Previously in Log24: Trudeau and the Story Theory of Truth.
Morerecent remarks by Trudeau —
Bible Stories for Skeptics
Review
About the Author
Product details 
Log24 on the above publication date — July 6, 2014 —
Previously in Log24: Trudeau and the Story Theory of Truth.
Morerecent remarks by Trudeau —
Bible Stories for Skeptics
Review
About the Author
Product details 
Log24 on the above publication date — July 6, 2014 —
The previous post suggests . . .
Jim Holt reviewing Edward Rothstein's Emblems of Mind: The Inner Life of Music and Mathematics in The New Yorker of June 5, 1995: "The fugues of Bach, the symphonies of Haydn, the sonatas of Mozart: these were explorations of ideal form, unprofaned by extramusical associations. Such 'absolute music,' as it came to be called, had sloughed off its motley cultural trappings. It had got in touch with its essence. Which is why, as Walter Pater famously put it, 'all art constantly aspires towards the condition of music.' The only art that can rival music for sheer etheriality is mathematics. A century or so after the advent of absolute music, mathematics also succeeded in detaching itself from the world. The decisive event was the invention of strange, nonEuclidean geometries, which put paid to the notion that the mathematician was exclusively, or even primarily, concerned with the scientific universe. 'Pure' mathematics came to be seen by those who practiced it as a free invention of the imagination, gloriously indifferent to practical affairs– a quest for beauty as well as truth." [Links added.] 
A line for James McAvoy —
"Pardon me boy, is this the Transylvania Station?"
See as well Worlds Out of Nothing , by Jeremy Gray.
"János Bolyai was a nineteenthcentury mathematician who
set the stage for the field of nonEuclidean geometry."
— Transylvania Now , October 26, 2018
From Coxeter and the Relativity Problem —
Desiring the exhilarations of changes:
The motive for metaphor, shrinking from
The weight of primary noon,
The A B C of being,
The ruddy temper, the hammer
Of red and blue, the hard sound—
Steel against intimation—the sharp flash,
The vital, arrogant, fatal, dominant X.
" There is a pleasantly discursive treatment
of Pontius Pilate’s unanswered question
‘What is truth?’ "
— Coxeter, 1987, introduction to Trudeau’s
The NonEuclidean Revolution
From this journal on December 13th, 2016 —
" There is a pleasantly discursive treatment
of Pontius Pilate’s unanswered question
‘What is truth?’ "
— Coxeter, 1987, introduction to Trudeau’s
The NonEuclidean Revolution
Also on December 13th, 2016 —
Earlier posts have discussed the "story theory of truth"
versus the "diamond theory of truth," as defined by
Richard Trudeau in his 1987 book The NonEuclidean Revolution.
In a New York Times opinion piece for tomorrow's print edition,*
novelist Dara Horn touched on what might be called
"the space theory of truth."
When they return to synagogue, mourners will be greeted
with more ancient words: “May God comfort you
among the mourners of Zion and Jerusalem.”
In that verse, the word used for God is hamakom —
literally, “the place.” May the place comfort you.
[Link added.]
The Source —
See Dara Horn in this journal, as well as Makom.
* "A version of this article appears in print on ,
on Page A23 of the New York edition with the headline:
American Jews Know This Story."
"When times are mysterious
Serious numbers
Will always be heard."
— Paul Simon,
"When Numbers Get Serious"
"There is a pleasantly discursive treatment of
Pontius Pilate's unanswered question 'What is truth?'"
— H. S. M. Coxeter, introduction to Richard J. Trudeau's remarks
on the "story theory" of truth as opposed to the "diamond theory"
of truth in The NonEuclidean Revolution (1987)
The deaths of Roth and Grünbaum on September 14th,
The Feast of the Holy Cross, along with Douthat's column
today titled "Only the Truth Can Save Us Now," suggest a
review of …
Background for the remarks of Koen Thas in the previous post —
Schumacher and Westmoreland, "Modal Quantum Theory" (2010).
Related material —
" There is a pleasantly discursive treatment
of Pontius Pilate’s unanswered question
‘What is truth?’ "
— Coxeter, 1987, introduction to Trudeau’s
The NonEuclidean Revolution
The whole truth may require an unpleasantly discursive treatment.
Example —
1. The reported death on Friday, Jan. 5, 2018, of a dancer
closely associated with George Balanchine
2. This journal on Friday, Jan. 5, 2018:
3. Illustration from a search related to the above dancer:
4. "Per Mare Per Terras" — Clan slogan above, illustrated with
what looks like a crossdagger.
"Unsheathe your dagger definitions." — James Joyce.
5. Discursive remarks on quantum theory by the above
Schumacher and Westmoreland:
6. "How much story do you want?" — George Balanchine
The passage from Lewis Carroll's Euclid and His Modern Rivals
in the previous post suggests two illustrations —
Click the Trudeau book for related Log24 posts.
These are Rothko's Swamps .
See a Log24 search for related meditations.
For all three topics combined, see Coxeter —
" There is a pleasantly discursive treatment
of Pontius Pilate’s unanswered question
‘What is truth?’ "
— Coxeter, 1987, introduction to Trudeau’s
The NonEuclidean Revolution
Update of 10 AM ET — Related material, with an elementary example:
Posts tagged "Defining Form." The example —
John Updike on Don DeLillo's thirteenth novel, Cosmopolis —
" DeLillo’s postChristian search for 'an order at some deep level'
has brought him to global computerization:
'the zerooneness of the world, the digital imperative . . . . ' "
— The New Yorker , issue dated March 31, 2003
On that date ….
Related remark —
" There is a pleasantly discursive treatment
of Pontius Pilate’s unanswered question
‘What is truth?’ "
— Coxeter, 1987, introduction to Trudeau’s
The NonEuclidean Revolution
The "points" and "lines" of finite geometry are abstract
entities satisfying only whatever incidence requirements
yield noncontradictory and interesting results. In finite
geometry, neither the points nor the lines are required to
lie within any Euclidean (or, for that matter, nonEuclidean)
space.
Models of finite geometries may, however, embed the
points and lines within non finite geometries in order
to aid visualization.
For instance, the 15 points and 35 lines of PG(3,2) may
be represented by subsets of a 4×4 array of dots, or squares,
located in the Euclidean plane. These "lines" are usually finite
subsets of dots or squares and not* lines of the Euclidean plane.
Example — See "4×4" in this journal.
Some impose on configurations from finite geometry
the rather artificial requirement that both points and lines
must be representable as those of a Euclidean plane.
Example: A CremonaRichmond pentagon —
A square version of these 15 "points" —
A 1905 square version of these 15 "points"
with digits instead of letters —
See Parametrizing the 4×4 Array
(Log24 post of Sept. 13, 2016).
Update of 8 AM ET Sunday, Sept. 25, 2016 —
For more illustrations, do a Google image search
on "the 2subsets of a 6set." (See one such search.)
* But in some models are subsets of the grid lines
that separate squares within an array.
"Again, in spite of that, we call this Friday good."
— T. S. Eliot, Four Quartets
From this journal on Orthodox Good Friday, 2016,
an image from New Scientist on St. Andrew's Day, 2015 —
From an old Dick Tracy strip —
See also meditations from this year's un Orthodox Good Friday
in a Tennessee weblog and in this journal —
" There is a pleasantly discursive treatment
of Pontius Pilate’s unanswered question
‘What is truth?’ ”
— Coxeter, 1987, introduction to Trudeau’s
The NonEuclidean Revolution
Toronto geometer H.S.M. Coxeter, introducing a book by Unitarian minister
Richard J. Trudeau —
"There is a pleasantly discursive treatment of Pontius Pilate’s
unanswered question ‘What is truth?’”
— Coxeter, 1987, introduction to Trudeau’s
The NonEuclidean Revolution
Another such treatment …
"Of course, it will surprise no one to find low standards
of intellectual honesty on the Tonight Show.
