(A review)
Friday, August 14, 2015
Discrete Space
Galois space:
Counting symmetries of Galois space:
The reason for these graphic symmetries in affine Galois space —
symmetries of the underlying projective Galois space:
Tuesday, May 28, 2019
Quaternion at Candlebrow
From a Groundhog Day post in 2009 —
The Candlebrow Conference The conferees had gathered here from all around the world…. Their spirits all one way or another invested in, invested by, the siegecraft of Time and its mysteries. "Fact is, our system of socalled linear time is based on a circular or, if you like, periodic phenomenon– the earth's own spin. Everything spins, up to and including, probably, the whole universe. So we can look to the prairie, the darkening sky, the birthing of a funnelcloud to see in its vortex the fundamental structure of everything–" "Um, Professor–"…. … Those in attendance, some at quite high speed, had begun to disperse, the briefest of glances at the sky sufficing to explain why. As if the professor had lectured it into being, there now swung from the swollen and lightpulsing clouds to the west a classic prairie "twister"…. … In the storm cellar, over semiliquid coffee and farmhouse crullers left from the last twister, they got back to the topic of periodic functions…. "Eternal Return, just to begin with. If we may construct such functions in the abstract, then so must it be possible to construct more secular, more physical expressions." "Build a time machine." "Not the way I would have put it, but if you like, fine." Vectorists and Quaternionists in attendance reminded everybody of the function they had recently worked up…. "We thus enter the whirlwind. It becomes the very essence of a refashioned life, providing the axes to which everything will be referred. Time no long 'passes,' with a linear velocity, but 'returns,' with an angular one…. We are returned to ourselves eternally, or, if you like, timelessly." "Born again!" exclaimed a Christer in the gathering, as if suddenly enlightened. Above, the devastation had begun. 
"As if the professor had lectured it into being . . . ."
See other posts now tagged McLuhan Time.
Monday, May 27, 2019
But Seriously . . .
I prefer the simple "four dots" figure
of the double colon:
For those who prefer stranger analogies . . .
Actors from "The Eiger Sanction" —
Doctor Strange on Mount Everest —
See as well this journal on the above Strange date, 2016/12/02,
in posts tagged Lumber Room.
Sunday, September 3, 2017
Sunday, May 21, 2017
Rota on Beauty
Tiptoe through the tulips with Rota and Erickson:
Attempts have been made to string together beautiful mathematical results and to present them in books bearing such attractive titles as The One Hundred Most Beautiful Theorems of Mathematics. Such anthologies are seldom found on a mathematician’s bookshelf. The beauty of a theorem is best observed when the theorem is presented as the crown jewel within the context of a theory. — GianCarlo Rota in Indiscrete Thoughts 
See also Martin Erickson in this journal . . .
Wednesday, April 12, 2017
Expanding the Spielraum
"Cézanne ignores the laws of classical perspective . . . ."
— Voorhies, James. “Paul Cézanne (1839–1906).”
In Heilbrunn Timeline of Art History . New York:
The Metropolitan Museum of Art, 2000–. (October 2004)
Some others do not.
This is what I called "the large Desargues configuration"
in posts of April 2013 and later.
Friday, August 19, 2016
Princeton University Press in 1947
From a review, in the context of Hollywood, of a Princeton
University Press book on William Blake from 1947 —
Thursday, August 11, 2016
The Large Desargues Configuration
(Continued from April 2013 and later)
This is what I called "the large Desargues configuration"
in posts of April 2013 and later.
Friday, November 27, 2015
Einstein and Geometry
(A Prequel to Dirac and Geometry)
"So Einstein went back to the blackboard.
And on Nov. 25, 1915, he set down
the equation that rules the universe.
As compact and mysterious as a Viking rune,
it describes spacetime as a kind of sagging mattress…."
— Dennis Overbye in The New York Times online,
November 24, 2015
Some pure mathematics I prefer to the sagging Viking mattress —
Readings closely related to the above passage —
Thomas Hawkins, "From General Relativity to Group Representations:
the Background to Weyl's Papers of 192526," in Matériaux pour
l'histoire des mathématiques au XXe siècle: Actes du colloque
à la mémoire de Jean Dieudonné, Nice, 1996 (Soc. Math.
de France, Paris, 1998), pp. 69100.
