The misleading image at right above is from the cover of
an edition of Charles Williams's classic 1931 novel
Many Dimensions published in 1993 by Wm. B. Eerdmans.
Compare and constrast —
Cover of a book by Douglas Hofstadter
The misleading image at right above is from the cover of
an edition of Charles Williams's classic 1931 novel
Many Dimensions published in 1993 by Wm. B. Eerdmans.
Compare and constrast —
Cover of a book by Douglas Hofstadter
Click for the pages below at Internet Archive.
Enveloping algebras also appeared later in the work on "crystal bases"
of Masaki Kashiwara. It seems highly unlikely that his work on enveloping
algebras, or indeed any part of his work on crystal bases, has any relation
to my own earlier notes.
A 1995 page by Kashiwara —
Kashiwara was honored with a Kyoto prize in 2018:
Kashiwara's 2018 Kyoto Prize diploma —
This post continues a post from yesterday on the square model of
PG(3,2) that apparently first appeared (presented as such*) in . . .
Cullinane, "Symmetry invariance in a diamond ring,"
Notices of the AMS , pp. A193194, Feb. 1979.
Yesterday's Wikipedia presentation of the square model was today
revised by yet another anonymous author —
Revision history accounting for the above change from yesterday —
The jargon "rm OR" means "remove original research."
The added verbiage about block designs is a smokescreen having
nothing to do with the subject, which is square representation
of the 35 points and lines.
* The 35 squares, each consisting of four 4element subsets, appeared earlier
in the Miracle Octad Generator (MOG) of R. T. Curtis (published in 1976).
They were not at that time presented as constituting a finite geometry,
either affine (AG(4,2)) or projective (PG(3,2)).
Tom Wolfe in The Painted Word (1975) —
“I am willing (now that so much has been revealed!)
to predict that in the year 2000, when the Metropolitan
or the Museum of Modern Art puts on the great
retrospective exhibition of American Art 194575,
the three artists who will be featured, the three seminal
figures of the era, will be not Pollock, de Kooning, and
Johnsbut Greenberg, Rosenberg, and Steinberg.
Up on the walls will be huge copy blocks, eight and a half
by eleven feet each, presenting the protean passages of
the period … a little ‘fuliginous flatness’ here … a little
‘action painting’ there … and some of that ‘all great art
is about art’ just beyond. Beside them will be small
reproductions of the work of leading illustrators of
the Word from that period….”
The above group of 322,560 permutations appears also in a 2011 book —
— and in 20132015 papers by Anne Taormina and Katrin Wendland:
The Institute for Advanced Study (IAS) at Princeton in its Fall 2015 Letter
revived "Beautiful Mathematics" as a title:
This ugly phrase was earlier used by Truman State University
professor Martin Erickson as a book title. See below.
In the same IAS Fall 2015 Letter appear the following remarks
by Freeman Dyson —
". . . a special case of a much deeper connection that Ian Macdonald
discovered between two kinds of symmetry which we call modular and affine.
The two kinds of symmetry were originally found in separate parts of science,
modular in pure mathematics and affine in physics. Modular symmetry is
displayed for everyone to see in the drawings of flying angels and devils
by the artist Maurits Escher. Escher understood the mathematics and got the
details right. Affine symmetry is displayed in the peculiar groupings of particles
created by physicists with highenergy accelerators. The mathematician
Robert Langlands was the first to conjecture a connection between these and
other kinds of symmetry. . . ." (Wikipedia link added.)
The adjective "modular" might aptly be applied to . . .
The adjective "affine" might aptly be applied to . . .
The geometry of the 4×4 square combines modular symmetry
(i.e., related to theta functions) with the affine symmetry above.
Hudson's 1905 discussion of modular symmetry (that of Rosenhain
tetrads and Göpel tetrads) in the 4×4 square used a parametrization
of that square by the digit 0 and the fifteen 2subsets of a 6set, but
did not discuss the 4×4 square as an affine space.
For the connection of the 15 Kummer modular 2subsets with the 16
element affine space over the twoelement Galois field GF(2), see my note
of May 26, 1986, "The 2subsets of a 6set are the points of a PG(3,2)" —
— and the affine structure in the 1979 AMS abstract
"Symmetry invariance in a diamond ring" —
For some historical background on the symmetry investigations by
Dyson and Macdonald, see Dyson's 1972 article "MIssed Opportunities."
For Macdonald's own use of the words "modular" and "affine," see
Macdonald, I. G., "Affine Lie algebras and modular forms,"
Séminaire N. Bourbaki , Vol. 23 (19801981), Talk no. 577, pp. 258276.
The above images were suggested in part by the birthdays
on Sept. 21, 2011, of Bill Murray and Stephen King.
More seriously, also in this journal on that date, from a post
titled Symmetric Generation —
As the Key to All Mythologies
For the theorem of the title, see "Diamond Theorem" in this journal.
"These were heavy impressions to struggle against,
and brought that melancholy embitterment which
is the consequence of all excessive claim: even his
religious faith wavered with his wavering trust in his
own authorship, and the consolations of the Christian
hope in immortality seemed to lean on the immortality
of the still unwritten Key to all Mythologies."
— Middlemarch , by George Eliot, Ch. XXIX
Related material from Sunday's print New York Times —
Sunday's Log24 sermon —
See also the LéviStrauss "Key to all Mythologies" in this journal,
as well as the previous post.
"Studies of spin½ theories in the framework of projective geometry
have been undertaken before." — Y. Jack Ng and H. van Dam,
February 20, 2009
For one such framework,* see posts from that same date
four years earlier — February 20, 2005.
* A 4×4 array. See the 1977, 1978, and 1986 versions by
Steven H. Cullinane, the 1987 version by R. T. Curtis, and
the 1988 ConwaySloane version illustrated below —
Cullinane, 1977
Cullinane, 1978
Cullinane, 1986
Curtis, 1987
Update of 10:42 PM ET on Sunday, June 19, 2016 —
The above images are precursors to …
Conway and Sloane, 1988
Update of 10 AM ET Sept. 16, 2016 — The excerpt from the
1977 "Diamond Theory" article was added above.
A less metaphysical approach to a "preform" —
From Wallace Stevens, "The Man with the Blue Guitar":
IX
And the color, the overcast blue
Of the air, in which the blue guitar
Is a form, described but difficult,
And I am merely a shadow hunched
Above the arrowy, still strings,
The maker of a thing yet to be made . . . .
"Arrowy, still strings" from the diamond theorem
See also "preforming" and the blue guitar
in a post of May 19, 2010.
Update of 7:11 PM ET:
More generally, see posts tagged May 19 Gestalt.
A Wikipedia edit today by David Eppstein, a professor
at the University of California, Irvine:
See the Fanoplane page before and after the Eppstein edit.
