Log24

Saturday, January 31, 2015

Spielraum

Filed under: General — Tags: — m759 @ 1:27 pm

From the concluding paragraph of a new book by
mathematician Michael Harris:

"A team of eminent scholars is completing a definitive
edition of Hausdorff’s collected works—'unique
in the annals of mathematical publishing'— with the
care befitting the literary figure he undoubtedly was….
he is honored as, perhaps, the first modern
mathematician to give a name to what we have been
calling the 'relaxed field'— he called it the
'Spielraum  of thought'— and as a mathematician
who never lost his sensitivity to his chosen field’s
problematic attractions while remaining fully aware that
every veil lifted only reveals another veil."

— Harris, Michael, Mathematics without Apologies:
Portrait of a Problematic Vocation  
(2015-01-18) 
(pp. 324-325). Princeton U. Press. Kindle Edition. 

Related material:  Spiel ist nicht Spielerei .

Friday, January 30, 2015

The Relaxed Field*

Filed under: General — m759 @ 7:00 pm

In memory of a dead poet —

"Relax," said the night man.
"We are programmed to receive."

* A phrase from a new book by mathematician
  Michael Harris, Mathematics without Apologies .

Wednesday, January 28, 2015

Snowpocalypse

Filed under: General — m759 @ 12:00 am

Columbia University physics writer Peter Woit dubbed
yesterday Snowpocalypse 2015 in New York City.

Woit used the day to ponder a new book by mathematician
Michael Harris, Mathematics without Apologies .

Related material: a search for Michael Harris in this journal.

That search includes…   

The above art includes an image of William Rubin,
former director of painting and sculpture at the
Museum of Modern Art. Rubin reportedly died on
January 22, 2006. See Log24 posts from that date.

Friday, May 23, 2014

The Iris Contingency

Filed under: General — m759 @ 10:00 pm

From a spring 2004 Michigan State University syllabus for the
T-Th course English 487, “The Twentieth Century English Novel”—

Tuesday, March 30: Murdoch
(her essay “ The Sublime and the Beautiful Revisited “)

Thursday, April 1: Murdoch

Related material from this journal—

Posts of Tuesday, March 30, 2004, and of Thursday, April 1, 2004.

For a related reference to the mathematician Michael Harris from
the Free-Floating Signs link in this afternoon’s 4:30 post, see
the  posts of Wednesday, March 31, 2004, the day intervening
between the above two class dates.

Sunday, November 24, 2013

Galois Groups and Harmonic Analysis

Filed under: General,Geometry — Tags: , , — m759 @ 9:29 am

“In 1967, he [Langlands] came up with revolutionary
insights tying together the theory of Galois groups
and another area of mathematics called harmonic
analysis. These two areas, which seem light years
apart
, turned out to be closely related.”

— Edward Frenkel, Love and Math, 2013

“Class field theory expresses Galois groups of
abelian extensions of a number field F
in terms of harmonic analysis on the
multiplicative group of [a] locally compact
topological ring, the adèle ring, attached to F.”

— Michael Harris in a description of a Princeton
mathematics department talk of October 2012

Related material: a Saturday evening post.

See also Wikipedia on the history of class field theory.
For greater depth, see Tate’s [1950] thesis and the book
Fourier Analysis on Number Fields .

Wednesday, March 6, 2013

Midnight in Pynchon*

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

"It is almost as though Pynchon wishes to
repeat the grand gesture of Joyce’s Ulysses…."

Vladimir Tasic on Pynchon's Against the Day

Related material:

Tasic's Mathematics and the Roots of Postmodern Thought  
and Michael Harris's "'Why Mathematics?' You Might Ask"

*See also Occupy Galois Space and Midnight in Dostoevsky.

Thursday, May 6, 2010

Infinite Jest

Filed under: General — m759 @ 10:31 pm

Leg-Pulling

Jim Holt, review of David Foster Wallace's book on infinity 'Everything and More'

Michael Harris in AMS Notices suggests David Foster Wallace may be pulling our legs in 'Everything and More'

"… to make the author manifestly unreliable"

Not to mention the reader.

Famous author hangs himself in the 2005 film 'Neverwas'

Related material —

But seriously…

Sunday, April 13, 2008

Sunday April 13, 2008

Filed under: General,Geometry — Tags: , , — m759 @ 7:59 am
The Echo
in Plato’s Cave

“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.”

