Conventionalism in Henri Poincaré and Marcel DuchampAuthor(s): Craig AdcockSource: Art Journal, Vol. 44, No. 3, Art and Science: Part II, Physical Sciences (Autumn, 1984),pp. 249-258Published by: College Art AssociationStable URL: http://www.jstor.org/stable/776825 .Accessed: 03/01/2011 10:40

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Conventionalism in Henri Poincare and Marcel Duchamp

By Craig Adcock

T he mathematical and philosophical writings of Henri Poincar6 (1854-

1912) were among the major influences on the development of Marcel Du- champ's art and thought.' Through Poincare, Duchamp learned the basic principles of at least three major branches of modern geometry: n-dimen- sional geometry, non-Euclidean geome- try, and topology.2 So important were these geometries to Duchamp that he incorporated numerous references to them in the iconography of his major works: the Bride Stripped Bare by Her Bachelors, Even (the Large Glass), 1915-23 (Fig. 1), Tu m', 1918 (Fig. 2), the ready-mades, and the satellite works associated with them. For Duchamp, Poincare's philosophical discussions of the conventional nature of geometry were a way of reinforcing his own spec- ulations about the provisional nature of aesthetics.

Many of Poincar6's most basic insights into complex areas of mathe- matics involved applications of geome- try. His brilliant work in his "qualitative theory of differential equations," for example, was predicated on topology. His work on automorphic functions of one complex variable (what he called Fuchsian functions) and his work on analytic functions of several complex variables (Abelian functions) also in- volved the application of geometrical techniques to problems that might otherwise have proven recalcitrant. Poincar6 argued that invention was often a matter of choice, involving put- ting things together that did not seem to belong together: "Among chosen combi- nations the most fertile will often be those formed of elements drawn from domains which are far apart."' He recalled that some of his own most important insights had involved just such unlikely combinations. In Science and Method, he writes that he had made breakthroughs in the study of Fuchsian functions because he had suddenly seen relationships between the transforma- tions he was using to specify differential equations and those that occurred in non-Euclidean geometry. He also recalls

Fig. 1 Duchamp, The Bride Stripped Bare by Her Bachelors, Even (the Large Glass), 1915-23, oil and lead wire on glass, 1091/4 x 691/8". Philadelphia Museum of Art, Bequest of Katherine S. Dreier.

Fall 1984 249

Fig. 2 Duchamp, Tu m', 1918, oil on canvas with long brush attached, 271/2 x 1223/4". Yale University Art Gallery, Bequest of Katherine S. Dreier.

that on another occasion, but with the same sense of immediacy, he had real- ized that "the arithmetic transforma- tions of indeterminate ternary quadratic forms were identical with those of non- Euclidean geometry."4

That Poincar6 could use geometries such as topology and non-Euclidean geometry and obtain useful results was important to his world view. Not too long before his time, only the postulates of plane and solid Euclidean geometry were believed to have any validity. The development of such mathematical con- structs as complex numbers, quater- nions, n-dimensional geometry, and non-Euclidean geometry during the course of the nineteenth century had forced mathematicians to reexamine a number of their fundamental assump- tions. No longer could they believe that mathematics represented a true picture of the world-that mathematical prog- ress was a matter of uncovering the hidden laws of nature. They were forced to admit that certain aspects of mathe- matics are the constructs of human rea- son.5 One could no longer deny the exis- tence of such mathematical entities as n-dimensional spaces with n > 3 or of non-Euclidean spaces in which Euclid's parallelism postulate does not hold. Such spaces may not have physical ana- logues, but they are mathematically no less real than three-dimensional Euclid- ean space.

Some of Poincare's most important mathematical discoveries involved methods of interrelating various geome- tries. In Science and Hypothesis, he describes, in accessible terms, a kind of dictionary through which Euclidean and non-Euclidean geometries could be translated from one to the other. He explains that if the theorems of a non- Euclidean geometry were translated according to the terms of his dictionary, as one might translate a passage from one language into another, one would arrive at the "theorems of the ordinary geometry."6 Moreover, such transla- tions would necessarily be consistent: if a contradiction were to occur between two theorems of the non-Euclidean

geometry, a contradiction would occur between the two Euclidean theorems from which they had been translated. But since "these translations are the- orems of ordinary geometry and no one doubts that the ordinary geometry is free from contradiction," questions regarding the legitimacy of non-Euclid- ean geometry should not arise.7 The important part of this argument for Poincar6 was that non-Euclidean geom- etries, because they could be interpreted in terms of the unquestioned proposi- tions of ordinary Euclidean geometry, were no longer just empty displays of logic but could be concretely useful in applied mathematics. He was quick to point out that he had obtained impor- tant results by applying Lobachevski's geometry to the integration of linear differential equations.8

p oincare's understanding of geome- try influenced his philosophical

insights. Because one geometry could be translated into another and because there was no way of determining if one geometry was more true than another, he argued that geometry was conven- tional. Poincar6 is the father of philo- sophical conventionalism. From such a position, there are numerous ways of describing the world; any one way can- not be said to be more true than any other-one way can only be said to be more useful than another under a cer- tain set of circumstances. Such argu- ments can lead to arbitrariness, as Poin- care himself was well aware. Since he believed that there was an external world and that it could be discovered through science, he eschewed extreme conventionalism, or "nominalism," as he called it.

Poincar6 argued that there were a number of different kinds of hypotheses. Some were verifiable facts and could be considered truths; some were useful in organizing research approaches but were unverifiable; and some were "dis- guised definitions" or "conventions." These last, conventional hypotheses were most likely to be encountered in mathematics and science. They were the

invention of human reason: "these con- ventions are the work of the free activity of our mind, which, in this domain, recognizes no obstacle. Here our mind can affirm, since it decrees; but let us understand that while these decrees are imposed upon our science, which, with- out them, would be impossible, they are not imposed upon nature."9

Poincar6 believed that the source of the general laws of mathematics and science lay in the rational human intel- lect; that they were rational was what gave them their rigor and their essential usefulness. But Poincar6 also asked whether, if made up in the mind, they were then also capable of being applied in random fashion. No, he said: "Experi- ment leaves us our freedom of choice, but it guides us by aiding us to discern the easiest way."10 In other words, experiment in the real world orders and directs one's choice of first principles. Some are useful in discovering the world, and some are not. Science pro- ceeds by discovering and making use of those conventional hypotheses that are productive and by discarding those that are unproductive.