Eppstein deleted my Dec. 6 Fano 3space image as well as
today's Fanoplane image. He apparently failed to read the
explanatory notes for both the 3space model and the
2space model. The research he refers to was original
(in 1979) but has been published for some time now in the
online Encyclopedia of Mathematics, as he could have
discovered by following a link in the notes for the 3space
model.
For a related recent display of ignorance, see Hint of Reality.
Happy darkest night.
The title was suggested by
http://benmarcus.com/smallwork/manifesto/.
The "O" of the title stands for the octahedral group.
See the following, from http://finitegeometry.org/sc/map.html —

An invariance of symmetry The diamond theorem on a 4x4x4 cube, and a sketch of the proof. 
831001  Portrait of O A table of the octahedral group O using the 24 patterns from the 2×2 case of the diamond theorem. 
831016  Study of O A different way of looking at the octahedral group, using cubes that illustrate the 2x2x2 case of the diamond theorem. 
840915  Diamonds and whirls Block designs of a different sort — graphic figures on cubes. See also the University of Exeter page on the octahedral group O. 
The words: "symplectic polarity"—
The images:
The Natural Symplectic Polarity in PG(3,2)
Symmetry Invariance in a Diamond Ring
The DiamondTheorem Correlation
Continued from earlier posts.
The Washington Post online yesterday: "Val Logsdon Fitch, the Nebraska rancher’s son who shared the Nobel Prize for detecting a breakdown in the overarching symmetry of physical laws, thus helping explain how the universe evolved after the Big Bang, died Feb. 5 in Princeton, N.J. He was 91. His death was confirmed by Princeton University, where he had been a longtime faculty member and led the physics department for several years. Dr. Fitch and his Princeton colleague James Cronin received the Nobel Prize in physics in 1980 for highenergy experiments conducted in 1964 that overturned fundamental assumptions about symmetries and invariances that are characteristic of the laws of physics." — By Martin Weil 
Fans of synchronicity may prefer some rather
ig Nobel remarks quoted here on the date
of Fitch's death:
"The Harvard College Events Board presents
Harvard Thinks Big VI, a night of big ideas
and thinking beyond traditional boundaries.
On Thursday February 5th at 8 pm in
Sanders Theatre …."
— Log24 post The Big Spielraum
The title refers to the previous post.
Click image for some context.
For further context, see some
mathematics from Halloween 1978.
See also May 12, 2014.
“The relevance of a geometric theorem is determined by what the theorem
tells us about space, and not by the eventual difficulty of the proof.”
— GianCarlo Rota discussing the theorem of Desargues
What space tells us about the theorem :
In the simplest case of a projective space (as opposed to a plane ),
there are 15 points and 35 lines: 15 Göpel lines and 20 Rosenhain lines.*
The theorem of Desargues in this simplest case is essentially a symmetry
within the set of 20 Rosenhain lines. The symmetry, a reflection
about the main diagonal in the square model of this space, interchanges
10 horizontally oriented (rowbased) lines with 10 corresponding
vertically oriented (columnbased) lines.
Vide Classical Geometry in Light of Galois Geometry.
* Update of June 9: For a more traditional nomenclature, see (for instance)
R. Shaw, 1995. The “simplest case” link above was added to point out that
the two types of lines named are derived from a natural symplectic polarity
in the space. The square model of the space, apparently first described in
notes written in October and December, 1978, makes this polarity clearly visible:
The webpage Galois.us, on Galois matrices , has been created as
a starting point for remarks on the algebra (as opposed to the geometry)
underlying the rings of matrices mentioned in AMS abstract 79TA37,
“Symmetry invariance in a diamond ring.”
See also related historical remarks by Weyl and by Atiyah.
From Northrop Frye's The Great Code: The Bible and Literature , Ch. 3: Metaphor I — "In the preceding chapter we considered words in sequence, where they form narratives and provide the basis for a literary theory of myth. Reading words in sequence, however, is the first of two critical operations. Once a verbal structure is read, and reread often enough to be possessed, it 'freezes.' It turns into a unity in which all parts exist at once, without regard to the specific movement of the narrative. We may compare it to the study of a music score, where we can turn to any part without regard to sequential performance. The term 'structure,' which we have used so often, is a metaphor from architecture, and may be misleading when we are speaking of narrative, which is not a simultaneous structure but a movement in time. The term 'structure' comes into its proper context in the second stage, which is where all discussion of 'spatial form' and kindred critical topics take their origin." 
Related material:
"The Great Code does not end with a triumphant conclusion or the apocalypse that readers may feel is owed them or even with a clear summary of Frye’s position, but instead trails off with a series of verbal winks and nudges. This is not so great a fault as it would be in another book, because long before this it has been obvious that the forward motion of Frye’s exposition was illusory, and that in fact the book was devoted to a constant reexamination of the same basic data from various closely related perspectives: in short, the method of the kaleidoscope. Each shake of the machine produces a new symmetry, each symmetry as beautiful as the last, and none of them in any sense exclusive of the others. And there is always room for one more shake."
— Charles Wheeler, "Professor Frye and the Bible," South Atlantic Quarterly 82 (Spring 1983), pp. 154164, reprinted in a collection of reviews of the book. 
For code in a different sense, but related to the first passage above,
see Diamond Theory Roullete, a webpage by Radamés Ajna.
For "the method of the kaleidoscope" mentioned in the second
passage above, see both the Ajna page and a webpage of my own,
Kaleidoscope Puzzle.
For the late mathematics educator Zoltan Dienes.
"There comes a time when the learner has identified
the abstract content of a number of different games
and is practically crying out for some sort of picture
by means of which to represent that which has been
gleaned as the common core of the various activities."
— Article by "Melanie" at Zoltan Dienes's website
Dienes reportedly died at 97 on Jan. 11, 2014.
From this journal on that date —
A star figure and the Galois quaternion.
The square root of the former is the latter.
Update of 5:01 PM ET Feb. 6, 2014 —
An illustration by Dienes related to the diamond theorem —
See also the above 15 images in …
… and versions of the 4×4 coordinatization in The 4×4 Relativity Problem
(Jan. 17, 2014).
A ReCode Project program from Radamés Ajna of São Paulo —
At the program's webpage, click the image to
generate random permutations of rows, columns,
and quadrants. Note the resulting image's ordinary
or colorinterchange symmetry.
The sixteendot square array in yesterday’s noon post suggests
the following remarks.
“This is the relativity problem: to fix objectively a class of
equivalent coordinatizations and to ascertain the group of
transformations S mediating between them.”