The 19thcentury algebraic theory of invariants is discussed
as what Weitzenböck called a guide "through the thicket
of formulas of general relativity."
Wallace Givens, "Tensor Coordinates of Linear Spaces," in
Annals of Mathematics Second Series, Vol. 38, No. 2, April 1937,
pp. 355385.
Tensors (also used by Einstein in 1915) are related to
the theory of line complexes in threedimensional
projective space and to the matrices used by Dirac
in his 1928 work on quantum mechanics.
For those who prefer metaphors to mathematics —
Rota fails to cite the source of his metaphor.

Monday, November 3, 2014
Wisconsin Death Trip*
Courtesy of Mira Sorvino.
Enter Madison :
From "Intruders," BBC America, Season 1, Episode 2, at 1:07 of 43:31.
"You sure know how to show a girl a good time."
* The title is a reference to a Wisconsinrelated Halloween post.
Friday, October 31, 2014
For the Late Hans Schneider
See a University of Wisconsin obituary for Schneider,
a leading expert on linear algebra who reportedly died
at 87 on Tuesday, October 28, 2014.
Some background on linear algebra and "magic" squares:
tonight's 3 AM (ET) post and a search in this
journal for Knight, Death, and the Devil.
Click image to enlarge.
Friday, November 30, 2012
Point
"….mirando il punto
a cui tutti li tempi son presenti"
— Dante, Paradiso , XVII, 1718
For instance…
Click image for higher quality.
Saturday, May 26, 2012
Harriot’s Cubes
See also Finite Geometry and Physical Space.
Related material from MacTutor—
The paper by J. W. Shirley, Binary numeration before Leibniz, Amer. J. Physics 19 (8) (1951), 452454, contains an interesting look at some mathematics which appears in the hand written papers of Thomas Harriot [15601621]. Using the photographs of the two original Harriot manuscript pages reproduced in Shirley’s paper, we explain how Harriot was doing arithmetic with binary numbers. Leibniz [16461716] is credited with the invention [16791703] of binary arithmetic, that is arithmetic using base 2. Laplace wrote:
However, Leibniz was certainly not the first person to think of doing arithmetic using numbers to base 2. Many years earlier Harriot had experimented with the idea of different number bases…. 
For a discussion of Harriot on the discretevs.continuous question,
see Katherine Neal, From Discrete to Continuous: The Broadening
of Number Concepts in Early Modern England (Springer, 2002),
pages 6971.
Saturday, April 28, 2012
Sprechen Sie Deutsch?
A Log24 post, "Bridal Birthday," one year ago today linked to
"The Discrete and the Continuous," a brief essay by David Deutsch.
From that essay—
"The idea of quantization—
the discreteness of physical quantities—
turned out to be immensely fruitful."
Deutsch's "idea of quantization" also appears in
the April 12 Log24 post Mythopoetic—
"Is Space Digital?" — Cover story, Scientific American "The idea that space may be digital — Physicist Sabine Hossenfelder "A quantization of space/time — Peter Woit in a comment 
It seems some clarification is in order.
Hossenfelder's "The idea that space may be digital"
and Woit's "a quantization of space/time" may not
refer to the same thing.
Scientific American on the concept of digital space—
"Space may not be smooth and continuous.
Instead it may be digital, composed of tiny bits."
Wikipedia on the concept of quantization—
Causal sets, loop quantum gravity, string theory,
and black hole thermodynamics all predict
a quantized spacetime….
For a purely mathematical approach to the
continuousvs.discrete issue, see
Finite Geometry and Physical Space.
The physics there is somewhat tongueincheek,
but the geometry is serious.The issue there is not
continuousvs.discrete physics , but rather
Euclideanvs.Galois geometry .
Both sorts of geometry are of course valid.
Euclidean geometry has long been applied to
physics; Galois geometry has not. The cited
webpage describes the interplay of both sorts
of geometry— Euclidean and Galois, continuous
and discrete— within physical space— if not
within the space of physics.