Poincar6 pointed out that some inves- tigators had gone too far in what they took to be the implications of the con- ventional nature of scientific principles: "they have wished to generalize beyond measure, and, at the same time, they have forgotten that liberty is not license. Thus they have reached what is called nominalism." Poincar6 suggested that such thinkers should "have asked them- selves if the savant is not the dupe of his own definitions and if the world he thinks he discovers is not simply created by his own caprice."" If science did operate according to the random inven- tion of descriptive models, it could never tell us anything about the natural world and the objects that occupy that world. "Still," Poincar6 added, "the things themselves are not what [science] can reach, as the naifve dogmatists think, but only the relations between things. Out- side of these relations there is no know- able reality."12

Geometry and mathematics were

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ways of articulating the relationships between objects of perception. They were not the same as experimental facts, but they were useful in picturing the world. Poincar6 argued that because the axioms of more than one kind of geome- try could be shown to be consistent, one could no longer believe that Euclidean geometry was the one true way of describing space and its inhabitants. He pointed out that

experience no doubt teaches us that the sum of the angles of a triangle is equal to two right angles; but this is because the triangles we deal with are too lit- tle; the difference, according to Lobachevski, is proportional to the surface of the triangle. Will this perhaps become sensible when we operate on larger triangles or when our measurements become more precise? The Euclidean ge- ometry would thus be only a provi- sional geometry."

What this provisional status implied for Poincare was that "one geometry cannot be more true than another; it can only be more convenient."14

Poincare's philosophy developed out of his mathematics. In particular, his attitudes were conditioned by his studies involving the relationships among dif- ferent kinds of geometry. One could invent the principles of geometry and develop them through logic. These prin- ciples were conventions and disguised definitions, but, nonetheless, they were "drawn from experimental laws."'5 In certain ways, Poincare's position was a precursor to logical positivism. In Science and Method, he argues that

a demonstration truly founded upon the principles of analytic logic will be composed of a series of propositions. Some, serving as premises, will be identities or defi- nitions; the others will be deduced from the premises step by step. But though the bond between each proposition and the following is immediately evident, it will not at first sight appear how we get from the first to the last, which we may be tempted to regard as a new truth. But if we replace succes- sively the different expressions therein by their definition and if this operation be carried as far as possible, there will finally remain only identities, so that all will reduce to an immense tautology. Logic therefore remains sterile unless made fruitful by intuition.l6 In his discussion of intuition in The

Value of Science, Poincar6 writes that even when scientists thought that their

research was free from intuitive ap- proaches they were deluding themselves. Innovation requires intuition: "Pure logic could never lead us to anything but tautologies; it could create nothing new; not from it alone can any science issue."" Poincar6 thought that mathe- matics and the other sciences progressed by something more than syllogistic arguments from first principles: they progressed by intuitive invention. One of the sources for original creation was "generalization by induction, copied, so to speak, from the procedures of the experimental sciences."18

D uchamp's philosophy also shares characteristics with positivism. In

his interview with Pierre Cabanne, Duchamp pointed out that

the Viennese logicians worked out a system wherein everything is, as far as I understand it, a tau- tology, that is, a repetition of premises. In mathematics, it goes from a very simple theorem to a very complicated one, but it's all in the first theorem. So, metaphysics: tautology; religion: tautology; everything is tautology, except black coffee because the senses are in control! The eyes see the black coffee, it's a truth; but the rest is always tautology.'9

Duchamp's reference here to the Vienna Circle is suggestive. Philipp Frank recalled that some of the most pressing questions that concerned him and his associates in Vienna, a group that included Otto Neurath and Hans Hahn, involved the relationship between exper- imental facts and scientific hypotheses: "In our opinion, the man who bridged the gap successfully was the French mathematician and philosopher Henri Poincar6. For us, he was a kind of Kant freed of the remnants of medieval scho- lasticism and anointed with the oil of modern science."20

One of the apparent discrepancies or problems in the development or progress of science concerned the role of scien- tific generalizations. Material science seemed to advance steadily with respect to the gradual accumulation of knowl- edge and facts acquired through experi- ment. But at the same time, grand the- ories or large-scale theoretical schemata were likely to be proven false and were then discarded by the scientific commu- nity. If a theory was soon to be thrown out, why advance it in the first place? What was the role of hypothesis in the advancement of science? An important part of Poincar6's philosophical inquiry addressed the latter question.

Duchamp was faced with a similar kind of problem within the operational

structure of the art world. How did progress in art proceed and what was the role of theory or aesthetics within that process? On the one hand, Duchamp had become disenchanted with practical matters of aesthetics: from personal experience, he had found that such things could be arbitrary and guided by suspect taste.21 On the other hand, he was intensely involved with making con- crete contributions to art, especially during his early years. He was like a scientist making discoveries in a labora- tory but not knowing how to unify his facts within an encompassing theory. Through his reading of Poincar6, Duchamp found a paradigm for articu- lating his art problems; science provided a metaphorical schema for defining the process of making art.

P oincar6's system of doubt does not imply a lack of faith in science or in

the ability of humans to do science. That one could not decide on the ultimate truth or falsity of certain kinds of hypo- theses did not mean that one had to retreat into arbitrariness. Duchamp's system had a similar kind of a reason- ableness about it. His system was careful, plotted, consistent, and subtle. What emerges from Duchamp's doubt is a careful skepticism-a skepticism that also characterizes the best scientists. What one sees in Duchamp, as in Poincar6, is a healthy willingness to question-a process that leads to new solutions. It was in this regard that Poincar6 was most important to Du- champ. It was a matter of one genius reinforcing another genius. It was a mat- ter of the best of human endeavor (science) reinforcing the best of human endeavor (art).

Poincar6 articulates his discussions of the provisional nature of different kinds of geometry by demonstrating their interrelatedness. By showing the con- nections between, say, metric geometry and projective geometry, between Eu- clidean and non-Euclidean geometry, and by then showing their various rela- tionships with topology, Poincar6 could show that how one chose one's geometry was a matter of convenience. Duchamp wanted to demonstrate the conventional nature of aesthetics-to show that how one chose one's art was also a matter of context.

