Related material from Log24 —

Posts tagged Cartier's Groundhog Day.

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* For the title, see a search for *Inside Man* .

Related material: The co-editor of *The Architecture of Modern Mathematics* —

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Yesterday's 11 AM post Mad Day concluded

with a link to a 2001 American Mathematical Society

article by Pierre Cartier that sums up the religion and

politics of many mathematicians…

"Here ends the infancy narrative of the gospel…."

"… while Simone Weil's Catholicism was violently

anti-Semitic (in 1942!), Grothendieck's Buddhism

bears a strong resemblance to the practices of

his Hasidic ancestors."

See also Simone Weil in *this* journal.

Note esp. a post of April 6, 2004 that provides

a different way of viewing Derrida's notion of

*inscription *.

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See “Pleasantly Discursive” in this journal.

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For the *Toro* , see Pierre Cartier in 2001 on the barber of Seville and

**“The evolution of concepts of space and symmetry.”**

For the *Torino *, see **. . .**

“… the ultimate goal of the present essay

which is to illustrate the historic

**evolution of the concepts of Space and Symmetry **“

— Pp. 157-158 of the above book.

See also Fré *et al.* , “The role of PSL(2,7) in M-theory”

(2018-2019) at http://arxiv.org/abs/1812.11049v2 ,

esp. Section 4, “Theory of the simple group PSL(2,7)”

on pages 11-27, and remarks on PSL(2,7) in *this* journal.

**Related material —**

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Click the ring for Pierre Cartier on the barber of Seville

and “The evolution of concepts of space and symmetry.”

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See *Ballet Blanc* in this journal.

For a darker perspective, click on the image below.

See also Cartier in The Hexagon of Opposition.

Happy birthday to Kirk Douglas.

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See also, from that same day, "24-Part Invention."

* The title is a reference to a 2001 article by Cartier on

"the evolution of concepts of space and symmetry" —

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Continued from yesterday.

The passage on Claude Chevalley quoted here

yesterday in the post Dead Reckoning was, it turns out,

also quoted by Peter Galison in his essay "Structure of Crystal,

Bucket of Dust" in *Circles Disturbed: The Interplay of
Mathematics and Narrative *(Princeton University Press, 2012,

ed. by Apostolos Doxiadis and Barry Mazur).

Galison gives a reference to his source:

"From 'Claude Chevalley Described by His Daughter (1988),'

in Michèle Chouchan, *Nicolas Bourbaki: Faits et légendes *

(Paris: Éditions du Choix, 1995), 36–40, translated and cited

in Marjorie Senechal, 'The Continuing Silence of Bourbaki:

An Interview with Pierre Cartier, June 18, 1997,'

*Mathematical Intelligencer* 1 (1998): 22–28."

Galison's essay compares Chevalley with the physicist

John Archibald Wheeler. His final paragraph —

"Perhaps, then, it should not surprise us too much if,

as Wheeler approaches the beginning-end of all things,

there is a bucket of Borelian dust. Out of this filth,

through the proposition machine of quantum mechanics

comes pregeometry; pregeometry makes geometry;

geometry gives rise to matter and the physical laws

and constants of the universe. At once close to and far

from the crystalline story that Bourbaki invoked,

Wheeler’s genesis puts one in mind of Genesis 3:19:

'In the sweat of thy face shalt thou eat bread, till thou

return unto the ground; for out of it wast thou taken:

for dust thou art, and unto dust shalt thou return.'"

See also posts tagged Wheeler.

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**Part I:**

The sermon, “God’s Architecture,” at Nassau Presbyterian

Church in Princeton on Sunday, Feb. 23, 2014. (This is the

“sermon” link in last Sunday’s 11 AM ET Log24 post.)

An excerpt:

“I wonder what God sees when God looks at our church.

Bear with me here because I’d like to do a little architectural

redesign. I look up at our sanctuary ceiling and I see buttons.

In those large round lights, I see buttons. I wonder what would

happen if we unbutton the ceiling, Then I wonder if we were to

unzip the ceiling, pull back the rooftop, and God were to look in

from above – What does God see? What pattern, what design,

what shape takes place?” — Rev. Lauren J. McFeaters

**Related material — All About Eve: **

A. The Adam and Eve sketch from the March 8 “Saturday Night Live”

B. “Katniss, get away from that tree!” —

C. Deconstructing God in last evening’s online *New York Times* .

**Part II:**

“Heavensbee!” in the above video, as well as Cartier’s Groundhog Day

and Say It With Flowers.

