On the Geometric Imaginary

Since this introduction cannot be a full history, let it be more anecdotal and suggestive of the elements in the field I want to bring into focus, and the problems they entail.

This field includes but is not limited to the “truth and beauty school of aesthetics.” For this school, as for idealistic philosophy, mathematics is the model of truth, grounding its aesthetic approach largely on composition, specifically on geometric order, through which it attempts to achieve perfect proportionality by deploying naturalized criteria like the golden section and dynamic rectangles.[i] Perspective follows the same logic, bringing the “real” into visuality. Most of the work of the European periods from the Renaissance through Modernism actually practiced, or are considered to have practice, these same strategies. Because these artists claim to be imitating natural truth, modeled on a Pythagorean, mathematical essentialism, they practice a variety of Platonic idealism; the truth of any proposition is deduced from a set of “self-evident” definitions and axioms, and the simplest proof is the most “elegant” or beautiful, and as such is a representative of the ideal truth. The epistemological strategy of the geometric imaginary here, is to establish “visual truth” through what might be called “visual deduction” of the “true” arrangement of the artwork’s parts to the ideal, perfectly expressed whole, as derived from already accepted aesthetic-geometric truths – those of the golden mean, dynamic rectangles and perspective. This process, when articulated in this way, does indeed qualify as a form of scientific-aesthetic. This art historical narrative is now hackneyed.

What is not, is to consider this aesthetic-geometric imaginary an epistemological method in practice that governs the invention/interpretation of content, as it’s form. Art historical interpretation speaks primarily in terms of iconography, of form, of the product, of the building or fresco that results from the application. It speaks in terms of the development of the visualization of the “real.” However, it has been pointed out in recent years that perspective is a disciplinary technology comparable to the disciplinary practices revealed by Foucault.[ii] While these accounts have done much to deconstruct the naturalization of perspective as constituting the “real” of visuality, they haven’t gone so far as to claim that this aesthetic-geometrical imaginary has the status of an epistemological strategy; in other words, blinded by the historical conditions that deny thought to artists, they have not interpreted these visual practices as a methods of producing visual form that cannot be reduced to the geometrical principles of perspective formulated by Brunelleschi, Alberti et. al.[iii] Nor can they be reduced solely to the tradition of platonic idealism and its truth and beauty axiom.

Though I must leave it without demonstration here, my claim is that the dominance of philosophical, historical discourses have failed to give this variety of visual epistemology a role sufficient to its impact in the construction of cultural world views; artists represent cultural knowledges, so goes the standard narrative, but not because of knowledge “inherent” to their epistemological practice, but merely because they arise from epistemologies of other practices. Artists are always reduced to illustrators of the principles of other disciplines.

Morris Kline’s description of Leonardo is typical:

Nevertheless, Leonardo did not fully grasp the true method of science. In fact, he had no methodology, nor any underlying philosophy. His work was that of a practical investigator of nature, motivated by aesthetic drives but otherwise undirected. (1972:224)

Coeval with the rise of perspective and its aesthetic-geometric productions, was another “tradition” that to my knowledge remains unacknowledged. One artist worked according to another principle, not reducible to those of the idealist, platonic school. Piero della Francesca is a prominent member of the Renaissance canon, but incorrectly assimilated to the tradition just described. It is true that his works use idealized forms. And though the spatial configurations of his paintings conformed to the laws of perspective, and he wrote one of the most widely used books on the subject for artists, his visualization practices also interfered with it. Flat vertical planes of walls interrupt the depiction of perspectivally ordered spatial volumes. He was not interested in the visualization of coherent wholes that subjected visual logic to a singular visualized perspective of “truth.” His works offered representations of disjunctive spaces and times out of sync with the perfectly constructed wholes of perspective. While not denied in his work, perspective is but one among other options in his aesthetic-geometric imaginary. Hence, his spatial investigations are far more complex than the geometrical reductivism in other work of this period.