— Simon Blackburn, Think (Oxford, 1999)

Michael Harris, mathematician at the University of Paris:

“… three ‘parts’ of tragedy identified by Aristotle that transpose to fiction of all types– plot (mythos), character (ethos), and ‘thought’ (dianoia)….”

— paper (pdf) to appear in Mathematics and Narrative, A. Doxiadis and B. Mazur, eds.

Mythos —

A visitor from France this morning viewed the entry of Jan. 23, 2006: “In Defense of Hilbert (On His Birthday).” That entry concerns a remark of Michael Harris.

A check of Harris’s website reveals a new article:

“Do Androids Prove Theorems in Their Sleep?” (slighly longer version of article to appear in Mathematics and Narrative, A. Doxiadis and B. Mazur, eds.) (pdf).

From that article:

“The word ‘key’ functions here to structure the reading of the article, to draw the reader’s attention initially to the element of the proof the author considers most important. Compare E.M. Forster in Aspects of the Novel:

[plot is] something which is measured not be minutes or hours, but by intensity, so that when we look at our past it does not stretch back evenly but piles up into a few notable pinnacles.”

Ethos —

“Forster took pains to widen and deepen the enigmatic character of his novel, to make it a puzzle insoluble within its own terms, or without. Early drafts of A Passage to India reveal a number of false starts. Forster repeatedly revised drafts of chapters thirteen through sixteen, which comprise the crux of the novel, the visit to the Marabar Caves. When he began writing the novel, his intention was to make the cave scene central and significant, but he did not yet know how:

When I began a A Passage to India, I knew something important happened in the Malabar (sic) Caves, and that it would have a central place in the novel– but I didn’t know what it would be… The Malabar Caves represented an area in which concentration can take place. They were to engender an event like an egg.”

E. M. Forster: A Passage to India, by Betty Jay

Dianoia —

Flagrant Triviality
or Resplendent Trinity?

“Despite the flagrant triviality of the proof… this result is the key point in the paper.”

— Michael Harris, op. cit., quoting a mathematical paper

Online Etymology Dictionary
:

flagrant
c.1500, “resplendent,” from L. flagrantem (nom. flagrans) “burning,” prp. of flagrare “to burn,” from L. root *flag-, corresponding to PIE *bhleg (cf. Gk. phlegein “to burn, scorch,” O.E. blæc “black”). Sense of “glaringly offensive” first recorded 1706, probably from common legalese phrase in flagrante delicto “red-handed,” lit. “with the crime still blazing.”

A related use of “resplendent”– applied to a Trinity, not a triviality– appears in the Liturgy of Malabar:

http://www.log24.com/log/pix08/080413-LiturgyOfMalabar.jpg

The Liturgies of SS. Mark, James, Clement, Chrysostom, and Basil, and the Church of Malabar, by the Rev. J.M. Neale and the Rev. R.F. Littledale, reprinted by Gorgias Press, 2002

On Universals and
A Passage to India:

 

“”The universe, then, is less intimation than cipher: a mask rather than a revelation in the romantic sense. Does love meet with love? Do we receive but what we give? The answer is surely a paradox, the paradox that there are Platonic universals beyond, but that the glass is too dark to see them. Is there a light beyond the glass, or is it a mirror only to the self? The Platonic cave is even darker than Plato made it, for it introduces the echo, and so leaves us back in the world of men, which does not carry total meaning, is just a story of events.”

 

— Betty Jay,  op. cit.

 

http://www.log24.com/log/pix08/080413-Marabar.jpg

Judy Davis in the Marabar Caves

In mathematics
(as opposed to narrative),
somewhere between
a flagrant triviality and
a resplendent Trinity we
have what might be called
“a resplendent triviality.”

For further details, see
A Four-Color Theorem.”

Sunday, March 12, 2006

Sunday March 12, 2006

Filed under: General,Geometry — Tags: , — m759 @ 1:00 pm

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.”

The image “http://www.log24.com/log/pix06/060312-PaulBamberg21.jpg” cannot be displayed, because it contains errors.

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.
They’re all in the math department.
They’re good.
They have this song called “i.”
You’d like it. Lowercase i.
They just stand there.
They don’t play anything for three minutes.

Imaginary number?

It’s a math joke.
You see why they’re way down the bill.