— Hermann Weyl, The Classical Groups ,
Princeton University Press, 1946, p. 16
The Galois tesseract appeared in an early form in the journal
Computer Graphics and Art , Vol. 2, No. 1, February 1977—
The 1977 matrix Q is echoed in the following from 2002—
A different representation of Cullinane’s 1977 square model of the
16point affine geometry over the twoelement Galois field GF(2)
is supplied by Conway and Sloane in Sphere Packings, Lattices and Groups
(first published in 1988) :
Here a, b, c, d are basis vectors in the vector 4space over GF(2).
(For a 1979 version of this vector space, see AMS Abstract 79TA37.)
See also a 2011 publication of the Mathematical Association of America —
Mathematics:
A review of posts from earlier this month —
Wednesday, September 4, 2013

Narrative:
Aooo.
Happy birthday to Stephen King.
Profile picture of "Jo Lyxe" (Josefine Lyche) at Vimeo—
Compare to an image of Vril muse Maria Orsitsch.
From the catalog of a current art exhibition
(25 May – 31 August, 2013) in Norway,
I DE LANGE NÆTTER —
Josefine Lyche
Keywords (to help place my artwork in the (See also the original catalog page.) 
Clearly most of this (the nonhighlighted parts) was taken
from my webpage Diamond Theory. I suppose I should be
flattered, but I am not thrilled to be associated with the
(apparently fictional) Vril Society.
For some background, see (for instance)
Conspiracy Theories and Secret Societies for Dummies .
Online biography of author Cormac McCarthy—
"… he left America on the liner Sylvania, intending to visit
the home of his Irish ancestors (a King Cormac McCarthy
built Blarney Castle)."
Two Years Ago:
Blarney in The Harvard Crimson—
Melissa C. Wong, illustration for "Atlas to the Text,"
by Nicholas T. Rinehart:
Thirty Years Ago:
NonBlarney from a rural outpost—
Illustration for the generalized diamond theorem,
by Steven H. Cullinane:
From an arXiv preprint submitted July 18, 2011,
and last revised on March 11, 2013 (version 4):
"By our construction, this vector space is the dual
of our hypercube F_{2}^{4} built on I \ O_{9}. The vector space
structure of the latter, to our knowledge, is first
mentioned by Curtis in [Cur89]. Hence altogether
our proposition 2.3.4 gives a novel geometric
meaning in terms of Kummer geometry to the known
vector space structure on I \ O_{9}."
[Cur89] reference:
R. T. Curtis, "Further elementary techniques using
the miracle octad generator," Proc. Edinburgh
Math. Soc. 32 (1989), 345353 (received on
July 20, 1987).
— Anne Taormina and Katrin Wendland,
"The overarching finite symmetry group of Kummer
surfaces in the Mathieu group M _{24 },"
arXiv.org > hepth > arXiv:1107.3834
"First mentioned by Curtis…."
No. I claim that to the best of my knowledge, the
vector space structure was first mentioned by me,
Steven H. Cullinane, in an AMS abstract submitted
in October 1978, some nine years before the
Curtis article.
Update of the above paragraph on July 6, 2013—
No. The vector space structure was described by
The vector space structure as it occurs in a 4×4 array 
See Notes on Finite Geometry for some background.
See in particular The Galois Tesseract.
For the relationship of the 1978 abstract to Kummer
geometry, see Rosenhain and Göpel Tetrads in PG(3,2).
Yesterday's post on the current Museum of Modern Art exhibition
"Inventing Abstraction: 19101925" suggests a renewed look at
abstraction and a fundamental building block: the cube.
From a recent Harvard University Press philosophical treatise on symmetry—
The treatise corrects Nozick's error of not crediting Weyl's 1952 remarks
on objectivity and symmetry, but repeats Weyl's error of not crediting
Cassirer's extensive 1910 (and later) remarks on this subject.
For greater depth see Cassirer's 1910 passage on Vorstellung :
This of course echoes Schopenhauer, as do discussions of "Will and Idea" in this journal.
For the relationship of all this to MoMA and abstraction, see Cube Space and Inside the White Cube.
"The sacramental nature of the space becomes clear…." — Brian O'Doherty
Last Wednesday's 11 PM post mentioned the
adjacencyisomorphism relating the 4dimensional
hypercube over the 2element Galois field GF(2) to
the 4×4 array made up of 16 square cells, with
opposite edges of the 4×4 array identified.
A web page illustrates this property with diagrams that
enjoy the Karnaugh property— adjacent vertices, or cells,
differ in exactly one coordinate. A brief paper by two German
authors relates the Karnaugh property to the construction
of a magic square like that of Dürer (see last Wednesday).
In a similar way (search the Web for Karnaugh + cube ),
vertex adjacency in the 6dimensional hypercube over GF(2)
is isomorphic to cell adjacency in the 4x4x4 cube, with
opposite faces of the 4x4x4 cube identified.
The above cube may be used to illustrate some properties
of the 64point Galois 6space that are more advanced
than those studied by enthusiasts of "magic" squares
and cubes.
See
Those who prefer narrative to mathematics may
consult posts in this journal containing the word "Cuber."
"And how do we keep our balance?
That I can tell you in one word!"
— Tevye in Fiddler on the Roof
"The object and characteristic of 'traditions,'
including invented ones, is invariance."
— Eric Hobsbawm, introduction (link added)
to The Invention of Tradition
"Math is all about questions and answers."
— Prof. John D. McCarthy, Michigan State U.,
Monday afternoon, October 1, 2012
"Who knows where madness lies?"
— Man of La Mancha
(linked to here Monday morning)
Background:
The Origin and Development of Erwin Panofsky's Theories of Art ,
Michael Ann Holly, doctoral thesis, Cornell University, 1981 (pdf, 10 MB)
Panofsky, Cassirer, and Perspective as Symbolic Form ,
Allister Neher, doctoral thesis, Concordia University, 2000
From the current Wikipedia article "Symmetry (physics)"—
"In physics, symmetry includes all features of a physical system that exhibit the property of symmetry—that is, under certain transformations, aspects of these systems are 'unchanged', according to a particular observation. A symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is 'preserved' under some change.
A family of particular transformations may be continuous (such as rotation of a circle) or discrete (e.g., reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries. Continuous symmetries can be described by Lie groups while discrete symmetries are described by finite groups (see Symmetry group)."….
"A discrete symmetry is a symmetry that describes noncontinuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square's original appearance."
Note the confusion here between continuous (or discontinuous) transformations and "continuous" (or "discontinuous," i.e. "discrete") groups .
This confusion may impede efforts to think clearly about some pure mathematics related to current physics— in particular, about the geometry of spaces made up of individual units ("points") that are not joined together in a continuous manifold.
For an attempt to forestall such confusion, see Noncontinuous Groups.
For related material, see Erlanger and Galois as well as the opening paragraphs of Diamond Theory—
Symmetry is often described as invariance under a group of transformations. An unspoken assumption about symmetry in Euclidean 3space is that the transformations involved are continuous.