Wednesday, March 21, 2012
Digital Theology
See also remarks on Digital Space and Digital Time in this journal.
Such remarks can, of course, easily verge on crackpot territory.
For some related pure mathematics, see Symmetry of Walsh Functions.
Monday, July 11, 2011
Thursday, April 28, 2011
Bridal Birthday
Catherine Elizabeth "Kate" Middleton, born 9 January 1982,
will marry Prince William of Wales on April 29th, 2011.
This suggests, by a very illogical and roundabout process
of verbal association, a search in this journal.
A quote from that search—
“‘Memory is nonnarrative and nonlinear.’
— Maya Lin in The Harvard Crimson , Friday, Dec. 2, 2005
A nonnarrative image from the same
general time span as the bride's birthday—
For some context, see Stevens + "The Rock" + "point A".
A post in that search, April 4th's Rock Notes, links to an essay
on physics and philosophy, "The Discrete and the Continuous," by David Deutsch.
See also the article on Deutsch, "Dream Machine," in the current New Yorker
(May 2, 2011), and the article's author, "Rivka Galchen," in this journal.
Galchen writes very well. For example —
Galchen on quantum theory—
"Our intuition, going back forever, is that to move, say, a rock, one has to touch that rock, or touch a stick that touches the rock, or give an order that travels via vibrations through the air to the ear of a man with a stick that can then push the rock—or some such sequence. This intuition, more generally, is that things can only directly affect other things that are right next to them. If A affects B without being right next to it, then the effect in question must be in direct—the effect in question must be something that gets transmitted by means of a chain of events in which each event brings about the next one directly, in a manner that smoothly spans the distance from A to B. Every time we think we can come up with an exception to this intuition—say, flipping a switch that turns on city street lights (but then we realize that this happens through wires) or listening to a BBC radio broadcast (but then we realize that radio waves propagate through the air)—it turns out that we have not, in fact, thought of an exception. Not, that is, in our everyday experience of the world. We term this intuition 'locality.' Quantum mechanics has upended many an intuition, but none deeper than this one." 
Monday, March 7, 2011
Punto
"Time it goes so fast
When you're having fun"
"….mirando il punto
a cui tutti li tempi son presenti"
– Dante, Paradiso , XVII, 1718
Monday, December 20, 2010
Contenders
Happy birthday to noir queen Audrey Totter. She starred in "The SetUp," a 1949 fight film.
"You sure know how to show a girl a good time."
— Renée Zellweger in "New in Town" (2009)
Tuesday, November 23, 2010
Art Object
There is more than one way
to look at a cube.
From Cambridge U. Press on Feb. 20, 2006 —
and from this journal on June 30, 2010 —
In memory of Wu Guanzhong, Chinese artist
who died in Beijing on June 25, 2010 —
See also this journal on Feb. 20, 2006
(the day The Cube was published).
Sunday, July 11, 2010
Philosophers’ Keystone
(Background— Yesterday's Quarter to Three,
A Manifold Showing, Class of 64, and Child's Play.)
Hermeneutics
Fans of Gregory Chaitin and Harry Potter
may consult Writings for Yom Kippur
for the meaning of yesterday's evening 673.
(See also Lowry and Cabbala.)
Fans of Elizabeth Taylor, Ava Gardner,
and the Dark Lady may consult Prime Suspect
for the meaning of yesterday's midday 17.
For some more serious background, see Dante—
"….mirando il punto
a cui tutti li tempi son presenti "
– Dante, Paradiso, XVII, 1718
“The symbol is used throughout the entire book
in place of such phrases as ‘Q.E.D.’ or
‘This completes the proof of the theorem’
to signal the end of a proof.”
— Measure Theory, by Paul R. Halmos, Van Nostrand, 1950
Halmos died on the date of Yom Kippur —
October 2, 2006.
Saturday, July 10, 2010
Class of 64
Samuel Beckett on Dante and Joyce:
"Another point of comparison is the preoccupation
with the significance of numbers."
"If I'd been out 'til quarter to three
Would you lock the door,
Will you still need me, will you still feed me,
When I'm sixtyfour?"