But we find a less trivial example if we enter the
hallowed halls of Harvard University. . . ."
— Neal Koblitz, "Mathematics as Propaganda"
Less pleasantly and less discursively —
"Funny how annoying a little prick can be."
— The late Garry Shandling
Material related to the title:
"Euclid's edifice loomed in my consciousness
as a marvel among sciences, unique in its clarity
and unquestionable validity."
—Richard J. Trudeau in
The NonEuclidean Revolution (1986)
"Euclid's edifice loomed in my consciousness as a marvel among
sciences, unique in its clarity and unquestionable validity."
—Richard J. Trudeau in The NonEuclidean Revolution (1986)
See also Edifice in this journal and last night's architectural post.
Update of Nov. 30, 2014 —
For further information on the geometry in
the remarks by Eberhart below, see
pp. 1617 of A Geometrical Picture Book ,
by Burkard Polster (Springer, 1998). Polster
cites a different article by Lemay.
A search for background to the exercise in the previous post
yields a passage from the late Stephen Eberhart:
The first three primes p = 2, 3, and 5 therefore yield finite projective planes with 7, 13, and 31 points and lines, respectively. But these are just the numbers of symmetry axes of the five regular solids, as described in Plato's Timaeus : The tetrahedron has 4 pairs of face planes and comer points + 3 pairs of opposite edges, totalling 7 axes; the cube has 3 pairs of faces + 6 pairs of edges + 4 pairs of comers, totalling 13 axes (the octahedron simply interchanges the roles of faces and comers); and the pentagon dodecahedron has 6 pairs of faces + 15 pairs of edges + 10 pairs of comers, totalling 31 axes (the icosahedron again interchanging roles of faces and comers). This is such a suggestive result, one would expect to find it dealt with in most texts on related subjects; instead, while "well known to those who well know such things" (as Richard Guy likes to quip), it is scarcely to be found in the formal literature [9]. The reason for the common numbers, it turns out, is that the groups of symmetry motions of the regular solids are subgroups of the groups of collineations of the respective finite planes, a face axis being different from an edge axis of a regular solid but all points of a projective plane being alike, so the latter has more symmetries than the former. [9] I am aware only of a series of inhouse publications by Fernand Lemay of the Laboratoire de Didactique, Faculté des Sciences de I 'Éducation, Univ. Laval, Québec, in particular those collectively titled Genèse de la géométrie IX.
— Stephen Eberhart, Dept. of Mathematics, 
Eberhart died of bone cancer in 2003. A memorial by his
high school class includes an Aug. 7, 2003, transcribed
letter from Eberhart to a classmate that ends…
… I earned MA’s in math (UW, Seattle) and history (UM, Missoula) where a math/history PhD program had been announced but canceled. So 1984 to 2002 I taught math (esp. nonEuclidean geometry) at C.S.U. Northridge. It’s been a rich life. I’m grateful. Steve 
See also another informative BRIDGES paper by Eberhart
on mathematics and the seven traditional liberal arts.
A post in honor of Évariste Galois (25 October 1811 – 31 May 1832)
From a book by Richard J. Trudeau titled The NonEuclidean Revolution —
See also “nonEuclidean” in this journal.
One might argue that Galois geometry, a field ignored by Trudeau,
is also “nonEuclidean,” and (for those who like rhetoric) revolutionary.
Continued from previous post and from Sept. 8, 2009.
Examination of the box's contents does not solve
the contents' real mystery. That requires knowledge
of the nonEuclidean geometry of Galois space.
In this case, without that knowledge, prattle (as in
today's online New York Times ) about creativity and
"thinking outside the box" is pointless.
New! Improved!
"Euclid's edifice loomed in my consciousness
as a marvel among sciences, unique in its
clarity and unquestionable validity."
—Richard J. Trudeau in
The NonEuclidean Revolution (First published in 1986)
Readers of this journal will be aware that Springer's new page
advertising Trudeau's book, pictured above, is a baitandswitch
operation. In the chapter advertised, Trudeau promotes what he
calls "the Diamond Theory of Truth" as a setup for his real goal,
which he calls "the Story Theory of Truth."
For an earlier use of the phrase "Diamond Theory" in
connection with geometry, see a publication from 1977.
A review of two theories of truth described
by a clergyman, Richard J. Trudeau, in
The NonEuclidean Revolution—
"But, I asked, is there a difference
between fiction and nonfiction?
'Not much,' she said, shrugging."
— New Yorker profile of tesseract
author Madeleine L'Engle
(Click image for some background.)
See also the links on a webpage at finitegeometry.org.
"In logic, the law of excluded middle (or the principle of excluded middle) is the third of the socalled three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is. The law is also known as the law (or principle) of the excluded third (or of the excluded middle), or, in Latin, principium tertii exclusi. Yet another Latin designation for this law is tertium non datur: 'no third (possibility) is given.'" 
"Clowns to the left of me, jokers to the right"
— Songwriter who died on January 4, 2011.
Online NY Times on the date of the songwriter's death—
"A version of this review appeared in print
on January 4, 2011, on page C6 of the New York edition."
"The philosopher Hubert Dreyfus and his former student
Sean Dorrance Kelly have a story to tell, and it is not
a pretty tale for us moderns. Ours is an age of nihilism,
they say, meaning not so much that we have nothing
in which to believe, but that we don’t know how to choose
among the various things to which we might commit
ourselves. Looking down from their perches at Berkeley
and Harvard, they see the 'human indecision that
plagues us all.'"
For an application of the excludedmiddle law, see
NonEuclidean Blocks and Deep Play.
Violators of the law may have trouble* distinguishing
between "Euclidean" and "nonEuclidean" phenomena
because their definition of the latter is too narrow,
based only on examples that are historically well known.
See the NonEuclidean Blocks footnote.
* Followers of the excludedmiddle law will avoid such
trouble by noting that "nonEuclidean" should mean
simply "not Euclidean in some way "— not necessarily
in a way contradicting Euclid's parallel postulate.
But see Wikipedia's defense of the standard, illogical,
usage of the phrase "nonEuclidean."
Postscript—
Tertium Datur
"Here I am, stuck in the middle with you."
Weblog posts of two prominent mathematicians today discussed
what appears to be a revolution inspired by the business practices
of some commercial publishers of mathematics.
My own concern is more with the socalled "NonEuclidean Revolution"
described by Richard Trudeau in a book of that title (Birkhäuser, 1987).
A 1976 document relevant to the concerns in the Trudeau book—
Though not as well known as another document discussing
"selfevident" truths, Cameron's remarks are also of some
philosophical interest.
They apply to finite geometry, a topic unknown to Euclid,
but nevertheless of considerable significance for the foundations
of mathematics.
"The hand of the creative artist, laid upon the major premise,
rocks the foundations of the world." — Dorothy Sayers
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—
Richard J. Trudeau, a mathematics professor and Unitarian minister, published in 1987 a book, The NonEuclidean Revolution , that opposes what he calls the Story Theory of truth [i.e., Quine, nominalism, postmodernism] to what he calls the traditional Diamond Theory of truth [i.e., Plato, realism, the Roman Catholic Church]. This opposition goes back to the medieval "problem of universals" debated by scholastic philosophers.
(Trudeau may never have heard of, and at any rate did not mention, an earlier 1976 monograph on geometry, "Diamond Theory," whose subject and title are relevant.)
From yesterday's Sunday morning New York Times—
"Stories were the primary way our ancestors transmitted knowledge and values. Today we seek movies, novels and 'news stories' that put the events of the day in a form that our brains evolved to find compelling and memorable. Children crave bedtime stories…."
— Drew Westen, professor at Emory University
From May 22, 2009—
The above ad is by Diamond from last night’s

For further details, see Saturday's correspondences 
Good question. See also
Chern died on the evening of Friday, Dec. 3, 2004 (Chinese time).