Diamond theory rejects this assumption, and in so doing reveals that Euclidean symmetry may itself be invariant under rather interesting groups of noncontinuous (and asymmetric) transformations. (These might be called noncontinuous groups, as opposed to socalled discontinuous (or discrete ) symmetry groups. See Weyl's Symmetry .)
For example, the affine group A on the 4space over the 2element field has a natural noncontinuous and asymmetric but symmetrypreserving action on the elements of a 4×4 array. (Details)
(Version first archived on March 27, 2002)
Update of Sunday, February 19, 2012—
The abuse of language by the anonymous authors
of the above Wikipedia article occurs also in more
reputable sources. For instance—
Some transformations referred to by Brading and Castellani
and their editees as "discrete symmetries" are, in fact, as
linear transformations of continuous spaces, themselves
continuous transformations.
This unfortunate abuse of language is at least made explicit
in a 2003 text, Mathematical Perspectives on Theoretical
Physics (Nirmala Prakash, Imperial College Press)—
"… associated[*] with any given symmetry there always exists
a continuous or a discrete group of transformations….
A symmetry whose associated group is continuous (discrete)
is called a continuous (discrete ) symmetry ." — Pp. 235, 236
[* Associated how?]
From math16.com—
Quotations on Realism

The story of the diamond mine continues
(see Coordinated Steps and Organizing the Mine Workers)—
From The Search for Invariants (June 20, 2011):
The conclusion of Maja Lovrenov's
"The Role of Invariance in Cassirer’s Interpretation of the Theory of Relativity"—
"… physical theories prove to be theories of invariants
with regard to certain groups of transformations and
it is exactly the invariance that secures the objectivity
of a physical theory."
— SYNTHESIS PHILOSOPHICA 42 (2/2006), pp. 233–241
Related material from Sunday's New York Times travel section—
Popular novelist Dan Brown is to speak at Chautauqua Institution on August 1.
This suggests a review of some figures discussed here in a note on Brown from February 20, 2004—
Related material: Notes from Nov. 5, 1981, and from Dec. 24, 1981.
For the lower figure in context, see the diamond theorem.
Recent piracy of my work as part of a London art project suggests the following.
From http://www.trussel.com/rls/rlsgb1.htm
The 2011 Long John Silver Award for academic piracy
goes to ….
Hermann Weyl, for the remark on objectivity and invariance
in his classic work Symmetry that skillfully pirated
the much earlier work of philosopher Ernst Cassirer.
And the 2011 Parrot Award for adept academic idealifting
goes to …
Richard Evan Schwartz of Brown University, for his
use, without citation, of Cullinane’s work illustrating
Weyl’s “relativity problem” in a finitegeometry context.
For further details, click on the above names.
The title of a recent contribution to a London artrelated "Piracy Project" begins with the phrase "The Search for Invariants."
A search for that phrase elsewhere yields a notable 1944* paper by Ernst Cassirer, "The Concept of Group and the Theory of Perception."
Page 20: "It is a process of objectification, the characteristic nature
and tendency of which finds expression in the formation of invariants."
Cassirer's concepts seem related to Weyl's famous remark that
“Objectivity means invariance with respect to the group of automorphisms.”
—Symmetry (Princeton University Press, 1952, page 132)
See also this journal on June 23, 2010— "Group Theory and Philosophy"— as well as some Math Forum remarks on Cassirer and Weyl.
Update of 6 to 7:50 PM June 20, 2011—
Weyl's 1952 remark seems to echo remarks in 1910 and 1921 by Cassirer.
See Cassirer in 1910 and 1921 on Objectivity.
Another source on Cassirer, invariance, and objectivity—
The conclusion of Maja Lovrenov's
"The Role of Invariance in Cassirer’s Interpretation of the Theory of Relativity"—
"… physical theories prove to be theories of invariants
with regard to certain groups of transformations and
it is exactly the invariance that secures the objectivity
of a physical theory."
— SYNTHESIS PHILOSOPHICA 42 (2/2006), pp. 233–241
A search in Weyl's Symmetry for any reference to Ernst Cassirer yields no results.
* Published in French in 1938.
The title was suggested by this evening's 4digit NY lottery number.
"… the rhetoric might be a bit over the top."
According to Amazon.com, 2198 (i.e., 2/1/98) was the publication
date of Geometry of Vector Sheaves , Volume I, by Anastasios Mallios.
Related material—
The question of S.S. Chern quoted here June 10: —
"What is Geometry?"— and the remark by Stevens that
accompanied the quotation—
"Reality is the beginning not the end,
Naked Alpha, not the hierophant Omega,
of dense investiture, with luminous vassals."
— Wallace Stevens,
“An Ordinary Evening in New Haven” VI
The work of Mallios in pure mathematics cited above seems
quite respectable (unlike his later remarks on physics).
His Vector Sheaves appears to be trying to explore new territory;
hence the relevance of Stevens's "Alpha." See also the phrase
"AInvariance" in an undated preprint by Mallios*.
For the evening 3digit number, 533, see a Stevens poem—
This meditation by Stevens is related to the female form of Mallios's Christian name.
As for the afternoon numbers, see "62" in The Beauty Test (May 23, 2007), Geometry and Death, and "9181" as the date 9/1/81.
* Later published in International Journal of Theoretical Physics , Vol. 47, No. 7, cover date 20080701
University Diaries praised today the late Robert Nozick's pedagogical showmanship.
His scholarship was less praiseworthy. His 2001 book Invariances: The Structure of the Objective World failed, quite incredibly, to mention Hermann Weyl's classic summary of the connection between invariance and objectivity. See a discussion of Nozick in The New York Review of Books of December 19, 2002—
"… one should mention, first and foremost, the mathematician Hermann Weyl who was almost obsessed by this connection. In his beautiful little book Symmetry he tersely says, 'Objectivity means invariance with respect to the group of automorphisms….'"
See also this journal on Dec. 3, 2002, and Feb. 20, 2007.
For some context, see a search on the word stem "objectiv" in this journal.
Excerpts from "The Concept of Group and the Theory of Perception,"
by Ernst Cassirer, Philosophy and Phenomenological Research,
Volume V, Number 1, September, 1944.
(Published in French in the Journal de Psychologie, 1938, pp. 368414.)
The grouptheoretical interpretation of the fundaments of geometry is,
from the standpoint of pure logic, of great importance, since it enables us to
state the problem of the "universality" of mathematical concepts in simple
and precise form and thus to disentangle it from the difficulties and ambigui
ties with which it is beset in its usual formulation. Since the times of the
great controversies about the status of universals in the Middle Ages, logic
and psychology have always been troubled with these ambiguities….
