Abstract
This paper
examines the intersection of symbolic logic, immersive experience
[VR] and concept visualization in the interpretation of the oeuvre
of Marcel Duchamp. Influenced by the mathematicians Henri Poincaré
and Élie Jouffret as well as his own intense practice of chess and
logic, Duchamp sought to merge the poetic and visceral nature of the
aesthetic experience with the logical and systematic character of
science. This convolution of elements as disparate as chance, 3d and
4d spacetime, linguistics, logic and authorship does not allow for
comfortable definitive explanation but rather one, like his work itself,
that engages simultaneous multidimensional thinking.
Duchamp questioned
the purpose of 'retinal art', art which is merely visually beautiful,
and examined the limitations of science as a singular method of interpreting
and communicating experience. The body of his work stands as a systematic
yet playful critique of deterministic reasoning. Using symbolic logic
to characterize the most common interpretations of Duchamp's work,
the author suggests that Concept Visualization in 3d immersive experience
offers a unique method for exploring and introducing the complex lattices
of interpretation, intention and concept in the work of Marcel Duchamp.
Introduction
click to enlarge



Figure 1

Marcel
Duchamp, Rotorelief (Optical Disks), 1935

Marcel Duchamp perhaps more than any other artist
in history challenged the definition of art. Throughout his life Duchamp
maintained an interest in science, mathematics, optics and art and more
than any other eminent artist of the twentieth century understood and
researched nonEuclidean geometry and the mathematics of higher dimensionality.
Born in 1887 in Blainville (SeineInfériuere) in Normandy to a notary's
family with a history of art, love of music literature and chess, Marcel
as well as his brothers Jacques Villon, Raymond (DuchampVillon) and his
sister Suzanne became artists. Among his scientific ventures included
the development of the illusionistic Rotorelief, (Fig.
1) spinning circular geometric patterns. Although Italian optical
scientist discovered and named this optical phenomenon "the stereokinetic
effect" in 1924, it is clear that Duchamp had discovered the phenomena
in early 1920. It is clear that Duchamp understood the mathematics of
this method of producing the illusion of volume. He wrote, " I only
had to use two circumferenceseccentricand make them turn on a third
center".
click to enlarge



Figure 2

Figure 3

Leonardo
da Vinci, La Gioconda [Mona Lisa],
150305, Musée du Louvre, Paris.

Marcel
Duchamp, L.H.O.O.Q., 1919

The history of art is filled with artists whose discoveries
and research were labeled or advocated as objet d' art and
whose scientific utility was not discovered until many years often
centuries later. However, it is the supposition of this author, that
no other artist cloaked his or her intentions in deception as a tactic
to subvert conventional interpretation. Noted and controversial Duchamp
scholar Rhonda Shearer has garnered attention by a stunning hypothesis
about the many realms of Duchamp's work. In 1919 Duchamp drew a supposed
impromptu mustache on a reproduction of Leonardo da Vinci's Mona Lisa (Fig. 2) and titled it L.H.O.O.Q.
(Fig. 3) When these letters are read aloud
they say "Elle a chaud au cul" or "She has a hot ass"
in French. Having created this work of art Duchamp stated that it
revealed a truth about his noted foregoer.
Duchamp championed the "readymade", a manufactured
object transformed into art merely by its selection and placement
in an aesthetic gallery or museum context. In so doing, Duchamp altered
the significance of the objet d' art as a precious commodity
created by the artist. Duchamp often maintained complex documentation
of the purchase or discovery of his "foundobjects". In
the case of L.H.O.O.Q., Duchamp asserts that it was purchased
in a postcard shop on Paris' rue de Rivoli. This notion that the art
object is defined and given value by its context not by an empirical
judgement of aesthetic value would transform the art of the twentieth
century, greatly influencing Conceptual Art and Postmodern movements.
Duchamp's assertion that art is a matter of selection and context
was perhaps a precursor to Baudrillard's Second Order of Simulacra.
According to Shearer, Duchamp had another more subversive
objective. She asserts that Duchamp's L.H.O.O.Q. was in fact
a creation of Duchampa composite photograph of himself taken in
1912 and a reproduction of Leonardo Da Vinci's Mona Lisa. Shearer's
research suggests that not only was L.H.O.O.Q. created by
Duchamp or fabricated to his specifications but so was the snow shovel
in In Advance of the Broken Arm (Fig. 4), the bird cage in Why Not Sneeze Rose Sélavy (Fig. 5), the ampoule in Ampoule Contenant 50cc d'air Paris (Ampoule containing 50cc of Paris air) (Fig.
6), and the urinal in Fountain (Fig. 7) signed R. Mutt (the famous object refused
exhibition in the Society of Independent Artists show in 1917). This
controversial theory is gaining greater attention in recent years,
although not without significant turmoil. The notion of the "readymade",
would remain safe according to Arthur Danto, art critic for the Nation,
and Thierry di Duve, author of Kant After Duchamp, as this
concept of "art object in context" has been an accepted
convention of art making and interpretation for three quarters of
a century. However if Shearer proves to be correct in her assertions
both Danto and the Immanent Duchamp scholar and author Francis Nauman
(Marcel Duchamp: The Art of Making Art in the Age of Mechanical
Reproduction) would find this a "grand act of deception."
click to enlarge