**Part III:**

*Humans’* architecture, as described (for instance) by architecture

theorist Anne Tyng, who reportedly died at 91 on Dec. 27, 2011.

See as well Past Tense and a post from the date of Tyng’s death.

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This journal a year ago yesterday—

“Some designs work subtly.

Others are successful through sheer force.”

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A perceptive review of *Missing Out: In Praise of the Unlived Life*—

**"Page 185: 'Whatever else we are, we are also mad.' "**

Related material— last night's Outside the Box and, from Oct. 22 last year—

"Some designs work subtly.

Others are successful through sheer force."

*Par exemple—*

See also Cartier in this journal.

The Cartier link leads to, among other things…

“A Mad Day’s Work: From Grothendieck to Connes and Kontsevich.

The Evolution of Concepts of Space and Symmetry,”

by Pierre Cartier, *Bulletin of the American Mathematical Society *,

Vol. 38 (2001) No. 4, pages 389-408

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From Penelope Green’s *New York Times * story on Wednesday night’s Cooper-Hewitt design awards gala (links added)—

“Then Mr. Wurman went into full curmudgeon mode, fiddling with the two mikes on the podium and questioning the format of the night.

‘We should have talked to each other longer,’ he said. ‘This is the least interesting part.’

When he was done, Gloria Nagy, his wife, recalled how he had critiqued Mrs. Obama’s speech during the awards luncheon in July. (The Huffington Post reported Mrs. Obama as saying Mr. Wurman was ‘quite dashing and sassy.’) Ms. Nagy said Mrs. Obama had teased her by offering condolences and asking how she put up with her husband. In answer, Ms. Nagy said, she flashed what she called her Hazardous Duty Prize, a blindingly huge diamond ring.

Some designs work subtly. Others are successful through sheer force.”

*Par exemple—*

See also Cartier in this journal.

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The new June/July issue of the *AMS Notices*

on a recent Paris exhibit of art and mathematics—

*Mathématiques, un dépaysement soudain*

Exhibit at the Fondation Cartier, Paris

October 21, 2011–March 18, 2012

… maybe walking

into the room was supposed to evoke the kind of

*dépaysement * for which the exhibition is named

(the word *dépaysement* refers to the sometimes

disturbing feeling one gets when stepping outside

of one’s usual reference points). I was with

my six-year-old daughter, who quickly gravitated

toward the colorful magnetic tiles on the wall that

visitors could try to fit together. She spent a good

half hour there, eventually joining forces with a

couple of young university students. I would come

and check on her every once in a while and heard

some interesting discussions about whether or not

it was worth looking for patterns to help guide the

placing of the tiles. The fifteen-year age difference

didn’t seem to bother anyone.

The tiles display was one of the two installations

here that offered the visitor a genuine chance to

engage in mathematical activity, to think about

pattern and structure while satisfying an aesthetic

urge to make things fit and grow….

The *Notices* included no pictures with this review.

A search to find out what sort of tiles were meant

led, quite indirectly, to the following—

The search indicated it is unlikely that these *Truchet* tiles

were the ones on exhibit.

Nevertheless, the *date* of the above French weblog post,

1 May 2011, is not without interest in the context of

today's previous post. (That post was written well *before
*I had seen the new

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J. M. Bernstein (previous post) has written of moving toward "a Marxist hermeneutic."

I prefer lottery hermeneutics.

Some background from Bernstein—

I would argue that at least sometimes, lottery numbers may be regarded, according to Bernstein's definition, as story statements. For instance—

Today's New York State Lottery— Midday 389, Evening 828.

For the significance of 389, see

“A Mad Day’s Work: From Grothendieck to Connes and Kontsevich.

The Evolution of Concepts of Space and Symmetry,”

by Pierre Cartier, *Bulletin of the American Mathematical Society*,

Vol. 38 (2001) No. 4, beginning on page **389**.

The philosophical import of page 389 is perhaps merely in Cartier's title (see previous post).

For the significance of 828, see 8/28, the feast of St. Augustine, in 2006.

See also Halloween 2007. (Happy birthday, Dan Brown.)

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"I was reading Durant's section on Plato, struggling to understand his theory of the ideal Forms that lay in inviolable perfection out beyond the phantasmagoria. (That was the first, and I think the last, time that I encountered that word.)" |

**Part I: Phantasmagoria
**

Photo by Phil Bray

Transcendence through spelling:

Richard Gere and Flora Cross

as father and daughter

in the film of *Bee Season*.