Our foregoing reflections on the concept of group permit us to define more
precisely what is involved in, and meant by, that "rule" which renders both
geometrical and perceptual concepts universal. The rule may, in simple
and exact terms, be defined as that group of transformations with regard to
which the variation of the particular image is considered. We have seen
above that this conception operates as the constitutive principle in the con
struction of the universe of mathematical concepts….
…Within Euclidean geometry,
a "triangle" is conceived of as a pure geometrical "essence," and this
essence is regarded as invariant with respect to that "principal group" of
spatial transformations to which Euclidean geometry refers, viz., displace
ments, transformations by similarity. But it must always be possible to
exhibit any particular figure, chosen from this infinite class, as a concrete
and intuitively representable object. Greek mathematics could not
dispense with this requirement which is rooted in a fundamental principle
of Greek philosophy, the principle of the correlatedness of "logos" and
"eidos." It is, however, characteristic of the modern development of
mathematics, that this bond between "logos" and "eidos," which was indis
soluble for Greek thought, has been loosened more and more, to be, in the
end, completely broken….
…This process has come to its logical
conclusion and systematic completion in the development of modern group
theory. Geometrical figures are no longer regarded as fundamental, as
date of perception or immediate intuition. The "nature" or "essence" of a
figure is defined in terms of the operations which may be said to
generate the figure. The operations in question are, in turn, subject to
certain group conditions….
…What we
find in both cases are invariances with respect to variations undergone by
the primitive elements out of which a form is constructed. The peculiar
kind of "identity" that is attributed to apparently altogether heterogen
eous figures in virtue of their being transformable into one another by means
of certain operations defining a group, is thus seen to exist also in the
domain of perception. This identity permits us not only to single out ele
ments but also to grasp "structures" in perception. To the mathematical
concept of "transformability" there corresponds, in the domain of per
ception, the concept of "transposability." The theory of the latter con
cept has been worked out step by step and its development has gone through
various stages….
…By the acceptance of
"form" as a primitive concept, psychological theory has freed it from the
character of contingency which it possessed for its first founders. The inter
pretation of perception as a mere mosaic of sensations, a "bundle" of simple
senseimpressions has proved untenable….
…In the domain of mathematics this state of affairs mani
fests itself in the impossibility of searching for invariant properties of a
figure except with reference to a group. As long as there existed but one
form of geometry, i.e., as long as Euclidean geometry was considered as the
geometry kat' exochen this fact was somehow concealed. It was possible
to assume implicitly the principal group of spatial transformations that lies
at the basis of Euclidean geometry. With the advent of nonEuclidean
geometries, however, it became indispensable to have a complete and sys
tematic survey of the different "geometries," i.e., the different theories of
invariancy that result from the choice of certain groups of transformation.
This is the task which F. Klein set to himself and which he brought to a
certain logical fulfillment in his Vergleichende Untersuchungen ueber neuere
geometrische Forschungen….
…Without discrimination between the
accidental and the substantial, the transitory and the permanent, there
would be no constitution of an objective reality.
This process, unceasingly operative in perception and, so to speak, ex
pressing the inner dynamics of the latter, seems to have come to final per
fection, when we go beyond perception to enter into the domain of pure
thought. For the logical advantage and peculiar privilege of the pure con –
cept seems to consist in the replacement of fluctuating perception by some
thing precise and exactly determined. The pure concept does not lose
itself in the flux of appearances; it tends from "becoming" toward "being,"
from dynamics toward statics. In this achievement philosophers have
ever seen the genuine meaning and value of geometry. When Plato re
gards geometry as the prerequisite to philosophical knowledge, it is because
geometry alone renders accessible the realm of things eternal; tou gar aei
ontos he geometrike gnosis estin. Can there be degrees or levels of objec
tive knowledge in this realm of eternal being, or does not rather knowledge
attain here an absolute maximum? Ancient geometry cannot but answer
in the affirmative to this question. For ancient geometry, in the classical
form it received from Euclid, there was such a maximum, a non plus ultra.
But modern group theory thinking has brought about a remarkable change
In this matter. Group theory is far from challenging the truth of Euclidean
metrical geometry, but it does challenge its claim to definitiveness. Each
geometry is considered as a theory of invariants of a certain group; the
groups themselves may be classified in the order of increasing generality.
The "principal group" of transformations which underlies Euclidean geome
try permits us to establish a number of properties that are invariant with
respect to the transformations in question. But when we pass from this
"principal group" to another, by including, for example, affinitive and pro
jective transformations, all that we had established thus far and which,
from the point of view of Euclidean geometry, looked like a definitive result
and a consolidated achievement, becomes fluctuating again. With every
extension of the principal group, some of the properties that we had taken
for invariant are lost. We come to other properties that may be hierar
chically arranged. Many differences that are considered as essential
within ordinary metrical geometry, may now prove "accidental." With
reference to the new groupprinciple they appear as "unessential" modifica
tions….
… From the point of view of modern geometrical systematization,
geometrical judgments, however "true" in themselves, are nevertheless not
all of them equally "essential" and necessary. Modern geometry
endeavors to attain progressively to more and more fundamental strata of
spatial determination. The depth of these strata depends upon the com
prehensiveness of the concept of group; it is proportional to the strictness of
the conditions that must be satisfied by the invariance that is a universal
postulate with respect to geometrical entities. Thus the objective truth
and structure of space cannot be apprehended at a single glance, but have to
be progressively discovered and established. If geometrical thought is to
achieve this discovery, the conceptual means that it employs must become
more and more universal….
The connection:
Abstract: "The Steiner system S(4,7,23) is constructed from the geometry of PG(3,2)."
Abstract: "The Steiner system S(5,8,24) is constructed from the geometry of PG(3,2)."
“The first credential
we should demand of a critic
is his ideograph of the good.”
— Ezra Pound,
How to Read
Music critic Bernard Holland in The New York Times on Monday, May 20, 1996:
The Juilliard’s
HalfCentury RipeningPhilosophers ponder the idea of identity: what it is to give something a name on Monday and have it respond to that name on Friday regardless of what might have changed in the interim. Medical science tells us that the body’s cells replace themselves wholesale within every seven years, yet we tell ourselves that we are what we were….
Schubert at the end of his life had already passed on to another level of spirit. Beethoven went back and forth between the temporal world and the world beyond right up to his dying day.
Exercise
Part I:
Apply Holland’s MondaytoFriday “idea of identity” to the lives and deaths during the week of Monday, Nov. 10 (“Frame Tales“), through Friday, Nov. 14, of a musician and a maker of music documentaries– Mitch Mitchell (d. Nov. 12) and Baird Bryant (d. Nov. 13).