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 self-government. 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 The image “http://www.log24.com/log/pix06/060312-Char-pi.jpg” cannot be displayed, because it contains errors. 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 non-euclidean 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 The image “http://www.log24.com/log/pix06/060312-Char-pi.jpg” cannot be displayed, because it contains errors. is injective.  In other words, The image “http://www.log24.com/log/pix06/060312-Char-pi.jpg” cannot be displayed, because it contains errors. can be identified algebraically with X, the variable par excellence.33

The image “http://www.log24.com/log/pix06/060312-X.jpg” cannot be displayed, because it contains errors.

More interestingly, one can ask what kind of object The image “http://www.log24.com/log/pix06/060312-Char-pi.jpg” cannot be displayed, because it contains errors. 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 Lévy-Leblond, 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

The image “http://www.log24.com/log/pix06/060312-pi.jpg” cannot be displayed, because it contains errors.

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 The image “http://www.log24.com/log/pix06/060312-Char-pi.jpg” cannot be displayed, because it contains errors. 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 non-Euclidean 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. 129-149.
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 The image “http://www.log24.com/log/pix06/060312-Char-pi.jpg” cannot be displayed, because it contains errors. 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 first-rate 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. 160-190 in A History of Scientific Thought, edited by Michel Serres, Blackwell Publishers (December 1995).

For the historically-challenged 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).

Tuesday, January 24, 2006

Tuesday January 24, 2006

Filed under: General — Tags: — m759 @ 7:00 am
ART WARS
for Michael Harris
(See previous entry.)
 

The image “http://www.log24.com/log/pix06/060124-Art.jpg” cannot be displayed, because it contains errors.

Related material:
A classic book in a postmodern
(“free-floating signs”) cover —

The image “http://www.log24.com/log/pix06/060124-Symmetry2.jpg” cannot be displayed, because it contains errors.

This is my Princeton Companion
to Mathematics
, from the days
when Princeton University Press
had higher scholarly standards.

Monday, January 23, 2006

Monday January 23, 2006

Filed under: General,Geometry — Tags: , — m759 @ 6:00 pm

In Defense of Hilbert
(On His Birthday)


Michael Harris (Log24, July 25 and 26, 2003) in a recent essay, Why Mathematics? You Might Ask (pdf), to appear in the forthcoming Princeton Companion to Mathematics:

“Mathematicians can… claim to be the first postmodernists: compare an art critic’s definition of postmodernism– ‘meaning is suspended in favor of a game involving free-floating signs’– with Hilbert’s definition of mathematics as ‘a game played according to certain simple rules with meaningless marks on paper.'”

Harris adds in a footnote:

“… the Hilbert quotation is easy to find but is probably apocryphal, which doesn’t make it any less significant.”

If the quotation is probably apocryphal, Harris should not have called it “Hilbert’s definition.”

For a much more scholarly approach to the concepts behind the alleged quotation, see Richard Zach, Hilbert’s Program Then and Now (pdf):

[Weyl, 1925] described Hilbert’s project as replacing meaningful mathematics by a meaningless game of formulas. He noted that Hilbert wanted to ‘secure not truth, but the consistency of analysis’ and suggested a criticism that echoes an earlier one by Frege: Why should we take consistency of a formal system of mathematics as a reason to believe in the truth of the pre-formal mathematics it codifies? Is Hilbert’s meaningless inventory of formulas not just ‘the bloodless ghost of analysis’?”

Some of Zach’s references:

[Ramsey, 1926] Frank P. Ramsey. Mathematical logic. The Mathematical Gazette, 13:185-94, 1926. Reprinted in [Ramsey, 1990, 225-244].

[Ramsey, 1990] Frank P. Ramsey. Philosophical Papers, D. H. Mellor, editor. Cambridge University Press, Cambridge, 1990

From Frank Plumpton Ramsey’s Philosophical Papers, as cited above, page 231:

“… I must say something of the system of Hilbert and his followers…. regarding higher mathematics as the manipulation of meaningless symbols according to fixed rules….
Mathematics proper is thus regarded as a sort of game, played with meaningless marks on paper rather like noughts and crosses; but besides this there will be another subject called metamathematics, which is not meaningless, but consists of real assertions about mathematics, telling us that this or that formula can or cannot be obtained from the axioms according to the rules of deduction….
Now, whatever else a mathematician is doing, he is certainly making marks on paper, and so this point of view consists of nothing but the truth; but it is hard to suppose it the whole truth.”

