
(Same entry, different title.)

I personally was at Chautauqua only one day this season — Friday, the 13th of August.  My stops of course included the Chautauqua Bookstore, where I purchased the following:

Human cultural activity is mostly what Walker Percy astutely called “symbol-mongering.”  Of the three books above, the central one offers the best symbols.

My own version of a
Chautauqua “Versus” symbol:

The Line,
by S. H. Cullinane

For further details, see the

entries of Aug. 15, 2004.

For an “arc” symbol, see

Loretta’s Rainbow.

Thursday, February 5, 2004

Thursday February 5, 2004

Filed under: General — m759 @ 12:00 pm

Affirmation of Place and Time:
East Coker and Grand Rapids

This morning’s meditation:

“Let us talk together with the courage, humor, and ardor of Socrates.

In that long conversation, we may find ourselves considering something Plato’s follower Plotinus said long ago about ‘a principle which transcends being,’ in whose domain one can ‘assert identity without the affirmation of being.’  There, ‘everything has taken its stand forever, an identity well pleased, we might say, to be as it is…. Its entire content is simultaneously present in that identity: this is pure being in eternal actuality; nowhere is there any future, for every then is a now; nor is there any past, for nothing there has ever ceased to be.’  Individuality and existence in space and time may be masks that our sensibilities impose on the far different face of quantum reality.”

— Peter Pesic, Seeing Double: Shared Identities in Physics, Philosophy, and Literature, MIT Press paperback, 2003, p. 145

A search for more on Plotinus led to sites on the Trinity, which in turn led to the excellent archives at Calvin College in Grand Rapids.

A search for the theological underpinnings of Calvin College led to the Christian Reformed church:

“Our emblem is
the cross in a triangle.”

The triangle, as a symbol of “the delta factor,” also plays an important role in the semiotic theory of Walker Percy.  A search for current material on Percy led back to one of my favorite websites, that of Percy expert Karey Perkins, and thus to the following paper:

The “East Coker” Dance
in T. S. Eliot’s Four Quartets:
An Affirmation of Place and Time

by Karey Perkins

For a rather different, but excellent, literary affirmation of place and time — in Grand Rapids, rather than East Coker — see, for instance, Michigan Roll, a novel by Tom Kakonis.

We may, for the purposes of this trinitarian meditation, regard Percy and Kakonis as speaking for the Son and Karey Perkins as a spokesperson for the Holy Spirit.  As often in my meditations, I choose to regard the poet Wallace Stevens as speaking perceptively about (if not for, or as) the Father.  A search for related material leads to a 1948 comment by Thomas McGreevy, who

“… wrote of Stevens’ ‘Credences of Summer’ (Collected Poems 376),

On every page I find things that content me, as ‘The trumpet of the morning blows in the clouds and through / The sky.’

A devout Roman Catholic, he added, ‘And I think my delight in it is of the Holy Spirit.’ (26 May 1948).”

An ensuing search for material on “Credences of Summer” led back, surprisingly, to an essay — not very scholarly, but interesting — on Stevens, Plotinus, and neoplatonism.

Thus the circle closed.

As previous entries have indicated, I have little respect for Christianity as a religion, since Christians are, in my experience, for the most part, damned liars.  The Trinity as philosophical poetry, is, however, another matter.  I respect Pesic’s speculations on identity, but wish he had a firmer grasp of his subject’s roots in trinitarian thought.  For Stevens, Percy, and Perkins, I have more than respect.

Saturday, July 26, 2003

Saturday July 26, 2003

Filed under: General,Geometry — Tags: — m759 @ 11:11 pm

The Transcendent
Signified

“God is both the transcendent signifier
and transcendent signified.”

— Caryn Broitman,
Deconstruction and the Bible

“Central to deconstructive theory is the notion that there is no ‘transcendent signified,’ or ‘logos,’ that ultimately grounds ‘meaning’ in language….”

— Henry P. Mills,
The Significance of Language,
Footnote 2

“It is said that the students of medieval Paris came to blows in the streets over the question of universals. The stakes are high, for at issue is our whole conception of our ability to describe the world truly or falsely, and the objectivity of any opinions we frame to ourselves. It is arguable that this is always the deepest, most profound problem of philosophy. It structures Plato’s (realist) reaction to the sophists (nominalists). What is often called ‘postmodernism’ is really just nominalism, colourfully presented as the doctrine that there is nothing except texts. It is the variety of nominalism represented in many modern humanities, paralysing appeals to reason and truth.”

Simon Blackburn, Think,
Oxford University Press, 1999, page 268

The question of universals is still being debated in Paris.  See my July 25 entry,

A Logocentric Meditation.

That entry discusses an essay on
mathematics and postmodern thought
by Michael Harris,
professor of mathematics
at l’Université Paris 7 – Denis Diderot.

A different essay by Harris has a discussion that gets to the heart of this matter: whether pi exists as a platonic idea apart from any human definitions.  Harris notes that “one might recall that the theorem that pi is transcendental can be stated as follows: the homomorphism Q[X] –> R taking X to pi is injective.  In other words, pi can be identified algebraically with X, the variable par excellence.”

Harris illustrates this with
an X in a rectangle:

For the complete passage, click here.

If we rotate the Harris X by 90 degrees, we get a representation of the Christian Logos that seems closely related to the God-symbol of Arthur C. Clarke and Stanley Kubrick in 2001: A Space Odyssey.  On the left below, we have a (1x)4×9 black monolith, representing God, and on the right below, we have the Harris slab, with X representing (as in “Xmas,” or the Chi-rho page of the Book of Kells) Christ… who is, in theological terms, also “the variable par excellence.”