Happy birthday to Sue Lyon (Night of the Iguana, 1964)
Sunday, September 2, 2007
Sunday September 2, 2007
Comment at the
nCategory Cafe
Re: This Week’s Finds in Mathematical Physics (Week 251)
On Spekkens’ toy system and finite geometry
Background–
 In “Week 251” (May 5, 2007), John wrote:
“Since Spekkens’ toy system resembles a qubit, he calls it a “toy bit”. He goes on to study systems of several toy bits – and the charming combinatorial geometry I just described gets even more interesting. Alas, I don’t really understand it well: I feel there must be some mathematically elegant way to describe it all, but I don’t know what it is…. All this is fascinating. It would be nice to find the mathematical structure that underlies this toy theory, much as the category of Hilbert spaces underlies honest quantum mechanics.”  In the nCategory Cafe ( May 12, 2007, 12:26 AM, ) Matt Leifer wrote:
“It’s crucial to Spekkens’ constructions, and particularly to the analog of superposition, that the statespace is discrete. Finding a good mathematical formalism for his theory (I suspect finite fields may be the way to go) and placing it within a comprehensive framework for generalized theories would be very interesting.”  In the ncategory Cafe ( May 12, 2007, 6:25 AM) John Baez wrote:
“Spekkens and I spent an afternoon trying to think about his theory as quantum mechanics over some finite field, but failed — we almost came close to proving it couldnt’ work.”
On finite geometry:
 In “Week 234” (June 12, 2006), John wrote:
“For a pretty explanation of M_{24}… try this:
… Steven H. Cullinane, Geometry of the 4 × 4 square,
http://finitegeometry.org/sc/16/geometry.html”
The actions of permutations on a 4 × 4 square in Spekkens’ paper (quantph/0401052), and Leifer’s suggestion of the need for a “generalized framework,” suggest that finite geometry might supply such a framework. The geometry in the webpage John cited is that of the affine 4space over the twoelement field.
Related material:
Sept. 5, 2007
See also arXiv:0707.0074v1 [quantph], June 30, 2007:
A fully epistemic model for a local hidden variable emulation of quantum dynamics,
by Michael Skotiniotis, Aidan Roy, and Barry C. Sanders, Institute for Quantum Information Science, University of Calgary. Abstract: "In this article we consider an augmentation of Spekkens’ toy model for the epistemic view of quantum states [1]…."
Hypercube from the Skotiniotis paper:
Reference:
Evidence for the epistemic view of quantum states: A toy theory,
Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, Canada N2L 2Y5 (Received 11 October 2005; revised 2 November 2006; published 19 March 2007.)
Tuesday, February 20, 2007
Tuesday February 20, 2007
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.
Friday, October 20, 2006
Friday October 20, 2006
at best be heuristically described
in terms that invoke some notion
of an ‘intelligent user standing
outside the system.'”
— GianCarlo Rota in
Indiscrete Thoughts, p. 152
The Devil’s Bible and
Nothing Nothings (Again).
The Context
One context for the Rota quote
is Paul Halmos’s remark, quoted
in today’s New York Times,
that mathematics is
“almost like being
in touch with God.”
Another context is
Log24 on Aug. 29, 2005.
Monday, October 31, 2005
Monday October 31, 2005
— Chiang Yee, Chinese Calligraphy,
quoted in Aspen no. 10, item 8
"'Burnt Norton' opens as a meditation on time. Many comparable and contrasting views are introduced. The lines are drenched with reminiscences of Heraclitus' fragments on flux and movement…. the chief contrast around which Eliot constructs this poem is that between the view of time as a mere continuum, and the difficult paradoxical Christian view of how man lives both 'in and out of time,' how he is immersed in the flux and yet can penetrate to the eternal by apprehending timeless existence within time and above it. But even for the Christian the moments of release from the pressures of the flux are rare, though they alone redeem the sad wastage of otherwise unillumined existence. Eliot recalls one such moment of peculiar poignance, a childhood moment in the rosegarden– a symbol he has previously used, in many variants, for the birth of desire. Its implications are intricate and even ambiguous, since they raise the whole problem of how to discriminate between supernatural vision and mere illusion. Other variations here on the theme of how time is conquered are more directly apprehensible. In dwelling on the extension of time into movement, Eliot takes up an image he had used in 'Triumphal March': 'at the still point of the turning world.' This notion of 'a mathematically pure point' (as Philip Wheelwright has called it) seems to be Eliot's poetic equivalent in our cosmology for Dante's 'unmoved Mover,' another way of symbolising a timeless release from the 'outer compulsions' of the world. Still another variation is the passage on the Chinese jar in the final section. Here Eliot, in a conception comparable to Wallace Stevens' 'Anecdote of the Jar,' has suggested how art conquers time:
Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness."