Part II:
Apply Holland’s “idea of identity” to last week (Monday, Nov. 17, through Friday, Nov. 21), combining it with Wigner’s remarks on invariance (discussed here on Monday) and with the “red dragon” (Log24, Nov. 15) concept of flight over “the Hump”– the Himalayas– and the 1991 documentary filmed by Bryant, “Heart of Tibet.”
Part III:
Discuss Parts I and II in the context of Eliot’s Four Quartets. (See Time Fold, The Field of Reason, and Balance.)
From the previous entry:
“If it’s a seamless whole you want,
pray to Apollo, who sets the limits
within which such a work can exist.”
— Margaret Atwood,
author of Cat’s Eye
Happy birthday
to the late
Eugene Wigner
… and a belated
Merry Christmas
to Paul Newman:
— Eugene Wigner, Nobel Prize Lecture, December 12, 1963
In memory of
Rudolf Arnheim,
who died on
Saturday, June 9
From the Wikipedia article on Gestalt psychology prior to its modification on May 31, 2007:
“Emergence, reification, multistability, and invariance are not separable modules to be modeled individually, but they are different aspects of a single unified dynamic mechanism.
For a mathematical example of such a mechanism using the cubes of psychologists’ block design tests, see Block Designs in Art and Mathematics and The Kaleidoscope Puzzle.”
The second paragraph of the above passage refers to my own work.
Some Gestaltrelated work of Arnheim:
Time of this entry:
1:06:18 AM ET.
Today is the 21st birthday of my note “The Relativity Problem in Finite Geometry.”
Some relevant quotations:
“This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.”
— Hermann Weyl, The Classical Groups, Princeton University Press, 1946, p. 16
Describing the branch of mathematics known as Galois theory, Weyl says that it
“… is nothing else but the relativity theory for the set Sigma, a set which, by its discrete and finite character, is conceptually so much simpler than the infinite set of points in space or spacetime dealt with by ordinary relativity theory.”
— Weyl, Symmetry, Princeton University Press, 1952, p. 138
Weyl’s set Sigma is a finite set of complex numbers. Some other sets with “discrete and finite character” are those of 4, 8, 16, or 64 points, arranged in squares and cubes. For illustrations, see Finite Geometry of the Square and Cube. What Weyl calls “the relativity problem” for these sets involves fixing “objectively” a class of equivalent coordinatizations. For what Weyl’s “objectively” means, see the article “Symmetry and Symmetry Breaking,” by Katherine Brading and Elena Castellani, in the Stanford Encyclopedia of Philosophy:
“The old and natural idea that what is objective should not depend upon the particular perspective under which it is taken into consideration is thus reformulated in the following grouptheoretical terms: what is objective is what is invariant with respect to the transformation group of reference frames, or, quoting Hermann Weyl (1952, p. 132), ‘objectivity means invariance with respect to the group of automorphisms [of spacetime].^{‘[22]}
22. The significance of the notion of invariance and its grouptheoretic treatment for the issue of objectivity is explored in Born (1953), for example. For more recent discussions see Kosso (2003) and Earman (2002, Sections 6 and 7).
References:
Born, M., 1953, “Physical Reality,” Philosophical Quarterly, 3, 139149. Reprinted in E. Castellani (ed.), Interpreting Bodies: Classical and Quantum Objects in Modern Physics, Princeton, NJ: Princeton University Press, 1998, pp. 155167.
Earman, J., 2002, “Laws, Symmetry, and Symmetry Breaking; Invariance, Conservation Principles, and Objectivity,’ PSA 2002, Proceedings of the Biennial Meeting of the Philosophy of Science Association 2002, forthcoming [Abstract/Preprint available online]
Kosso, P., 2003, “Symmetry, objectivity, and design,” in K. Brading and E. Castellani (eds.), Symmetries in Physics: Philosophical Reflections, Cambridge: Cambridge University Press, pp. 410421.
Weyl, H., 1952, Symmetry, Princeton, NJ: Princeton University Press.
See also
Archives Henri Poincaré (research unit UMR 7117, at Université Nancy 2, of the CNRS)–
“Minkowski, Mathematicians, and the Mathematical Theory of Relativity,” by Scott Walter, in The Expanding Worlds of General Relativity (Einstein Studies, volume 7), H. Goenner, J. Renn, J. Ritter and T. Sauer, editors, Boston/Basel: Birkhäuser, 1999, pp. 4586–
“Developing his ideas before Göttingen mathematicians in April 1909, Klein pointed out that the new theory based on the Lorentz group (which he preferred to call ‘Invariantentheorie’) could have come from pure mathematics (1910: 19). He felt that the new theory was anticipated by the ideas on geometry and groups that he had introduced in 1872, otherwise known as the Erlangen program (see Gray 1989: 229).”
References:
Gray, Jeremy J. (1989). Ideas of Space. 2d ed. Oxford: Oxford University Press.
Klein, Felix. (1910). “Über die geometrischen Grundlagen der Lorentzgruppe.” Jahresbericht der deutschen MathematikerVereinigung 19: 281300. [Reprinted: Physikalische Zeitschrift 12 (1911): 1727].
Related material: A pathetically garbled version of the above concepts was published in 2001 by Harvard University Press. See Invariances: The Structure of the Objective World, by Robert Nozick.
Serious
"I don't think the 'diamond theorem' is anything serious, so I started with blitzing that."
— Charles Matthews at Wikipedia, Oct. 2, 2006
"The 'seriousness' of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects. We may say, roughly, that a mathematical idea is 'significant' if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas."
— G. H. Hardy, A Mathematician's Apology
Matthews yesterday deleted references to the diamond theorem and related material in the following Wikipedia articles:
Affine group
Reflection group
Symmetry in mathematics
Incidence structure
Invariant (mathematics)
Symmetry
Finite geometry
Group action
History of geometry
This would appear to be a fairly large complex of mathematical ideas.
See also the following "large complex" cited, following the above words of Hardy, in Diamond Theory:
Affine geometry, affine planes, affine spaces, automorphisms, binary codes, block designs, classical groups, codes, coding theory, collineations, combinatorial, combinatorics, conjugacy classes, the Conwell correspondence, correlations, design theory, duads, duality, error correcting codes, exceptional groups, finite fields, finite geometry, finite groups, finite rings, Galois fields, generalized quadrangles, generators, geometry, GF(2), GF(4), the (24,12) Golay code, group actions, group theory, Hadamard matrices, hypercube, hyperplanes, hyperspace, incidence structures, invariance, Karnaugh maps, Kirkman's schoolgirl problem, Latin squares, Leech lattice, linear groups, linear spaces, linear transformations, Mathieu groups, matrix theory, Meno, Miracle Octad Generator, MOG, multiply transitive groups, octads, the octahedral group, orthogonal arrays, outer automorphisms, parallelisms, partial geometries, permutation groups, PG(3,2), polarities, PolyaBurnside theorem, projective geometry, projective planes, projective spaces, projectivities, ReedMuller codes, the relativity problem, Singer cycle, skew lines, sporadic simple groups, Steiner systems, symmetric, symmetry, symplectic, synthemes, synthematic, tesseract, transvections, Walsh functions, Witt designs.