[Weyl, 1925] Hermann Weyl. Die heutige Erkenntnislage in der Mathematik. Symposion, 1:1-23, 1925. Reprinted in: [Weyl, 1968, 511-42]. English translation in: [Mancosu, 1998a, 123-42]….

[Weyl, 1968] Hermann Weyl. Gesammelte Abhandlungen, volume 1, K. Chandrasekharan, editor. Springer Verlag, Berlin, 1968.

[Mancosu, 1998a] Paolo Mancosu, editor. From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s. Oxford University Press, Oxford, 1998.

From Hermann Weyl, “Section V: Hilbert’s Symbolic Mathematics,” in Weyl’s “The Current Epistemogical Situation in Mathematics,” pp. 123-142 in Mancosu, op. cit.:

“What Hilbert wants to secure is not the truth, but the consistency of the old analysis.  This would, at least, explain that historic phenomenon of the unanimity amongst all the workers in the vineyard of analysis.
To furnish the consistency proof, he has first of all to formalize mathematics.  In the same way in which the contentual meaning of concepts such as “point, plane, between,” etc. in real space was unimportant in geometrical axiomatics in which all interest was focused on the logical connection of the geometrical concepts and statements, one must eliminate here even more thoroughly any meaning, even the purely logical one.  The statements become meaningless figures built up from signs.  Mathematics is no longer knowledge but a game of formulae, ruled by certain conventions, which is very well comparable to the game of chess.  Corresponding to the chess pieces we have a limited stock of signs in mathematics, and an arbitrary configuration of the pieces on the board corresponds to the composition of a formula out of the signs.  One or a few formulae are taken to be axioms; their counterpart is the prescribed configuration of the pieces at the beginning of a game of chess.  And in the same way in which here a configuration occurring in a game is transformed into the next one by making a move that must satisfy the rules of the game, there, formal rules of inference hold according to which new formulae can be gained, or ‘deduced,’ from formulae.  By a game-conforming [spielgerecht] configuration in chess I understand a configuration that is the result of a match played from the initial position according to the rules of the game.  The analogue in mathematics is the provable (or, better, the proven) formula, which follows from the axioms on grounds of the inference rules.  Certain formulae of intuitively specified character are branded as contradictions; in chess we understand by contradictions, say, every configuration which there are 10 queens of the same color.  Formulae of a different structure tempt players of mathematics, in the way checkmate configurations tempt chess players, to try to obtain them through clever combination of moves as the end formula of a correctly played proof game.  Up to this point everything is a game; nothing is knowledge; yet, to use Hilbert’s terminology, in ‘metamathematics,’ this game now becomes the object of knowledge.  What is meant to be recognized is that a contradiction can never occur as an end formula of a proof.  Analogously it is no longer a game, but knowledge, if one shows that in chess, 10 queens of one color cannot occur in a game-conforming configuration.  One can see this in the following way: The rules are teaching us that a move can never increase the sum of the number of queens and pawns of one color.  In the beginning this sum = 9, and thus– here we carry out an intuitively finite [anschaulich-finit] inference through complete induction– it cannot be more than this value in any configuration of a game.  It is only to gain this one piece of knowledge that Hilbert requires contentual and meaningful thought; his proof of consistency proceeds quite analogously to the one just carried out for chess, although it is, obviously, much more complicated.
It follows from our account that mathematics and logic must be formalized together.  Mathematical logic, much scorned by philosophers, plays an indispensable role in this context.”

Constance Reid says it was not Hilbert himself, but his critics, who described Hilbert’s formalism as reducing mathematics to “a meaningless game,” and quotes the Platonist Hardy as saying that Hilbert was ultimately concerned not with meaningless marks on paper, but with ideas:

“Hilbert’s program… received its share of criticism.  Some mathematicians objected that in his formalism he had reduced their science to ‘a meaningless game played with meaningless marks on paper.’  But to those familiar with Hilbert’s work this criticism did not seem valid.
‘… is it really credible that this is a fair account of Hilbert’s view,’ Hardy demanded, ‘the view of the man who has probably added to the structure of significant mathematics a richer and more beautiful aggregate of theorems than any other mathematician of his time?  I can believe that Hilbert’s philosophy is as inadequate as you please, but not that an ambitious mathematical theory which he has elaborated is trivial or ridiculous.  It is impossible to suppose that Hilbert denies the significance and reality of mathematical concepts, and we have the best of reasons for refusing to believe it: “The axioms and demonstrable theorems,” he says himself, “which arise in our formalistic game, are the images of the ideas which form the subject-matter of ordinary mathematics.”‘”

— Constance Reid in Hilbert-Courant, Springer-Verlag, 1986 (The Hardy passage is from “Mathematical Proof,” Mind 38, 1-25, 1929, reprinted in Ewald, From Kant to Hilbert.)