Figure 4

Figure 5

Marcel
Duchamp, In Advance of the Broken Arm, 1915/64

Marcel
Duchamp, Why Not Sneeze Rrose Sélvey?, 1921



Figure 6

Figure 7

Marcel
Duchamp, Paris Air, 1919/49

Marcel
Duchamp, Fountain, 1917/64

Duchamp's work,
particularly that which displays his keen interest in science and
mathematics is also garnering attention outside of the disciplines
of art history and art criticism. New York University Physicist Jonathan
Williams postulates that Duchamp's deep play with physics or what
Duchamp and the playwright Alfred Jarry referred to as "pataphysics",
was a systematic way of satirizing early 20th century deterministic
systems of scientific thinking. Duchamp, according to Williams, began
this direction through his investigations of nonEuclidean geometry,
fourth dimensional spacetime, electromagnetism, and radiation. Duchamp's
playful explorations of these areas seems to be a harbinger, of sorts,
for some of the foundations of quantum mechanical theory such as Heisenberg
uncertainty principle and the Erwin Schrodinger equations.
Duchamp's systematic
critiques were not limited to the scientific thinking of the day but
also confronted modes of artistic production. Duchamp's disinterest
in what he referred to as "retinal art" or art which solely
engaged the reproduction of visual experience was methodically deconstructed
and supplanted by an art which focused on the grey matter or existed
in the realm of pure intellect. "All through the nineteenth century
the phrase 'bête comme un peintre' or 'as stupid as a painter'",
Duchamp said. "And it was true that kind of painter who just
puts down what he sees is stupid." For Duchamp, traditional art
making merely copied itself in kind of mobius strip of mimesis and
selfreflection simply cloning itself over and over again.
As the myth goes,
Duchamp gave up art in favor of playing chess throughout the world.
"All chess players are artists but not all artists are chess
players." Duchamp used chess as a kind of model for much of his
work, using it in his explorations of physics, mathematics and logic.
This does not mean that he solely engaged in a form of sublime mathematically
derived art. He continued his love of semantics, word games and humor
throughout his life one sees this in L.H.O.O.Q. as well as
his frequent use of the alter ego Rrose Sélavy (Eros is Life). Any
complete and singular interpretation of Marcel Duchamp's work is quite
impossible as it is a lattice manifesting complex interweaving of
intentions; mathematics, science, logic, art, consumer critique word
play and alter ego.
Parsing the Oeuvre
The human propensity for
binary oppositional thinking has been studied by psychologists and
linguist and is evident in a great portion of western philosophy from
Diogenes Laertus 200AD (lives and Opinions of Eminent Philosophers)
to the present. Nowhere is this more apparent than in the discipline
of art history, particularly as practiced in the first half of the
twentieth century. Often art history has been essentially a historiography
of connoisseurship, determining why one body of work by one artist
is necessarily better than another body of work by another artist.
This reasoning was held together by the supposition of progress, in
the inevitable evolution of one mode of artistic production to another.
Though not the subject of this study it is notable that when one examines
art historiography (or any historiography for that matter) events,
intentions and outcomes are often limited to singular reasoning. The
academy, when confronted by opposing singular reasoning on any particular
individual or subject matter, results in imbroglio. Critical interpretation
of Duchamp's body of work and his intentions is vested in this sort
of competition between singular theories.
Duchamp consistently
plays with these battles often examining them through his own prism.
As Duchamp advocates the value of the "readymade" he simultaneously
describes his intent to define existence "through slightly distending
the laws of physics and chemistry." By using his own form of
