This essay develops a new way of thinking about the cultural relationships among and within the sciences and the arts through a new understanding of the term postmodernism that at once derives from literary theory and the mathematical discipline of topology. While topology forms the main vertebra of this connective approach in its capacity as the mathematics of connectivity, quantum mechanics and non-Euclidean geometry -- the atlas and axis of this spinal column -- form the context through which this “postmodern” approach will develop. However, in order to position topology as a “postmodern” branch of mathematics, some brief explanations are in order: first, regarding postmodernism, and finally regarding topology.

Cultural Topology: an Introduction to Postmodern Mathematics

Brent M. Blackwell

<1> “Postmodern” and “Mathematics” are two terms that are not usually grouped together, especially among mathematicians. Mathematics is a well-defined pursuit, complete with its own home, exhaustive sets of protocols, and a body of willing participants eager to disseminate its contents to all those who wish to receive it. Postmodernism, on the other hand, is more ephemeral. Though a growing group of scholars in the humanities and social sciences have been referring to themselves as “postmodernists” since the 1980’s, by and large postmodernism does not yet carry the same “weight” as mathematics. Outside of passing hallway conversations or classroom exercises, postmodernism has no identifiable home, no clear definition, and certainly no consensual set of protocols. Generally speaking, postmodernism and mathematics are two very different kinds of things. One cannot point to the pages of a book or a carton of milk and exclaim, “There, you see that? That’s postmodernism,” like it was just mis-marked and placed in the wrong aisle.

<2> However, over the last decade, there have been many attempts to place the larger areas that claim ownership of these disciplines, namely the arts and sciences, into a dialogue with each other [1]. The volatile union of these two sets has produced what is now called the science wars [2]. The feeling is that if the parent companies begin negotiations, perhaps their prodigal subsidiaries will be less apt to quarrel and fight over property rights and copyright infringements. Many of the recent debates have centered on the ways in which the arts have used and/or misused the sciences in their highly public arguments. The criticism (mostly from scientists themselves) is that many in the emerging area of science and technology studies (STS) or cultural studies of science (CSS) base their critique on uninformed assumptions about what science is or how it is done. In 1996, theoretical physicist Alan Sokal, like Hephaestus, rigged his own net, hopping to catch an unsuspecting adulteress. His fraudulent article taught a lesson that will not soon be forgotten [3]. My essay, while perhaps a partial or provisional response to this kind of criticism, seeks to reconfigure the long-boiling issue between the arts and the sciences over the term “postmodernism” in a different way. In brief, then, let me summarize the form of this issue.

<3> Many scholars in the growing area of STS and CSS in general, tend to frame science as a product of culture like the media or religion; that is, they tend to view science as a part of culture rather than apart from it. And as such, so the argument goes, science can be analyzed as a cultural institution. The history of science is rife with those who have conducted their experiments under this assumption. Unfortunately, only a select few in either science or cultural studies have developed a means through which the two “traditionally” separate areas can be combined in such a way as to either avoid simplifying the inherent complexity in both, or in such a way as to not make one an instrumental product of the other [4]. And the “war,” as I see it, has generally taken one of these two forms, both of which tacitly support the age-old dichotomy in one of its more vitriolic manifestations.

<4> For example, “science” rightly concedes Steve Fuller’s characterization of post war programs like Harvard’s General Education in Science as a “remedy […] to normalize the role of science in society so that it not seem [sic] irretrievably alien” (52). And while he is also careful to emphasize that such programs were intended to train future managers of scientists who themselves, had little scientific training, his work borders on hypocrisy in light of Norman Levitt’s criticism of his misunderstanding of set theory [5]. Fuller’s harkening to C.P. Snow’s famous imperative that the science curriculum should adapt to include courses that promote an understanding of the cultural and social contexts in which it finds itself, is matched by Levitt’s argument that the liberal arts should do the same [6]. In “Mathematics as the Stepchild of Contemporary Culture,” Levitt suggests that today, the “liberal arts” is something of a misnomer as the more “rigorous” subjects of the traditional liberal education, like mathematics, have been all but abandoned since the nineteenth century (49).

<5> So, within the generalized miscommunication of the maelstrom of cries from both sides, lies two “kinds” of thought that have not always been so distinct, according to Levitt, certainly not to Aristotle. Fortunately for those who feel a stronger connection between science and culture than one of incidence or accidence, one of wholesale misappropriation or even misunderstanding, there has been a steady growth in approaches over the last decade that seek a new kind of connection -- a connection I will call a “postmodern” connection, even in spite of the almost unholy connotations this term now caries in academic debate [7]. The work of Floyd Merrell in semiotics, Arkady Plotnitsky in literary theory, Andrew Pickering in Sociology, comparative scholar Brian Rotman, and poet Jonathan Holden stand out in this regard. And it must be readily conceded that my use of the term postmodernism, while deriving from the sociolinguistic “poststructural” theories of the late 1960’s, is not reducible to them.

<6> This essay develops a new way of thinking about the cultural relationships among and within the sciences and the arts through a new understanding of the term postmodernism that at once derives from literary theory and the mathematical discipline of topology. While topology forms the main vertebra of this connective approach in its capacity as the mathematics of connectivity, quantum mechanics and non-Euclidean geometry -- the atlas and axis of this spinal column -- form the context through which this “postmodern” approach will develop. However, in order to position topology as a “postmodern” branch of mathematics, some brief explanations are in order: first, regarding postmodernism, and finally regarding topology.

I. Introduction: Avoiding a Definition of Postmodernism

<7> For the last forty years, hardly a year has gone by in either the humanities or the social sciences in which the question “What is postmodernism?” has not been raised on one formal level or another. And the single greatest deterrent to overcoming the quarrel between the sciences and the humanities is the inherent resistance of this term to definition. Year after year, science asks for a clear and precise definition of “postmodernism,” and neither the social sciences nor the humanities can offer one, since any and all definitions of postmodernism lead to tautologies. To use the language of science, these contradictions arise because the “nature” of postmodernism varies depending on who is conducting the experiment and in which lab the experiment is performed. The kind of “postmodernism” titrated in sociology laboratories is intrinsically different from the kind brewed in creative design laboratories. So, more like a quantum phenomenon (which will be explained in due course), the kinds of experimental determinations that are used to observe postmodernism actually determine the “nature” of it. Therefore, since “definition” is no longer defined within the axiomatics of postmodernism, we must ask different kinds of questions.

<8> First, we must determine the nature of the attempt since this determines whether or not the term is to be celebrated (as in the social sciences) or mourned (as in the humanities). And second, we must determine for whom it is defined, be those scholars in a highly specialized field, French Neo-fascist youth covering their faces with kefiyeh at a pro-Le Pen rally, or fans of the Simpsons television show. Admittedly, mathematics is more equipped to deal with an entity like postmodernism since mathematical entities need not be defined in order to be utilized. Even the most cutting-edge research in intuitionism (the dominant philosophy of contemporary mathematics) has yet to provide a sufficient definition of “number.” By and large, mathematicians are content as Platonists. The sinuous axiomatics of postmodernism contains many contending propositions, however, it is significant that no postulate can be found. So rather than define the concept of postmodernism, a development of some central propositions would better suit the endeavors of this essay [8].

<9> For the purposes of this essay, the term “postmodernism” will come to designate a certain set of propositions that I will develop alongside an account of its conceptual history according to the humanities and social sciences. In avoiding definition, which would imply a sense of conceptual completeness or totality, postmodernism derives from certain propositions, which as mathematics teaches us, are always open to debate. These propositions derive directly from accepted conventions and assertions from within cultural studies in general and literary studies in particular. The only assumption on my part is an audience with a vested interest in cultural intersections among the sciences and the humanities. And it is the nature of this intersection that I understand to be “postmodern” in the general sense that neither the sciences nor the humanities are thought of in absolute or complete ways. This amounts to identifying C.P. Snow’s famous “two cultures” thesis as no longer applicable to the current state of affairs in either.

