Friday, February 3, 2017

A Fable of Art Criticism

Filed under: General — Tags: , — m759 @ 6:42 PM

Part I —

Part II —

Part III —

"Let us examine our domain." — Character in a Balzac novel

Ashton reportedly died on Monday, Jan. 30, 2017.
See some more-scholarly remarks by Ernst Cassirer
on "the domain of perception" quoted here on that date.

Monday, January 30, 2017

Devotional Space

Filed under: General — m759 @ 4:16 PM

Quotations by and for an artist who reportedly died
on Sunday, January 15, 2017 —

"What drives my vision is a need to locate
a 'genetically felt' devotional space
in which a simultaneous multiplicity
of disparate realities coexists."

— The late Ciel Bergman, in her webpage
     "Artist's Statement"

"Once a registered nurse who worked in a hospital
psychiatric ward, Ms. Bergman was a struggling
single mom of two when she couldn’t resist the pull
of her art. In 1969, she entered a painting in the
Jack London Invitational, an art contest in Oakland,
and won first prize. This compelled her to enroll at
the San Francisco Art Institute, where she earned
her master of fine arts with honors in painting."

Sam Whiting in the San Francisco Chronicle

See also Oakland in this journal and
"Only a peculiar can enter a time loop."

"The peculiar kind of 'identity' that is attributed to
apparently altogether heterogeneous 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."

— Ernst Cassirer, quoted here on
     Midsummer Eve (St. John's Eve), 2010

Thursday, September 29, 2016


Filed under: General — Tags: — m759 @ 10:30 PM

Cassirer vs. Heidegger at Harvard —

A remembrance for Michaelmas —

A version of Heidegger's "Sternwürfel " —

From Log24 on the upload date for the above figure —

Reading for Michaelmas 2016

Filed under: General — m759 @ 12:00 AM

When Philosophy Mattered

A review of

Continental Divide :  Heidegger, Cassirer, Davos
By Peter E. Gordon
(Harvard University Press, 426 pp., $39.95)

The reviewer: David Nirenberg in The New Republic .
The review, dated January 13, 2011, ran in the
February 3, 2011, issue of the magazine.

Wednesday, December 17, 2014

For Rilke’s Panther

Filed under: General — m759 @ 12:00 PM

The title refers to yesterday evening's remarks titled
"Free the Philosophical Beast" in The Stone , a NY Times  weblog. 

The January 2015 issue of the Notices of the American Mathematical Society
has an article by Michael J. Barany.  From November 2012 remarks
by Barany :

"A highlight of the workshop was Cathryn Carson’s interpretation
of the transcendental phenomenology and historicism of Husserl,
Heidegger, Cassirer, and a few others, launched from a moving
reflection on the experience of reading Kuhn."

See Carson's paper "Science as Instrumental Reason: Heidegger, Habermas,
Heisenberg," Continental Philosophy Review  (2010) 42483–509.

Related material: Monday's Log24 posts Rota on Husserl and Annals of Perception.

Saturday, February 22, 2014


Filed under: General — Tags: — m759 @ 12:00 PM

The title was suggested by a 1921 article
by Hermann Weyl and by a review* of
a more recent publication —

The above Harvard Gazette  piece on Davos is
from St. Ursula's Day, 2010. See also this  journal
on that date.

See as well a Log24 search for Davos.

A more interesting piece by Peter E. Gordon
(author of the above Davos book) is his review
of Charles Taylor's A Secular Age .
The review is titled

"The Place of the Sacred
in the Absence of God

(The place of the sacred is not, perhaps, Davos,
but a more abstract location.)

* Grundlagenkrise  was a tag for a Jan. 13, 2011,
  review in The New Republic  of Gordon's
  book on Cassirer and Heidegger at Davos.

Thursday, December 27, 2012

Object Lesson

Filed under: General,Geometry — Tags: , — m759 @ 3:17 AM

Yesterday's post on the current Museum of Modern Art exhibition
"Inventing Abstraction: 1910-1925" suggests a renewed look at
abstraction and a fundamental building block: the cube.

From a recent Harvard University Press philosophical treatise on symmetry—

The treatise corrects Nozick's error of not crediting Weyl's 1952 remarks
on objectivity and symmetry, but repeats Weyl's error of not crediting
Cassirer's extensive 1910 (and later) remarks on this subject.