Kubrick’s
monolith

Harris’s
slab

For a more serious discussion of deconstruction and Christian theology, see

Walker Percy’s Semiotic.

Wednesday, October 23, 2002

Wednesday October 23, 2002

Filed under: General — m759 @ 8:00 pm

Bright Star

From the website of Karey Lea Perkins:

“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…”

Walker Percy, The Message in the Bottle, Farrar, Strauss, and Giroux, 1975

Today’s New York Times story on Richard Helms, together with my reminiscences in the entry that follows it below, suggest the following possibility for symbol-mongering:

Compare the 16-point star of the C.I.A.

with the classic 8-point star of Venus:

This comparison is suggested by the Spanish word “Lucero” (the name, which means “Bright Star,” of the girl in Cuernavaca mentioned two entries down) and by the following passage from Robert A. Heinlein‘s classic novel, Glory Road:

    “I have many names. What would you like to call me?”

    “Is one of them ‘Helen’?”

    She smiled like sunshine and I learned that she had dimples. She looked sixteen and in her first party dress. “You are very gracious. No, she’s not even a relative. That was many, many years ago.” Her face turned thoughtful. “Would you like to call me ‘Ettarre’?”

    “Is that one of your names?”

    “It is much like one of them, allowing for different spelling and accent. Or it could be ‘Esther’ just as closely. Or ‘Aster.’ Or even ‘Estrellita.’ ”

    ” ‘Aster,’ ” I repeated. “Star. Lucky Star!”

The C.I.A. star above is from that organization’s own site.  The star of Venus (alias Aster, alias Ishtar) is from Symbols.com, an excellent site that has the following variations on the Bright Star theme:

Ideogram for light Alchemical sign
Greek “Aster” Babylonian Ishtar
Phoenician Astarte Octagram of Venus
Phaistos Symbol Fortress Octagram

See also my notes The Still Point and the Wheel and Midsummer Eve’s Dream.  Both notes quote Robinson Jeffers:

“For the essence and the end
Of his labor is beauty…
one beauty, the rhythm of that Wheel,
and who can behold it is happy
and will praise it to the people.”

— Robinson Jeffers, “Point Pinos and Point Lobos,”
quoted at the end of The Cosmic Code,
by Heinz Pagels, Simon & Schuster, 1982

Place the eightfold star in a circle, and you have the Buddhist Wheel of Life:

Friday, September 13, 2002

Friday September 13, 2002

Filed under: General — m759 @ 2:24 pm

Meditation for Friday the 13th

The 1946 British film below (released as “Stairway to Heaven” in the U.S.) is one of my favorites.  I saw it as a child. Since costar Kim Hunter died this week (on 9/11), and since today is Friday the 13th, the following material seems relevant.

Kim Hunter in 1946

R.A.F pilot
and psychiatrist 

Alan McGlashan

Alan McGlashan has practiced as a psychiatrist in London for more than forty years.  He also served as a pilot for the R.A.F. (with MC and Croix de Guerre decorations). 

The doctor in “A Matter of Life and Death” addresses a heavenly court on behalf of his patient, R.A.F pilot David Niven:

In the film, David Niven is saved by mistake from a fated death and his doctor must argue to a heavenly court that he be allowed to live. 

In a similar situation, I would want Dr. Alan McGlashan, a real-life psychiatrist, on my side.  For an excerpt from one of my favorite books, McGlashan’s The Savage and Beautiful Country,

click here.

As Walker Percy has observed (see my Sept. 7 note, “The Boys from Uruguay”), a characteristic activity of human beings is what Percy called “symbol-mongering.”  In honor of today’s anniversary of the births of two R.A.F. fighter pilots,

Sir Peter Guy Wykeham-Barnes (b. 1915) and author

Roald Dahl (b. 1916),

here is one of the better symbols of the past century:

The circle is of course a universal symbol, and can be made to mean just about whatever one wants it to mean.  In keeping with Clint Eastwood’s advice, in the soundtrack album for “Midnight in the Garden of Good and Evil,” to “accentuate the positive,” here are some positive observations on a circle from the poet (and perhaps saint) Dante, who died on the night of September 13-14:

In the sun, Dante and Beatrice find themselves surrounded by a circle of souls famous for their wisdom on earth. They appear as splendid lights and precious jewels who dance and sing as they lovingly welcome two more into their company. Their love for God is kindled even more and grows as they find more individuals to love. Among the blessed souls are St. Thomas Aquinas and one of his intellectual “enemies”, Siger of Brabant, a brilliant philosopher at the University of Paris, some of whose teachings were condemned as heretical. Conflicts and divisions on earth are now forgotten and absorbed into a communal love song and dance “whose sweetness and harmony are unknown on earth and whose joy becomes one with eternity.”

Dante compares their dance and song to God’s bride on earth, the Church, when she answers the morning bells to rise from bed and “woo with matins song her Bridegroom’s love.” Some critics consider this passage the most “spiritually erotic” of all the one hundred cantos of the Comedy. It is the ending of Canto 10, verses 139-148.

— Fr. James J. Collins, “The Spiritual Journey with Dante V,” Priestly People October 1997

The above material on Dante is from the Servants of the Paraclete website.

For more on the Paraclete, see

The Left Hand of God.

See also the illustration in the note below.

Saturday, September 7, 2002

Saturday September 7, 2002

Filed under: General — m759 @ 4:44 pm

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.

  1. From Peggy Noonan’s column of Friday, Sept. 6:

    The cross, the heart, and the flag.

  2. From Karey Perkins’s website:

    A brain, a heart, and courage

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…

  1. Saul of Tarsus, who later assumed an alias.

  2. Paul Newman, whose performance in “The Verdict” continues, indirectly, to trouble Cardinal Law of Boston.

  3. 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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