II. The Postmodern Theorem

<10> My understanding of postmodernism derives directly from the genealogy we in the humanities have constructed: one usually traced through a group of French academics from the late 1960’s who worked together to forge the theoretical groundwork from which the term comes [9]. However, the form of my narrative follows mathematical practice.

Postmodern Porism: the conceptual history of Postmodernism is a collective, rather than individual development.

Suffice it to say that as a general rule in the humanities and many of the social sciences, the birth of “postmodernism” dates back to the moment Jacques Derrida presented his paper “Structure, Sign, and Play in the Discourse of the Human Sciences,” at the John Hopkins Conference on “The Language of Criticism and the Sciences of Man” in 1966. However, as New Testament narrators like Matthew and Luke have taught us, the account of the nativity varies depending on the narrator. While remaining a product of the mid to late 1960’s, the actual date of the birth tends to vary: as early (and specific) as July 25, 1965 in Newport, Rhode Island at 8:45 p.m., and as late (and vague) as the Watergate scandal between 1972 and 1974 [10].

Proposition One: Postmodernism is performative.

Corollary: Postmodernism has an epistemological, rather than ontological status.

<11> Though the humanities and social sciences have used the term for thirty years, for the hard sciences: physics, chemistry, and many include mathematics in this list, the term postmodernism is a bit of a misnomer, in so far as their practitioners do not tend to use it when describing their work. Likewise, it is also telling that when these hard sciences had their respective “postmodern” revolutions in the earlier part of the twentieth century, they each thought of the revolution in similar terms: as a radical paradigm shift.

Proposition Two: Postmodernism designates a radical conceptual shift in the established structures of interpretation of a given system of thought.

For example, mathematics began its “postmodern revolution” at the turn of the twentieth century with the development of mathematical philosophy, which culminated in 1931 when Kurt Gödel’s “On formally Undecidable Propositions” brought the three contending mathematical philosophies at the time to their knees (Logicism, Intuitionism, and Formalism). Gödel’s humbling discovery that within any given area of mathematics (he chose the still young field of logic), there will always be certain propositions that cannot be proven true or false using the rules of that branch of mathematics alone, caused a major revolution in every branch of mathematics, the most profound of which was in logic itself [11]. Unequivocally accepted by all branches of mathematics, Gödel’s proof forced mathematics to come to terms with its own foundations, its own raison d’etre.

Proposition Three: As a method of analysis, Postmodernism interrogates the fundamental assumptions upon which extant structures of thought are predicated.

Proposition Four: Postmodern methods of analysis demonstrate the ultimate undecidability of a given system’s truth-values.

<13> In answering one of mathematics’ oldest questions, Gödel’s proof caused a revolution in the way mathematicians thought about their subject. In Infinity and the Mind (1982), Rudy Rucker characterizes the nature of Gödel's answer as political and “libratory” when he claims, “to understand the essentially labyrinthine nature of the castle [mathematics] is, somehow, to be free of it” (165).

Proposition Five: The aims of Postmodernism are political.

Since Descartes, a significant portion of Western thought has been occupied with the struggle between the scratch-and-sniff, British brand of Empiricism and the I-believe-it-in-spite-of-not-seeing-it, Occidental brand of Rationalism. Gödel’s answer to this dilemma, Rucker speculates, is that “rational thought can never penetrate to the final ultimate truth” (165, emphasis his). Mathematics was one of the earliest disciplines to face its own mortality and survive to tell the tale. A decade before, a new breed of physicist placed the mirror up to classical, Newtonian physics and found similar results, among them that their methods of utilizing measuring devices like mirrors needed a radical re-conceptualization. A decade later, fascism would force political scientists to re-conceptualize how they defined even the most basic of political categories like right wing and left, nationalism and sovereignty.

Proposition Six: The political dimension of postmodernism aims at internal change, rather than external.

Corollary: The political agenda of Postmodernism aims towards its practitioners, rather than its consumers.

The humanities, unfortunately, were fashionably late to this dance with the absolute. And to be sure, this is precisely how particular disciplines such as literary studies conceptualized their own axiomatics: as an absolute beyond which there was only opacity and nothingness [14]. Arriving thirty years late to the dance had its price, as literary studies started down a path of neurosis over the devaluation of their medium (language), occasionally creating compulsive rituals to ease the obsessive anxiety of the loss [15]. After a pirouette, mathematics and physics resumed their work and developed new areas of inquiry in spite of their own disciplinary devaluation such as nuclear physics and category theory.

<14> Of these “postmodern” areas to explode after Gödel’s Proof, topology eagerly embraced this inconsistent logic. In fact, this inconsistent, internal logic is the very fuel of topologic analysis. For example, Douglass Hoffstadter reminds us of Gödel’s Second Theorem, which asserts that only propositions asserting their own consistency are themselves, inconsistent (696). And as a kind of response to Gödel, algebraic topology, for example, transforms the logical concept of consistency into the spatial concept of connectivity. And since consistency is not provable within any given axiomatic, topology is able to transform the nature of the problem: from one of logic to one of space. And as I shall show, mathematical space contains an infinite number of “axiomatics,” which Felix Hausdorff conveniently first calls “neighborhoods” in 1914 [16]. A kind of patchwork space, topologic analysis can combine incongruent, even contradictory axiomatics by bounding them within a single topologic field. Though Gödel’s theorem does not coincide with the initial body of work on early topological analysis found in Bernhard Reimann, Henri Poincaré, Hausdorff, or L.E.J. Brouwer, such early work seems to utilize a certain understanding of his ideas. In the next section, a selective history of topology is developed to help illustrate these connections. The first example is the most well known and most misunderstood of topological oddities, the Möbius band.

III. A Brief History of Topology

A. Part One: The Patchwork of Topology

<15> As a discipline of mathematics, topology has existed for only seventy years, though antecedents go back centuries. The history of topology, like so many areas of mathematics, more closely resembles a patchwork quilt than a neatly bound tapestry. Leonard Euler discovered the first topological property in 1750, but the second was not discovered for another century [17]. The first real work in topology did not appear until the turn of the twentieth century when Brouwer, considered the father of modern topology, unified some of Poincaré’s work on differential equations and published the first fixed point theorem in 1909. Little comfort to the non-mathematician, topologists themselves have a hard time defining what it is they do. Though it started as a kind of geometry (colloquially referred to as a rubber-sheet geometry), today topologists are an eclectic lot spanning the gamut of mathematical pursuit from analysis (calculus), to algebra, and geometry. In Experiments in Topology, Steven Barr seems to have his hand on the pulse of the early process of disciplinization when he claims that topology is “curiously hard to define” (1). Recognizing an overt process at work, he recognizes an inherent “postmodern” tendency when he claims that topology is “its own goal” (though, of course, he does not use this term) (2).

<16> It is perhaps not surprising, then, that the story of topology reads more like literature than history, more like Lewis Carroll’s Through the Looking Glass (1872) than Edward Gibbon’s The Decline and Fall of the Roman Empire (1776-1788): it occurs in fits and spurts, rather than along a linear progression of discreet bundles [18]. And while it is a relatively simple task to characterize topology as the study of continuity to non-mathematicians or as the formal study of the features of geometrical figures that remain invariant under spatial transformations to mathematicians, neither fully captures the “sense” of topology.