From the morning of that day (also Chinese time)—
i.e. , the evening of the preceding day here— some poetry.
For the title, see Palm Sunday.
"There is a pleasantly discursive treatment of
Pontius Pilate's unanswered question 'What is truth?'" — H. S. M. Coxeter, 1987
From this date (April 22) last year—
Richard J. Trudeau in The NonEuclidean Revolution , chapter on "Geometry and the Diamond Theory of Truth"– "… Plato and Kant, and most of the philosophers and scientists in the 2200year interval between them, did share the following general presumptions: (1) Diamonds– informative, certain truths about the world– exist. Presumption (1) is what I referred to earlier as the 'Diamond Theory' of truth. It is far, far older than deductive geometry." Trudeau's book was published in 1987. The nonEuclidean* figures above illustrate concepts from a 1976 monograph, also called "Diamond Theory." Although nonEuclidean,* the theorems of the 1976 "Diamond Theory" are also, in Trudeau's terminology, diamonds. * "NonEuclidean" here means merely "other than Euclidean." No violation of Euclid's parallel postulate is implied. 
Trudeau comes to reject what he calls the "Diamond Theory" of truth. The trouble with his argument is the phrase "about the world."
Geometry, a part of pure mathematics, is not about the world. See G. H. Hardy, A Mathematician's Apology .
The following has rather mysteriously appeared in a search at Google Scholar for "Steven H. Cullinane."
[HTML] Romancing the NonEuclidean Hyperspace 
This turns out to be a link to a search within this weblog. I do not know why Google Scholar attributes the resulting web page to a journal article by "AB Story" or why it drew the title from a post within the search and applied it to the entire list of posts found. I am, however, happy with the result— a Palm Sunday surprise with an eclectic mixture of styles that might please the late Robert de Marrais.
I hope the late George Temple would also be pleased. He appears in "Romancing" as a resident of Quarr Abbey, a Benedictine monastery.
The remarks by Martin Hyland quoted in connection with Temple's work are of particular interest in light of the Pope's Christmas remark on mathematics quoted here yesterday.
Your mission, should you choose to accept it…
See also "Mapping Music" from Harvard Magazine , Jan.Feb. 2007—
"Life inside an orbifold is a nonEuclidean world"
— as well as the cover story "The Shape of Music" from Princeton Alumni Weekly ,
Feb. 9, 2011, and "Bead Game" + music in this journal (click, then scroll down).
Those impressed by the phrase "nonEuclidean" may also enjoy
NonEuclidean Blocks and Pilate Goes to Kindergarten.
The "Bead Game" + music search above includes, notably, a passage describing a
sort of nonEuclidean abacus in the classic 1943 story "Mimsy Were the Borogoves."
For a visually related experience, see the video "Chord Geometries Demo: Chopin
on a Mobius Strip" at a music.princeton.edu web page.
* Motto of the American Mathematical Society, said to be also the motto of Plato's Academy.
Recent posts (Church Logic and Church Narrative) have discussed finite geometry as a type of nonEuclidean geometry.
For those who prefer nonfinite geometry, here are some observations.
"A characteristic property of hyperbolic geometry
is that the angles of a triangle add to less
than a straight angle (half circle)." — Wikipedia
From To Ride Pegasus, by Anne McCaffrey, 1973:
“MaryMolly luv, it’s going to be accomplished in steps, this establishment of the Talented in the scheme of things. Not society, mind you, for we’re the original nonconformists…. and Society will never permit us to integrate. That’s okay!” He consigned Society to insignificance with a flick of his fingers. “The Talented form their own society and that’s as it should be: birds of a feather. No, not birds. Winged horses! Ha! Yes, indeed. Pegasus… the poetic winged horse of flights of fancy. A bloody good symbol for us. You’d see a lot from the back of a winged horse…”
“Yes, an airplane has blind spots. Where would you put a saddle?” Molly had her practical side.
On the practical side:
The above chapel is from a Princeton Weekly Bulletin story of October 6th, 2008.
Related material: This journal on that date.
Thanks to David Lavery for the following dialogue on the word "narrative" in politics—
"It's like – does this fit into narrative?
It's like, wait, wait, what about a platform? What about, like, ideas?
What about, you know, these truths we hold to be selfevident?
No, it's the narrative."
"Is narrative a fancy word for spin?"
Related material —
Church Logic (Log24, October 29) —
What sort of geometry
is the following?
"What about, you know, these truths we hold to be selfevident?"
Some background from Cambridge University Press in 1976 —
Commentary —
The Church Logic post argues that Cameron's implicit definition of "nonEuclidean" is incorrect.
The fourpoint, sixline geometry has as lines "all subsets of the point set" which have cardinality 2.
It clearly satisfies Euclid's parallel postulate. Is it, then, not nonEuclidean?
That would, according to the principle of the excluded middle (cf. Church), make it Euclidean.
A definition from Wikipedia that is still essentially the same as it was when written on July 14, 2003—
"Finite geometry describes any geometric system that has only a finite number of points. Euclidean geometry, for example, is not finite, because a Euclidean line contains infinitely many points…."
This definition would seem to imply that a finite geometry (such as the fourpoint geometry above) should be called non Euclidean whether or not it violates Euclid's parallel postulate. (The definition's author, unlike many at Wikipedia, is not anonymous.)
See also the rest of Little Gidding.
"The law of excluded middle is the logical principle in
accordance with which every proposition is either true or
false. This principle is used, in particular, whenever a proof
is made by the method of reductio ad absurdum . And it is
this principle, also, which enables us to say that the denial of
the denial of a proposition is equivalent to the assertion of
the proposition."
— Alonzo Church, "On the Law of Excluded Middle,"
Bulletin of the American Mathematical Society ,
Vol. 34, No. 1 (Jan.–Feb. 1928), pp. 75–78
It seems reasonable to define a Euclidean geometry as one describing what mathematicians now call a Euclidean space.
What sort of geometry
is the following?
Four points and six lines,
with parallel lines indicated
by being colored alike.
Consider the proposition "The finite geometry with four points and six lines is nonEuclidean."
Consider its negation. Absurd? Of course.
"NonEuclidean," therefore, does not apply only to geometries that violate Euclid's parallel postulate.
The problem here is not with geometry, but with writings about geometry.
"In the plainest terms, nonEuclidean geometry
took something that was rather simple and straightforward
(Euclidean geometry) and made it endlessly more difficult."
Had the Greeks investigated finite geometry before Euclid came along, the reverse would be true.
"Many of the finite simple groups can be described as symmetries of finite geometries, and it remains a hot topic in group theory to expand our knowledge of the Classification of Finite Simple Groups using finite geometry."
— Finite geometry page at the Centre for the Mathematics of
Symmetry and Computation at the University of Western Australia
(Alice Devillers, John Bamberg, Gordon Royle)
For such symmetries, see Robert A. WIlson's recent book The Finite Simple Groups.
The finite simple groups are often described as the "building blocks" of finite group theory.
At least some of these building blocks have their own building blocks. See NonEuclidean Blocks.
For instance, a set of 24 such blocks (or, more simply, 24 unit squares) appears in the Miracle Octad Generator (MOG) of R.T. Curtis, used in the study of the finite simple group M_{24}.
(The octads of the MOG illustrate yet another sort of mathematical blocks— those of a block design.)
Barnes & Noble has an informative new review today of the recent Galois book Duel at Dawn.
It begins…
"In 1820, the Hungarian noble Farkas Bolyai wrote an impassioned cautionary letter to his son Janos:
'I know this way to the very end. I have traversed this bottomless night, which extinguished all light and joy in my life… It can deprive you of your leisure, your health, your peace of mind, and your entire happiness… I turned back when I saw that no man can reach the bottom of this night. I turned back unconsoled, pitying myself and all mankind. Learn from my example…'
Bolyai wasn't warning his son off gambling, or poetry, or a poorly chosen love affair. He was trying to keep him away from nonEuclidean geometry."