— F. O. Matthiessen,
The Achievement of T.S. Eliot,
Oxford University Press, 1958,
as quoted in On "Burnt Norton"
Monday, August 29, 2005
Monday August 29, 2005
Date: Sun, 28 Aug 2005 12:30:40 0400 From: Alf van der Poorten AM Subject: Vale George Szekeres and Esther Klein Szekeres Members of the Number Theory List will be sad to learn that George and Esther Szekeres both died this morning. George, 94, had been quite ill for the last 23 days, barely conscious, and died first at 06:30. Esther, 95, died a half hour later. Both George Szekeres and Esther Klein will be recalled by number theorists as members of the group of young Hungarian mathematicians of the 1930s including Turan and Erdos. George and Esther's coming to Australia in the late 40s played an important role in the invigoration of Australian Mathematics. George was also an expert in group theory and relativity; he was my PhD supervisor. Emeritus Professor 
AVE
"Hello! Kinch here. Put me on to Edenville. Aleph, alpha: nought, nought, one."
"A very short space of time through very short times of space…. — James Joyce, Ulysses, Proteus chapter A very short space of time through very short times of space…. "It is demonstrated that spacetime should possess a discrete structure on Planck scales." — Peter Szekeres, abstract of Discrete SpaceTime 
Peter Szekeres is the son of George and Esther Szekeres.
"At present, such relationships can at best be heuristically described in terms that invoke some notion of an 'intelligent user standing outside the system.'"
— GianCarlo Rota in Indiscrete Thoughts, p. 152
Thursday, August 25, 2005
Thursday August 25, 2005
Train of Thought
Part I: The 24Cell
From S. H. Cullinane,
Visualizing GL(2,p),
March 26, 1985–
From John Baez, “This Week’s Finds in Mathematical Physics (Week 198),” September 6, 2003: Noam Elkies writes to John Baez:
The enrapturing discoveries of our field systematically conceal, like footprints erased in the sand, the analogical train of thought that is the authentic life of mathematics – GianCarlo Rota 
Like footprints erased in the sand….
“Hello! Kinch here. Put me on to Edenville. Aleph, alpha: nought, nought, one.”
“A very short space of time through very short times of space….
Am I walking into eternity along Sandymount strand?”
— James Joyce, Ulysses, Proteus chapter
A very short space of time through very short times of space….
“It is demonstrated that spacetime should possess a discrete structure on Planck scales.”
— Peter Szekeres, abstract of Discrete SpaceTime
“A theory…. predicts that space and time are indeed made of discrete pieces.”
— Lee Smolin in Atoms of Space and Time (pdf), Scientific American, Jan. 2004
“… a fundamental discreteness of spacetime seems to be a prediction of the theory….”
— Thomas Thiemann, abstract of Introduction to Modern Canonical Quantum General Relativity
“Theories of discrete spacetime structure are being studied from a variety of perspectives.”
— Quantum Gravity and the Foundations of Quantum Mechanics at Imperial College, London
The above speculations by physicists
are offered as curiosities.
I have no idea whether
any of them are correct.
Related material:
Stephen Wolfram offers a brief
History of Discrete Space.
For a discussion of space as discrete
by a nonphysicist, see John Bigelow‘s
Space and Timaeus.
in a Discrete Space
physics, there are of course many
purely mathematical discrete spaces.