This begins to make sense once we know the following fact. Della Francesca was the only Italian of his day to invent a new mathematical concept. He invented a form of mathematics, in anticipation of what would eventually become, with Liebniz and Newton, the integral calculus. It was both a descriptive and constructive method determining the gradient of the curvature of a body. He used it to construct the gradient of curvature of the human head.[iv] The implication here is that his interest in the curvature of bodies, the very particular curvature of very particular bodies, pointed toward a geometric practice not reducible to planar, Euclidean geometric methods of construction of curves based on the circle, and on the assemblages of circular arcs to produced more complex curves. His system was at once constructive, and numerical. To the degree that it was numerical, it moved away from the assumed standards of the geometric imaginary of his day, based as they were on the ideal of ‘self-evidently’ intuited spatial perspectives. Arithmetic techniques retreat from the privileged arena of sight, and from the hegemony of geometry, tending toward the formalisms of algebraic unrepresentability. Francesca’s epistemological practices must be considered in contrast to the strict linear logic of perspective, generated by the grided determinations of vanishing points. He imagined a more complex spatial world then other artists of his time. One that begins to question the very possibility of representation. Hence, his planar interruptions of deep perspectival space is in league with his pursuit of the curve and its methods of constructing them. It is thus to him, that I want to ascribe the historical emergence of an epistemological process uniquely formative of aesthetic knowledge, in its geometric form.

After della Francesca, we skip more than 250 years to the work of the Jesuit priest and mathematics professor at Pavia, Gerolamo Saccheri, whose works fall in with attempts to prove the truth of Euclid’s fifth postulate, the parallel postulate, through the long accepted technique of the reductio ad absurdum (the method of demonstrating the correctness of a proposition by reducing its assumed opposite to contradiction). In his attempt to prove the fifth postulate, Saccheri produced a series of theorems following from a line of thought that did not lead to contradiction. The results, however, were too bizarre for him to accept, and he simply pronounced them false. This account also follows a well-worn historical narrative. Saccheri’s work led to several moments in the history of mathematics that need recapitulation here. These moments illustrate the process by which a radical geometrical imaginary emerged, and was then again repressed. The fact of this repression is interesting by itself, but the reasons are far more compelling.

In 1763, the German mathematician Georg Klugel

made the remarkable observation that the certainty with which men accepted the truth of the Euclidean parallel axiom was based on experience. This observation introduced for the first time the thought that experience rather than self-evidence substantiated the axioms. (Kline 1972: 868)

Klugel’s work was taken up in 1766 (not published until 1786) by Johann Lambert, who realized:

that any body of hypotheses which did not lead to contradictions offered a possible geometry. Such a geometry would be a valid logical structure even though it might have little to do with real figures. (Kline 1972: 868)

We witness here a double epistemological extension of mathematics; extension to “experience” as the ground of “certainty,” and the extension of what was considered properly “logical” beyond “real” figures. But as Kline points out, Lambert’s recognition, while remarkable, doesn’t make the leap from logical consistency as mathematical statements, to applications to physical space. This was achieved by Carl Gauss beginning in 1799 when he reports in a letter to Bolyai that he has begun “to doubt the truth of geometry itself.” But it was not until 1813 that he used the term non-Euclidean geometry. In 1817, in a letter to Olbers, he writes:

I am becoming more and more convinced that the [physical] necessity for our [Euclidean] geometry cannot be proved, at least not by human reason nor for human reason. Perhaps in another life we will be able to obtain insight into the nature of space, which is now unattainable. Until then we must place geometry not in the same class with arithmetic, which is purely a priori, but with mechanics. (Kline 1972: 872)

These three historical moments, when combined, lead to, for this moment in history (the early 19th century), a radical philosophical position; one for which the very premise of logical truth, self-evidence (mathematical reason) is insufficient; for which “experience” is the ground of axioms; for which a multiplicity of geometries based on noncontradictory sets of axioms exists; and for which geometry, no longer founded solely on a priori principles, must be treated physically as a subset of mechanics. The implication is that non-Euclidean geometries are equally applicable to physical space as to abstract, logical space; that Euclidean space is not the only form of geometry that may account for physical space. The transfer to mechanics, and proof of Gauss’s claim, required almost another century.