For a less dark view (obtained by simply redefining "nonEuclidean" in a more logical way*) see NonEuclidean Blocks and Finite Geometry and Physical Space.
* Finite geometry is not Euclidean geometry— and is, therefore, nonEuclidean
in the strictest sense (though not according to popular usage), simply because
Euclidean geometry has infinitely many points, and a finite geometry does not.
(This more logical definition of "nonEuclidean" seems to be shared by
at least one other person.)
And some finite geometries are nonEuclidean in the popularusage sense,
related to Euclid's parallel postulate.
The sevenpoint Fano plane has, for instance, been called
"a nonEuclidean geometry" not because it is finite
(though that reason would suffice), but because it has no parallel lines.
(See the finite geometry page at the Centre for the Mathematics
of Symmetry and Computation at the University of Western Australia.)
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….
From a post by Ivars Peterson, Director
of Publications and Communications at
the Mathematical Association of America,
at 19:19 UTC on June 19, 2010—
Exterior panels and detail of panel,
Michener Gallery at Blanton Museum
in Austin, Texas—
Peterson associates the fourdiamond figure
with the Pythagorean theorem.
A more relevant association is the
fourdiamond view of a tesseract shown here
on June 19 (the same date as Peterson's post)
in the "Imago Creationis" post—
This figure is relevant because of a
tesseract sculpture by Peter Forakis—
This sculpture was apparently shown in the above
building— the Blanton Museum's Michener gallery—
as part of the "Reimagining Space" exhibition,
September 28, 2008January 18, 2009.
The exhibition was organized by
Linda Dalrymple Henderson, Centennial Professor
in Art History at the University of Texas at Austin
and author of The Fourth Dimension and
NonEuclidean Geometry in Modern Art
(Princeton University Press, 1983;
new ed., MIT Press, 2009).
For the sculptor Forakis in this journal,
see "The Test" (December 20, 2009).
"There is such a thing
as a tesseract."
— A Wrinkle in TIme
“Nuvoletta in her lightdress, spunn of sisteen shimmers,
was looking down on them, leaning over the bannistars….
Fuvver, that Skand, he was up in Norwood’s sokaparlour….”
— Finnegans Wake
To counteract the darkness of today’s 2:01 AM entry—
Part I— Artist Josefine Lyche describes her methods—
A— “Internet and hard work”
B— “Books, both fiction and theory”
Part II— I, too, now rely mostly on the Internet for material. However, like Lyche, I have Plan B— books.
Where I happen to be now, there are piles of them. Here is the pile nearest to hand, from top to bottom.
(The books are in no particular order, and put in the same pile for no particular reason.)
Lyche probably could easily make her own list of what Joyce might call “sisteen shimmers.”
Romancing the
NonEuclidean Hyperspace
Backstory —
Mere Geometry, Types of Ambiguity,
Dream Time, and Diamond Theory, 1937
For the 1937 grid, see Diamond Theory, 1937.
The grid is, as Mere Geometry points out, a nonEuclidean hyperspace.
For the diamonds of 2010, see Galois Geometry and Solomon’s Cube.
“Mere anarchy is loosed upon the world”
From a document linked to here on April 30, Walpurgisnacht–
“…the Golden Age, or Dream Time, is remote only from the rational mind. It is not accessible to euclidean reason….”
“The utopia of the Grand Inquisitor ‘is the product of “the euclidean mind” (a phrase Dostoyevsky often used)….'”
“The purer, the more euclidean the reason that builds a utopia, the greater is its selfdestructive capacity. I submit that our lack of faith in the benevolence of reason as the controlling power is well founded. We must test and trust our reason, but to have faith in it is to elevate it to godhead.”
“Utopia has been euclidean, it has been European, and it has been masculine. I am trying to suggest, in an evasive, distrustful, untrustworthy fashion, and as obscurely as I can, that our final loss of faith in that radiant sandcastle may enable our eyes to adjust to a dimmer light and in it perceive another kind of utopia.”
“You will recall that the quality of static perfection is an essential element of the noninhabitability of the euclidean utopia….”
“The euclidean utopia is mapped; it is geometrically organized, with the parts labeled….”
— Ursula K. Le Guin, “A NonEuclidean View of California as a Cold Place to Be”
San Francisco Chronicle today—
“A May Day rally in Santa Cruz erupted into chaos Saturday night….”
“Had Goodman Brown fallen asleep in the forest,
and only dreamed a wild dream of a witchmeeting?”
McLuhan in Space by Richard Cavell—
As the word "through" in the title of Through the Vanishing Point^{ □} hints… key reference points for McLuhan and Parker in writing Through the Vanishing Point were the "Alice" books.
[The footnote symbol ^{ □} here is mine.]
Alice Rae, McLuhan's Unconscious, doctoral dissertation, School of History and Politics, University of Adelaide, May 2008—
What McLuhan calls the "unconscious"' is more often named by him as Logos, "acoustic space" or the "media environment," and I trace the debts that these concepts owe not only to Freud and Jung, but to Aristotle, St. Thomas Aquinas, gestalt theory, art theory, Henri Bergson, Pierre Teilhard de Chardin, Wyndham Lewis, Siegfried Giedion, Harold Innis, the French symbolist poets of the late nineteenth century and the British modernists of the early twentieth.
The declaration section of the thesis is dated November 19, 2008.
Related material— Halloween 2005 and The Gospel According to Father Hardon.
^{□} A work suggested by Ander Monson's new Vanishing Point . (See April 17 and April 23, together with the April 22 picture of a nonEuclidean point □ in the context of "The Seventh Symbol.")
Stanford Encyclopedia of Philosophy —
“Mereology (from the Greek μερος, ‘part’) is the theory of parthood relations: of the relations of part to whole and the relations of part to part within a whole. Its roots can be traced back to the early days of philosophy, beginning with the Presocratics….”
A nonEuclidean* approach to parts–
Corresponding nonEuclidean*
projective points —
Richard J. Trudeau in The NonEuclidean Revolution, chapter on “Geometry and the Diamond Theory of Truth”–
“… Plato and Kant, and most of the philosophers and scientists in the 2200year interval between them, did share the following general presumptions:
(1) Diamonds– informative, certain truths about the world– exist.
(2) The theorems of Euclidean geometry are diamonds.
Presumption (1) is what I referred to earlier as the ‘Diamond Theory’ of truth. It is far, far older than deductive geometry.”
Trudeau’s book was published in 1987. The nonEuclidean* figures above illustrate concepts from a 1976 monograph, also called “Diamond Theory.”
Although nonEuclidean,* the theorems of the 1976 “Diamond Theory” are also, in Trudeau’s terminology, diamonds.
* “NonEuclidean” here means merely “other than Euclidean.” No violation of Euclid’s parallel postulate is implied.
Deep Play:
Mimzy vs. Mimsy
From a 2007 film, "The Last Mimzy," based on
the classic 1943 story by Lewis Padgett
"Mimsy Were the Borogoves"–
As the above mandala pictures show,
the film incorporates many New Age fashions.
The original story does not.
A more realistic version of the story
might replace the mandalas with
the following illustrations–
For a commentary, see "NonEuclidean Blocks."
(Here "nonEuclidean" means simply
other than Euclidean. It does not imply any
violation of Euclid's parallel postulate.)
Truth, Geometry, Algebra
The following notes are related to A Simple Reflection Group of Order 168.
1. According to H.S.M. Coxeter and Richard J. Trudeau
“There is a pleasantly discursive treatment of Pontius Pilate’s unanswered question ‘What is truth?’.”