<17> My “selective” history of topology tries to characterize the sense by describing some of its detail. By focusing on some of the detail of Möbius’ discovery of the second topological property in 1865, a disciplinary narrative develops: one that reveals the latent postmodern tendencies in certain kinds of mathematical thinking. To be sure, standard classifications (such as the distinction between topology and geometry) are necessary, not only to provide a familiar correlative for the non-mathematician, but more importantly, to provide a context, as topology can also be described as the mathematics of context. As postmodern cultural theory has shown, context determines the conditions under which meaning is produced. But the proofs of “postmodern” science such as quantum mechanics demonstrate that a rigorous analysis of context reveals that not only are the most basic of concepts (such as protons and electrons) without any meaning, they lack any real existence

B. Part Two: Möbius and His Band

<18> In 1865, August Möbius, an admittedly mediocre German mathematician, began work on a strange one-sided, two-dimensional figure that now bears his name (though Johann Listing actually published his findings on the “Möbius” band four years earlier). What he discovered is that when he tried to align triangles along the surface of the band, the orientation of the triangles had changed by the time he finished covering the surface. He described this “property” of one sidedness as “non-orientability”. What this means is that when an orientable figure (one with bilateral symmetry like our bodies, for instance) traverses the length of the one-dimensional figure, the right and left sides of it become inverted as if viewed in a mirror. For example, if a person wears an eye-patch on his/her right eye, then whether they live in Finland or New Zealand, face East or West, or stand on their head, their right eye is always the one covered. This is a simple demonstration of the orientability of the shape of the earth, an ellipsoid. The only exception to this is the reflection of the one-eyed person in a mirror, in which the eye patch seems reversed. The Möbius band is a perpetual mirror world in which it is impossible to tell which image is real and which is a reflection. If our friend with the eye-patch over his/her right eye traveled along a Möbius band, eventually the patch would switch to the left eye.

<19> Now at this point, a ready objection comes from cultural theorists who argue that perspective can determine one’s reality. Even on an ellipsoid like the earth, “up” is relative, especially to the New Zealanders. But orientability is different from mathematical perspective in that orientability is invariant in spite of perspective. Orientation is an internal property of a figure, while perspective is an exterior orientation applied to it. As a kind of internal perspective, the orientation only changes when the internal nature of the figure is changed, while exterior perspective changes as the external circumstances change and in spite of internal orientation. From the anterior, a pyramid not only appears to be a square, but even the dimension appears different (two rather than three). As the position of the observer changes, so does the perspective. However, the internal orientation of the pyramid does not change. A Möbius band is an example of a figure (or a space) without orientability.

<20> A circle seems non-orientable as well, with no inherent top or bottom (only a relatively defined one), but when half of the diameter is colored red and the other half blue, the orientation of a circle becomes obvious (and remains even in spite of any color-coding). However, a Möbius band is not orientable since there is only one side: coloring only the “top” half of the band red (which is relative in terms of perspective) becomes impossible as eventually, the entire figure will be colored red as well. The top of a Möbius band is the same as the bottom, just as the inside is also the outside [19]. Without orientation, perspective itself is still possible because the fundamental distinction between self and other, between here and there remains (from a distance, the Möbius band appears to be a figure-eight). But without orientability, there is no inside or outside, as one eventually becomes the other. Two-dimensionality (such as a sheet of paper with infinite thinness) and three-dimensionality as kinds of spaces have a consistent oreintability. And here it is helpful to think of space as an open, unbound geometrical figure. Möbius happened onto an intrinsically different kind of space than the usual, three-dimensional version we seem to inhabit. As such, he discovered the second topological property: for the same kinds of space, orientability is invariant [20].

<21> Topological properties like orientability are thus the most fundamental characteristics of figures and spaces as they are invariant across drastic (albeit allowable) kinds of spatial transformations. In a sense, these properties form the core nature of all topologically similar figures, whereas geometrical properties vary even among similar figures. For example, a triangle and a circle are the same kind of topological figure because their topological properties are the same, even though their geometrical qualities are drastically different. By following the rules of topological transformation, a triangle can be “rounded” into a circle, which does not change its intrinsic qualities. Only the extrinsic quantities (distances, angles, degrees, volumes, dimensions, etc.) change during topological transformations.

C. Part Three: Topology vs. Geometry

<22> Topology is like geometry: it is concerned with the shape of space and how we understand (visualize) figures within different kinds of spaces. However, whereas geometry is concerned with quantitative properties of geometrical structures such as lengths, degrees, and areas, topology is concerned with their qualitative properties. But despite this difference, both disciplines share a concern for the kinds of transformations that are allowed. Euclidean geometry -- the familiar, planar world high school students explore with varying degrees of success -- is the most restrictive of geometries. The only acceptable transformations are turning and flipping. Two objects are considered “equal” in Euclidean space if they can be exactly mapped one on top of the other (every point on one figure corresponds with every point on the other). Even though two triangles may have the same interior angles (they are “similar”), if the lengths of their sides are different, then they are not “congruent” triangles. In this sense, topology is a much less restrictive kind of geometry. For example, the concept of distance is crucial for topology, but actual distances (numerical values) are irrelevant. More than turning and flipping, topology allows for all continuous transformations such as bending and stretching, but discontinuous transformations such as cutting and puncturing are not allowed. A two-dimensional disk and a three-dimensional pyramid are topologically the same kinds of figures since continuous transformations can change one into the other.

<23> Conceptually, topology is the study of continuity. Rather than measuring angles and distances, topology looks for holes and sutures. Ironically, while Euclidean geometry focuses on minute and precise ratios, it is most efficient when dealing with the most simple of figures. In order for Euclidean geometry to determine the surface area of a complex figure such as a coffee mug, it must first break down the complexity into simple elements (essentially a cylinder and a torus -- the donut shape of the handle). On the other hand, topology is able to analyze very complex structures in their totality, without disassociation. In fact, as we shall see, topology is in a position to make qualitative statements about figures only because objects are understood in their complex totality.

<24> While both geometry and topology study figures and objects (real, impossible, or imaginary), each has its own set of rules and operations that are allowed. And it is this axiomatic that distinguishes geometry from topology. As I suggested, the fundamental relationship in geometry is congruence through an exact mapping of points. To be sure, geometry has developed many theorems that allow one to avoid such an exact mapping, but the mapping itself is essential. If this map cannot be produced through deduction or induction, theorems or postulates, then there is no congruence, only difference. In this way, geometry is concerned only with manipulating the extrinsic properties of a figure, that is, with its variable properties.

<25> Topology, on the other hand, is interested only in the intrinsic or permanent properties of figures. The concept of distance, while crucial in topology is only important in so far as it remains consistent as a concept. For geometry, only the value of distance is important. The nature of that value is irrelevant. For example, geometry is not concerned with the nature of roundness, only the value or “kind” of that roundness (elliptical, hyperbolic, circular, etc.). In topology, extrinsic determinations of distance or degree do not fundamentally change the nature of a topological object or its reliance on a consistent definition of distance [21]. As a kind of rigorous, qualitative geometry, early pioneers in the field of topology began by distinguishing among types of spaces (again, open and unbound figures), rather than among different perceptions of those spaces (such as projective geometry).

<26> Many early topologists were trained as geometers, and much of their work evolved out of a dissatisfaction with the kind of space Euclid envisioned. As my first example demonstrated, Möbius discovered that the shape of space determines what kind of orientation is possible. In the 1850’s, Riemann discovered that while our reality seemed to be Euclidean, it was not. Riemann was among a growing number of mathematicians who became dissatisfied with some of Euclid’s assumptions about the nature of space, particularly his parallel postulate [22]. Keeping the rest of Euclid’s system in tact, figures like Bolyai, Lobachevsky, and Riemann were able to create consistent geometries free of contradictions in which the parallel postulate did not hold. Early non-Euclidean geometries thus discovered that the shape of space determines the kind of geometry that is possible within it.

<27> In thinking about the order backwards (space determining the kinds of figures that are possible within it), non-Euclidean geometers were able to challenge the way mathematics thought about its objects of study as well. Unlike geometry, which can only compare quantitative differences (since sameness is so narrowly defined), topology actually compares the qualitative similarities between objects. In geometry, complexity is only understood as an instrument and extension of simplicity. Geometry distinguishes between circular and elliptical to arrive at quantitative statements about their differences. Topology understands complexity in itself, in order to develop a detailed and formal method for dealing with the qualitative similarities among objects. In this way, topology is the basis for geometry since the nature of roundness (in a formal and rigorous way) forms the basis for concepts such as circular and elliptical. As a kind of “postmodern” geometry, topology analyzes the nature of the ground upon which its own self-construction lies. Even when dealing with the same “object,” geometry and topology approach that object in a radically different way. A common illustration to distinguish the two pursuits is the example of the London Underground. Geometry can determine the distance from Piccadilly Square to King’s Cross, while topology is more interested in how many different trains the trip will take.