Tu m', 1918 (see Fig. 2), clearly shows that approach. The painting can be taken as a demonstration of the prin- ciples discussed by Poincar6, and the philosophical implications of the paint- ing for art are similar to those that Poincar6 articulated for science. Tu m' is susceptible to interpretation from the point of view of several different geome- tries. The cast shadows or projected

Fall 1984 251

shadows (ombres portedes) are refer- ences to projective geometry. Perspec- tive is a subcategory of projective geom- etry, and part of the subject matter of Tu m' is perspective. On one level, Duchamp was dealing with a traditional two-dimensional picture surface that represents a three-dimensional space. This is already n-dimensional geometry with n = 2 and n = 3. But Duchamp was also interested in situations where n > 3. Such situations are considerably more complex.

Because it could be thought of as an analogy for projection, the concept of a cast shadow was important in Du- champ's thinking about n-dimensional geometry. Part of the imagery of Tu m' consists of shadows cast by three of his ready-mades: the Bicycle Wheel, the Corkscrew, and the Hat Rack. The Corkscrew has survived only as a shadow on Tu m'. The shadows of the three ready-mades are outstretched across the surface of the canvas with the Bicycle Wheel on the left and the spiral of the Corkscrew leading out from the axle of the wheel to the center of the painting. A sign-painter's hand emerges from the handle of the Corkscrew and points towards the shadow of the Hat Rack on the right. These shadows are analogies: if a three-dimensional object casts a two-dimensional shadow, then by analogy a four-dimensional object would cast a three-dimensional shadow.

In one of his notes for the Large Glass in A l'Infinitif, Duchamp used that anal- ogy: "The shadow cast by a four-dimen- sional figure on our space is a three- dimensional shadow."22 He then refers to a passage is Esprit Pascal Jouffret's Trait eblementaire de geometrie it quatre dimensions: "See Jouffret, Geom. of 4 dim., page 186, last 3 lines." In those lines, Jouffret suggests that "in this regard [conceptualizing the fourth di- mension]," one would do well to "con- sider the horizontal shadow that attaches itself to you as you walk along in the sun and that, long or short, wide or narrow, repeats your movements as if it under- stood you, although it is only an empty semblance."23 Jouffret uses the analogy of the cast shadow to introduce a discus- sion of flat-beings. Since it is difficult to envision higher-dimensional spaces, it might be useful to envision lower-dimen- sional spaces. By imagining what it would be like to live in a flat, two- dimensional plane, one could better appreciate the relationships between three-dimensional beings and four- dimensional spaces.

In Tu m', three-dimensional ready- mades become flat, two-dimensional shadows. They become, in a sense, flat- beings. In one of his notes for the Large Glass included in the Green Box,

Fig. 3 Duchamp, "Shadows of Ready-mades," photograph taken in Duchamp's studio, 33 West 67 Street, New York, 1918. Collection, Mme Marcel Duchamp, Villiers-sous-Grez.

Duchamp discusses the "cast shadows of ready-mades."24 The inclusion of this note in the Green Box indicates that he intended that the imagery both of Tu m' and of the Large Glass involve shadows cast from ready-mades. A photograph taken by Duchamp in his New York apartment in 1918 (Fig. 3) reinforces this idea. The forms of the shadows are very similar to the flat shapes of the Bride in the upper panel of the Large Glass.

The implications of n-dimensional pro- jection were explained by Duchamp on several occasions. He told George Heard Hamilton and Richard Hamilton:

anything that has three-dimen- sional form is a projection in our world from a four-dimensional world, and my Bride, for example, would be a three-dimensional pro- jection of a four-dimensional Bride. All right. Then, since it's on the glass it's flat, and so my Bride is a two-dimensional representa- tion of a three-dimensional Bride, who also would be a four-dimen- sional projection on a three- dimensional world of the Bride.25

In his interview with Pierre Cabanne, Duchamp connected this kind of discus- sion of projective geometry with the notion of cast shadows:

Since I found that one could make a cast shadow from a three-dimen- sional thing, any object whatso-

ever... I thought that, by simple intellectual analogy, the fourth dimensional could project an object of three dimensions. . ... "The Bride" in the "Large Glass" was based on this, as if it were the projection of a four-dimensional object.26

In Duchamp's thinking, Tu m', the Large Glass, the notes, the ready- mades, their cast shadows, and their geometrical implications were all tightly interconnected.

The same may be said of their philo- sophical implications. Tu m' is a meta- painting and perhaps metaphysical. Du- champ may have intended his shadows as Platonic references; he may have wanted to suggest that the work of art was a shadow of a shadow. If so, the reasons would not have been Plato's, but Duchamp's. More precisely, they would have been Duchamp's as developed out of Poincar6. By 1918, Duchamp had come to feel that painting was largely empty semblance. He was interested in the idea behind the work of art, not in order to affirm the first principles of a valuational aesthetics but in order to cast doubt on those first principles. Tu m' clearly involves geometry. The purpose of this involvement is less clear, but the connec- tions between various geometries and their conventionalist implications are what Duchamp used in the unification of his art. He had learned the geometry and the philosophy from Poincar6. In Du- champ's descriptions of art, as in Poin- car6's descriptions of the world, the rela- tionships between objects were more accessible than the objects themselves.

rojective analogies need not be confined to discussing the interrela-

tionships of different n-dimensional Eu- clidean spaces. Explanations of non- Euclidean geometry can also involve interdimensional analogies such as flat- beings.27 The flat-being represented by the shadow of the Bicycle Wheel in Tu m' may be a reference to non-Euclidean geometry. The circular rim of the wheel, the circumference of the circle, if rotated around one of its diameters would gener- ate a sphere, which is an analytic Euclid- ean surface that can serve as a model for the elliptic non-Euclidean geometry of Riemann. The wheel would generate a figure much like the one on the right- hand side of a diagram from Jouffret's Traitb (Fig. 4).28

The diagram accompanies a descrip- tion of Riemann's geometry. The draw- ing might also serve as an illustration for Poincar6's explanation in Science and Hypothesis. In order to discuss Rie- mann's geometry, Poincar6 introduces the notion of "beings with no thickness."