For greater depth see Cassirer's 1910 passage on Vorstellung :

IMAGE- Ernst Cassirer on 'representation' or 'Vorstellung' in 'Substance and Function' as 'the riddle of knowledge'

This of course echoes Schopenhauer, as do discussions of "Will and Idea" in this journal.

For the relationship of all this to MoMA and abstraction, see Cube Space and Inside the White Cube.

"The sacramental nature of the space becomes clear…." — Brian O'Doherty

Sunday, December 23, 2012

In a Nutshell…

Filed under: General — m759 @ 1:00 AM

The Kernel of the Concept of the Object

according to the New York Lottery yesterday—

From 4/27

From 11/24

IMAGE- Agent Smith from 'The Matrix,' 1999

A page numbered 176

A page numbered 187

Sunday, March 25, 2012

Compare and Contrast

Filed under: General,Geometry — Tags: — m759 @ 9:00 AM

IMAGE- Escher, 'Fishes and Scales'

IMAGE- Cullinane, 'Invariance'


The Origin and Development of Erwin Panofsky's Theories of Art ,
Michael Ann Holly, doctoral thesis, Cornell University, 1981 (pdf, 10 MB)

Panofsky, Cassirer, and Perspective as Symbolic Form ,
Allister Neher, doctoral thesis, Concordia University, 2000

Monday, November 21, 2011

Random Reference

Filed under: General,Geometry — m759 @ 12:00 PM

IMAGE- NY Evening Lottery Nov. 20, 2011: 245 and 0182

Joseph T. Clark, S. J., Conventional Logic and Modern Logic:
A Prelude to Transition
  (Philosophical Studies of the American
Catholic Philosophical Association, III) Woodstock, Maryland:
Woodstock College Press, 1952—

Alonzo Church, "Logic: formal, symbolic, traditional," Dictionary of Philosophy  (New York: Philosophical Library, 1942), pp. 170-182. The contents of this ambitious Dictionary are most uneven. Random reference to its pages is dangerous. But this contribution is among its best. It is condensed. But not dense. A patient and attentive study will pay big dividends in comprehension. Church knows the field and knows how to depict it. A most valuable reference.

Another book to which random reference is dangerous


For greater depth, see "Cassirer and Eddington on Structures,
Symmetry and Subjectivity" in Steven French's draft of
"Symmetry, Structure and the Constitution of Objects"

Saturday, September 24, 2011

Kernels of Being

Filed under: General,Geometry — Tags: — m759 @ 10:29 PM

For the Pope in Germany

"We wish to see Jesus. For somehow we know, we suspect, we intuit, that if we see Jesus we will see what Meister Eckhart might call “The Divine Kernel of Being”— that Divine Spark of God’s essence, God’s imago Dei, the image in which we are created. We seem to know that in seeing Jesus we just might find something essential about ourselves."

—The Reverend Kirk Alan Kubicek, St. Peter’s at Ellicott Mills, Maryland, weblog post of Saturday, March 28, 2009, on a sermon for Sunday, March 29, 2009

See also this journal in March 2009.

Related non-theology—

Weyl on coordinate systems, Cassirer on the kernel of being, and A Study in Art Education.

Wednesday, September 21, 2011

Symmetric Generation

Filed under: General,Geometry — Tags: , , , , — m759 @ 2:00 PM

Suggested by yesterday's Relativity Problem Revisited and by Cassirer on Objectivity

From Symmetric Generation of Groups , by R.T. Curtis (Cambridge U. Press, 2007)—

"… we are saying much more than that G M 24 is generated by
some set of seven involutions, which would be a very weak
requirement. We are asserting that M 24 is generated by a set
of seven involutions which possesses all the symmetries of L3(2)
acting on the points of the 7-point projective plane…."
Symmetric Generation , p. 41

"It turns out that this approach is particularly revealing and that
many simple groups, both sporadic and classical, have surprisingly
simple definitions of this type."
Symmetric Generation , p. 42

See also (click to enlarge)—


Cassirer's remarks connect the concept of objectivity  with that of object .

The above quotations perhaps indicate how the Mathieu group M 24 may be viewed as an object.