D. Part Four: Topology as a Pirouette

<28> As I hope the breadth of my examples have shown, mathematics is different from most disciplines in that it has not so much had a single, definitive postmodern revolution (in spite of Gödel’s achievement). On the contrary, twentieth century mathematics is composed almost entirely of them. The history of mathematics, like the history of science according to Thomas S. Kuhn, is composed of revolutions. When the consequences of disciplinary momentum reach their limit and are articulated, as in Gödel’s proof or Derrida’s essay, the shape of the space in which that discipline has functioned becomes visible to those within it for the first time. And it is the job of this first-time audience to carefully describe what it is they have seen. Werner Heisenberg’s autobiographical writings provide a contemporaneous correlative to this postmodern shift in physics.

<29> In Physics and Beyond: Encounters and Conversations (1971), Heisenberg tells the story of his battle with tremendous hay fever that forced him to leave his studies with Max Born in May 1925. He traveled to Heligoland, hoping the sea air would ease his swollen face. Sitting on his balcony, he mused, “I had ample opportunity to reflect on Bohr’s remark that part of infinity seems to lie within the grasp of those who look across the sea” (60). Thanks to the post-classical discoveries of Heisenberg and his contemporaries, the classical space of traditional physics came into focus during the 1920’s for the first time since Newton painted the original picture of it two hundred years earlier. In Gödelian fashion, Heisenberg deduced that one’s position with respect to the sea determines the way in which the sea is interpreted. The sea of quantum physical reality is only understandable as the sea from the land, that is, from classical reality (the “outside” of the sea). To utilize the Zen-like rhetoric of The Matrix, Heisenberg understands that a shore-less sea is not a sea. As soon as one leaves the orientability of the land and enters the quantum sea, the classical laws of the land no longer apply. For Heisenberg in atomic physics, Gödel in logic, and even Melville in literature, this moment of self-realization can only come from a return, a return to the outside from the inside. Extrapolating the infinite from the finite, whether in mathematics or literature, not only changes the concept of the infinite, but the concept of the finite as well.

<30> The local histories of each branch of mathematics all contain this moment of self-realization -- this recognition of one’s place within a larger system. Set theory has the tragic story of Georg Cantor who, like the protagonist in a Marlowe play, was so castigated and vilified by his contemporaries for trying to understand that which was previously held to be ineffable, that he eventually lost his mind [23]. As Kuhn notes, Copernicus, Boyle, and Galileo all faced this same fate: confronting the limits and inconsistencies of their respective systems of thought. The kinds of inconsistencies that Gödel and Derrida deconstruct, however, are particularly “postmodern” because they develop from the kinds of meta-disciplinary actions described in the first part of this essay. These kinds of en-disciplined actions (for example, attempts to define a “philosophy of” -- almost unique to the twentieth century) were unavailable to Galileo or Copernicus [24]. In terms of conceptual progression, philosophies tend to develop late in the history of disciplines, and are often followed quickly by “postmodern” revolutions.

<31> I previously described this disciplinary self-confrontation in terms of a pirouette: a sort of dance with a virtual twin, in the guise of disciplinary raison d’etre. Like looking in a Möbian mirror, lost innocence is indicated in the flawed visage of the now evil twin of disciplinary oblivion, almost out of the pages of an Edgar Allen Poe tale. However, the miasma filling Poe’s tales obscures reality, as the protagonists are trapped by their narrative construction and are denied the perspective of the reader, outside the story. With the anticipated consequences, Poe’s protagonists, like William Wilson, often find themselves lying on a floor covered in shards of broken mirrors, bloody from self-inflicted wounds. It is from this confrontation that early twentieth century mathematics and science walked away: bloody, but on the road to recovery. Unfortunately, as the narrative of Derrida’s particular revolution demonstrates, literary studies has read too much Poe.

<32> Plotting the path of this pirouette and describing the kind of dance that ensues is perhaps a job best suited for either vector calculus or choreography. Both can plot the trajectory this pirouette of disciplinary momentum takes (either towards oblivion or redemption), but topology is able to mathematically assess not only the direction of the trajectory, but its destination as well [25]. In a combinatorial way, topology is a science of relation, developing formal propositions about the mathematical context of this “complex” object. The pirouette is complex in that within its boundaries lie different kinds of mathematical objects (a trajectory in a vector space and the topologic properties of the trajectory itself). As such, complex topological objects are not geometrical; they are not composites of simples. As the study of boundaries, topology widens the scope of the definition of object to include its context (what topology refers to as the “embedding space”). In this way, an object is not distinct from its context. Topologically speaking, without a context, an object is just a collection of geometric pieces that are only understandable in terms of their quantitative, transient properties.

IV. Topology as Postmodern Science

<33> Early in my essay, I noted that conventional wisdom aligns mathematics with the hard sciences like physics and chemistry. One is hard pressed to find a university or college in which mathematics is not housed in one of many schools of “science”. This classification, however, needs some qualification. Technically speaking, mathematicians are correct when they declare that mathematics is not a science. Science deals with empirical objects (or at least their empirical traces) from protozoa to pulsars, population growth to radioactive decay. However, mathematics is fundamentally abstract and numbers -- arguably the “essence” of mathematics -- do not exist in the material world [26]. In learning to count, we only come to understand the concept of number through the concept of empirical object. This connection, mathematics proves, is only accidental [27]. No amount of scrutiny over two apples will yield any further proof into the existence of number, only apples (and philosophers like Bishop Berkeley have skillfully contested this assumption as well). In fact, even mathematics has only a primitive understanding of number. Until the last century, the concept of “two” was only understandable in terms of what it was not (three, five, a square, an integral, etc.) [28]. While Intuitionists have taken on the challenge of defining number in itself through a primitive understanding of before and after (two is the successor of one, the predecessor of three), they cannot avoid the fundamentally abstract nature of their subject matter.

<34> However, mathematics is a science in the very specific sense that it analyzes its objects in a scientific way. This way, the method, according to the protocols of modern science, is what makes a traditional discipline like geology and a relatively recent discipline like psychology “scientific”. While the traditional conception of scientific method has gone through many forms from Aristotle to Kuhn, one of the most original versions is found in Robert M. Pirsig’s Zen and the Art of Motorcycle Maintenance (1974). During his motorcycle trip across Montana, Pirsig philosophizes on “the same ghost that Phaedrus pursued -- rationality itself, that dull, complex, classical ghost of underlying form” (92). After a careful explanation of deduction and induction, he succinctly lays out the classical form of scientific method: “(1) statement of problem, (2) hypothesis as to the cause of the problem, (3) experiments designed to test each hypothesis, (4) predicted results of the experiments, (5) observed results of the experiments and (6) conclusions from the results of the experiments” (93). From quantum physicists studying the ghost data of particle traces and the shadows of wave equations to field botanists taking cross sections of infected apple blossom stamens and atrophied pistols, the method remains the same. For the scientists, the proof is not in the pudding (or the motorcycle engine), but in following the steps. And this proof has nothing to do with “truth” [29].

<35> Mathematics is no different in this respect. Though the method is often hidden in the symbols, mathematics is a virtual (and admittedly, synthetic) science since the objects of its analysis are its own creation and do not exist, properly speaking. However, in the early part of the cold war when science was starting to reap the benefits from its own exploitation at the hands of the state, Karl Popper, one of the most prolific natural and social philosophers of the last century proposed a humble alternative to the increasingly capital-driven scientific method. In “Philosophy of Science: a Personal Report,” he notes that the concept of refutation is what distinguishes science from what he calls “pseudo-science”. What makes science “scientific,” according to Popper, is not its reliance on a concept of truth, but its demonstrability.