"absurd mathematics" he simultaneously critiques scientific
thinking, logic, the definition of art, the artistic mode of production
and aesthetic interpretation. Duchamp writes:
Calcul par l'absurde mathématique algébrique 
SiA=intention 10.
B=Crainte 5
C=Desir
on a une première équation
C=ab
et une 2 équation
C=A x B
Math ces 2 éq. Sont absurdes
C=50 / C = 5 / 2C = 55 / 2=27.5
Si A = 10 / B = 5 / c = 27.5
Si A = a = 9 / B=a/ 3 =3/ C +33/ 2=16.5 
This formula
demonstrates the limits of scientific reasoning by illustrating its
inability to explicate immeasurable, highly personalized, data. By
inserting variable equivalencies for intention, dread (crainte) and
desir (desire) Duchamp demonstrates the limits of deterministic mathematics
and hints at his attempts to demonstrate the breakdown of rational
models of examining reality. Jonathan Williams in his article
Pata or Quantum the End of Deterministic Physics, likens
this propensity in Duchamp's work to the Schrodinger equations which
demonstrated that the behaviors, qualities and position of quantum
particles needed to be expressed in terms of statistical probabilities
thus rendering vacant the possibility of determining that a subatomic
particle had any fixed quality at any given point in time. Duchamp's
continued fascination with illustrating this notion also points out
his interest in expressing the difficulty that systems of logic have
in making distinct calculations of human emotional variables. As one
can see when examining the diversity of Duchamp's work and intentions,
traditional art historical and art critical interpreters were and
in many cases remain befuddled.
####PAGES####
Symbolic Logic and Visualizing Concept in the Work of Marcel Duchamp
The seemingly
innate propensity of the human mind to binary, "this not that
thinking" may be fruitfully illustrated by using symbolic logic
in an interpretation of the intentions of Marcel Duchamp. With symbolic
logic as a form of information or concept visualization one can demonstrate
the difficulty and visual complexity of examining Duchamp's work using
a deterministic system of equivalencies.
First let us
use the most oft sited interpretations of Duchamp's oeuvre and their
corresponding significances and assign to each a variable. The following
is by no means an exhaustive codification of the many interpretations
of Marcel Duchamp's works it is essentially a categorization of a
few of the major theories.
A. The use of the "readymade" or "found object" asserts
that by altering the context of a commonplace object it can become
art.
D. The statement A allows that Duchamp in challenging the definition of the art object by exalting the primacy of the idea over the creative
act he subverted the modernist convention of the artist/object and
viewer relationships
B. The work was an exploration of the mathematics of uncertainty pioneered
by Henri Poincaré and the study of the fourth dimensional space theorized
by Élie Jouffret (Traité Élémentaire de Géometrie à Quatre Dimensions
1903).
H. The statement B allows that Duchamp called into question the discipline boundaries between art and science and destroys the notion of the
artist as creator of 'retinal art' or the aesthetic object.
C. The work
was engineered to be reassembled by the patron or viewer, who followed
complex, often informed by chance, instructions. This process is evident
in Duchamp's assemblage book works such as La Bôite Verte.
P. The statement C allows that Duchamp transformed the
boundaries between producer and consumer in the art market and engaged
the artist/manufacturer and viewer in the process of creation.
We will next
make a formula that contextualizes more precisely the relationship
between the upper level referent variables A, B and C (in this case
those variables that refer to the condition of Duchamp's work rather
than his intentions). Supplanting a corresponding lower case Greek
letters, A becoming a(alpha), B becoming b(beta), and C becomes g(gamma)
the following is the rule, universal quantifier or binding of variables
governing their relationships. (Fig. 8)