"This is the moment which I call epiphany. First we recognise that the object is one  integral thing, then we recognise that it is an organised composite structure, a thing  in fact: finally, when the relation of the parts is exquisite, when the parts are adjusted to the special point, we recognise that it is that  thing which it is. Its soul, its whatness, leaps to us from the vestment of its appearance. The soul of the commonest object, the structure of which is so adjusted, seems to us radiant. The object achieves its epiphany."

— James Joyce, Stephen Hero

For a simpler object "which possesses all the symmetries of L3(2) acting on the points of the 7-point projective plane…." see The Eightfold Cube.

For symmetric generation of L3(2) on that cube, see A Simple Reflection Group of Order 168.

Saturday, September 17, 2011


Filed under: General — m759 @ 12:00 PM

The previous two posts, Baggage and The Uploading, suggest
a review of Wroclaw's native son Ernst Cassirer.

Wednesday, August 10, 2011


Filed under: General,Geometry — m759 @ 12:25 PM

From math16.com

Quotations on Realism
and the Problem of Universals:

"It is said that the students of medieval Paris came to blows in the streets over the question of universals. The stakes are high, for at issue is our whole conception of our ability to describe the world truly or falsely, and the objectivity of any opinions we frame to ourselves. It is arguable that this is always the deepest, most profound problem of philosophy. It structures Plato's (realist) reaction to the sophists (nominalists). What is often called 'postmodernism' is really just nominalism, colourfully presented as the doctrine that there is nothing except texts. It is the variety of nominalism represented in many modern humanities, paralysing appeals to reason and truth."
— Simon Blackburn, Think, Oxford University Press, 1999, page 268

"You will all know that in the Middle Ages there were supposed to be various classes of angels…. these hierarchized celsitudes are but the last traces in a less philosophical age of the ideas which Plato taught his disciples existed in the spiritual world."
— Charles Williams, page 31, Chapter Two, "The Eidola and the Angeli," in The Place of the Lion (1933), reprinted in 1991 by Eerdmans Publishing

For Williams's discussion of Divine Universals (i.e., angels), see Chapter Eight of The Place of the Lion.

"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 Non-Euclidean 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 Non-Euclidean Revolution, Birkhauser Boston, 1987, pages 256 and 259

Trudeau's confusion seems to stem from the nominalism of W. V. Quine, which in turn stems from Quine's appalling ignorance of the nature of geometry. Quine thinks that the geometry of Euclid dealt with "an emphatically empirical subject matter" — "surfaces, curves, and points in real space." Quine says that Euclidean geometry lost "its old status of mathematics with a subject matter" when Einstein established that space itself, as defined by the paths of light, is non-Euclidean. Having totally misunderstood the nature of the subject, Quine concludes that after Einstein, geometry has become "uninterpreted mathematics," which is "devoid not only of empirical content but of all question of truth and falsity." (From Stimulus to Science, Harvard University Press, 1995, page 55)
— S. H. Cullinane, December 12, 2000

The correct statement of the relation between geometry and the physical universe is as follows:

"The contrast between pure and applied mathematics stands out most clearly, perhaps, in geometry. There is the science of pure geometry, in which there are many geometries: projective geometry, Euclidean geometry, non-Euclidean geometry, and so forth. Each of these geometries is a model, a pattern of ideas, and is to be judged by the interest and beauty of its particular pattern. It is a map or picture, the joint product of many hands, a partial and imperfect copy (yet exact so far as it extends) of a section of mathematical reality. But the point which is important to us now is this, that there is one thing at any rate of which pure geometries are not pictures, and that is the spatio-temporal reality of the physical world. It is obvious, surely, that they cannot be, since earthquakes and eclipses are not mathematical concepts."
— G. H. Hardy, section 23, A Mathematician's Apology, Cambridge University Press, 1940

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—

"Exhibit A is certainly Ljubljana…."

Tuesday, June 21, 2011

Truth, Beauty, Bullshit

Filed under: General — m759 @ 1:16 PM

This post is for the Stonehenge solstice crowd, who might,
like the London artist Steve Richards, confuse bullshit
with scholarship and inspire the same confusion
in others.