Harris concludes his essay with a footnote giving an unsourced Weyl quotation he found on a web page of David Corfield:

“.. we find ourselves in [mathematics] at exactly that crossing point of constraint and freedom which is the very essence of man’s nature.”

One source for the Weyl quotation is the above-cited book edited by Mancosu, page 136.  The quotation in the English translation given there:

“Mathematics is not the rigid and petrifying schema, as the layman so much likes to view it; with it, we rather stand precisely at the point of intersection of restraint and freedom that makes up the essence of man itself.”

Corfield says of this quotation that he’d love to be told the original German.  He should consult the above references cited by Richard Zach.

For more on the intersection of restraint and freedom and the essence of man’s nature, see the Kierkegaard chapter cited in the previous entry.

Wednesday, March 31, 2004

Wednesday March 31, 2004

Filed under: General — Tags: , , , — m759 @ 12:25 am

To Be

A Jesuit cites Quine:

"To be is to be the value of a variable."

— Willard Van Orman Quine, cited by Joseph T. Clark, S. J., in Conventional Logic and Modern Logic: A Prelude to Transition,  Woodstock, MD: Woodstock College Press, 1952, to which Quine contributed a preface.

Quine died in 2000 on Xmas Day.

From a July 26, 2003, entry,
The Transcendent Signified,
on an essay by mathematician
Michael Harris:

Kubrick's
monolith

Harris's
slab

From a December 10, 2003, entry:

Putting Descartes Before Dehors

      

"Descartes déclare que c'est en moi, non hors de moi, en moi, non dans le monde, que je pourrais voir si quelque chose existe hors de moi."

ATRIUM, Philosophie

For further details, see ART WARS.

The above material may be regarded as commemorating the March 31 birth of René Descartes and death of H. S. M. Coxeter.

For further details, see

Plato, Pegasus, and the Evening Star.

Wednesday, August 6, 2003

Wednesday August 6, 2003

Filed under: General — Tags: , , — m759 @ 10:23 am

Postmodern
Postmortem

“I had a lot of fun with this audacious and exasperating book. … [which] looks more than a little like Greil Marcus’s Lipstick Traces, a ‘secret history’ tracing punk rock through May 1968….”

— Michael Harris, Institut de Mathématiques de Jussieu, Université Paris 7, review of Mathematics and the Roots of Postmodern Thought, by Vladimir Tasic, Notices of the American Mathematical Society, August 2003

For some observations on the transgressive  predecessors of punk rock, see my entry Funeral March of July 26, 2003 (the last conscious day in the life of actress Marie Trintignant — see below), which contains the following:

“Sky is high and so am I,
If you’re a viper — a vi-paah.”
The Day of the Locust,
    by Nathanael West (1939)

As I noted in another another July 26 entry, the disease of postmodernism has, it seems, now infected mathematics.  For some recent outbreaks of infection in physics, see the works referred to below.

Postmodern Fields of Physics: In his book The Dreams of Reason, H. R. Pagels focuses on the science of complexity as the most outstanding new discipline emerging in recent years….”

— “The Semiotics of ‘Postmodern’ Physics,” by Hans J. Pirner, in Symbol and Physical Knowledge: The Conceptual Structure of Physics, ed. by M. Ferrari and I.-O. Stamatescu, Springer Verlag, August 2001 

For a critical look at Pagels’s work, see Midsummer Eve’s Dream.  For a less critical look, see The Marriage of Science and Mysticism.  Pagels’s book on the so-called “science of complexity” was published in June 1988.  For more recent bullshit on complexity, see

The Critical Idiom of Postmodernity and Its Contributions to an Understanding of Complexity, by Matthew Abraham, 2000,

which describes a book on complexity theory that, besides pronouncements about physics, also provides what “could very well be called a ‘postmodern ethic.’ “

The book reviewed is Paul Cilliers’s Complexity and Postmodernism: Understanding Complex Systems.