"Every aspect of the *alef's*

construction has been

Divinely designed

to teach us something."

— Alef– The Difference Between

Exile And Redemption,

by Rabbi Aaron L. Raskin

**Related material–**

Art Theory for Yom Kippur

and

Log24 entries, Nov. 2005.

*The Alphabet Versus the Goddess:
The Conflict Between Word and Image*.

See also the references

to Zelazny's *Eye of Cat*

in the Nov. 2005 entries

as well as

today's previous entry—

with the Norton Simon motto

"Hunt for the best"– and…

"Photography has always involved waiting…. the photographer is understood to be waiting for the right convergence of subject, lighting and frame before clicking the shutter– waiting for what a master of the genre, Henri Cartier-Bresson, famously called 'the decisive moment.' Lee Friedlander, another great street photographer, compared this anticipatory state to the hunting alertness of a 'one-eyed cat.' The metaphor of the hunt has seeped into the essential language of photography."

— Arthur Lubow in *The New York Times*, Feb. 25, 2007

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**One Ring to Rule Them All**

In memory of J. R. R. Tolkien, who died on this date, and in honor of Israel Gelfand, who was born on this date.

Leonard Gillman on his collaboration with Meyer Jerison and Melvin Henriksen in studying rings of continuous functions:

“The triple papers that Mel and I wrote deserve comment. Jerry had conjectured a characterization of *beta X* (the Stone-Cech compactification of *X*) and the three of us had proved that it was true. Then he dug up a 1939 paper by Gelfand and Kolmogoroff that Hewitt, in his big paper, had referred to but apparently not appreciated, and there we found Jerry’s characterization. The three of us sat around to decide what to do; we called it the ‘wake.’ Since the authors had not furnished a proof, we decided to publish ours. When the referee expressed himself strongly that a title should be informative, we came up with *On a theorem of Gelfand and Kolmogoroff concerning maximal ideals in rings of continuous functions*. (This proved to be my second-longest title, and a nuisance to refer to.) Kolmogoroff died many years ago, but Gelfand is still living, a vigorous octogenarian now at Rutgers. A year or so ago, I met him at a dinner party in Austin and mentioned the 1939 paper. He remembered it very well and proceeded to complain that the only contribution Kolmogoroff had made was to point out that a certain result was valid for the complex case as well. I was intrigued to see how the giants grouse about each other just as we do.”

— Leonard Gillman: An Interview

This clears up a question I asked earlier in this journal….

On the mathematician Kolmogorov: “It turns out that he DID prove one basic theorem that I take for granted, that a compact hausdorff space is determined by its ring of continuous functions (this ring being considered without any topology) — basic discoveries like this are the ones most likely to have their origins obscured, for they eventually come to be seen as mere common sense, and not even a theorem.” — Richard Cudney, Harvard ’03, writing at Xanga.com as rcudney on May 14, 2003 That this theorem is Kolmogorov’s is news to me. See - the discussion of the Gelfand representation theorem on p. 397, “A Mad Day’s Work: From Grothendieck to Connes and Kontsevich,” by Pierre Cartier, Bulletin of the American Mathematical Society, Vol. 38 (2001) No. 4, pp. 389-408,
- a remark on Gelfand’s work on page 467 of the above AMS Bulletin issue,
- V. S. Varadarajan‘s discussion of the “Hilbert-Gel’fand principle” on page 11 of “The Concept of a Supermanifold” and the “Gel’fand Principle” on page 11 of “What Is the Geometry of Physical Space?,” and
- the excellent 1963 textbook
*Introduction to Topology and Modern Analysis,*by George F. Simmons, chapters 13 and 14.
The above references establish that Gelfand is usually cited as the source of the theorem Cudney discusses. Gelfand was a student of Kolmogorov’s in the 1930’s, so who discovered what when may be a touchy question in this case. A reference that seems relevant: I. M. Gelfand and A. Kolmogoroff, “On rings of continuous functions on topological spaces,” Doklady Akad. Nauk SSSR There ARE some references that indicate Kolmogorov may have done some work of his own in this area. See here (“quite a few duality theorems… including those of Banaschewski, Morita, Gel’fand-Kolmogorov and Gel’fand-Naimark”) and here (“the classical theorems of M. H. Stone, Gelfand & Kolmogorov”). Any other references to Kolmogorov’s work in this area would be of interest. Naturally, any discussion of this area should include a reference to the pioneering work of M. H. Stone. I recommend the autobiographical article on Stone in McGraw-Hill Modern Men of Science, Volume II, 1968. |