See Visible Mathematics, continued
(Aug. 4, 2005):
Saturday, June 4, 2005
Saturday June 4, 2005
The 4×4 Square:
French Perspectives
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
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 U. Press, 1952
Metaphor and Algebra…
“Perhaps every science must
start with metaphor
and end with algebra;
and perhaps without metaphor
there would never have been
any algebra.”
Max Black, Models and Metaphors, 1962
For metaphor and
algebra combined, see
in a diamond ring,”
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.
More on Max Black…
“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.
Tuesday, September 28, 2004
Tuesday September 28, 2004
3:33:33 PM
Romantic Interaction, continued…
The Rhyme of Time
From American Dante Bibliography for 1983:
Freccero, John. "Paradiso X: The Dance of the Stars" (1968). Reprinted in Dante in America … (q.v.), pp. 345371. [1983] Freccero, John. "The Significance of terza rima." In Dante, Petrarch, Boccaccio: Studies in the Italian Trecento … (q.v.), pp. 317. [1983] Interprets the meaning of terza rima in terms of a temporal pattern of past, present, and future, with which the formal structure and the thematics of the whole poem coordinate homologically: "both the verse pattern and the theme proceed by a forward motion which is at the same time recapitulary." Following the same pattern in the three conceptual orders of the formal, thematical, and logical, the autobiographical narrative too is seen "as forward motion that moves towards its own beginning, or as a form of advance and recovery, leading toward a final recapitulation." And the same pattern is found especially to obtain theologically and biblically (i.e., historically). By way of recapitulation, the author concludes with a passage from Augustine's Confessions on the nature of time, which "conforms exactly to the movement of terza rima." Comes with six diagrams illustrating the various patterns elaborated in the text. 
From Rachel Jacoff's review of Pinsky's translation of Dante's Inferno:
"John Freccero's Introduction to the translation distills a compelling reading of the Inferno into a few powerful and immediately intelligible pages that make it clear why Freccero is not only a great Dante scholar, but a legendary teacher of the poem as well."
From The Undivine Comedy, Ch. 2, by Teodolinda Barolini (Princeton University Press, 1992):
"… we exist in time which, according to Aristotle, "is a kind of middlepoint, uniting in itself both a beginning and an end, a beginning of future time and an end of past time."* It is further to say that we exist in history, a middleness that, according to Kermode, men try to mitigate by making "fictive concords with origins and ends, such as give meaning to lives and to poems." Time and history are the media Dante invokes to begin a text whose narrative journey will strive to imitate– not escape– the journey it undertakes to represent, "il cammin di nostra vita." * Aristotle is actually referring to the moment, which he considers indistinguishable from time: "Now since time cannot exist and is unthinkable apart from the moment, and the moment is a kind of middlepoint, uniting as it does in itself both a beginning and an end, a beginning of future time and an end of past time, it follows that there must always be time: for the extremity of the last period of time that we take must be found in some moment, since time contains no point of contact for us except in the moment. Therefore, since the moment is both a beginning and an end there must always be time on both sides of it" (Physics 8.1.251b1826; in the translation of R. P. Hardie and R. K. Gaye, in The Basic Works of Aristotle, ed. Richard McKeon [New York: Random House, 1941]). 
From Four Quartets:
And the pool was filled with water out of sunlight,
And the lotos rose, quietly, quietly,
The surface glittered out of heart of light,
And they were behind us, reflected in the pool.
Then a cloud passed, and the pool was empty.
Go, said the bird, for the leaves were full of children,
Hidden excitedly, containing laughter.
Go, go, go, said the bird: human kind
Cannot bear very much reality.
Time past and time future
What might have been and what has been
Point to one end, which is always present.
Friday, February 20, 2004
Friday February 20, 2004
Finite Relativity
Today is the 18th birthday of my note
“The Relativity Problem in Finite Geometry.”
That note begins with a quotation from Weyl:
“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
Here is another quotation from Weyl, on the profound branch of mathematics known as Galois theory, which he says
“… 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
This second quotation applies equally well to the much less profound, but more accessible, part of mathematics described in Diamond Theory and in my note of Feb. 20, 1986.