A search for related material on Cilliers yields the following:

Janis Joplin, Postmodernist

” …’all’ is ‘one,’ … the time is ‘now’ and … ‘tomorrow never happens,’ …. as Janis Joplin says, ‘it’s all the same fucking day.’

It appears that ‘time,’ … the linear, independent notion of ‘time’ that our culture embraces, is an artifact of our abstract thinking …

The problem is that ‘tomorrow never happens’ …. Aboriginal traditionalists are well aware of this topological paradox and so was Janis Joplin. Her use of the expletive in this context is therefore easy to understand … love is never having to say ‘tomorrow.’ “

Web page citing Paul Cilliers

“That’s the dumbest thing I ever heard.”

— Ryan O’Neal in “What’s Up, Doc?”

A more realistic look at postmodernism in action is provided by the following news story:

Brutal Death of an Actress Is France’s Summertime Drama

By JOHN TAGLIABUE

The actress, Marie Trintignant, died Friday [Aug. 1, 2003] in a Paris hospital, with severe head and face injuries. Her rock star companion, Bertrand Cantat, is confined to a prison hospital….

According to news reports, Ms. Trintignant and Mr. Cantat argued violently in their hotel room in Vilnius in the early hours of [Sunday] July 27 at the end of a night spent eating and drinking….

In coming months, two films starring Ms. Trintignant are scheduled to debut, including “Janis and John” by the director Samuel Benchetrit, her estranged husband and the father of two of her four children. In it, Ms. Trintignant plays Janis Joplin.

New York Times of Aug. 5, 2003

” ‘…as a matter of fact, as we discover all the time, tomorrow never happens, man. It’s all the same f…n’ day, man!’ –Janis Joplin, at live performance in Calgary on 4th July 1970 – exactly four months before her death. (apologies for censoring her exact words which can be heard on the ‘Janis Joplin in Concert’ CD)”

Janis Joplin at FamousTexans.com

All of the above fits in rather nicely with the view of science and scientists in the C. S. Lewis classic That Hideous Strength, which I strongly recommend.

For those few who both abhor postmodernism and regard the American Mathematical Society Notices

as a sort of “holy place” of Platonism, I recommend a biblical reading–

Matthew 24:15, CEV:

“Someday you will see that Horrible Thing in the holy place….”

See also Logos and Logic for more sophisticated religious remarks, by Simone Weil, whose brother, mathematician André Weil, died five years ago today.

Saturday, July 26, 2003

Saturday July 26, 2003

Filed under: General,Geometry — Tags: — m759 @ 11:11 pm

The Transcendent
Signified

“God is both the transcendent signifier
and transcendent signified.”

— Caryn Broitman,
Deconstruction and the Bible

“Central to deconstructive theory is the notion that there is no ‘transcendent signified,’ or ‘logos,’ that ultimately grounds ‘meaning’ in language….”

— Henry P. Mills,
The Significance of Language,
Footnote 2

“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

The question of universals is still being debated in Paris.  See my July 25 entry,

A Logocentric Meditation.

That entry discusses an essay on
mathematics and postmodern thought
by Michael Harris,
professor of mathematics
at l’Université Paris 7 – Denis Diderot.

A different essay by Harris has a discussion that gets to the heart of this matter: whether pi exists as a platonic idea apart from any human definitions.  Harris notes that “one might recall that the theorem that pi is transcendental can be stated as follows: the homomorphism Q[X] –> R taking X to pi is injective.  In other words, pi can be identified algebraically with X, the variable par excellence.”

Harris illustrates this with
an X in a rectangle:

For the complete passage, click here.

If we rotate the Harris X by 90 degrees, we get a representation of the Christian Logos that seems closely related to the God-symbol of Arthur C. Clarke and Stanley Kubrick in 2001: A Space Odyssey.  On the left below, we have a (1x)4×9 black monolith, representing God, and on the right below, we have the Harris slab, with X representing (as in “Xmas,” or the Chi-rho page of the Book of Kells) Christ… who is, in theological terms, also “the variable par excellence.”

Kubrick’s
monolith

Harris’s
slab

For a more serious discussion of deconstruction and Christian theology, see

Walker Percy’s Semiotic.

Friday, July 25, 2003

Friday July 25, 2003

Filed under: General — Tags: , , — 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.

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