Figure 8

The above statement allows that there can only be a single interpretation
of the work of Duchamp. In accordance with the current highly polarized
arguments about his work the equation illustrates that, of the contemporary
hypothesis, only one can be correct. The next equations maintain that
there can only be a single derived intention from the overall statement
of the condition of Duchamp's work. In other words the statement:
"The work is an exploration of the mathematics of uncertainty
pioneered by Henri Poincaré and the study of the fourth dimensional
space theorized by Élie Jouffret" may only be linked to the intention,
"Duchamp calls into question the discipline boundaries between
art and science and destroys the notion of the artist as creator of
'retinal art' or the aesthetic object". We may express this with
symbolic logic in the following: (Fig. 9)

Figure 9

The next section
of this exploration of the work of Marcel Duchamp through symbolic
logic will determine the consistency of each of the condition/intention
hypotheses. First we will examine the consistency of the argument
A, if and only if, D:(Fig. 10)

Figure 10

The statement
A if and only if D proves logically consistent. Next
we have the hypothesis B, if and only if, H: (Fig.
11)

Figure 11

Having proved the statement B if and only if H consistent
we address the theory C if and only if P:(Fig. 12)

Figure 12

Having proved the consistency of all three hypotheses given the a
priori context that only one of them is correct where are we left
in this complex visual analysis of Duchamp's intentions? His intimate
knowledge of Henri Poincaré's theories and his use of chance in the
construction of many of his key works would seem to indicate that
Duchamp had been quietly challenging the notion of deterministic reasoning
in both the interpretation of art and physical and experiential phenomena.
As Poincaré suggests in 1895;
Experiment
has revealed a multitude of facts which can be summed up in the
following statement: it is impossible to detect the absolute motion
of matter, or rather the relative motion of ponderable matter
with respect to the ether; all that one can exhibit is the motion
of ponderable matter with respect to ponderable matter. 
In short, Poincaré
asserts that all quantifiable and qualifiable information pertaining
to any phenomena can only be measured relative to other qualified
and quantified data. As the first to elucidate this "principle
of relativity" Poincaré discerned that all explicit information
about any physical phenomena in motions is best expressed in the form
of a probability. Poincaré's critique of determinism extends to other
disciplines as well as he states, "The science of history is
built out of bricks; but an accumulation of historical facts is no
more a science than a pile of bricks is a house." This kind of
reasoning is the bedrock of semiotics (meaning in language is ascertain
through the relationship between the symbol and its meaning relative
to the culture that produced it). It has also been used to critique
symbolic logic. The discipline itself relies on abstract patterns,
its meaning determined not from the symbols themselves but from the
relationship between the marks and other patterns and more significantly
cultural meanings.
Duchamp's connection
to logic is most clearly noted in two of his most significant areas
of concern: chance and chess. As a chess master Duchamp was, on several
occasions, a member of the French championship chess team. For Duchamp
chess was an organized, integrated and ordered whole, composed of
rule based interactions wherein outcomes were as influenced by unquantifiable
elements such as guile or desire as by systematic reasoning. This
led Duchamp to assert that complexity in any system was inherently
nondeterministic. We see this questioning of aggregation, perhaps
more clearly, in his use of chance in aesthetic production.
click to enlarge


Figure 13

Marcel
Duchamp, Three Standard Stoppages, 191314

Duchamp's subversion
of deterministic systems through chance finds vent in his Trois StoppagesÉtalon
[Three Standard Stoppages], 1913 (Fig.
13), He first measures three sections of thread each precisely
one meter in length and drops them from a height of exactly one meter.
He then uses the curves of the threads to produce three templates cut
from a straight edge to produce the work. The templates are then enshrined
in a box and become the piece Trois StoppagesÉtalon. This work
and activity albeit unusual, is a subversion of the concept of immutable
standards of measurement, thus questioning the validity of a system
based on a platinumiridium bar stored in a Parisian vault. His attention
to chance not only posited an alternative to early twentieth century
"laws" of science but also undermined early twentieth century
conventions about aesthetic production.
Le Penseur MultiDimensionnelle
Duchamp's work
Nude Descending a Staircase (1912) was first displayed at the
Cubist Exhibition at the Damau Gallery in Barcelona and later at the
Armory Show (New York, 1913). This painting took the observational
cubist penchant of displaying an object from multiple spatial vantagepoints
and added a temporal element by rendering a nude figure in motion.
This work explored the conceptual possibility of 2d painting, which
displayed and illustrates a 3 dimensional figure traversing time.
The piece arrives at a visceral form of multidimensional cognition.
Partial inspired by his interest in chronophotography and the mathematics
of Henri Poincaré Nude Descending a Staircase is perhaps
his last clear attempt to use a traditional modality of retinal art
to express a conceptual or gray matter art. It is also his first widely
exhibited work to express his interest in the merger of science and
art.
His continued
interest in multiple dimensions, though I cannot prove this, is probably
where we may find the solution or at least a map to a clear understanding
of his work. Though we may never have a concise definition of "what
his work was about" Duchamp may have left us clues as to how
we may begin to "make sense" of his intentions.
click to enlarge