Sunday, November 2, 2003
Sunday November 2, 2003
All Souls' Day
at the Still Point
From remarks on Denis Donoghue's Speaking of Beauty in the New York Review of Books, issue dated Nov. 20, 2003, page 48:
"The Russian theorist Bakhtin lends his august authority to what Donoghue's lively conversation has been saying, or implying, all along. 'Beauty does not know itself; it cannot found and validate itself — it simply is.' "
From The Bakhtin Circle:
"Goethe's imagination was fundamentally chronotopic, he visualised time in space:
Time and space merge … into an inseparable unity … a definite and absolutely concrete locality serves at the starting point for the creative imagination… this is a piece of human history, historical time condensed into space….
Dostoevskii… sought to present the voices of his era in a 'pure simultaneity' unrivalled since Dante. In contradistinction to that of Goethe this chronotope was one of visualising relations in terms of space not time and this leads to a philosophical bent that is distinctly messianic:
Only such things as can conceivably be linked at a single point in time are essential and are incorporated into Dostoevskii's world; such things can be carried over into eternity, for in eternity, according to Dostoevskii, all is simultaneous, everything coexists…. "
Bakhtin's notion of a "chronotope" was rather poorly defined. For a geometric structure that might well be called by this name, see Poetry's Bones and Time Fold. For a similar, but somewhat simpler, structure, see Balanchine's Birthday.
From Four Quartets:
"At the still point, there the dance is."
From an essay by William H. Gass on Malcolm Lowry's classic novel Under the Volcano:
"There is no o'clock in a cantina."
Saturday, November 1, 2003
Saturday November 1, 2003
Symmetry in Diamond Theory:
Robbing Peter to Pay Paul
"Groups arise in most areas of pure and applied mathematics, usually as a set of operators or transformations of some structure. The appearance of a group generally reflects some kind of symmetry in the object under study, and such symmetry may be considered one of the fundamental notions of mathematics."
"Counterchange is sometimes known as Robbing Peter to Pay Paul."
Paul Robeson in 
For a look at the Soviet approach
to counterchange symmetry, see
The Kishinev School of Discrete Geometry.
The larger cultural context:
See War of Ideas (Oct. 24),
The Hunt for Red October (Oct. 25),
On the Left (Oct. 25), and
ART WARS for Trotsky's Birthday (Oct. 26).
Monday, March 10, 2003
Monday March 10, 2003
ART WARS:
Art at the Vanishing Point
Two readings from The New York Times Book Review of Sunday,
2003 are relevant to our recurring "art wars" theme. The essay on Dante by Judith Shulevitz on page 31 recalls his "point at which all times are present." (See my March 7 entry.) On page 12 there is a review of a novel about the alleged "high culture" of the New York art world. The novel is centered on Leo Hertzberg, a fictional Columbia University art historian. From Janet Burroway's review of What I Loved, by Siri Hustvedt:
"…the 'zeros' who inhabit the book… dramatize its speculations about the self…. the spectator who is 'the true vanishing point, the pinprick in the canvas.'''
Here is a canvas by Richard McGuire for April Fools' Day 1995, illustrating such a spectator.
For more on the "vanishing point," or "point at infinity," see
Connoisseurs of ArtSpeak may appreciate Burroway's summary of Hustvedt's prose: "…her real canvas is philosophical, and here she explores the nature of identity in a structure of crystalline complexity."
For another "structure of crystalline
complexity," see my March 6 entry,
For a more honest account of the
New York art scene, see Tom Wolfe's
The Painted Word.
Friday, March 7, 2003
Friday March 7, 2003
Lovely, Dark and Deep
On this date in 1923, "Stopping by Woods on a Snowy Evening," by Robert Frost, was published. On this date in 1999, director Stanley Kubrick died. On this date in 1872, Piet Mondrian was born.
"….mirando il punto
a cui tutti li tempi son presenti"
— Dante, Paradiso, XVII, 1718
Chez Mondrian
Kertész, Paris, 1926
6:23 PM Friday, March 7:
From Measure Theory, by Paul R. Halmos, Van Nostrand, 1950:
"The symbol is used throughout the entire book in place of such phrases as 'Q.E.D.' or 'This completes the proof of the theorem' to signal the end of a proof."