— Coxeter, 1987, introduction to Trudeau’s The NonEuclidean Revolution
1.1 Trudeau’s Diamond Theory of Truth
1.2 Trudeau’s Story Theory of Truth
2. According to Alexandre Borovik and Steven H. Cullinane
2.1 Coxeter Theory according to Borovik
2.1.1 The Geometry–
Mirror Systems in Coxeter Theory
2.1.2 The Algebra–
Coxeter Languages in Coxeter Theory
2.2 Diamond Theory according to Cullinane
2.2.1 The Geometry–
Examples: Eightfold Cube and Solomon’s Cube
2.2.2 The Algebra–
Examples: Cullinane and (rather indirectly related) Gerhard Grams
Summary of the story thus far:
Diamond theory and Coxeter theory are to some extent analogous– both deal with reflection groups and both have a visual (i.e., geometric) side and a verbal (i.e., algebraic) side. Coxeter theory is of course highly developed on both sides. Diamond theory is, on the geometric side, currently restricted to examples in at most three Euclidean (and six binary) dimensions. On the algebraic side, it is woefully underdeveloped. For material related to the algebraic side, search the Web for generators+relations+”characteristic two” (or “2“) and for generators+relations+”GF(2)”. (This last search is the source of the Grams reference in 2.2.2 above.)
Today's previous entry quoted a review by Edward Rothstein of Jung's The Red Book. The entry you are now reading quotes a review by Jim Holt of a notable book by Rothstein:
The Golden Book—
Cover illustration— Arithmetic and Music,
Borgia Apartments, The Vatican
Jim Holt reviewing Edward Rothstein's Emblems of Mind: The Inner Life of Music and Mathematics in The New Yorker of June 5, 1995:
Advent
"The fugues of Bach, the symphonies of Haydn, the sonatas of Mozart: these were explorations of ideal form, unprofaned by extramusical associations. Such 'absolute music,' as it came to be called, had sloughed off its motley cultural trappings. It had got in touch with its essence. Which is why, as Walter Pater famously put it, 'all art constantly aspires towards the condition of music.'
The only art that can rival music for sheer etheriality is mathematics. A century or so after the advent of absolute music, mathematics also succeeded in detaching itself from the world. The decisive event was the invention of strange, nonEuclidean geometries, which put paid to the notion that the mathematician was exclusively, or even primarily, concerned with the scientific universe. 'Pure' mathematics came to be seen by those who practiced it as a free invention of the imagination, gloriously indifferent to practical affairs– a quest for beauty as well as truth."
Related material: Hardy's Apology, NonEuclidean Blocks, and The Story Theory of Truth.
See also Holt on music and emotion:
"Music does model… our emotional life… although
the methods by which it does so are 'puzzling.'"
Also puzzling: 2010 AMS Notices.
From Peter J. Cameron's
Parallelisms of Complete Designs (pdf)–
"…the Feast of Nicholas Ferrar
is kept on the 4th December."
Cameron's is the usual definition
of the term "nonEuclidean."
I prefer a more logical definition.
From the Bulletin of the American Mathematical Society, Jan. 26, 2005:
What is known about unit cubes
by Chuanming Zong, Peking University
Abstract: Unit cubes, from any point of view, are among the simplest and the most important objects in ndimensional Euclidean space. In fact, as one will see from this survey, they are not simple at all….
From Log24, now:
What is known about the 4×4×4 cube
by Steven H. Cullinane, unaffiliated
Abstract: The 4×4×4 cube, from one point of view, is among the simplest and the most important objects in ndimensional binary space. In fact, as one will see from the links below, it is not simple at all.
The Klein Correspondence, Penrose SpaceTime, and a Finite Model
Related material:
Monday’s entry Just Say NO and a poem by Stevens,
Unitarian Universalist Origins: Our Historic Faith—
“In sixteenthcentury Transylvania, Unitarian congregations were established for the first time in history.”
Gravity’s Rainbow–
“For every kind of vampire, there is a kind of cross.”
Unitarian minister Richard Trudeau—
“… I called the belief that
(1) Diamonds– informative, certain truths about the world– exist
the ‘Diamond Theory’ of truth. I said that for 2200 years the strongest evidence for the Diamond Theory was the widespread perception that
(2) The theorems of Euclidean geometry are diamonds….
As the news about nonEuclidean geometry spread– first among mathematicians, then among scientists and philosophers– the Diamond Theory began a long decline that continues today.
Factors outside mathematics have contributed to this decline. Euclidean geometry had never been the Diamond Theory’s only ally. In the eighteenth century other fields had seemed to possess diamonds, too; when many of these turned out to be manmade, the Diamond Theory was undercut. And unlike earlier periods in history, when intellectual shocks came only occasionally, received truths have, since the eighteenth century, been found wanting at a dizzying rate, creating an impression that perhaps no knowledge is stable.
Other factors notwithstanding, nonEuclidean geometry remains, I think, for those who have heard of it, the single most powerful argument against the Diamond Theory*– first, because it overthrows what had always been the strongest argument in favor of the Diamond Theory, the objective truth of Euclidean geometry; and second, because it does so not by showing Euclidean geometry to be false, but by showing it to be merely uncertain.” —The NonEuclidean Revolution, p. 255
H. S. M. Coxeter, 1987, introduction to Trudeau’s book—
“There is a pleasantly discursive treatment of Pontius Pilate’s unanswered question ‘What is truth?’.”
As noted here on Oct. 8, 2008 (A Yom Kippur Meditation), Coxeter was aware in 1987 of a more technical use of the phrase “diamond theory” that is closely related to…
The Sept. 8 entry on nonEuclidean* blocks ended with the phrase “Go figure.” This suggested a MAGMA calculation that demonstrates how Klein’s simple group of order 168 (cf. Jeremy Gray in The Eightfold Way) can be visualized as generated by reflections in a finite geometry.
* i.e., other than Euclidean. The phrase “nonEuclidean” is usually applied to only some of the geometries that are not Euclidean. The geometry illustrated by the blocks in question is not Euclidean, but is also, in the jargon used by most mathematicians, not “nonEuclidean.”
NonEuclidean
Blocks
Passages from a classic story:
… he took from his pocket a gadget he had found in the box, and began to unfold it. The result resembled a tesseract, strung with beads…. Tesseract "Your mind has been conditioned to Euclid," Holloway said. "So this– thing– bores us, and seems pointless. But a child knows nothing of Euclid. A different sort of geometry from ours wouldn't impress him as being illogical. He believes what he sees."
"Are you trying to tell me that this gadget's got a fourth dimensional extension?" Paradine demanded. "Hardening of the thoughtarteries," Jane interjected. Paradine was not convinced. "Then a baby could work calculus better than Einstein? No, I don't mean that. I can see your point, more or less clearly. Only–" "Well, look. Let's suppose there are two kinds of geometry– we'll limit it, for the sake of the example. Our kind, Euclidean, and another, which we'll call x. X hasn't much relationship to Euclid. It's based on different theorems. Two and two needn't equal four in it; they could equal y, or they might not even equal. A baby's mind is not yet conditioned, except by certain questionable factors of heredity and environment. Start the infant on Euclid–" "Poor kid," Jane said. Holloway shot her a quick glance. "The basis of Euclid. Alphabet blocks. Math, geometry, algebra– they come much later. We're familiar with that development. On the other hand, start the baby with the basic principles of our x logic–" "Blocks? What kind?" Holloway looked at the abacus. "It wouldn't make much sense to us. But we've been conditioned to Euclid." — "Mimsy Were the Borogoves," Lewis Padgett, 1943 
For the intuitive basis of one type of nonEuclidean* geometry– finite geometry over the twoelement Galois field– see the work of…
Friedrich Froebel
(17821852), who
invented kindergarten.
His "third gift" —
Begettings of
the Broken Bold
Thanks for the following
quotation (“Non deve…
nella testa“) go to the
weblog writer who signs
himself “Conrad H. Roth.”