[note: Gauss is to geometry as Kepler was to astronomy]

In his essay, “General Investigations of Curved Surfaces,” Gauss considered a surface as a space in itself. This is a radical departure from any previous mathematical conceptions of space, still under the Newtonian dominance of absoluteness, homogeneity, coextensivity. Gauss showed that all of the properties of a surface space can be determined by the second derivative, the process by which the rate of curvature is determined. In other words, for mathematical purposes, one can forget that the surface lies in a 3D space. If one takes the “straight lines” of this surface as geodesics, (the shortest line between two points on a surface, not necessarily “straight,” as is the case of the spherical geometry), then the geometry becomes non-Euclidean. If the sphere is considered to be located in 3D space, then its geometry is Euclidean (i.e., the shortest distance between 2 points is a straight line not lying on the surface). This is as far as Gauss went. An even more radical step was taken by his student, Bernard Riemann, in a lecture delivered in 1854 entitled: “On the Hypotheses which Lie at the Bases of Geometry.” Riemann clearly realized that spherical geometry implies a geometry unique to the characteristic of its surface conditions. What is of significance to us here is the shift from an absolute hegemony of Euclidean space to a fracturing of space into myriad varieties.

With this concept, we arrive at a mathematical correlate to concept of multiplicity as understood by Nietzsche and Deleuze. We must also recognize that this movement came about through a metonymic process equivalent to Derrida’s concept of differance. The concepts of intrinsic and extrinsic geometries arises in a way analogous to the use of “inside” and “outside” in deconstruction. The intrinsic geometry is the surface geometry of the sphere independent of it’s location in 3D space. Extrinsic geometry would be the 3D space in which spherical geometry exists. But there is no logical necessity which compels us to “zoom out” to a privileged Euclidean container. Euclidean space is one species of space among many. [And it’s not clear that its experiential, corporeal space because this may be a culturally entrained belief] Indeed, in “physical” terms, Euclidean space becomes a special case of non-Euclidean spaces. One main result is: the same surface can have different geometries. A second implication is the reverse of this: the Pythagorean formula is only one functional expression of the second derivative; in other words, da2 + db2 + dc2 could be replaced by other functional expressions, thereby determining within the conventional frame of Cartesian rectangular coordinates, a non-Euclidean geometry. This is what Riemann did. We may see this as a significant, break with mathematical essentialism, one which undermines the Platonic, idealist philosophical schools that predicate “truth” on the model of mathematical certainty. Again, the significance is that the concept of “whole,” or “unity,” is simply eliminated. There is no whole, only parts, which are at most unities only in signification, or under the semiotic dominance of interpretive trajectories. This is a clear alliance with Nietzsche’s “perspectivism.” What has yet to be determined, however, is the impact on aesthetics.

In the service of being brief, I will summarize three further points to illustrate the historical reasons that non-Euclidean concepts have had so little impact. Interestingly, at least according to Kline, the reason lies with mathematicians themselves on the one hand, and on the other with physicists, who are responsible for the erroneous popularization of the legendary difficulty of relativity theory (the cliché, “only three people in the world understood the theory at its inception”).

Kline points out that Helmholtz, in his fundamental paper, “On the Facts Which Underlie Geometry,”

showed that if the motions of rigid bodies are to be possible in a space then Riemann’s expression for ds (the derivative) in a space of constant curvature is the only one possible. (1972: 921)

Helmholtz’s point was that for a mathematical description of observed physical motions of the types of bodies physics specified, then the concept of constant curvature was a necessary condition; and that it was exactly this that Riemannian geometry provided. Hence, of the myriad possible geometries, only this was adequate for physical description of observed phenomena. What happens next is characteristic of the power and hold of the Euclidean imaginary. It is re-naturalized by very influential mathematicians and scientific institutions. Despite Helmholtz’s paper, as Kline points out:

Another reason for the loss of interest in the non-Euclidean geometries was their seeming lack of relevance to the physical world….. Cayley, Klein, and Poincare, though they considered this matter, affirmed that we would not ever need to improve on or abandon Euclidean geometry…. In fact, most mathematicians regarded non-Euclidean geometry as a logical curiosity. (1972: 921)

[If Newton had not synthesized Kepler’s unproven speculations about planetary motion, with Galileo’s terrestrial kinetics, thus unifying the mathematical physics of both earth and heaven, then perhaps Kepler’s mathematics may have remained a curiosity.]