IMAGE- Motto of Forgotten Books, with pirated quotation from Shakespeare that might be appropriate for London's 'Piracy Project'

The image, apparently an epigraph put there
by the author, is from the Forgotten Books edition
of Cassirer's Substance and Function:
And Einstein's Theory of Relativity

This is a scanned copy of the 1923 original.
The egg-figure above, however, is from the publisher's
prefatory notes and not  from the original.

A check of other Forgotten Books publications
shows that the motto and the Bacon
attribution are those of Forgotten Books and
not  of the authors they reprint — in particular,
not  of Ernst Cassirer, who would probably
be dismayed to have this nonsense associated
with his work.

Why nonsense? The attribution to Francis Bacon is
false. The lines are from "The Phoenix and the Turtle"
by William Shakespeare.

Requiem for a Publisher

Filed under: General — m759 @ 2:09 AM

In memory of A. Whitney Ellsworth, first publisher of
The New York Review of Books , who died at 75
on Saturday—


The Review  has sometimes been cited in this journal.

See also posts from the date of Ellsworth's death—

Piracy Project

Filed under: General,Geometry — Tags: , — m759 @ 2:02 AM

Recent piracy of my work as part of a London art project suggests the following.


           From http://www.trussel.com/rls/rlsgb1.htm

The 2011 Long John Silver Award for academic piracy
goes to ….

Hermann Weyl, for the remark on objectivity and invariance
in his classic work Symmetry  that skillfully pirated
the much earlier work of philosopher Ernst Cassirer.

And the 2011 Parrot Award for adept academic idea-lifting
goes to …

Richard Evan Schwartz of Brown University, for his
use, without citation, of Cullinane’s work illustrating
Weyl’s “relativity problem” in a finite-geometry context.

For further details, click on the above names.

Monday, June 20, 2011

The Search for Invariants

Filed under: General,Geometry — m759 @ 9:29 AM

The title of a recent contribution to a London art-related "Piracy Project" begins with the phrase "The Search for Invariants."

A search for that phrase  elsewhere yields a notable 1944* paper by Ernst Cassirer, "The Concept of Group and the Theory of Perception."

Page 20: "It is a process of objectification, the characteristic nature
and tendency of which finds expression in the formation of invariants."

Cassirer's concepts seem related to Weyl's famous remark that

“Objectivity means invariance with respect to the group of automorphisms.”
Symmetry  (Princeton University Press, 1952, page 132)

See also this journal on June 23, 2010— "Group Theory and Philosophy"— as well as some Math Forum remarks on Cassirer and Weyl.

Update of 6 to 7:50 PM June 20, 2011—

Weyl's 1952 remark seems to echo remarks in 1910 and 1921 by Cassirer.
See Cassirer in 1910 and 1921 on Objectivity.

Another source on Cassirer, invariance, and objectivity—

The conclusion of Maja Lovrenov's 
"The Role of Invariance in Cassirer’s Interpretation of the Theory of Relativity"—

"… physical theories prove to be theories of invariants
with regard to certain groups of transformations and
it is exactly the invariance that secures the objectivity
of a physical theory."

— SYNTHESIS PHILOSOPHICA 42 (2/2006), pp. 233–241

A search in Weyl's Symmetry  for any reference to Ernst Cassirer yields no results.

* Published in French in 1938.

Tuesday, March 1, 2011

Women’s History Month

Filed under: General — Tags: — m759 @ 12:00 PM

Susanne for Suzanne

From pages 7-8 of William York Tindall’s Literary Symbolism  (Columbia U. Press, 1955)—

                                     ... According to Cassirer's Essay 
on Man, as we have seen, art is a symbolic form, parallel in respect 
of this to religion or science. Each of these forms builds up a universe 
that enables man to interpret and organize his experience; and each 
is a discovery, because a creation, of reality. Although similar in func- 
tion, the forms differ in the kind of reality built. Whereas science
builds it of facts, art builds it of feelings, intuitions of quality, and 
the other distractions of our inner life— and in their degrees so do 
myth and religion. What art, myth, and religion are, Cassirer con- 
fesses, cannot be expressed by a logical definition. 

Nevertheless, let us see what Clive Bell says about art. He calls 
it "significant form," but what that is he is unable to say. Having 
no quarrel with art as form, we may, however, question its signifi- 
cance. By significant he cannot mean important in the sense of 
having import, nor can he mean having the function of a sign; 
for to him art, lacking reference to nature, is insignificant. Since, 
however, he tells us that a work of art "expresses" the emotion of 
its creator and "provokes" an emotion in its contemplator,he seems 
to imply that his significant means expressive and provocative. The 
emotion expressed and provoked is an "aesthetic emotion," contem- 
plative, detached from all concerns of utility and from all reference. 