… Yesterday I took leave of my Captain, with a promise of visiting him at Bologna on my return. He is a true A PAPAL SOLDIER’S IDEAS OF PROTESTANTS 339 representative of the majority of his countrymen. Here, however, I would record a peculiarity which personally distinguished him. As I often sat quiet and lost in thought he once exclaimed “Che pensa? non deve mai pensar l’uomo, pensando s’invecchia;” which being interpreted is as much as to say, “What are you thinking about: a man ought never to think; thinking makes one old.” And now for another apophthegm of his; “Non deve fermarsi l’uomo in una sola cosa, perche allora divien matto; bisogna aver mille cose, una confusione nella testa;” in plain English, “A man ought not to rivet his thoughts exclusively on any one thing, otherwise he is sure to go mad; he ought to have in his head a thousand things, a regular medley.” Certainly the good man could not know that the very thing that made me so thoughtful was my having my head mazed by a regular confusion of things, old and new. The following anecdote will serve to elucidate still more clearly the mental character of an Italian of this class. Having soon discovered that I was a Protestant, he observed after some circumlocution, that he hoped I would allow him to ask me a few questions, for he had heard such strange things about us Protestants that he wished to know for a certainty what to think of us. 
Notes for Roth:
The title of this entry,
“Begettings of the Broken Bold,”
is from Wallace Stevens’s
“The Owl in the Sarcophagus”–
This was peace after death, the brother of sleep, The inhuman brother so much like, so near, Yet vested in a foreign absolute, Adorned with cryptic stones and sliding shines, An immaculate personage in nothingness, With the whole spirit sparkling in its cloth, Generations of the imagination piled In the manner of its stitchings, of its thread, In the weaving round the wonder of its need, And the first flowers upon it, an alphabet By which to spell out holy doom and end, A bee for the remembering of happiness. Peace stood with our last blood adorned, last mind, Damasked in the originals of green, A thousand begettings of the broken bold. This is that figure stationed at our end, Always, in brilliance, fatal, final, formed Out of our lives to keep us in our death.... 
Related material:
Some further context:
Roth’s entry of Nov. 3, 2006–
“Why blog, sinners?“–
and Log24 on that date:
“First to Illuminate.”
Pilate Goes
to Kindergarten
“There is a pleasantly discursive
treatment of Pontius Pilate’s
unanswered question
‘What is truth?’.”
— H. S. M. Coxeter, 1987,
introduction to Trudeau’s
remarks on the “Story Theory“
of truth as opposed to the
“Diamond Theory” of truth in
The NonEuclidean Revolution
Consider the following question in a paper cited by V. S. Varadarajan:
E. G. Beltrametti, “Can a finite geometry describe physical spacetime?” Universita degli studi di Perugia, Atti del convegno di geometria combinatoria e sue applicazioni, Perugia 1971, 57–62.
Simplifying:
“Can a finite geometry describe physical space?”
Simplifying further:
“Yes. Vide ‘The Eightfold Cube.'”
This journal on October 8, 2008, at noon: “There is a pleasantly discursive treatment of Pontius Pilate’s unanswered question ‘What is truth?'” Trudeau’s 1987 book uses the phrase “diamond theory” to denote the philosophical theory, common since Plato and Euclid, that there exist truths (which Trudeau calls “diamonds”) that are certain and eternal– for instance, the truth in Euclidean geometry that the sum of a triangle’s angles is 180 degrees. Insidehighered.com on “Future readers may consider Updike our era’s Mozart; Mozart was once written off as a tooprolific composer of ‘charming nothings,’ and some speak of Updike that way.” — Comment by BPJ 
Updike died on January 27.
On the same date,
Mozart was born.
Requiem
Mr. Best entered, tall, young, mild, light. He bore in his hand with grace a notebook, new, large, clean, bright. — James Joyce, Ulysses, 
Serious Numbers
A Yom Kippur
Meditation
"When times are mysterious
Serious numbers
Will always be heard."
— Paul Simon,
"When Numbers Get Serious"
"There is a pleasantly discursive treatment of Pontius Pilate's unanswered question 'What is truth?'"
— H. S. M. Coxeter, introduction to Richard J. Trudeau's remarks on the "story theory" of truth as opposed to the "diamond theory" of truth in The NonEuclidean Revolution
Trudeau's 1987 book uses the phrase "diamond theory" to denote the philosophical theory, common since Plato and Euclid, that there exist truths (which Trudeau calls "diamonds") that are certain and eternal– for instance, the truth in Euclidean geometry that the sum of a triangle's angles is 180 degrees. As the excerpt below shows, Trudeau prefers what he calls the "story theory" of truth–
"There are no diamonds. People make up stories about what they experience. Stories that catch on are called 'true.'"
(By the way, the phrase "diamond theory" was used earlier, in 1976, as the title of a monograph on geometry of which Coxeter was aware.)
What does this have to do with numbers?
Pilate's skeptical tone suggests he may have shared a certain confusion about geometric truth with thinkers like Trudeau and the slave boy in Plato's Meno. Truth in a different part of mathematics– elementary arithmetic– is perhaps more easily understood, although even there, the existence of what might be called "nonEuclidean number theory"– i.e., arithmetic over finite fields, in which 1+1 can equal zero– might prove baffling to thinkers like Trudeau.
Trudeau's book exhibits, though it does not discuss, a less confusing use of numbers– to mark the location of pages. For some philosophical background on this version of numerical truth that may be of interest to devotees of the Semitic religions on this evening's High Holiday, see Zen and Language Games.
For uses of numbers that are more confusing, see– for instance– the new website The Daily Beast and the old website Story Theory and the Number of the Beast.
“There is a pleasantly discursive treatment
of Pontius Pilate’s unanswered question
‘What is truth?'”
— H. S. M. Coxeter, 1987, introduction to
Richard J. Trudeau’s remarks on
the “Story Theory” of truth
as opposed to
the “Diamond Theory” of truth
in The NonEuclidean Revolution
A Serious Position
“‘Teitelbaum,’ in German,
is ‘date palm.'”
— Generations, Jan. 2003
“In Hasidism, a mystical brand
of Orthodox Judaism, the grand rabbi
is revered as a kinglike link to God….”
— Today’s New York Times obituary
of Rabbi Moses Teitelbaum,
who died on April 24, 2006
(Easter Monday in the
Orthodox Church)
From Wikipedia, an unsigned story:
“In 1923 Alfred Teitelbaum and his brother Wacław changed their surnames to Tarski, a name they invented because it sounded very Polish, was simple to spell and pronounce, and was unused. (Years later, he met another Alfred Tarski in northern California.) The Tarski brothers also converted to Roman Catholicism, the national religion of the Poles. Alfred did so, even though he was an avowed atheist, because he was about to finish his Ph.D. and correctly anticipated that it would be difficult for a Jew to obtain a serious position in the new Polish university system.”
A Circle of Quiet
From the Harvard Math Table page:
“No Math table this week. We will reconvene next week on March 14 for a special Pi Day talk by Paul Bamberg.”
Transcript of the movie “Proof”–
Some friends of mine are in this band. They’re playing in a bar on Diversey, way down the bill, around… I said I’d be there. Great. Imaginary number? It’s a math joke. 
From the April 2006 Notices of the American Mathematical Society, a footnote in a review by Juliette Kennedy (pdf) of Rebecca Goldstein’s Incompleteness:
^{4} There is a growing literature in the area of postmodern commentaries of [sic] Gödel’s theorems. For example, Régis Debray has used Gödel’s theorems to demonstrate the logical inconsistency of selfgovernment. For a critical view of this and related developments, see Bricmont and Sokal’s Fashionable Nonsense [13]. For a more positive view see Michael Harris’s review of the latter, “I know what you mean!” [9]….
[9] MICHAEL HARRIS, “I know what you mean!,” http://www.math.jussieu.fr/~harris/Iknow.pdf.