Cayley was the president of the British Association for the Advancement of Science, and in 1883, at about the same time Nietzsche was writing Thus Spake Zarathrustra, claimed in an address to the BAAS:

… that non-Euclidean spaces were a priori a mistaken idea, but non-Euclidean geometries were acceptable because they resulted merely from a change in the distance function in Euclidean space. (1972: 922)

Cayley thereby threw the weight of the prestigious BAAS behind the suppression of the more radical thought, thus re-founding “common sense” on the basis of traditional Euclidean notions. What is compelling here, and can only be briefly indicated, is the role played by institutions, prestigious reputations, models of the physical world, and an ideological commitment to methodological blindness. It is here that a discussion of Kuhn’s turn toward the role of aesthetics in scientific motivations will be useful, independently of whether his position on paradigm shifts is adequate or not. What it is important to note here is that the availability of a non-Euclidean imaginary arose briefly in the mid 19th century, only to be lost because of its demotion to the status of a “logical curiosity,” a position essentially equivalent to Saccheri’s during the Renaissance.

That non-Euclidean geometry, and in particular, its Riemannian version, was retrieved and given renewed credibility by the work of Einstein is well known. However, the language with which he conceptualized this shift deserves to be revisited.

The only justification for our concepts and system of concepts is that they serve to represent the complex of our experiences; beyond this they have no legitimacy. I am convinced that the philosophers have had a harmful effect upon the progress of scientific thinking in removing certain fundamental concepts from the domain of empiricism, where they are under our control, to the intangible heights of the a priori. (1955: 2) [emphasis mine]


In this sense, we cannot speak of space in the abstract, but only of the “space belonging to a body A.” The earth’s crust plays such a dominant role in our daily life in judging the relative positions of bodies that it has led to an abstract conception of space which certainly cannot be defended. In order to free ourselves from this fatal error we shall speak only of “bodies of reference,” or “space of reference.” (1955: 3)

Not only does Einstein retrieve the Gauss-Riemann-Hemholtz non-Euclidiean break with mathematical abstraction, and its embodiment in physics, but his acknowledgement of this historical recovery is significant. It signals an end, scientifically, not only to the dominance of Newtonian conceptions of absolute space and time, but also, in general, to the idealism of the truth and beauty schools of science, mathematics, philosophy and because based on these, to aesthetics as it has been held hostage by philosophy. Einstein’s clear allusion to the damage Kant and his followers have done to science, (though as Kline has pointed out, mathematicians and physicisits share equal blame), while creating an entirely new conception and language for spacetime, has had little impact on the reception of Kant’s aesthetic theories. This is in part because of the legendary, if erroneous, conception that relativity theory is just too difficult for anyone but geniuses to understand. This in turn, as astrophysicist, Chandresekhar points out, has had and equally pernicious effect on scientific thinking. Yet, Einstein’s condemnation of philosophy is consistent with, and “foundational” to, Adorno’s call for the right proportion between practical experience and philosophic contemplation.

Any conception of aesthetic knowledge must thus be consistent with the challenge to Kant’s essentializing of Euclidean space and time. The question I wish to pose is how to develop aesthetic knowledge on the basis of non-Euclidian, experientially and physically determined, geometric grounds? If the requirements of relativity, of speaking only of bodies and spaces of reference, in spacetime frames, is imposed on aesthetic interpretation and aesthetic production, what would be the results?

[note: of course the term, experience, is a wash or sorts, as indefinable. but it is a place holder for something non-ideal, non-conceptual, non-abstract. or, ‘experience’ needs further definition though expanding it beyond the corporeal, beyond ‘sensation’, and therefore, beyond, ‘aesthetics’. bakhtin/whitehead/bachelard/stein, therefore…]