<36> Distinguishing between the pseudo-scientific theories of Marxism and Freudian psychoanalysis, which attempt to explain phenomena in terms of theory, Popper juxtaposes physical science as the antithesis of explanation. As theories popular during Popper’s youth in Vienna, Marxism and psychoanalysis, rather than gaining strength through their confirmations and verifications in the everyday world are only weakened by them (35). According to Popper, “irrefutability is not a virtue of a theory (as people often think) but a vice” (36). Thus every test of a theory, according to him, is an attempt to falsify it. Taking this line of reasoning to its conclusion reveals that the results of any given scientific experiment (he used Einstein’s gravitational theory) have no meaning outside of their demonstration. Though he does not use the phrase, science is performative according to his model. And because the heart of science lies in its repeatability, then framing the conditions of the method -- the hypothesis -- becomes more important to the disciplinarity of science than the results.

<37> Pirsig makes the same claim: it is the hypothetical nature of a theory that makes an experiment scientific (94). The concept of truth, while relevant to the results of science, does not figure into its essence. In fact, as Kuhn again has noted, science progresses through revolutions that arise, often, from failed experiments. The success or failure of a scientific hypothesis does not determine scientific “value” any more than it determines “truth”. Scientific value is determined by proper adherence to the methods and protocols of science. But Popper’s method in his essay is as revolutionary as his argument. Rather than describing the scientific method that begins with the statement of the problem, hypothesis, experimentation, predictions, observations, and conclusions (like Pirsig), Popper actually describes the transformation of the term “scientific” from the standpoint of his own hypothesis. This transformation, I suggest, is postmodern in the way that I have described above not only because this transformation is performative, but because this shift in the way disciplinary ideology is constructed lead those within it to a crisis of faith of sorts in their own methods. I want to return to the story of quantum mechanics one last time in order to illustrate the re-fortifying way in which the sciences engaged the conditions of their own demise.

A. From the ashes of Newtonian Mechanics

<38> As I suggested in the introduction to this essay, quantum mechanics was the first science to face its own mortality in the early twentieth century. I also suggested that this coming to terms with one’s mortality in terms of disciplinary protocols constituted a “postmodern” shift in the way that discipline operates. In the early part of the last century physicists like Bohr, Heisenberg, and Dirac were not only discovering the practical limits of their own experimental devices and techniques (and thus the limits of the classical conceptions of reality they produce), they were also discovering the conceptual limits of classical thinking. Light was re-defined as a “quantum” phenomenon because it appeared to have an indeterminate nature, at least in the sense that different experimental situations produced different pictures of it.

<39> In “Complementarity, Idealization, and the Limits of Classical Conceptions of Reality,” Arkady Plotnitsky describes Bohr’s theory of complementarity as a method designed to connect both versions of reality (the classical and the post-classical) without synthesizing them. Bohr’s theory suggests that both classical and what he calls “modern” theories of reality must be employed in mutually exclusive, yet equally necessary ways for a comprehensive description of quantum phenomena (137). However, this theory arises from the mutually exclusive experimental arrangements that, while producing equal results, lead to different conceptions of quantum phenomena [30]. Therefore, due to these experimental arrangements, no complete configuration of quantum phenomena is possible, since in order to measure one state of a given physical system (the momentum of a photon, for example), another state must be sacrificed (the position). As Bohr himself explains in a late essay “Unity of Knowledge,”

The recognition that the interaction between the measuring tools and the physical systems under investigation constitutes an integral part of quantum phenomena has only revealed an unsuspected limitation of the mechanical conception of nature, as characterized by attribution of separate properties to physical systems, but has forced us, in the ordering of experience, to pay proper attention to the conditions of observation (74)

Delineating an understanding of postmodern, rather than “modern” science, Bohr’s essay enunciates a tension shared by many of his colleagues in physics at the time. This growing tension arose from the radical discrepancy between the results of quantum mechanical experiments and the classical frame of reference in which they were observed and understood. Bohr was the first to openly embrace the possibility that classical conceptions of reality simply were not sufficient to describe quantum phenomena such as light. The experiments produced the correct results, so without human or methodological error, Bohr had no choice but to accept the conclusion that reality was unlike anything he had anticipated.

<40> The history of mathematics is so rife with these instances in which classical reality is insufficient to describe mathematical hypotheses that mathematics can be said to have been a postmodern science since the late eighteenth century with the discovery of imaginary numbers [31]. Unlike most sciences, which derive a picture of reality based on the objects contained therein, most areas of mathematics are more interested in the objects themselves. Quantum mechanics is a unique combination of both pursuits. The mathematical results of quantum mechanical experiments forced it to deviate from the trend in science, since the objects of its study were no longer “objects” in the classical sense. The mathematics (Schrödinger’s wave equations and Heisenberg’s uncertainty relations) forced the science in quantum mechanics to change its course as well since, as Plotnitsky notes, quantum mathematics “represents the interaction between quantum objects” rather than the objects themselves (138). More than an overly mathematical science, quantum mechanics not only changed the way science interacted with its objects, but it also changed the way mathematicians approached their own.

<41> The post-war work in topology that has spread to nearly every corner of mathematics, moves the discipline in this direction as well: towards a redefining of its own scope. The sporadic, even nomadic history of topology forms an archipelago of similar theories all struggling with the same problem: how to mathematically describe that which is not describable in classical mathematical terms. Many of these early conundrums derived from problems in geometry. Many of the lingering Greek conundrums such as squaring the circle and trisecting the angle remained possibilities until topology proved them to be unsolvable. Until the nineteenth century (recent according to the standards of mathematical history), the meaning behind mathematical demonstration was defined according to the skill and acuity of the individual practitioner and their ability to construct areas, lengths, and volumes [32]. Topology changed the conditions of the game by re-defining what it means to construct, suggesting that perhaps extrinsic properties were no longer sufficient to describe a figure. Since topology analyzes the qualitative properties of figures, it becomes the first branch of mathematics to be in a position to make meaningful, qualitative statements about the “culture” of their realities as well.

V. Conclusion

A star’s a reward, so I’ve been told,
This custom passed on from days of old.

--Inge Auerbacher

<42> As a kind of postmodern science of culture, the methods of topological analysis, when applied to cultural, rather than mathematical objects, provides a rigorous, yet unabashedly humble investigation of the nature of cultural relation since the claims to consistency are no longer hidden in the assumptions of the system. When viewed as a massively complex, topological object, the system of cultural relations is far from simple. Yet, a topological approach this complexity on the local level -- the mathematical “neighborhood” -- without reducing it to simple, quantitative functions of larger systems. The topologic method systematically and formally interrogates how an object is defined, and in doing so, provides the des res and ad hoc grounds from which a concept of relation emerges. And since Gödel’s Second Theorem demonstrates that only the axiomatic propositions that claim consistency are inconsistent, topology (like so many branches of mathematics to blossom after Gödel’s revolution) frames and contextualizes the conditions under which such claims are made. The extrinsic conditions that have traditionally bound certain cultural objects in terms of race or gender, for example, can be reconfigured topologically in non-totalizing, non-classical ways.

<43> Contrary to some recent interpretations, Gödel did not prove that there is no underlying truth behind mathematical exegesis. Mathematical reasoning has always functioned on the suspension of disbelief, on intuitive hypothesis. The assumption that Gödel’s proof exposes is that such truth cannot be consistently proven. Similar to quantum mechanical experiments with light, the conditions of mathematical proof determine what kind of truth is deducible. Basic and undeniable operations such as addition and subtraction are defined differently depending on the kinds of numbers that are being utilized. If mathematical truth is contextual in this way, if “starness” or “triangleness” is as much a function of the branch of mathematics that is doing the defining as “Jewishness” or “homosexualness,” is to the institutions that have defined them in the past, then perhaps mathematics is closer to cultural studies than we think [33].