[13] ALAN SOKAL and JEAN BRICMONT, Fashionable Nonsense, Picador, 1999.
Following the trail marked by Ms. Kennedy, we find the following in Harris’s paper:
“Their [Sokal’s and Bricmont’s] philosophy of mathematics, for instance, is summarized in the sentence ‘A mathematical constant like doesn’t change, even if the idea one has about it may change.’ ( p. 263). This claim, referring to a ‘crescendo of absurdity’ in Sokal’s original hoax in Social Text, is criticized by anthropologist Joan Fujimura, in an article translated for IS*. Most of Fujimura’s article consists of an astonishingly bland account of the history of noneuclidean geometry, in which she points out that the ratio of the circumference to the diameter depends on the metric. Sokal and Bricmont know this, and Fujimura’s remarks are about as helpful as FN’s** referral of Quine’s readers to Hume (p. 70). Anyway, Sokal explicitly referred to “Euclid’s pi”, presumably to avoid trivial objections like Fujimura’s — wasted effort on both sides.^{32} If one insists on making trivial objections, one might recall that the theorem
that p is transcendental can be stated as follows: the homomorphism Q[X] –> R taking X to is injective. In other words, can be identified algebraically with X, the variable par excellence.^{33}
More interestingly, one can ask what kind of object was before the formal definition of real numbers. To assume the real numbers were there all along, waiting to be defined, is to adhere to a form of Platonism.^{34} Dedekind wouldn’t have agreed.^{35} In a debate marked by the accusation that postmodern writers deny the reality of the external world, it is a peculiar move, to say the least, to make mathematical Platonism a litmus test for rationality.^{36} Not that it makes any more sense simply to declare Platonism out of bounds, like LévyLeblond, who calls Stephen Weinberg’s gloss on Sokal’s comment ‘une absurdité, tant il est clair que la signification d’un concept quelconque est évidemment affectée par sa mise en oeuvre dans un contexte nouveau!’^{37} Now I find it hard to defend Platonism with a straight face, and I prefer to regard the formula
as a creation rather than a discovery. But Platonism does correspond to the familiar experience that there is something about mathematics, and not just about other mathematicians, that precisely doesn’t let us get away with saying ‘évidemment’!^{38}
^{32} There are many circles in Euclid, but no pi, so I can’t think of any other reason for Sokal to have written ‘Euclid’s pi,’ unless this anachronism was an intentional part of the hoax. Sokal’s full quotation was ‘the of Euclid and the G of Newton, formerly thought to be constant and universal, are now perceived in their ineluctable historicity.’ But there is no need to invoke nonEuclidean geometry to perceive the historicity of the circle, or of pi: see Catherine Goldstein’s ‘L’un est l’autre: pour une histoire du cercle,’ in M. Serres, Elements d’histoire des sciences, Bordas, 1989, pp. 129149.
^{33} This is not mere sophistry: the construction of models over number fields actually uses arguments of this kind. A careless construction of the equations defining modular curves may make it appear that pi is included in their field of scalars.
^{34} Unless you claim, like the present French Minister of Education [at the time of writing, i.e. 1999], that real numbers exist in nature, while imaginary numbers were invented by mathematicians. Thus would be a physical constant, like the mass of the electron, that can be determined experimentally with increasing accuracy, say by measuring physical circles with ever more sensitive rulers. This sort of position has not been welcomed by most French mathematicians.
^{35 } Cf. M. Kline, Mathematics The Loss of Certainty, p. 324.
^{36} Compare Morris Hirsch’s remarks in BAMS April 94.
^{37} IS*, p. 38, footnote 26. Weinberg’s remarks are contained in his article “Sokal’s Hoax,” in the New York Review of Books, August 8, 1996.
^{38} Metaphors from virtual reality may help here.”
* Earlier defined by Harris as “Impostures Scientifiques (IS), a collection of articles compiled or commissioned by Baudouin Jurdant and published simultaneously as an issue of the journal Alliage and as a book by La Découverte press.”
** Earlier defined by Harris as “Fashionable Nonsense (FN), the North American translation of Impostures Intellectuelles.”
What is the moral of all this French noise?
Perhaps that, in spite of the contemptible nonsense at last summer’s Mykonos conference on mathematics and narrative, stories do have an important role to play in mathematics — specifically, in the history of mathematics.
Despite his disdain for Platonism, exemplified in his remarks on the noteworthy connection of pi with the zeta function in the formula given above, Harris has performed a valuable service to mathematics by pointing out the excellent historical work of Catherine Goldstein. Ms. Goldstein has demonstrated that even a French nominalist can be a firstrate scholar. Her essay on circles that Harris cites in a French version is also available in English, and will repay the study of those who, like Barry Mazur and other Harvard savants, are much too careless with the facts of history. They should consult her “Stories of the Circle,” pp. 160190 in A History of Scientific Thought, edited by Michel Serres, Blackwell Publishers (December 1995).
For the historicallychallenged mathematicians of Harvard, this essay would provide a valuable supplement to the upcoming “Pi Day” talk by Bamberg.
For those who insist on limiting their attention to mathematics proper, and ignoring its history, a suitable Pi Day observance might include becoming familiar with various proofs of the formula, pictured above, that connects pi with the zeta function of 2. For a survey, see Robin Chapman, Evaluating Zeta(2) (pdf). Zeta functions in a much wider context will be discussed at next May’s politically correct “Women in Mathematics” program at Princeton, “Zeta Functions All the Way” (pdf).
Adapted from
illustration below:
“There is a pleasantly discursive treatment of Pontius Pilate’s unanswered question ‘What is truth?'”
— H. S. M. Coxeter, 1987, introduction to Richard J. Trudeau’s remarks on the “Story Theory” of truth as opposed to the “Diamond Theory” of truth in The NonEuclidean Revolution
“A new epistemology is emerging to replace the Diamond Theory of truth. I will call it the ‘Story Theory’ of truth: There are no diamonds. People make up stories about what they experience. Stories that catch on are called ‘true.’ The Story Theory of truth is itself a story that is catching on. It is being told and retold, with increasing frequency, by thinkers of many stripes*….”
— Richard J. Trudeau in
The NonEuclidean Revolution
“‘Deniers’ of truth… insist that each of us is trapped in his own point of view; we make up stories about the world and, in an exercise of power, try to impose them on others.”
— Jim Holt in The New Yorker.
Exercise of Power:
Show that a white horse–
a figure not unlike the
symbol of the mathematics
publisher Springer–
is traced, within a naturally
arranged rectangular array of
polynomials, by the powers of x
modulo a polynomial
irreducible over a Galois field.
This horse, or chess knight–
“Springer,” in German–
plays a role in “Diamond Theory”
(a phrase used in finite geometry
in 1976, some years before its use
by Trudeau in the above book).
Related material
On this date:
In 1490, The White Knight
(Tirant lo Blanc )–
a major influence on Cervantes–
was published, and in 1910
the Mexican Revolution began.
Illustration:
Zapata by Diego Rivera,
Museum of Modern Art,
New York
“First published in the Catalan language in Valencia in 1490…. Reviewing the first modern Spanish translation in 1969 (Franco had ruthlessly suppressed the Catalan language and literature), Mario Vargas Llosa hailed the epic’s author as ‘the first of that lineage of Godsupplanters– Fielding, Balzac, Dickens, Flaubert, Tolstoy, Joyce, Faulkner– who try to create in their novels an allencompassing reality.'”
— H. S. M. Coxeter, introduction to
Richard J. Trudeau’s
The NonEuclidean Revolution
“People have always longed for truths about the world — not logical truths, for all their utility; or even probable truths, without which daily life would be impossible; but informative, certain truths, the only ‘truths’ strictly worthy of the name. Such truths I will call ‘diamonds’; they are highly desirable but hard to find….The happy metaphor is Morris Kline’s in Mathematics in Western Culture (Oxford, 1953), p. 430.”