Attempting to explain Bell's significant form, Roger Fry, equally 
devoted to Whistler and art for art's sake, says that Flaubert's "ex- 
pression of the idea" is as near as he can get to it, but neither Flaubert 
nor Fry tells what is meant by idea. To "evoke" it, however, the artist 
creates an "expressive design" or "symbolic form," by which the 
spirit "communicates its most secret and indefinable impulses." 

Susanne Langer,who occupies a place somewhere between Fry 
and Cassirer, though nearer the latter, once said in a seminar that a 
work of art is an "unassigned syntactical symbol." Since this defini- 

End of page 7 

tion does not appear in her latest book, she may have rejected it, but 
it seems far more precise than Fry's attempt. By unassigned she prob- 
ably intends insignificant in the sense of lacking sign value or fixed 
reference; syntactical implies a form composed of parts in relation- 
ship to one another; and a symbol, according to Feeling and Form, 
is "any device whereby we are enabled to make an abstraction." Too 
austere for my taste, this account of symbol seems to need elaboration, 
which, to be sure, her book provides. For the present, however, taking 
symbol to mean an outward device for presenting an inward state, 
and taking unassigned and syntactical as I think she uses them, let 
us tentatively admire her definition of the work of art.



Oh, the red leaf looks to the hard gray stone
To each other, they know what they mean

— Suzanne Vega, “Song in Red and Gray

Tuesday, July 20, 2010

The Corpse Express

Filed under: General,Geometry — m759 @ 2:02 AM

See Malcolm Lowry's "A corpse will be transported by express!" in this journal.

From June 23

"When Plato regards 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."

— Ernst Cassirer, Philosophy and Phenomenological Research,
   Volume V, Number 1, September, 1944.


June 23, Midsummer Eve, was the date of death for Colonel Michael Cobb.

Cobb, who died aged 93, was "a regular Army officer who in retirement produced
the definitive historical atlas of the railways of Great Britain." — Telegraph.co.uk, July 19

As for geometry, railways, and things eternal, see parallel lines converging
in Tequila Mockingbird and Bedlam Songs.

Station of the Rock Island Line

The Rock Island Line’s namesake depot 
in Rock Island, Illinois

See also Wallace Stevens on "the giant of nothingness"
in "A Primitive Like an Orb" and in Midsummer Eve's Dream

At the center on the horizon, concentrum, grave
And prodigious person, patron of origins.

Saturday, June 26, 2010

Plato’s Logos

Filed under: General,Geometry — m759 @ 9:00 AM

“The present study is closely connected with a lecture* given by Prof. Ernst Cassirer at the Warburg Library whose subject was ‘The Idea of the Beautiful in Plato’s Dialogues’…. My investigation traces the historical destiny of the same concept….”

* See Cassirer’s Eidos und Eidolon : Das Problem des Schönen und der Kunst in Platons Dialogen, in Vorträge der Bibliothek Warburg II, 1922/23 (pp. 1–27). Berlin and Leipzig, B.G. Teubner, 1924.

— Erwin Panofsky, Idea: A Concept in Art Theory, foreword to the first German edition, Hamburg, March 1924

On a figure from Plato’s Meno

IMAGE- Plato's diamond and finite geometry

The above figures illustrate Husserl’s phrase  “eidetic variation”
a phrase based on Plato’s use of eidos, a word
closely related to the word “idea” in Panofsky’s title.

For remarks by Cassirer on the theory of groups, a part of
mathematics underlying the above diamond variations, see
his “The Concept of Group and the Theory of Perception.”

Sketch of some further remarks—


The Waterfield question in the sketch above
is from his edition of Plato’s Theaetetus
(Penguin Classics, 1987).

The “design theory” referred to in the sketch
is that of graphic  design, which includes the design
of commercial logos. The Greek  word logos
has more to do with mathematics and theology.

“If there is one thread of warning that runs
through this dialogue, from beginning to end,
it is that verbal formulations as such are
shot through with ambiguity.”