I stand tall and proud
My voice shouts in silence loud:
“I am a real person still,
No one can break my spirit or will!”
I am a star! [34]

Works Cited

Auerbacher, Inge. I am a Star. New York: Puffin, 1986.

Barr, Stephen. Experiments in Topology. New York: Dover, 1989.

Benacerraf, Paul and Hilary Putnam, eds. Philosophy of Mathematics: Selected Readings. Cambridge: Cambridge UP, 1983.

Bohr, Niels. “The Unity of Knowledge,” The Philosophical Writings Volume II: Essays 1933-1957 on Atomic Physics and Human Knowledge. Woodbridge, Connecticut: Ox Bow Press, 1987: 67-82.

Derrida, Jacques. “Structure, Sign, and Play in the Discourse of the Human Sciences,” In Writing and Difference, translated by Alan Bass. Chicago: University of Chicago Press, 1978, 278-293.

---. “White Mythology: Metaphor in the Text of Philosophy,” In The Margins of Philosophy, translated by Alan Bass.  Chicago: U of C Press, 1982, 207-271.

Eaves, Howard. An Introduction to the History of Mathematics. New York: Saunders Harcourt Brace, 1992.

Euclid. The Thirteen Books of the Elements, translated by Sir Thomas L. Heath. New York: Dover, 1956. 3 Vols.

Foucault, Michel. Discipline and Punish: the Birth of the Prison, translated by Alan Sheridan. New York: Vintage, 1995.

Fuller, Steve. “Does Science put and End to History, or History to Science? Or, Why Being Pro-Science is Harder than You Think,” In Ross, 29-60.

Gamow, George. Thirty Years that Shook Physics: the Story of Quantum Theory. New York: Dover, 1966.

Gödel, Kurt. On Formally Undecidable Propositions of ‘Principia Mathematica’ and Other Related Systems, translated by B. Meltzer. New York: Dover, 1992.

Heisenberg, Werner. Physics and Beyond: Encounters and Conversations, translated by Arnold J. Pomerans. New York: Harper & Row, 1971.

Henle, Michael. A Combinatorial Introduction to Topology. New York: Dover, 1979.

Hoffstadter, Douglass R. Gödel, Escher, Bach: an Eternal Golden Braid. New York: Vintage, 1989.

Kuhn, Thomas S. The Structure of Scientific Revolutions. Chicago: U of C Press, 1962.

Levitt, Norman. “Mathematics as the Stepchild of Contemporary Culture,” in The Flight from Science and Reason, edited by Paul R. Gross, Norman Levitt, and Martin W,. Lewis. Baltimore: Johns Hopkins UP, 1996, 39-53.

Nelkin, Dorothy. “The Science Wars: Responses to a Marriage Failed,” in Ross, 114-122.

Pirsig, Robert M. Zen and the Art of Motorcycle Maintenance. New York: William Morrow, 1974.

Plotnitsky, Arkady. “Complementarity, Idealization, and the Limits of Classical Reality,” in Mathematics, Science and Postclassical Theory, edited by Barbara Hernstein Smith and Arkady Plotnitsky. Durham: Duke UP, 1997, 134-172.

Poincaré, Henri. Science and Hypothesis. New York: Dover, 1952.

Popper, Karl R. “Philosophy of Science: a Personal Report,” Conjectures and Refutations: the Growth of Scientific Knowledge. New York: Basic, 1962, 33-65.

Ross, Andrew, ed. Science Wars. Durham: Duke UP, 1996.

Rucker, Rudy. Infinity and the Mind: the Science and Philosophy of the Infinite. Princeton: Princeton UP, 1995.

Shapiro, Stewart. Philosophy of Mathematics: Structure and Ontology. New York: Oxford UP, 1997.

Snow, C. P. The Two Cultures: and a Second Look. New York: Mentor, 1964.

Notes

[1] I use the terms “art” and “science” here purely out of convention. The so-called “arts” often comprise as much social science (communication studies, psychology, sociology, anthropology, political science, etc.) as they do traditional humanities (English and foreign language arts, history, philosophy, theater and performing arts, etc.) [^]

[2] The so-called “science wars” have been a fixture in the academy since the late 1980’s. Fueled by what Dorothy Nelkin has called a failed “marriage contract” between science and the state after WWII in which science became increasingly political and value-laden, rather than impartial and value-less, social critics of science (Marxists, feminists, postmodernists, anthropologists, rhetoricians, literary theorists, etc.) began to question the ways in which science claims to know what it knows (116). What ensued in the early 1990’s was a heated backlash from scientists themselves who felt threatened by these critiques, which culminated in the publication of Norman Levitt and Paul Gross’ Higher Superstitions: the Academic Left and its Quarrels with Science (1994). Gross, Levitt, and Martin W. Lewis’ The Flight from Science and Reason (1996) soon followed. Shortly thereafter, Andrew Ross edited Science Wars (1996), a response to this backlash from across the humanities and social sciences. I would refer readers to these collections, as they are the most convenient collections of disciplinary responses to the debate. [^]

[3] In 1996, SocialText published Alan Sokal’s “Transgressing the Boundaries: Towards a Transformative Hermeneutics of Quantum Gravity.” Sokal claims that the article was a hoax, intended to lampoon the New Left and their theories of cultural relativism and their assertion that reality is socially constructed. For further reading about this hoax and the effects it had across the disciplines, see Gross and Levitt’s Higher Superstition (1997); Noretta Koertge, Ed. A House Built on Sand: Exposing Postmodernist Myths about Science (Oxford UP, 1998); and Paul A. Boghossian “What the Sokal Hoax Ought to Teach us: the Pernicious Consequences and Internal Contradictions of ‘Postmodernists’ Relativism.” (Times Literary Supplement, Commentary December 13, 1996: 14-15). [^]

[4] I use the term “traditionally” cautiously as it is clear that while many cultural institutions may perpetuate the dichotomy, a broader history of Western culture shows otherwise. A dominant line can be traced from Aristotle through figures like Galileo, Da Vinci, Leibniz, Einstein, Fenyman and Gould that contradicts this traditional dichotomy. In fact, albeit in general terms, Levitt has located the moment of disjunction between the sciences and the arts relatively recently, in the eighteenth century (41). [^]

[5] In “Mathematics as the Stepchild of Contemporary Culture,” Levitt criticizes Fuller’s attempt to characterize set theory as superior to “number” in his essay “Can Science Studies be Spoken in a Civil Tongue?” (53). [^]

[6] Novelists and critic C. P. Snow first presented a series of lectures at Harvard University in 1959 in which he makes the claim that society in general tends to separate the sciences and arts in terms of their methods, practice, theory, reception, and purpose. Though Snow ultimately argues that such divisions should be dismantled, his heuristic of “the two cultures” has survived his intentions and led many over the years to support the division. [^]

[7] Without providing an exegesis of the troubled and twisted fate of this term, it is sufficient to say that “postmodernism,” certainly more than any other term of the last forty years, has caused more headaches than perhaps it is worth. Thumbing through Gross, Levitt, and Lewis’ collection will provide ample support for this claim. However, I intend to recoup some of the lost affect in this essay by characterizing postmodernism in a less “ambiguous” way. [^]

[8] To be sure, the history of the concept in the humanities and the social sciences is overflowing with such attempts to define the term: to locate primitive postulates. Some critics have carefully defined postmodernism in relation to modernism, while others broaden the scope of the definition to include political and poetic aspects as well. See, for example, Ihab Hassan’s The Postmodern Turn: Essays in Postmodern Theory and Culture (Columbus: Ohio State UP, 1987); Linda Hutcheon’s The Poetics of Postmodernism: History, Theory, Fiction (New York: Routledge, 1988) and The Politics of Postmodernism (New York: Routledge, 1989); and Andreas Huyssen’s After the Great Divide: Modernism, Mass Culture, Postmodernism (Bloomington: Indiana UP, 1986). However, recent cultural critics are realizing that their term has very few, if any, clearly definable borders. Any attempt to utilize a term like postmodernism (or fascism for that matter) must carefully contextualize the particular local history of the concept. In literary studies, postmodernism once contained only an aesthetic dimension, while today, it contains an equally political one. [^]