— Richard J. Trudeau,
The NonEuclidean Revolution,
Birkhauser Boston,
1987, pages 114 and 117
“A new epistemology is emerging to replace the Diamond Theory of truth. I will call it the ‘Story Theory’ of truth: There are no diamonds. People make up stories about what they experience. Stories that catch on are called ‘true.’ The Story Theory of truth is itself a story that is catching on. It is being told and retold, with increasing frequency, by thinkers of many stripes…. My own viewpoint is the Story Theory…. I concluded long ago that each enterprise contains only stories (which the scientists call ‘models of reality’). I had started by hunting diamonds; I did find dazzlingly beautiful jewels, but always of human manufacture.”
— Richard J. Trudeau,
The NonEuclidean Revolution,
Birkhauser Boston,
1987, pages 256 and 259
An example of
the story theory of truth:
Actress Gwyneth Paltrow (“Proof”) was apparently born on either Sept. 27, 1972, or Sept. 28, 1972. Google searches yield “about 193” results for the 27th and “about 610” for the 28th.
Those who believe in the “story theory” of truth may therefore want to wish her a happy birthday today. Those who do not may prefer the contents of yesterday’s entry, from Paltrow’s other birthday.
Mathematics and Narrative
continued
"There is a pleasantly discursive treatment of Pontius Pilate's unanswered question 'What is truth?'"
— H. S. M. Coxeter, 1987, introduction to Richard J. Trudeau's remarks on the "Story Theory" of truth as opposed to the "Diamond Theory" of truth " in The NonEuclidean Revolution
"I had an epiphany: I thought 'Oh my God, this is it! People are talking about elliptic curves and of course they think they are talking mathematics. But are they really? Or are they talking about stories?'"
— An organizer of last month's "Mathematics and Narrative" conference
"A new epistemology is emerging to replace the Diamond Theory of truth. I will call it the 'Story Theory' of truth: There are no diamonds. People make up stories about what they experience. Stories that catch on are called 'true.' The Story Theory of truth is itself a story that is catching on. It is being told and retold, with increasing frequency, by thinkers of many stripes*…."
— Richard J. Trudeau in The NonEuclidean Revolution
"'Deniers' of truth… insist that each of us is trapped in his own point of view; we make up stories about the world and, in an exercise of power, try to impose them on others."
— Jim Holt in this week's New Yorker magazine. Click on the box below.
* Many stripes —
"What disciplines were represented at the meeting?"
"Apart from historians, you mean? Oh, many: writers, artists, philosophers, semioticians, cognitive psychologists – you name it."
— An organizer of last month's "Mathematics and Narrative" conference
Kevin Baker in 2001 on
E. L. Doctorow’s City of God:
David Van Biema in Time Magazine
(May 2, 2005, p. 43)
on Augustine’s City of God:
This year’s April –
theme is “Mathematics and the Cosmos.”
For my own views on this theme as it applies
to education, see Wag the Dogma.
For some other views, see this year’s
Mathematics Awareness Month site.
One of the authors at that site,
which is mostly propaganda
for the religion of Scientism,
elsewhere quotes
an ignorant pedagogue:
Damned nonsense. See Math16.com.
Indiana Jones 
In memory of Bernard Williams,
Oxford philosopher, who died Tuesday, June 10, 2003.
“…in… Truth and Truthfulness [September, 2002], he sought to speak plainly, and took on the postmodern, politically correct notion that truth is merely relative…”
“People have always longed for truths about the world — not logical truths, for all their utility; or even probable truths, without which daily life would be impossible; but informative, certain truths, the only ‘truths’ strictly worthy of the name. Such truths I will call ‘diamonds’; they are highly desirable but hard to find….
A new epistemology is emerging to replace the Diamond Theory of truth. I will call it the ‘Story Theory’ of truth: There are no diamonds. People make up stories about what they experience. Stories that catch on are called ‘true.’ The Story Theory of truth is itself a story that is catching on. It is being told and retold, with increasing frequency, by thinkers of many stripes…. My own viewpoint is the Story Theory….”
— Richard J. Trudeau, The NonEuclidean Revolution, Birkhauser Boston, 1987
Today is the feast day of Saint Jorge Luis Borges (b. Buenos Aires, August 24, 1899 – d. Geneva, June 14, 1986).
From Borges’s “The Aleph“:
“The Faithful who gather at the mosque of Amr, in Cairo, are acquainted with the fact that the entire universe lies inside one of the stone pillars that ring its central court…. The mosque dates from the seventh century; the pillars come from other temples of preIslamic religions…. Does this Aleph exist in the heart of a stone?”
(“Los fieles que concurren a la mezquita de Amr, en el Cairo, saben muy bien que el universo está en el interior de una de las columnas de piedra que rodean el patio central…. la mezquita data del siglo VII; las columnas proceden de otros templos de religiones anteislámicas…. ¿Existe ese Aleph en lo íntimo de una piedra?”)
From The Hunchback of Notre Dame:
Un cofre de gran riqueza
Hallaron dentro un pilar,
Dentro del, nuevas banderas
Con figuras de espantar.*
* A coffer of great richness
In a pillar’s heart they found,
Within it lay new banners,
With figures to astound.See also the figures obtained by coloring and permuting parts of the above religious symbol.
Lena Olin and Harrison Ford
in “Hollywood Homicide“
ART WARS:
Toward Eternity
April is Poetry Month, according to the Academy of American Poets. It is also Mathematics Awareness Month, funded by the National Security Agency; this year's theme is "Mathematics and Art."
Some previous journal entries for this month seem to be summarized by Emily Dickinson's remarks:
"Because I could not stop for Death–
He kindly stopped for me–
The Carriage held but just Ourselves–
And Immortality.
Math Awareness Month April is Math Awareness Month.

An Offer He Couldn't Refuse Today's birthday: Francis Ford Coppola is 64.
From a note on geometry of April 28, 1985: 
The Eight Today, the fourth day of the fourth month, plays an important part in Katherine Neville's The Eight. Let us honor this work, perhaps the greatest bad novel of the twentieth century, by reflecting on some properties of the number eight. Consider eight rectangular cells arranged in an array of four rows and two columns. Let us label these cells with coordinates, then apply a permutation.
The resulting set of arrows that indicate the movement of cells in a permutation (known as a Singer 7cycle) outlines rather neatly, in view of the chess theme of The Eight, a knight. This makes as much sense as anything in Neville's fiction, and has the merit of being based on fact. It also, albeit rather crudely, illustrates the "Mathematics and Art" theme of this year's Mathematics Awareness Month. The visual appearance of the "knight" permutation is less important than the fact that it leads to a construction (due to R. T. Curtis) of the Mathieu group M_{24} (via the Curtis Miracle Octad Generator), which in turn leads logically to the Monster group and to related "moonshine" investigations in the theory of modular functions. See also "Pieces of Eight," by Robert L. Griess. 
An Offer He Couldn't Refuse
Today's birthday: Francis Ford Coppola is 64.
"There is a pleasantly discursive treatment
of Pontius Pilate's unanswered question
'What is truth?'."
— H. S. M. Coxeter, 1987, introduction to Richard J. Trudeau's remarks on the "Story Theory" of truth as opposed to the "Diamond Theory" of truth in The NonEuclidean Revolution
"Then came From Here to Eternity. Sinatra lobbied hard for the role, practically getting on his knees to secure the role of the street smart punk G.I. Maggio. He sensed this was a role that could revive his career, and his instincts were right. There are lots of stories about how Columbia Studio head Harry Cohn was convinced to give the role to Sinatra, the most famous of which is expanded upon in the horse's head sequence in The Godfather. Maybe no one will know the truth about that. The one truth we do know is that the feisty New Jersey actor won the Academy Award as Best Supporting Actor for his work in From Here to Eternity. It was no looking back from then on."
From a note on geometry of April 28, 1985:
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