— Rosemary Desjardins, The Rational Enterprise:
Logos in Plato’s Theaetetus
, SUNY Press, 1990

Related material—

(Click to enlarge.)


Wednesday, June 23, 2010

Group Theory and Philosophy

Filed under: General,Geometry — Tags: — m759 @ 5:01 PM

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. 368-414.)

The group-theoretical 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
sense-impressions 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 non-Euclidean
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 group-principle they appear as "unessential" modifica-

                 … 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….

Thursday, December 16, 2004

Thursday December 16, 2004

Filed under: General — Tags: — m759 @ 3:00 AM

Nothing Nothings

Background: recent Log24 entries (beginning with Chorus from the Rock on Dec. 5, 2004) and Is Nothing Sacred? (quotations compiled on March 9, 2000).

From an obituary of Paul Edwards, a writer on philosophy, in this morning's New York Times:

"Heidegger's Confusions, a collection of Professor Edwards's scholarly articles, was published last month by Prometheus."

Edwards, born in Vienna in 1923 to Jewish parents, died on December 9.

Some sites I visited earlier this evening, before reading of Edwards's death:

  • " 'Nothingness itself nothings' — with these words, uttered by Martin Heidegger in the early 1930s, the incipient (and now-familiar) split between analytic and continental philosophy began tearing open. For Rudolf Carnap, a leader of the Vienna Circle [Wiener Kreis] of logical empiricists and a strident advocate of a new, scientific approach to philosophy, this Heideggerian proposition exemplified 'a metaphysical pseudo-sentence,' meaningless and unable to withstand any logical analysis. Heidegger countered that Carnap’s misplaced obsession with logic missed the point entirely."
    Review of A Parting of the Ways: Carnap, Cassirer, and Heidegger
  • "Death and Metaphysics," by Peter Kraus, pp. 98-111 in Death and Philosophy, ed. by Jeff Malpas and Robert Solomon.  Heidegger's famous phrase (misquoted by Quine in Gray Particular in Hartford) "Das Nichts selbst nichtet" is discussed on page 102.

Friday, July 25, 2003

Friday July 25, 2003

Filed under: General — m759 @ 5:24 PM

For Jung’s 7/26 Birthday:
A Logocentric Meditation

Leftist academics are trying to pull a fast one again.  An essay in the most prominent American mathematical publication tries to disguise a leftist attack on Christian theology as harmless philosophical woolgathering.

In a review of Vladimir Tasic’s Mathematics and the Roots of Postmodern Thought, the reviewer, Michael Harris, is being less than candid when he discusses Derrida’s use of “logocentrism”:

“Derrida uses the term ‘logocentrism’… as ‘the metaphysics of phonetic writing’….”

Notices of the American Mathematical Society, August 2003, page 792

We find a rather different version of logocentrism in Tasic’s own Sept. 24, 2001, lecture “Poststructuralism and Deconstruction: A Mathematical History,” which is “an abridged version of some arguments” in Tasic’s book on mathematics and postmodernism:

“Derrida apparently also employs certain ideas of formalist mathematics in his critique of idealist metaphysics: for example, he is on record saying that ‘the effective progress of mathematical notation goes along with the deconstruction of metaphysics.’

Derrida’s position is rather subtle. I think it can be interpreted as a valiant sublation of two completely opposed schools in mathematical philosophy. For this reason it is not possible to reduce it to a readily available philosophy of mathematics. One could perhaps say that Derrida continues and critically reworks Heidegger’s attempt to ‘deconstruct’ traditional metaphysics, and that his method is more ‘mathematical’ than Heidegger’s because he has at his disposal the entire pseudo-mathematical tradition of structuralist thought. He has himself implied in an interview given to Julia Kristeva that mathematics could be used to challenge ‘logocentric theology,’ and hence it does not seem unreasonable to try looking for the mathematical roots of his philosophy.”

The unsuspecting reader would not know from Harris’s review that Derrida’s main concern is not mathematics, but theology.  His ‘deconstruction of metaphysics’ is actually an attack on Christian theology.

From “Derrida and Deconstruction,” by David Arneson, a University of Manitoba professor and writer on literary theory:

Logocentrism: ‘In the beginning was the word.’ Logocentrism is the belief that knowledge is rooted in a primeval language (now lost) given by God to humans. God (or some other transcendental signifier: the Idea, the Great Spirit, the Self, etc.) acts a foundation for all our thought, language and action. He is the truth whose manifestation is the world.”