Earendil_Silmarils:
Les Anamorphoses:
“Pour construire un dessin en perspective,
le peintre trace sur sa toile des repères:
la ligne d’horizon (1),
le point de fuite principal (2)
où se rencontre les lignes de fuite (3)
et le point de fuite des diagonales (4).”
_______________________________
Serge Mehl,
Perspective &
Géométrie Projective:
“… la géométrie projective était souvent
synonyme de géométrie supérieure.
Elle s’opposait à la géométrie
euclidienne: élémentaire…
La géométrie projective, certes supérieure
car assez ardue, permet d’établir
de façon élégante des résultats de
la géométrie élémentaire.”
Similarly…
Finite projective geometry
(in particular, Galois geometry)
is certainly superior to
the elementary geometry of
quiltpattern symmetry
and allows us to establish
de façon élégante
some results of that
elementary geometry.
Other Related Material…
from algebra rather than
geometry, and from a German
rather than from the French:
“This is the relativity problem:
to fix objectively a class of
equivalent coordinatizations
and to ascertain
the group of transformations S
mediating between them.”
— Hermann Weyl,
The Classical Groups,
Princeton U. Press, 1946
Evariste Galois
Weyl also says that the profound branch
of mathematics known as Galois theory
“Perhaps every science must
start with metaphor
and end with algebra;
and perhaps without metaphor
there would never have been
any algebra.”
For metaphor and
algebra combined, see
A.M.S. abstract 79TA37,
Notices of the
American Mathematical Society,
February 1979, pages A193, 194 —
the original version of the 4×4 case
of the diamond theorem.
“When approaching unfamiliar territory, we often, as observed earlier, try to describe or frame the novel situation using metaphors based on relations perceived in a familiar domain, and by using our powers of association, and our ability to exploit the structural similarity, we go on to conjecture new features for consideration, often not noticed at the outset. The metaphor works, according to Max Black, by transferring the associated ideas and implications of the secondary to the primary system, and by selecting, emphasising and suppressing features of the primary in such a way that new slants on it are illuminated.”
— Paul Thompson, University College, Oxford,
The Nature and Role of Intuition
in Mathematical Epistemology
That intuition, metaphor (i.e., analogy), and association may lead us astray is well known. The examples of French perspective above show what might happen if someone ignorant of finite geometry were to associate the phrase “4×4 square” with the phrase “projective geometry.” The results are ridiculously inappropriate, but at least the second example does, literally, illuminate “new slants”– i.e., diagonals– within the perspective drawing of the 4×4 square.
Similarly, analogy led the ancient Greeks to believe that the diagonal of a square is commensurate with the side… until someone gave them a new slant on the subject.
Old School Tie
"We are introduced to John Nash, fuddling flatfooted about the Princeton courtyard, uninterested in his classmates' yammering about their various accolades. One chap has a rather unfortunate sense of style, but rather than tritely insult him, Nash holds a patterned glass to the sun, [director Ron] Howard shows us refracted patterns of light that take shape in a punch bowl, which Nash then displaces onto the neckwear, replying, 'There must be a formula for how ugly your tie is.' "
— Draft of
Computing with Modal Logics
(pdf), by Carlos Areces
and Maarten de Rijke
… diamonds and boxes are upper and lower adjoints of Galois connections…."
Evariste Galois
"Perhaps every science must
start with metaphor
and end with algebra;
and perhaps without metaphor
there would never have been
any algebra."
— attributed, in varying forms
(1, 2, 3), to Max Black,
Models and Metaphors, 1962
For metaphor and
algebra combined, see
"Symmetry invariance
in a diamond ring,"
A.M.S. abstract 79TA37,
Notices of the Amer. Math. Soc.,
February 1979, pages A193, 194 —
the original version of the 4×4 case
of the diamond theorem.
Battle of Gods and Giants
In checking the quotations from Dante in the previous entry, I came across the intriguing site Gigantomachia:
"A gigantomachia or primordial battle between the gods has been retold in myth, cult, art and theory for thousands of years, from the Egyptians to Heidegger. This site will present the history of the theme. But it will do so in an attempt to raise the question of the contemporary relevance of it. Does the gigantomachia take place today? Where? When? In what relation to you and me?"
Perhaps atop the Empire State Building?
(See An Affair to Remember and Empire State Building to Honor Fay Wray.)
Perhaps in relation to what the late poet Donald Justice called "the wood within"?
Perhaps in relation to T. S. Eliot's "The Waste Land" and the Feast of the Metamorphosis?
Or perhaps not.
Perhaps at Pergamon:
Perhaps at Pergamon Press:
"What modern painters are trying to do,
if they only knew it, is paint invariants."
— James J. Gibson in Leonardo
(Vol. 11, pp. 227235.
Pergamon Press Ltd., 1978)
An example of invariant structure:
The three line diagrams above result from the three partitions, into pairs of 2element sets, of the 4element set from which the entries of the bottom colored figure are drawn. Taken as a set, these three line diagrams describe the structure of the bottom colored figure. After coordinatizing the figure in a suitable manner, we find that this set of three line diagrams is invariant under the group of 16 binary translations acting on the colored figure.
A more remarkable invariance — that of symmetry
This sort of mathematics illustrates the invisible "form" or "idea" behind the visible twocolor pattern. Hence it exemplifies, in a way, the conflict described by Plato between those who say that "real existence belongs only to that which can be handled" and those who say that "true reality consists in certain intelligible and bodiless forms."
For further details, see a section on Plato in the Gigantomachia site.
Parable
“A comparison or analogy. The word is simply a transliteration of the Greek word: parabolé (literally: ‘what is thrown beside’ or ‘juxtaposed’), a term used to designate the geometric application we call a ‘parabola.’…. The basic parables are extended similes or metaphors.”
— http://religion.rutgers.edu/nt/
primer/parable.html
“If one style of thought stands out as the most potent explanation of genius, it is the ability to make juxtapositions that elude mere mortals. Call it a facility with metaphor, the ability to connect the unconnected, to see relationships to which others are blind.”
— Sharon Begley, “The Puzzle of Genius,” Newsweek magazine, June 28, 1993, p. 50
“The poet sets one metaphor against another and hopes that the sparks set off by the juxtaposition will ignite something in the mind as well. Hopkins’ poem ‘Pied Beauty’ has to do with ‘creation.’ “
— Speaking in Parables, Ch. 2, by Sallie McFague
“The Act of Creation is, I believe, a more truly creative work than any of Koestler’s novels…. According to him, the creative faculty in whatever form is owing to a circumstance which he calls ‘bisociation.’ And we recognize this intuitively whenever we laugh at a joke, are dazzled by a fine metaphor, are astonished and excited by a unification of styles, or ‘see,’ for the first time, the possibility of a significant theoretical breakthrough in a scientific inquiry. In short, one touch of genius—or bisociation—makes the whole world kin. Or so Koestler believes.”