A response by Richard Cudney:

“In regard to your entry, it is largely correct. The paper by Kolmogorov and Gelfand that you refer to is the one that I just read in his collected works. So, I suppose my entry was unfair to Gelfand. You’re right, the issue of credit is a bit touchy since Gelfand was his student. In a somewhat recent essay, Arnol’d makes the claim that this whole thread of early work by Gelfand may have been properly due to Kolmogorov, however he has no concrete proof, having been but a child at the time, and makes this inference based only on his own later experience as Kolmogorov’s student. At any rate, I had known about Gelfand’s representation theorem, but had not known that Kolmogorov had done any work of this sort, or that this theorem in particular was due to either of them. And to clarify-where I speak of the credit for this theorem being obscured, I speak of my own experience as an algebraic geometer and not a functional analyst. In the textbooks on algebraic geometry, one sees no explanation of why we use Spec A to denote the scheme corresponding to a ring A. That question was answered when I took functional analysis and learned about Gelfand’s theorem, but even there, Kolmogorov’s name did not come up. This result is different from the Gelfand representation theorem that you mention-this result concerns algebras considered without any topology(or norm)-whereas his representation theorem is a result on Banach algebras. In historical terms, this result precedes Gelfand’s theorem and is the foundation for it-he starts with a general commutative Banach algebra and reconstructs a space from it-thus establishing in what sense that the space to algebra correspondence is surjective, and hence by the aforementioned theorem, bi-unique. That is to say, this whole vein of Gelfand’s work started in this joint paper. Of course, to be even more fair, I should say that Stone was the very first to prove a theorem like this, a debt which Kolmogorov and Gelfand acknowledge. Stone’s paper is the true starting point of these ideas, but this paper of Kolmogorov and Gelfand is the second landmark on the path that led to Grothendieck’s concept of a scheme(with Gelfand’s representation theorem probably as the third). As an aside, this paper was not Kolmogorov’s first foray into topological algebra-earlier he conjectured the possibility of a classification of locally compact fields, a problem which was solved by Pontryagin. The point of all this is that I had been making use of ideas due to Kolmogorov for many years without having had any inkling of it.” |

**Common Sense**

On the mathematician Kolmogorov:

“It turns out that he DID prove one basic theorem that I take for granted, that a compact hausdorff space is determined by its ring of continuous functions (this ring being considered without any topology) — basic discoveries like this are the ones most likely to have their origins obscured, for they eventually come to be seen as mere common sense, and not even a theorem.”

— Richard Cudney, Harvard ’03, writing at Xanga.com as rcudney on May 14, 2003

That this theorem is Kolmogorov’s is news to me.

See

- the discussion of the Gelfand representation theorem on p. 397, “A Mad Day’s Work: From Grothendieck to Connes and Kontsevich,” by Pierre Cartier, Bulletin of the American Mathematical Society, Vol. 38 (2001) No. 4, pp. 389-408,

- a remark on Gelfand’s work on page 467 of the above AMS Bulletin issue,

- V. S. Varadarajan‘s discussion of the “Hilbert-Gel’fand principle” on page 11 of “The Concept of a Supermanifold” and the “Gel’fand Principle” on page 11 of “What Is the Geometry of Physical Space?,” and

- the excellent 1963 textbook
*Introduction to Topology and Modern Analysis,*by George F. Simmons, chapters 13 and 14.

The above references establish that Gelfand is usually cited as the source of the theorem Cudney discusses. Gelfand was a student of Kolmogorov’s in the 1930’s, so who discovered what when may be a touchy question in this case. A reference that seems relevant: I. M. Gelfand and A. Kolmogoroff, “On rings of continuous functions on topological spaces,” Doklady Akad. Nauk SSSR **22** (1939), 11-15. This is cited by Gillman and Jerison in the classic *Rings of Continuous Functions*.

There ARE some references that indicate Kolmogorov may have done some work of his own in this area. See here (“quite a few duality theorems… including those of Banaschewski, Morita, Gel’fand-Kolmogorov and Gel’fand-Naimark”) and here (“the classical theorems of M. H. Stone, Gelfand & Kolmogorov”).

Any other references to Kolmogorov’s work in this area would be of interest.

Naturally, any discussion of this area should include a reference to the pioneering work of M. H. Stone. I recommend the autobiographical article on Stone in McGraw-Hill Modern Men of Science, Volume II, 1968.

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