252 Art Journal

He then argues that if such beings were cognizant and if they lived on the sur- face of a sphere, they would not invent ordinary geometry: "First it is clear they will attribute to space only two dimen- sions; what will play for them the role of the straight line will be the shortest path from one point to another on the sphere, that is to say, an arc of a great circle; in a word, their geometry will be the spheri- cal geometry." He goes on to say that "Riemann's geometry is spherical ge- ometry extended to three dimensions."29 The shadow of the Bicycle Wheel in Tu m' is essentially made up of one- dimensional line segments. They can be thought of as projections from a two- dimensional, non-Euclidean surface like the one in Jouffret's illustration. The left-hand side of Jouffret's diagram, which resembles a bicycle wheel, can be taken as a projection of the right-hand diagram. The sphere, in its turn, can be taken as a projection from a three- dimensional, non-Euclidean space.

Geometers often point out that there is a conceptual correspondence between a three-dimensional, non-Euclidean Rie- mannian space and a four-dimensional Euclidean hypersphere." Each can be thought of as a sphere with an extra dimension. Poincar6 had this correspon- dence in mind when, in the introduction to his discussion of his Euclidean-non- Euclidean dictionary, he said that people who were used to thinking about four- dimensional geometry would have no dif- ficulty in extending the two-dimensional models of non-Euclidean geometry into three dimensions.31

Evidence that Duchamp was inter- ested in the relationships between Euclidean and non-Euclidean geometry, specifically in regard to Poincar6's use of Lobachevski's geometry, is provided by his Unhappy Ready-Made, 1919 (Fig. 5), and his original photograph of the book hanging from a Paris balcony (Fig. 6). In the latter, no "geometry" is visible at all. Duchamp chose the dia- grams as ready-made geometry and then retouched the photograph with compass and ruler. What he chose is significant. The drawing depicts orthog- onal circles and demonstrates one of Euclid's theorems.32 Various Euclidean theorems and their corresponding dia- grams often begin discussions of paral- lelism in Lobachevski's geometry.33 Spe- cifically, Euclidean orthogonal circles are often used by way of comparison in explanations of Poincar6's method of demonstrating the consistency of Loba- chevski's non-Euclidean geometry. Poincar6 first used this approach in the context of his work on automorphic (or Fuchsian) functions.34

Poincar6 begins by defining the fun- damental plane as a circle. He then

Fig. 2

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.: iiii;?:::~L~i-::?aiC-i~i:~4 aa'~c~g-? ib~~i -:.~~ii~ ~;,i~,is~~iri~irxgi ."'-' ak'fi--ii i-iii-i~-i ::-j__ .: --ii~ii?ell :

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Fig. 4 Illustration from E. Jouffret, Traite blekmentaire de g omi trie 'i quatre dimensions, p. 16.

Fig. 5. Duchamp, Unhappy. Ready-made, 1919, from the Box-in-a-Valise, Paris, 1941-42. Philadelphia Museum of Art, The Louise and Walter Arensberg Collection.

Fig. 6. Duchamp, Unretouched photograph of Unhappy Ready-made, c. 1919. Philadelphia Museum of Art, Marcel Duchamp Archives, Gift of Dr. William A. Camfield.

defines straight lines as being either straight lines passing through the center of the fundamental plane or arcs of circles cutting the fundamental plane orthogonally. Angles remain angles. Distance becomes logarithmic. Two tan- gent arcs that intersect on the funda- mental plane are parallel. What Poin- car6's system amounts to is a way of relating the two geometries: he shows that the axioms and theorems of Loba- chevski's geometry are special cases of

Euclidean geometry.35 Thus, the non- Euclidean geometry is consistent if the Euclidean geometry is consistent. Poin- car6's model is the one that he later described in more general terms as a "dictionary" in Science and Hypothe- sis.36 If one were to translate the dia- grams of orthogonal circles that Du- champ used for his ready-made geome- try book according to Poincar6's rules, one would arrive at an arrangement that looks very different: the Euclidean dia-

Fall 1984 253

Figs. 7 and 8 Duchamp, 3 Standard Stoppages, 1913-14, assemblage: three threads glued to three painted canvas strips, 51/4 x 471/4", each mounted on a glass panel, and wooden templates shaped along one edge to match the curves of the threads; the whole fitted into a wood box, 111/8 x 507/8". New York, Collection The Museum of Modern Art, Katherine S. Dreier Bequest.

grams would be transformed into non- Euclidean diagrams. Duchamp may have intended the deformations caused in his book by hanging it from a balcony as references to such transformations. The curved pages may have been refer- ences to the curvature involved in non- Euclidean geometry.37

The two-dimensional models for non- Euclidean geometry require surfaces of

constant curvature. Poincar6 explained this concept thus:

Consider any figure on a surface. Imagine this figure traced on a flexible and inextensible canvas applied over this surface in such a way that when the canvas is dis- placed and deformed, the various lines of this figure can change their form without changing their

length. In general, this flexible and inextensible figure cannot be dis- placed without leaving the surface; but there are certain particular sur- faces for which such a movement would be possible; these are the surfaces of constant curvature.

He goes on to point out that such sur- faces can be derived for both branches of non-Euclidean geometry: "The two- dimensional geometries of Riemann and Lobachevski are thus correlated to the Euclidean geometry."38

P oincare's remarks about a canvas bring Tu m' to mind. The buckling

around the trompe l'oeil tear suggests that the two-dimensional plane surface of the canvas has become a two-dimen- sional curved surface. This kind of deformation can be taken as a reference to non-Euclidean geometry. The shad- ows can be taken as references to the figures projected onto that surface. The points of an n-dimensional configuration can be projected onto an n - 1 dimen- sional configuration, as when three- dimensional objects are projected onto two-dimensional perspective or isomet- ric drawings. In a less complete sense, a two-dimensional shadow is a projection of a three-dimensional object.