Figure 14

Front
view of the postcard in the White Box, 1967

Rhonda Shearer and
Stephen J. Gould In their article Boats
and Deckchairs present the most profound example of Duchamp's
trickery and play with the multidimensional mathematics of Henri Poincaré
and Élie Jouffret. Inside Duchamp's 1967 piece White Box Francis
Nauman discovered a "commercial" postcard (1914) (Fig.
14). The postcard displays on its front three boats floating
on a placid lake or river and on the reverse some writing. Nauman categorized this discovery as a "random notation" written on a "found
object" citing that "on the verso of a postcard, Duchamp notes
'a possible means by which the fourth dimension could be visually established
through the optical illusion of two deck chairs'." This note was
accompanied by an illustration of parallel lines bisected by a perpendicular.
The true nature of this object has never been addressed by art historians
as the work could be safely categorized as one of Duchamp's "readymades."
The piece is
in fact, an original painting not a commercial postcard. The curious
parallel and perpendicular lines on the back are in fact obscure instructions.
Duchamp's fascination with rotation and relative vantagepoints indicates
that a new dimension may be experienced through altering ones position
relative to an object. When the postcard is turned 90º to the right
the boats become an orthogonal rendering of deckchairs viewed from
a bird's eye vantagepoint. The mysterious "random note"
on the verso is a plea to adjust your perspective when viewing the
postcard but also, when correlated with the image from the front the
piece becomes a profound statement about the relationship between
the second, third and fourth dimensions.
Like E. A. Abbot's
famous book Flatland (1885) whose main character, a
square, is shockingly introduced to the third dimension, Duchamp has
demonstrated for us that one can examine from a three dimensional
vantage point all sides of a two dimensional object. In turning the
postcard we are taking a clearly twodimensional image and viewing
it from the third dimension wherein the objects in question become
something entirely different. In postcard he begs the analogy:
that when viewed from the fourth dimension a threedimensional object
may be seen from all sides. From years of singular interpretations
of Duchamp's oeuvre art historians have safely ignored Duchamp's multiple
interpretations: one obvious and the others subversive. When the work
is proclaimed (by the artist himself) and interpreted as a "readymade"
the hidden intention with all of its possible significance is obscured.
Duchamp disavowed
models of reasoning, which relied on singular definitions. This kind
of one or twodimensional interpretation is inherently flawed when
attempting to ascertain his intentions. With this in mind, Duchamp's
work requires that any conceptual model of his intentions necessitates
threedimensional thinking and thusly is well suited to threedimensional
visualization.
Immersive Experience and Concept Visualization
As demonstrated,
the use of symbolic logic as a means to visualizing concept in the
work of Marcel Duchamp is extremely difficult. Though I have not examined
the use of more advanced forms of symbolic logic (I am not a logician)
it is apparent that the data, as envisaged, is not of the highest
utility.
click to enlarge


Figure. 15

Screen
still from the author's Immersive Duchamp Concept World an
interactive virtual reality computer art piece.


Figure. 16

Screen
still from the author's Immersive Duchamp Concept World an
interactive virtual reality computer art piece.

Clearly,
Duchamp had multiple intentions and the existence of seemingly inconsistent
hypotheses about his work point more to the human propensity for dualistic
thinking rather than to grasping a more pluralistic possibility. Engaging
data that is not quantifiable and highly subjective is difficult to
manage logically and exceedingly difficult to graph. However if we create
a 3d cartographic form of the logic equations introduced earlier in
this paper, we make the data more intuitive and thus cognitively manageable.
(Figs. 15 & 16) Using interactive
virtual reality software[the software we use in this example is the
Glass Virtual Reality Engine, created by the author] one can have an
immersive experience of the main theories about Duchamp's work.
The virtual reality
computer art piece Immersive Duchamp Concept World, presents
the theories concerning the artist's work. At the center of the virtual
space is the entrance point to the world. The immersant or
viewer may follow the map which branches off to various nodal points.
Each of these nodal points represents a single theory. From the vantagepoint
of the theory the immersant sees the other possible theories through
a fog and translucent sheets, they are barely visible, as the immersant/viewer
has chosen an alternate path (Fig. 15).
In Immersive Duchamp Concept World the immersant is also introduced
to various interactive media; readings of Duchamp's Notes as well
as to still images and animations of his work and to the writings
of Henri Poincaré. If the immersant chooses to fly above the object
it is from this vantagepoint the viewer sees all of the theories as
a totality (Fig. 16). This totality,
is essentially a relativistic rather than a fixed deterministic system
as the viewer governs the experience. This model for information visualization
does not stand in opposition to symbolic logic, however it does allow
a form of concept visualization that merges reason quantification,
qualification and the visceral.
Conclusion
The body of work
produced by Marcel Duchamp was a programmatic, if playful, undermining
of deterministic thinking. He demolished arbitrary discipline boundaries
between artist, scientist and mathematician. His clues to altering
our perspective were equally pertinent to viewing and understanding
his oeuvre as they were to viewing individual works of art. His implicit
and explicit call for altering our vantagepoint relative to his intentions
inherently calls into question modernist singular interpretations.
Yet, through the use of concept visualization, we can create more
exploratory modes of information visualization; modes which allow
for simultaneous multiple dimensional thinking. In an immersive environment
the viewer can experience a panorama of Duchamp's intentions, one
that does not enforce strict rules of consistency, but nonetheless
leads us to comprehension of a polydynamic yet visceral logic.
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Figs. 1, 3~7, 13, 14
©2003 Succession Marcel Duchamp, ARS, N.Y./ADAGP, Paris. All rights reserved.