— Henry David Aiken, The Metaphysics of Arthur Koestler, New York Review of Books, Dec. 17, 1964
For further details, see
Speaking in Parables:
A Study in Metaphor and Theology
by Sallie McFague
Fortress Press, Philadelphia, 1975
Introduction
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
“Perhaps every science must start with metaphor and end with algebra; and perhaps without metaphor there would never have been any algebra.”
— attributed, in varying forms (1, 2, 3), to Max Black, Models and Metaphors, 1962
For metaphor and algebra combined, see
“Symmetry invariance in a diamond ring,” A.M.S. abstract 79TA37, Notices of the Amer. Math. Soc., February 1979, pages A193, 194 — the original version of the 4×4 case of the diamond theorem.
The Da Vinci Code
and Symbology at Harvard
The protagonist of the recent bestseller The Da Vinci Code is Robert Langdon, "a professor of Religious Symbology at Harvard University." A prominent part in the novel is played by the wellknown Catholic organization Opus Dei. Less well known (indeed, like Langdon, nonexistent) is the academic discipline of "symbology." (For related disciplines that do exist, click here.) What might a course in this subject at Harvard be like?
Harvard Crimson, April 10, 2003: While Opus Dei members said that they do not refer to their practices of recruitment as "fishing," the Work’s founder does describe the process of what he calls "winning new apostles" with an aquatic metaphor. Point #978 of The Way invokes a passage in the New Testament in which Jesus tells Peter that he will make him a "fisher of men." The point reads:

Exercise for Symbology 101:
Describe the symmetry
in each of the pictures above.
Show that the second picture
retains its underlying structural
symmetry under a group of
322,560 transformations.
Having reviewed yesterday's notes
on Gombrich, Gadamer, and Panofsky,
discuss the astrological meaning of
the above symbols in light of
today's date, February 20.
Extra credit:
Relate the above astrological
symbolism to the fourdiamond
symbol in Jung's Aion.
Happy metaphors!
Diamond theory is the theory of affine groups over GF(2) acting on small square and cubic arrays. In the simplest case, the symmetric group of degree 4 acts on a twocolored diamond figure like that in Plato's Meno dialogue, yielding 24 distinct patterns, each of which has some ordinary or colorinterchange symmetry .
This symmetry invariance can be generalized to (at least) a group of order approximately 1.3 trillion acting on a 4x4x4 array of cubes.
The theory has applications to finite geometry and to the construction of the large Witt design underlying the Mathieu group of degree 24.
Symmetry and a Trinity
From a web page titled Spectra:
"What we learn from our whole discussion and what has indeed become a guiding principle in modern mathematics is this lesson:
Whenever you have to do with a structureendowed entity S try to determine its group of automorphisms, the group of those elementwise transformations which leave all structural relations undisturbed. You can expect to gain a deep insight into the constitution of S in this way. After that you may start to investigate symmetric configurations of elements, i.e., configurations which are invariant under a certain subgroup of the group of all automorphisms . . ."
— Hermann Weyl in Symmetry, Princeton University Press, 1952, page 144
"… any color at all can be made from three different colors, in our case, red, green, and blue lights. By suitably mixing the three together we can make anything at all, as we demonstrated . . .
Further, these laws are very interesting mathematically. For those who are interested in the mathematics of the thing, it turns out as follows. Suppose that we take our three colors, which were red, green, and blue, but label them A, B, and C, and call them our primary colors. Then any color could be made by certain amounts of these three: say an amount a of color A, an amount b of color B, and an amount c of color C makes X:
Now suppose another color Y is made from the same three colors:
Then it turns out that the mixture of the two lights (it is one of the consequences of the laws that we have already mentioned) is obtained by taking the sum of the components of X and Y:
It is just like the mathematics of the addition of vectors, where (a, b, c ) are the components of one vector, and (a', b', c' ) are those of another vector, and the new light Z is then the "sum" of the vectors. This subject has always appealed to physicists and mathematicians."
— According to the author of the Spectra site, this is Richard Feynman in Elementary Particles and the Laws of Physics, The 1986 Dirac Memorial Lectures, by Feynman and Steven Weinberg, Cambridge University Press, 1989.
These two concepts — symmetry as invariance under a group of transformations, and complicated things as linear combinations (the technical name for Feynman's sums) of simpler things — underlie much of modern mathematics, both pure and applied.
Symmetry, Invariance, and Objectivity
The book Invariances: The Structure of the Objective World, by Harvard philosopher Robert Nozick, was reviewed in the New York Review of Books issue dated June 27, 2002.
On page 76 of this book, published by Harvard University Press in 2001, Nozick writes:
"An objective fact is invariant under various transformations. It is this invariance that constitutes something as an objective truth…."
Compare this with Hermann Weyl's definition in his classic Symmetry (Princeton University Press, 1952, page 132):
"Objectivity means invariance with respect to the group of automorphisms."
It has finally been pointed out in the Review, by a professor at Göttingen, that Nozick's book should have included Weyl's definition.
I pointed this out on June 10, 2002.
For a survey of material on this topic, see this Google search on "nozick invariances weyl" (without the quotes).
Nozick's omitting Weyl's definition amounts to blatant plagiarism of an idea.
Of course, including Weyl's definition would have required Nozick to discuss seriously the concept of groups of automorphisms. Such a discussion would not have been compatible with the current level of philosophical discussion at Harvard, which apparently seldom rises above the level of cocktailparty chatter.
A similarly low level of discourse is found in the essay "Geometrical Creatures," by Jim Holt, also in the issue of the New York Review of Books dated December 19, 2002. Holt at least writes well, and includes (if only in parentheses) a remark that is highly relevant to the Nozickvs.Weyl discussion of invariance elsewhere in the Review:
"All the geometries ever imagined turn out to be variations on a single theme: how certain properties of a space remain unchanged when its points get rearranged." (p. 69)
This is perhaps suitable for intelligent but ignorant adolescents; even they, however, should be given some historical background. Holt is talking here about the Erlangen program of Felix Christian Klein, and should say so. For a more sophisticated and nuanced discussion, see this web page on Klein's Erlangen Program, apparently by JeanPierre Marquis, Département de Philosophie, Université de Montréal. For more by Marquis, see my later entry for today, "From the Erlangen Program to Category Theory."


Example:





Initial Xanga entry. Updated Nov. 18, 2006.
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