Some further background, putting my July 23 entry on Lévi-Strauss and structuralism in the proper context:

Part I.  The Roots of Structuralism

“Literary science had to have a firm theoretical basis…”

Part II.  Structuralism/Poststructuralism

“Most [structuralists] insist, as Levi-Strauss does, that structures are universal, therefore timeless.”

Part III.  Structuralism and
             Jung’s Archetypes

Jung’s “theories, like those of Cassirer and Lévi-Strauss, command for myth a central cultural position, unassailable by reductive intellectual methods or procedures.”

And so we are back to logocentrism, with the Logos — God in the form of story, myth, or archetype — in the “central cultural position.”

What does all this have to do with mathematics?  See

Plato’s Diamond,

Rosalind Krauss on Art –

“the Klein group (much beloved of Structuralists)”

Another Michael Harris Essay, Note 47 –

“From Krauss’s article I learned that the Klein group is also called the Piaget group.”

and Jung on Quaternity:
      Beyond the Fringe –

“…there is no denying the fact that [analytical] psychology, like an illegitimate child of the spirit, leads an esoteric, special existence beyond the fringe of what is generally acknowledged to be the academic world.”

What attitude should mathematicians have towards all this? 

Towards postmodern French
  atheist literary/art theorists –

Mathematicians should adopt the attitude toward “the demimonde of chic academic theorizing” expressed in Roger Kimball’s essay, Feeling Sorry for Rosalind Krauss.

Towards logocentric German
  Christian literary/art theorists –

Mathematicians should, of course, adopt a posture of humble respect, tugging their forelocks and admitting their ignorance of Christian theology.  They should then, if sincere in their desire to honestly learn something about logocentric philosophy, begin by consulting the website

The Quest for the Fiction of an Absolute.

For a better known, if similarly disrespected, “illegitimate child of the spirit,” see my July 22 entry.

Thursday, October 31, 2002

Thursday October 31, 2002

Filed under: General,Geometry — m759 @ 11:07 PM


From The Unknowable (1999), by Gregory J. Chaitin, who has written extensively about his constant, which he calls Omega:

"What is Omega? It's just the diamond-hard distilled and crystallized essence of mathematical truth! It's what you get when you compress tremendously the coal of redundant mathematical truth…" 

Charles H. Bennett has written about Omega as a cabalistic number.

Here is another result with religious associations which, historically, has perhaps more claim to be called the "diamond-hard essence" of mathematical truth: The demonstration in Plato's Meno that a diamond inscribed in a square has half the area of the square (or that, vice-versa, the square has twice the area of the diamond).

From Ivars Peterson's discussion of Plato's diamond and the Pythagorean theorem:

"In his textbook The History of Mathematics, Roger Cooke of the University of Vermont describes how the Babylonians might have discovered the Pythagorean theorem more than 1,000 years before Pythagoras.

Basing his account on a passage in Plato's dialogue Meno, Cooke suggests that the discovery arose when someone, either for a practical purpose or perhaps just for fun, found it necessary to construct a square twice as large as a given square…."

From "Halving a Square," a presentation of Plato's diamond by Alexander Bogomolny, the moral of the story:

SOCRATES: And if the truth about reality is always in our soul, the soul must be immortal….

From "Renaissance Metaphysics and the History of Science," at The John Dee Society website:

Galileo on Plato's diamond:

"Cassirer, drawing attention to Galileo's frequent use of the Meno, particularly the incident of the slave's solving without instruction a problem in geometry by 'natural' reason stimulated by questioning, remarks, 'Galileo seems to accept all the consequences drawn by Plato from this fact…..'"

Roger Bacon on Plato's diamond:

"Fastening on the incident of the slave in the Meno, which he had found reproduced in Cicero, Bacon argued from it 'wherefore since this knowledge (of mathematics) is almost innate and as it were precedes discovery and learning or at least is less in need of them than other sciences, it will be first among sciences and will precede others disposing us towards them.'"

It is perhaps appropriate to close this entry, made on All Hallows' Eve, with a link to a page on Dr. John Dee himself.

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