In one of his notes for the Large Glass, first published by Matta in his magazine Instead, Duchamp says that "after the Bride," he wanted "to make pictures using cast shadows." Tu m' was apparently one of the results. In the note, he goes on to say that by means of n - 1 dimensional shadows projected by n-dimensional objects, "first on a plane, second on a surface of such or such curvature, third on several transparent surfaces,..,. one can obtain a hypophy- sical analysis of the successive transfor- mations of objects (in their form con- tour)."39 The reference here to both plane and curved surfaces suggests that Duchamp was thinking in terms of both metric and projective geometry and in terms of both Euclidean and non- Euclidean geometry. Depending upon the surface chosen-plane, positively curved, negatively curved--the result- ing geometries could be quite different.

In another of his notes, this one from the Green Box, Duchamp talks about the shadows cast by "2, 3, 4 ready- mades 'brought together.'" He dis- cusses the shadows as being three- dimensional ready-mades "having be- come" two-dimensional projections of ready-mades. "Take these 'having be- comes,' "he says, "and make from them a tracing without of course changing their position in relation to each other in the original projection."40 Duchamp traced the shadows of the ready-mades

254 Art Journal

onto the surface of Tu m' with pencil. This technique (calquer) involves the "projection" of three-dimensional ob- jects onto two-dimensional surfaces. The French term is interesting because a calque is not only a "tracing" but also an "imitation" or "close copy." The ready-mades come out of the molds of mass-production techniques as close copies. The calque made from the shadow of a ready-made is itself a com- monplace reproduction: a shadow can be cast by "any object whatsoever."

In a metaphorical sense, the fourth dimension may have been for Duchamp a device for flattening works of art into shadows-"three times removed from the true." Images on picture surfaces and shadows on picture planes are n- dimensional projections of n + 1 dimen- sional objects. They are geometrical "sections" in the same sense that Alberti used the term in his discussion of per- spective. In Tu m', Duchamp is dealing with a two-dimensional picture surface, a Renaissance mirror surface as it were, and three-dimensional objects projected onto that surface, both as shadows and as perspective renderings. In the center of the painting, the conventional symbol of a hand emerges from the handle of the Corkscrew and points towards both a shadow and an unconventional per- spective arrangement. Overlapping the traced shadow of the Hat Rack is an open-ended, transparent box. One end of the box is made up of a white rectangle drawn in perspective. From each corner of this rectangle, two curved lines are drawn parallel to the picture plane with the templates of the 3 Standard Stop- pages (Figs. 7 and 8). The volume suggests a curved region of space. The double edges drawn with the tem- plates are perhaps meant to cast doubt on measuring devices. They suggest the provisional nature of geometry. Fur- ther, they suggest that there is an ad libitum freedom involved in choosing between alternatives.

Tu m' becomes an essay in making- in art making, hypothesis making, and aesthetic judgment making. Tradition- ally, the picture plane had been an area in which reality took place; it had been a window opening onto a space con- structed according to the principles of perspective-a subcategory of pro- jective geometry. Duchamp tore a hole through that area. Similarly, the inven- tion of non-Euclidean geometries had torn a hole through the intellectual structure of reality. The Euclidean space that had been depicted in pictures was called into question by the invention of these geometries. Truth, geometric reality, was ripping apart. It could be held together only by conventional con- structs, by safety pins.

T he iconography of Tu m' involves both the philosophy of art and the

philosophy of science. Part of Poincar6's questioning of the foundations of geom- etry concerned the sources of axioms and theorems. Were they experimental truths? He argued that "we constantly reason as if geometric figures behaved like solids," but such apparent connec- tions with what we see in the world do not justify an assumption that geometry is experimental. He goes on to say that "the properties of light and its recti- linear propagation have also given rise to some of the propositions of geometry, and in particular those of projective geometry, so that from this point of view one would be tempted to say that metric geometry is the study of solids, and projective, that of light."4' But there is a fundamental problem with such ap- proaches: if geometry were based on experiment or measurement, it could never be exact. Because there are no rigorously rigid solids in nature, an experimentally based geometry could only be approximate.42

Several aspects of Tu m' involve the approximate nature of measuring de- vices and the distinctions between met- ric geometry and projective geometry. Poincar6 explained that

metric geometry is based on the notion of distance; in it, two fig- ures are considered as equivalent when they are "equal" in the sense which mathematicians assign to this word. Projective geometry is based on the notion of the straight line. For two figures to be consid- ered as equivalent in projective geometry it is not necessary that they be equal; it is sufficient that they correspond to each other by means of a projective transforma- tion; that is, that one be the pro- jection of the other.43 Classical perspective is subsumed

under projective geometry. In one of his notes, Duchamp says that "by perspec- tive (or other conventional means) the lines, the drawing, are 'strained' and lose the nearly of the 'always possible' with moreover the irony to have chosen the body or original object which inevi- tably becomes according to this perspec- tive (or other convention)."44 In a per- spective transformation, lengths and angles are "strained" and "inevitably become" according to the principles of the transformation. Duchamp's "other convention" may have been topology because, in it, transformations are more nearly "always possible."

Poincar6's remarks about metric and projective geometries introduce an ex- amination of analysis situs, a geometry now called topology. He explains that

in this discipline two figures are equivalent every time it is possible to have one correspond to the other by means of a continuous defor- mation, whatever the law gov- erning the deformation may be, provided that continuity is main- tained. Thus, a circle is equivalent to an ellipse or even to any type of closed curve, but it is not equiva- lent to a line segment because the segment is not a closed figure. A sphere is equivalent to any convex surface whatever, but it is not equivalent to a torus because in the torus there is a hole and there is none in a sphere. Let us imagine a pattern of any kind and the copy of this pattern drawn by a clumsy draftsman. The proportions are distorted, straight lines drawn by a trembling hand have undergone distressing deviations and result in disproportional curves. From the point of view of metric geometry, and even from that of projective geometry, the two figures are not equivalent; but on the contrary they are equivalent from the point of view of analysis situs.45 Topological operations are carried