[9] These academics include, but are not limited to: Jacques Derrida, Jean-François Lyotard, Michel Foucault, Roland Barthes, and Gilles Deleuze. Jean Baudrillard, though arguably the current CEO of the postmodernism industry today, should also be included in this short list of usual suspects. Though an interesting story, I will not explicate the full history of the birth of postmodernism in order to expedite my analysis. There are many fine summaries and surveys of the term postmodernism and its canonical history available for easy consumption. As reference, I would draw the reader’s attention to Steven Best and Douglas Kellner’s Postmodern Theory (New York: Guilford, 1991), which provides readable, detailed, if selective, summaries of the major figures and their principle works (their references to Derrida are scant, while their references to Barthes are almost non-existent). Partly due to the abundance of such fine works, the critical exercise of tracing the developments of postmodernism in literary studies has all but gone the way of the dodo. [^]

[10] The Newport date refers to the moment Bob Dylan plugged-in and joined the Paul Butterfield Band on the main stage of the Newport Folk Festival for an electric rendition of “Like a Rolling Stone,” which signaled one of the greatest musical paradigm shifts of the last century. The Watergate Scandal is more vague as it unfolded over two years. Initially, the burglars broke into the Democratic Offices in the Watergate Hotel in 1972. But Richard Nixon made his first public speech about the incident in April 1973. He did not resign until 1974. [^]

[11] This shift was so profound in logic that the discipline of mathematics never actually opened their doors to this problem child. Now a marginalized discipline, the prodigal field of logic is mostly housed in philosophy departments when it exists at all. Logic (also called symbolic logic) is distinct from the field of mathematical logic, which developed many years later. Mathematical logic is a specialized area of intuitionist mathematics devoted to redefining the whole of mathematics in intuitionistic terms. In his Introduction to the History of Mathematics (Saunders Harcourt Brace 1992), Howard Eaves describes Intuitionism as a dominant philosophy of mathematics that “asserts that mathematics should be built on finite constructive methods derived from the intuitively given sequence of natural numbers” (631). The major consequence of this theory is that mathematical entities must be proven to exist in a finite number of steps. According to intuitionist thinking, assuming that an entity does not exist is not sufficient to prove that a contradiction will arise -- a standard mathematical approach. [^]

[12] In Gödel, Escher, Bach: an Eternal Golden Braid (1979), Douglass Hoffstadter characterizes this “critical point” as the “kiss of death” in which the reflexive representation of mental structures, for example, becomes impossible through the tools that system uses to interpret (697). [^]

[13] Though it took many years for Derrida’s theory to have its full effect (he published six books about this theory which he called “deconstruction” in as many years), and while others at the conference presented equally radical theories (c.f. Roland Barthes), his theory alone was considered responsible, by many, for what was being called the “death of literature”. [^]

[14] It is important to note that while figures like Derrida, Barthes, Lyotard, and Deleuze were often cited as the perpetrators of such thinking, their writing -- admittedly, often dense and obtuse -- proposed strategies for overcoming this kind of resignation. Many of their early critics were quick to combat what was called the exhaustion of language, without recognizing the libratory aspect of their theories. For example, in “White Mythology: Metaphor in the Text of Philosophy,” Derrida proclaims that philosophy’s quest for truth is doomed through the use of symbolic language. But he is careful to situate this claim within the essential undecidability of language, as truth must be reduced to a relative function of the language in which it is conveyed. [^]

[15] Such rituals, if I may be so bold as to characterize them in this way, often took the form of eulogies for the impending doom of literary studies that was supposed to follow. In addition to Roland Barthes’ short piece “The Death of the Author,” which started this death fixation, other notable book-length works in this genre include Geoffrey Hartman’s Criticism in the Wilderness: the Study of Literature Today (New Haven: Yale UP, 1980); Alvin Kernan’s The Death of Literature (New Haven: Yale UP, 1990); Richard Schwartz’ After the Death of Literature (Carbondale, Il: Southern Illinois UP, 1997; and Mark Bauerlein’s Literary Criticism: an Autopsy (Philadelphia: Penn UP, 1997). [^]

[16] Felix Hausdorff defined the concept of the topological “neighborhood” as a logically defined space, without a consideration for a geometric sense of distance. As will be shown, space itself has undergone many transformations since Euclid’s first attempt to codify and systematize it as early as the eighth century BCE. As a concept, it is significant that mathematical space neither functions according to an enlightened logic of progress or a Darwinian logic of natural selection. Each historical attempt to redefine space has only extended our understanding of the irreducible multiplicity of spatial concepts, each necessary for a comprehensive description of certain phenomena. [^]

[17] In 1750, Leonard Euler discovered the first topological property, called the Euler number. For every closed, unbound, and finite polyhedron with v vertices, e edges, and f faces, the equation v – e + f = 2. What he discovered is that all polyhedra share a common property. [^]

[18] Lewis Carroll, while primarily remembered for his fantastical children’s stories, was also a first-rate mathematician, publishing books on Euclidean geometry, analysis, and logic. [^]

[19] French topologists Camille Jordan first developed the concept of homotopy, from which this distinction between inside and outside comes. According to Michael Henle, a Jordan curve is “a closed path” that “divides the plane into two pieces, an inside and an outside” (79). As he notes, this apparently simple notion gets more complicated when the Jordan curves become more complex. The artwork of M.C. Escher and Oscar Reutersvärd are examples of the “impossible” perspective complex Jordan curves can cause. Reutersvärd is considered the father of impossible objects, of which the symbol for a recycled product is the most familiar example. An impossible object is an object composed of real parts, but combined in an impossible way. Escher’s work is also based on impossible objects (a continuous staircase or aqueduct in which the top connects with the base in such a way as to suggest the “impossibility” of the top becoming the bottom), which are all derivative of the Möbius band. [^]

[20] This discussion of perspective and orientability will sound familiar to those familiar with Einstein’s General Theory of Relativity in which perspective actually determines such formerly immutable concepts such as time and gravity. According to fellow physicist George Gamow, “Einstein was probably the first to realize the important fact that the basic notions and the laws of nature, however well established, were valid only within the limits of observation and did not necessarily hold beyond them” (106). [^]

[21] We discover in topology that changing the concept of distance changes the topology of a figure. For example, while the definition of distance is consistent among congruent topological objects, it is not consistent across all objects. The concept of topological distance is consistent between a circle and a triangle and between a donut and a coffee cup respectively, but between these two classes of topological objects, the concept of distance is different. [^]

[22] Euclid’s fifth postulate of Book I of the Elements (the parallel postulate) states that through a given point, only one parallel line can be drawn to a given line. In the 1820’s Janoš Bolyai and Nicolai Lobachevsky were the first to discover a consistent geometry, based on Euclid’s, in which more than one parallel line could be drawn through any given point. In the 1850’s Georg Riemann discovered a geometry in which no parallel lines could be drawn through a given point. Though Felix Kline first called these geometries hyperbolic and elliptical, respectively, they tend to be grouped with others like them today into what mathematicians call non-Euclidean geometry. [^]

[23] Cantor’s story is tragic in part, because his discoveries that the real numbers are not countable, and thus form a kind of infinity that is “larger” than the infinity of the natural or algebraic numbers occurred the generation before the revolution took place. Mainstream mathematics at the time (led by Leopold Kronecker) judged his work misguided and nonsense. And though most cutting-edge mathematicians quickly gravitated towards his work by the end of his life, the damage of frustration and rejection had already been done. After a long history of depression and hospitalization, Cantor died on the doorstep of the postmodern mathematical revolution in 1918. Cantor’s contribution to the revolution in mathematical thinking cannot be stressed enough. His creation of set theory has made nearly all of twentieth century mathematics possible. Warren Christopher Dauben’s GeorgCantor: His Mathematics and Philosophy of the Infinite (Princeton: Princeton UP, 1990), while requiring some mathematical training for the reader, is nonetheless the most comprehensive biography of Cantor in print. [^]

[24] Of course, this is not to suggest that a discipline of science did not exist during Galileo’s time. It certainly did. However, individual practitioners of a given discipline did not consciously frame their work as disciplinary. Well aware that he was discovering certain “laws” of nature, Galileo in no way understood these laws as either emerging from an established set of disciplinary protocols, or as forming a new protocol in the way that someone like Einstein did. Michel Foucault’s Discipline and Punish (1972) traces the genealogy of the concept of disciplinarity. According to him, the transition from medieval ideas of torture to the “enlightened” theory of punishment and incarceration signaled the birth of the concept of disciplinarity that emerged in the eighteenth century and dominated nineteenth century Europe.