It’s worth pointing out that Kant’s Critique of Pure Reason was written in the years 1769-80 and published in 1781 at the same moment as the works of Klugel and Lambert, also published in Germany. The Critique of Judgment was published in Berlin in 1790, after Gauss had come to entertain the fact that non-Euclidean geometries could account for physical space as well as Euclidean spaces. Because Kant’s original training was as a physicist, it’s not unreasonable to assume that he would have been aware of these developments. And yet, he doesn’t take them into consideration, and sets the course for aesthetics through a geometric imaginary that naturalized Euclidean space and time, the two most fundamental concepts of his project of Critique. Thus, we are faced with the chiasmic historical trajectories of a mathematics and physics turning to a non-Euclidean geometric imaginary, briefly, on the one hand, and on the other, with an aesthetics mired in a Cartesian/Newtonian geometric imaginary able to maintain its cultural hegemony firstly, because of the mathematical coup that derailed the non-Euclidean from emerging; and secondly, because the non-Euclidean then became hostage to a relativistic physics accessible only to a tiny elite. [there were no popular books written for women about einstein as there were about newton…] A turn to a non-Euclidean “imaginary” is therefore a turn to something which either does not yet exist, or exists but is not recognized as such, or cannot exist.

Counter-intuitively, the aesthetic component of the aesthetic knowledge must therefore derive from mathematics and physics rather than from art. And the knowledge component must derive from art, because some artists are indeed investigating the non-Euclidean that some domains of math and science still eschew, or perform investigations that might at least lead to what a non-Euclidean-aesthetic might be. This requires a radical break not only from Kant, but from a naturalized privileging of sensory perception as Euclidean. The project is to begin a process by which an “aesthetics” based on a non-Euclidean geometry may begin to be “imagined.” Such an imagined aesthetic knowledge will be quite different than the truth and beauty school of traditional, Kantian aesthetics. It may best arise, I claim, in the gaps opened by the space of differance opened by Mallarme; but entered by Roussel, Artaud and Kafka in a “literary” appropriation based not on narrative, but on a challenge to master narratives, to signification, and to language itself. Roussel used a mechanico-chance method to produce his “novels,” an extra-linguistic process through which language is produced. Similarly, Artuad sought in his Theater of Cruelty, not “psychological states” of identification between audience and actors, but “spiritual states” expressed “between gesture and language,” based on bodies moving in relations that suggest meanings that language prohibits. Kafka sought a writing which mobilized the non-Germanic components of German through mobilizing the non-written elements of Yiddish and the dialectical components of the Chech German dialect, mobilizing systems of extra-semiotic exchange. It is in this work, that a literary representation may diverge into a non-literary art practice situated and oriented with reference to what Derrida, speaking of Artaud, calls the closure of representation.[v] This is a complex term with several meanings; here I will only hint that it refers to the mathematically phrased problem of subtracting unity from multiplicity; to the necessity of limiting theory to a metonymic range. In Einstein’s language, we may speak “only of the space belonging to a body A.” We can only recover the non-Euclidean potential repressed by Kant’s work, however, within a critique of science that begins with the deconstructive project. Still, we must be careful not to privilege the non-Euclidean over the Euclidean. We must recover the non-Euclidean moment of the late 19th century, and use it to re-erase the hegemony of the Euclidean imaginary today.

Methodologically, by stepping “outside” science and into art, we may criticize science; and, by stepping “outside” of art and into mathematics, we may criticize art. Art, in its dominant expression, continues to aid the Euclidean imaginary; one need only look at Modernism’s commitment to pure, positivistic conceptualism in any of its many guises, based in a rigorous reduction to Euclidean elements. The flat plane, more than anything, is the symbol of standardization, the absolute, the rational, embodied in its most ideal form by the grid. (Cubism was one of the few exceptions to the Euclidian rule. This will seem untrue to some, but, my claim can be supported.) What could be more Cartesian? And this commitment has not lessened during the post-modern period, though formalism is not the driving force. It is because art and science are so unconsciously intertwined, that I pursue the pendulous trajectories between both discourses.

[i] There is a large body of work that goes by the general term, sacred geometry, that addresses these issues. Much of it speculative, popular, and often of a new age sensibility, though some of it is scholarly work as well. See Bruno (1967), Dunlap (1997), Ghyka (1946, 1952, 1958) Hambidge (1920, 1924, 1962), Hartel (1988), Huntley (1970), Lesser (1957), Pennick (1980), Vajda (1989).

[ii] Crary (1990), Foster, (1988).

[iii] Baxandall (1988), Panofsky (1990).

[iv] See, Baxandall (1988). Today, his mathematical techniques are still used by the airline industry in the design of airplane bodies.

[v] “The Theater of Cruelty and the Closure of Representation,” in Derrida (1978: 232).

On the Geometric Imaginary

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