out as if the bodies undergoing the transformations were made of rubber. So long as continuity is maintained, a geometrical figure can be stretched into any number of configurations. A circle is topologically equivalent to a square and a torus is topologically equivalent to a coffee cup, because if the circle were a rubber band, it could be stretched into the shape of the square, just as a rubber- sheet torus could be stretched into the shape of a coffee cup. With these points in mind, any number of Duchamp's "de- formations" can be interpreted as topo- logical transformations. The flat and curved pages of the geometry book can be taken as topologically equivalent con- figurations, or homeomorphisms, as can the flat and curved surface implied by the trompe l'oeil buckling on Tu m'. From a topological point of view, the curved line segments in 3 Standard Stoppages (see Figs. 7 and 8) are equiv- alent to straight lines or to each other. As the individual pieces of string fell through space, they underwent any number of transformations. Topologi- cally, the transformations were ad libi- tum or h son gr&. Such variations inter- ested Duchamp because of their philo- sophical implications. He believed that the meaning of a work of art was elastic and ad libitum: "An oeuvre by itself doesn't exist, it's an optical illusion. It's only made to be seen by the people who look at it. The poor medium is only gratuitous. You could invent a false artist. Whatever happens could have

Fall 1984 255

been completely different."46 For Du- champ, aesthetics, critical interpreta- tion, art history, and artistic intention were topological.

The line segments in 3 Standard Stoppages are topological homeomor- phisms in one dimension. At the lower left of Tu m' are three curved lines drawn with the templates of the 3 Stan- dard Stoppages. They are at the edges of three overlapping curved surfaces that suggest the configurations that would have been generated by the curved line segments of the strings as they fell through space. These three surfaces are topological homeomor- phisms in two dimensions. The three "Draft Pistons" in the upper panel of the Large Glass (see Fig. 1) are also topo- logical homeomorphisms in two dimen- sions. Duchamp hung flat, two-dimen- sional pieces of cloth in front of a win- dow and allowed the wind to distort their shapes. He then photographed the pieces of cloth (Fig. 9) and used the forms as templates for determining the irregular holes in the ipanouissement of the Bride. As a next logical step in this n-dimensional sequence, Duchamp's ready-made Traveler's Folding Item (Fig. 10) can be taken as a reference to topological homeomorphisms in three dimensions. The work looks like a rub- ber-sheet cube. If taken as a metaphor for a topological solid, it would become a three-manifold with boundary. The two- dimensional deformed surfaces of the three "Draft Pistons" can be taken as analogues for the deformed surfaces that would be generated by the one- dimensional deformed lines of the 3 Standard Stoppages displacing them- selves through space. The Traveler's Folding Item can, in its turn, be taken as an analogue for the configuration that would be generated by the two-dimen- sional "Draft Pistons" displacing them- selves through space. The next step towards a displacement into the fourth dimension would be, in Duchamp's term, "hypophysical."

Traveler's Folding Item is a type- writer cover and thus associated with writing. Both the three "Draft Pistons" and the 3 Standard Stoppages are also associated with writing. In one of his notes, Duchamp refers to the "Draft Pistons" as "alphabetic units," and it is through them that the "commands" of the Bride are telegraphed to the Bache- lors in the lower panel of the Large Glass. Duchamp explains that the Bride's instructions have "their alphabet and terms governed by the orientation of the three Draft Pistons."47 Thus, communi- cation in the Large Glass proceeds by a topologically deformed sign system. In another of his notes, Duchamp suggests that geometrically deformed symbols

Fig. 9 Duchamp, "Draft Piston," 1914, photograph, 231/8 x 191/16". Collection, Mme Marcel Duchamp, Villiers-sous-Grez.

lin

!i::iiiiiiiiiiiiiiiiiiiiiiiiililiiiiiiii i:R,

mom ? . ...... 'n Xx~

Fig. 10 Duchamp, Traveler's Folding Item, 1916, H. 23". Sarasota, Fla., John and Mable Ringling Museum of Art, Gift of Mary Sisler Foundation/Mrs. William T. Sisler.

256 Art Journal

could be constructed using the 3 Stan- dard Stoppages. He says that one could "compose a schematic sign designating each of [the so-called abstract words in a Larousse dictionary]." This sign could "be composed with the standard stops."48 Duchamp's desire to construct a "topo- logical language" reflects his attitude concerning meaning: he believed that meaning could be stretched into any number of configurations, as if it were made of rubber.

Topological interpretations are in keeping with Duchamp's interests in n- dimensional and non-Euclidean geome- tries. The various geometries can be interrelated, as Duchamp had learned from Poincar&: non-Euclidean geometry is a metric geometry, and metric geome- try can be subsumed under projective geometry; projective geometry can be n-dimensional, and non-Euclidean geom- etry can be n-dimensional; Riemann's geometry is commensurate with topol- ogy. Poincar6 points out that "[analysis situs] gives rise to a series of theorems just as closely interconnected as those of Euclid; and it is from this set of proposi- tions that Riemann constructed one of the most remarkable and abstract theo- ries of pure analysis."49 Topology can also be n-dimensional, as Poincar6 reminds us: "there is an analysis situs of more than three dimensions."50

p oincar6 demonstrated that one's choice of geometry was conven-

tional. Duchamp demonstrated that one's choice of art was conventional. Through conventional means, he trans- formed objects that were not art into art objects. He did it by placing ready- mades within the art context and by then saying that they were not works of art. In spite of his disavowals, they became works of art. Poincar6 argued that one could not know objects; one could know only the relationships between objects. Duchamp concurred. When asked what determined his choice of ready-mades, Duchamp replied that it

depended on the object. In general, I had to beware of its "look." It's very difficult to choose an object, because, at the end of fifteen days, you begin to like it or to hate it. You have to approach something with an indifference, as if you had no aesthetic emotion. The choice of ready-mades is always based on visual indifference and, at the same time, on the total absence of good or bad taste.5'

Duchamp claimed that he had chosen the ready-mades precisely because they were visually neutral-he neither liked them nor disliked them-and that their

beauty was the last thing that he had had on his mind when he chose them.52 But these works were far from uninter- esting. He said that he had thrown the ready-mades into the face of the art public as a challenge and that now they admired them for their aesthetic beauty. This challenge was not just a joke, a schoolboy blague. It was a way of ques- tioning the underpinnings of art criti- cism and art history. Duchamp said that "taste is momentary," that if one waited fifty years, taste changed. He argued that "if one is logical, one doubts the history of art."53

Duchamp's doubt began early in his career. He remembered being turned around by the small mindedness that had caused the rejection of Nude Descending a Staircase from the Inde- pendents exhibition in 1912:

Cubism had lasted two or three years, and they already had an absolutely clear, dogmatic line on it, foreseeing everything that might happen. I found that nafvely fool- ish. So, that cooled me off so much that, as a reaction against such behavior coming from artists whom I had believed to be free, I got a job. I became a librarian at the Sainte-Genevieve Library in Paris.54

Shocked into reevaluating the premises involved in interpretation and aesthetic judgment, Duchamp must have been attracted to the alternative intellectual tradition represented by mathematics and science; it was in his new job in the library that he probably began reading Poincar6 and found in him a kindred spirit.

