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)

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)

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

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 multi dimensional 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.

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 two-dimensional 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 three-dimensional 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 "ready-made" 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 two-dimensional 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 three-dimensional thinking and thusly is well suited to three-dimensional 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.

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 poly-dynamic yet visceral logic.