Duchamp called art "une dedale illogique"--an illogical labyrinth.55 He believed that one lost oneself within the laybrinth, retraced one's steps, and com- ing around new corners that obscured old vistas, perceived new ones. Poincare argued that the approach to mathemat- ics and science also involved intricate pathways: truth was hidden within a labyrinth, but, nonetheless, one provided oneself with maps, with hypotheses, and proceeded inward. Both Poincare and Duchamp were aided by the philosophi- cal thread of conventionalism: it pro- vided the scientist and the artist with a skepticism that prevented mistaking a corner of the labyrinth for its center; it provided them with an abiding doubt about the veil of certainty that seems to fall across human consciousness too eas- ily and too quickly.

Duchamp took Poincar&'s advice and put things together that did not seem to belong together. He chose ready-mades and put them together with art objects; he chose intellectual disciplines-mathe-

matics and the philosophy of science- and put them together with aesthetics. His connections were metaphors for the importance of relationship over any cate- gory of individual objecthood. Given cer- tain relationships, the objects could be anything whatsoever. Duchamp used conventionalism in his arrangements to show that "everything could have been completely different." His strategy rep- resents a substantial connection between art and science. Duchamp said that "we don't speak about science because we don't know the language, but everyone speaks about art."56 Duchamp himself was an exception. He knew a great deal about the language of science. Through Poincar6, he had learned the philosophi- cal implications of advanced geometries, and he used what he had learned to speak about art. And because what he said has been so influential, those geom- etries and the world views that they en- gendered, have had fundamental effects on art, art theory, and the revisionism of art history.

Notes 1 Duchamp was probably familiar with at least

four of Poincar6's general works: La Science et

l'hypothese, Paris, Flammarion, 1902; Science et methode, Paris, Flammarion, 1904; La Val- eur de la science, Paris, Flammarion, 1908; Dernibres pensbes, Paris, Flammarion, 1913 (English translations: The Foundations of Science: Science and Hypothesis, The Val- ue of Science, Science and Method, trans. George Bruce Halsted, New York, The Science Press, 1921; Mathematics and Science: Last Essays, trans. John W. Bolduc, New York, Dover, 1963).

2 In addition to Poincar6's books, the major sources for Duchamp's mathematical knowl- edge were two works of Esprit Pascal Jouffret: Traite elKmentaire de geombtrie i' quatre dimensions et introduction ia la gbombtrie a n dimensions, Paris, Gauthier-Villars, 1903; and Mclanges de geombtrie 'i quatre dimensions, Paris, Gauthier-Villars, 1906. Jouffret's books are straightforward mathematical textbooks and, although they contain discussions of numerous geometrical operations that Du- champ made use of, they were probably not as important for the development of his philoso- phy as were Poincare's books. For more detailed discussions of Duchamp's sources, see: Craig Adcock, Marcel Duchamp's Notes from the Large Glass: An N-Dimensional Analysis, Ann Arbor, UMI Research Press, 1983, pp. 29-39; and Linda Dalrymple Henderson, The Fourth Dimension and Non-Euclidean Geome- try in Modern Art, Princeton, Princeton Uni- versity Press, 1983, pp. 117-30.

3 Poincar6, Foundations of Science, p. 386.

4 Ibid., pp. 387-88.

5 For a discussion of these philosophical changes, see: Morris Kline, Mathematical Thought from Ancient to Modern Times, New York, Oxford University Press, 1972, pp. 1023-39.

Fall 1984 257

6 Poincar6, Foundations of Science, pp. 56-60; Nikolai Ivanovich Lobachevski's most accessi- ble essay is his Geometrische Untersuchungen zur Theorie der Parallellinien, Berlin, 1840; a translation by George Bruce Halsted (Geomet- rical Researches on the Theory of Parallels) is included in Non-Euclidean Geometry by Roberto Bonola, New York, Dover, 1955, after

p. 268; Bernhard Riemann's most famous

paper is his "Uber die Hypothesen welche der Geometrie zu Grunde liegen," Abhandlungen der Ki5niglichen Gesellschaft der Wissen-

schaften zu Gbttingen, 13 (1868), pp. 1-20.

7 Poincar6, Foundations of Science, p. 60.

8 Ibid., these were the Fuchsian functions men- tioned earlier.

9 Ibid., p. 28.

10 Ibid.

11 Ibid.

12 Ibid.

13 Ibid., p. 63.

14 Ibid., p. 65.

15 Ibid., p. 125.

16 Ibid., p. 483.

17 Ibid., pp. 214-15.

18 Ibid., pp. 215-16.

19 Pierre Cabanne, Dialogues with Marcel

Duchamp, trans. Ron Padgett, New York, Vik-

ing, 1971, p. 107.

20 Philipp Frank, Modern Science and Its Philos-

ophy, New York, Braziller, 1955, p. 8; Frank's discussion was brought to my attention by John

Philip Paul, "An Analysis and Evaluation of Henri Poincare's Cosmology, Epistemology and Philosophy of Science," Ph.D. Disserta- tion, Marquette University, 1969, p. 165.

21 In his interview with Cabanne (cited n. 19), p. 31, Duchamp pointed out that the "'Nude

Descending a Staircase' had been refused by the Independents in 1912." He added that the rejection was an occurrence that "helped liber- ate me completely from the past, in the per- sonal sense of the word. I said, 'All right, since it's like that, there's no question of joining a

group-I'm going to count on no one but

myself, alone.' " Throughout the rest of his life, Duchamp remained suspicious of taste. In his interview with Katherine Kuh, The Artist's Voice: Talks with Seventeen Artists, New York, Harper & Row, 1960, pp. 91-92, he said, "I consider taste--bad or good--the greatest enemy of art."

22 Marcel Duchamp, Salt Seller: The Writings of Marcel Duchamp (Marchand du Sel), ed. Michel Sanouillet and Elmer Peterson, New York, Oxford University Press, 1973, pp. 89-90.

23 Jouffret, Traitt (cited n. 2), pp. 186-87, my translation.

24 Duchamp (cited n. 22), p. 33.

25 Interview with George Heard Hamilton and Richard Hamilton, "Marcel Duchamp Speaks,"

broadcast by the BBC, third program in the series "Art, Anti-Art," 1959, quoted in Arturo Schwarz, The Complete Works of Marcel Duchamp, New York, Abrams, 1970, p. 23.

26 Interview with Cabanne (cited n. 19), p. 40.

27 In his interview with Cabanne, p. 39, Duchamp credits Gaston de Pawlowski with being a pop- ularizer of the fourth dimension who had

explained "that there are flat beings who have only two dimensions, etc." and who had "ex-

plained measurements, straight lines, curves, etc." Pawlowski's only mention of flat-beings occurs in his book, Voyage au pays de la

quatribme dimension, Paris, Eugene Fasquelle, 1912, pp. 27-28, in the context of a discussion of non-Euclidean geometry: flat-beings living on the surface of a sphere would believe that the angles of a triangle would add up to more than 180 degrees, etc. Jean Clair, Marcel Duchamp ou le grand fictif"

Essai mythana- lyse du Grand Verre, Paris, Galilee, 1975, has discussed Duchamp's dependence on Pawlows- ki. Both Henderson (cited n. 2), pp. 119, 128, nn. 7, 31, 33, and Adcock (cited n. 2), pp. 33-34, argue that Clair's case for Pawlowski's being a major influence on Duchamp is over- stated. Pawlowski was without doubt part of Duchamp's intellectual background, as he him- self says, but Duchamp's knowledge of geome- try is far more sophisticated than anything he could have found in Pawlowski.

28 For an interpretation of Jouffret's diagram in relation to non-Euclidean geometry, the Bicy- cle Wheel, and the "Oculist Witnesses" in the Large Glass, see: Craig Adcock, "Geometrical Complication in the Art of Marcel Duchamp," Arts Magazine 58 (January 1984), pp. 105-9.

29 Poincar6, Foundations of Science, p. 57.

30 See, for example: H. S. M. Coxeter, Non- Euclidean Geometry, Toronto, University of Toronto Press, 1965, p. 12.

31 Poincar6, Foundations of Science, p. 59.

32 This is pointed out by Henderson (cited n. 2), p. 160. The diagram that Duchamp uses shows tangents drawn from an axis through the points of intersection. Such tangents are equal.

33 See, for example: Bonola (cited n. 6), pp. 250-64.

34 Between 1882 and 1884, Poincare published five landmark papers on automorphic functions in Acta Mathematica. For this particular result, see: Acta Mathematica 1 (1882), pp. 1-62, esp. p. 8 and p. 52; reprinted in Henri Poincar6, Oeuvres, Paris, Gauthier-Villars, 1954, II, pp. 108-68.

35 See: Kline (cited n. 5), pp. 916-17.

36 Poincar6, Foundations of Science, pp. 58-60.

37 Henderson (cited n. 2), p. 160.

38 Poincar6, Foundations of Science, pp. 58-59.

39 Duchamp (cited n. 22), p. 72.

40 Ibid., p. 33.

41 Poincar6, Foundations of Science, p. 64.

42 Ibid., pp. 64-65.

43 Poincar6, Mathematics and Science: Last Essays, pp. 57-58.

44 Duchamp (cited n. 22), p. 36.

45 Poincar6, Mathematics and Science: Last

Essays, pp. 58-59.

46 Dore Ashton, "An Interview with Marcel

Duchamp," Studio International, 171 (June 1966), p. 246.

47 Duchamp (cited n. 22), p. 36.

48 Ibid., pp. 31-32.

49 Poincar6, Mathematics and Science: Last

Essays, pp. 58-59.

50 Ibid., p. 43.

51 Interview with Cabanne (cited n. 19), p. 48.

52 See: Duchamp's interviews with Kuh (cited n. 21), pp. 91-92; with Jeanne Siegel, "Some Late Thoughts of Marcel Duchamp," Arts

Magazine, 43 (December 1968-January 1969), p. 21; and with Francis Roberts, "I

Propose to Strain the Laws of Physics," Art News, 67 (December 1968), p. 62.

53 Interview with Otto Hahn, "Marcel Du-

champ," L'Express (Paris), no. 684 (July 23, 1964), p. 22. The original passage in French is worth quoting at greater length: "Le gofit est

momentan6, c'est une mode. Mais ce que l'on considere comme une forme esth6tique est

d6barrass6 du gout. On attend donc cinquante ans, et la mode disparait. Les choses prennent alors un sens. En fin de compte, c'est une

entourloupette: une autre forme de gout. Ce qui ne l'6tait pas sur le moment le devient plus tard. Si on est logique, on doute de l'histoire de

l'art.... Le public est victime d'un v6ritable complot 6bahi. Les critiques parlent de la

'v6rit6 de l'art' comme on dit 'la v6rit6 de la religion.' Les gens suivent comme des moutons de Panurge. Moi, je n'accepte pas, c'est inexis- tant. Ce sont des voiles invent6s. Cela n'exista

pas plus qu'en religion. D'ailleurs, je ne crois en rien, car croire donne lieu a un mirage."

54 Interview with Cabanne (cited n. 19), p. 17.

55 Interview with Robert Lebel, "Marcel Du-

champ, maintenant et ici," L'Oeil (Paris), no. 149 (May 1967), p. 20.

56 Interview with Ashton (cited n. 46), p. 247.

Craig Adcock is Assistant Professor of Art History at The Florida State University, Tallahassee.

258 Art Journal