Following this argument, what makes something like Gödel’s proof postmodern is the fact that it specifically targeted disciplinary activities rather than a given mathematical assertion. His proof in no way changed the content of mathematical activity, only its form. Gödel’s proof buried the philosophy of Formalism, developed by David Hilbert in the early part of the twentieth century. Formalism, Logicism, and Intuitionism were mathematics’ first attempt at devising formalized philosophies of their subject. Briefly, Formalism viewed mathematics as a collection of formal systems of symbols. Logicism was the product of Bertrand Russell who thought that mathematics was actually a branch of logic and could be reduced to it. Finally, L.E.J. Brouwer developed Intuitionism. According to this philosophy, mathematics is build from concrete constructions of “natural” numbers. The outcome of each school of thought has been drastically different. Always on the fringes of mathematical activity, Logicism retreated to philosophy departments with the birth of Positivism. However, Intuitionism has steadily grown over the last hundred years, greatly influencing new areas of mathematics like topology and group theory.

Similarly, the philosophy of language had existed in a recognizable form since William James and Charles Sander Peirce before Derrida’s deconstruction of it one hundred years later caused the postmodern revolution in the humanities and social sciences. However, unlike Gödel, Derrida’s reception has been much more colorful. There are still contingents of scholars who consider Derrida the angel of death some forty years later. [^]

[25] For example, it is apparent that mathematics, on the whole, embraced the radical inconsistency Gödel’s proof uncovered, whereas literary studies nearly collapsed in the 1980’s over Derrida’s “proof” that language is insufficient to fully capture that which it represents (and I am greatly simplifying Derrida’s argument). As I have already suggested, a large part of the last forty years in literary studies has been spent ritualizing over-obsessive responses to the inadequacies of language. Harold Bloom’s The Anxiety of Influence: a Theory of Poetry (1973) was the first attempt at disciplinary therapy for literary studies. [^]

[26] It is important to distinguish between mathematical existence and the existence of material objects. Topologists Henri Poincaré clarifies this distinction in Science and Hypothesis (1905) as one of proof. Mathematical entities exist, he tells us, if they do not contain a contradiction in their definition (44). Thus, a mathematical entity is not assumed to exist simply because it is defined (he juxtaposes this to John Stuart Mill’s rather stringent assertion that definitions must contain an axiom). Every mathematical entity must be proven to exist through a demonstration that its definition does not lead to a contradiction.

Of course, this stance is not universally accepted in mathematics. The dominant semantic philosophy of mathematics today is the model theory of Tarski and the closely related variable model theory of Quine. Both philosophies contend that mathematical objects exist (as sets). These theories have led to many dilemmas surrounding the relationship between mathematical language, scientific language, and everyday language, which has been a topic of debate for the last century (c.f. Russell, Whitehead, Wittgenstein). Paul Benacerraf developed the contending theory that numbers do not exist in 1965, which was reprinted in Philosophy of Mathematics (1983) co-edited with Hilary Benacerraf. [^]

[27] A quick proof of this accidental correspondence between numbers and objects can be shown through simple number theory itself. In number theory, we discover many different kinds of numbers, only the most basic of which correspond with the kinds of numbers most of us manipulate in a given day (say, to balance our checkbooks, for example). The transcendental numbers π and e follow different rules of arithmetic than natural or rational numbers; for example, they are not the sum of two other numbers. The infinite number ω (the set of infinite ordinal numbers: {0,1,2,3…}) is a kind of number in which, when added to (or even multiplied by itself) is itself: ω + ω = ω [^]

[28] Again, there have been many attempts to define the natural numbers in terms of sets. As Stewart Shapiro has noted, both John von Neuman and Ernst Zermelo have developed consistent, yet mutually exclusive theories of number, neither of which lead to a unified understanding of the concept of number (5). Shapiro’s argument is somewhat technical (hinging, for example, on whether a space in a structure is an object, or whether or not “to construct” is equal to “there is”), but his “Structuralist” philosophy is based on the same kind of question topology asks: what it means to be an object (5). [^]

[29] For example, while it may seem that the statement “My cat’s name is Rudy,” relies on a concept of truth about catness in general or in the specific catness contained in an individual cat, the connection, again is incidental. It just so happens that empirical evidence exists to prove the validity of this statement. The same cannot be said for more tenuous inferences such as the statement “Goblin sharks mate in the same way other sharks do,” or “The core of Saturn is composed of metallic, liquid hydrogen”. While we assume that Goblin Sharks mate like other sharks, no one has ever witnessed a mating. Likewise, our predictions about the core of Saturn are based on an even more tenuous chain of inferences starting with the measurement of the concentration of methane and ammonia in Saturn’s atmosphere to our interpretation of that data millions of miles away. [^]

[30] The ubiquitous example found throughout the sciences and humanities is light. As Bohr, Heisenberg, Einstein and even Popper have explained, quantum theory suggests that light behaves at times, like a particle, and at other times, like a wave. Since we are incapable of actually observing the nature of light, its actual state is indeterminate. Therefore, the choice of experimental apparatus used to “observe” light actually determines the nature of the light we observe. [^]

[31] The first “imaginary” number, i, was so called because its definition (i² = -1), according to the rules of arithmetic, was impossible. But late eighteenth and early nineteenth century analysis found themselves using this concept to solve very “real” problems, such as partial differential equations. [^]

[32] It has only been in the last two hundred years that the oldest conundrums of mathematics have been proven to be impossible through topology. The oldest of these: doubling the cube, squaring the circle, and trisecting the angle were already problems for Hypocrites in the fifth century BCE. It took well over two thousand years to prove that each problem was impossible to solve according to the rules of geometry (Wantzel published the first proof in 1837). [^]

[33]The first of the so-called Nuremberg Race Laws, the “Law for the Protection of German Blood and German Honor” was ratified in 1935. The Nazis did not define Jewishness in either religious or cultural terms, but according to biological determinism. Under Nazi law, a Jew is anyone with at least one Jewish grandparent. Likewise, under Halakhah (orthodox Jewish law), practiced in Israel and throughout orthodox Jewish communities worldwide, biology, again, determines Jewishness, as a Jew is anyone born of a Jewish mother. The issue of consistency takes on an ominous tone with an issue such as Jewishness, as the Nazi definition of a Jew is analogous to the definition accoriding to orthodox Jewish law. In this case, careful explication of the context under which consistency claims are established is crucial for determining the nature of the claims. Nazi Germany treated homosexuality in a similar way.

In 1935, the Nazis amended paragraph 175 of the Civil Code to include “lewd and lascivious” acts between men under the punishment of imprisonment. The following year, Reichsfhurer Heinrich Himmler created the “Reich Central Office for the Combating of Homosexuality and Abortion” under the aegis of the Gestapo. Thus, under the Nazi regime, homosexuality as a biological act had civil and certainly cultural consequences, as sexual “deviants” were those who functioned outside of the matrix of biological reproduction. Though lesbians were never the targets of legal precedent, the fact that abortion was included with homosexuality underscores the biological foundation of reproduction upon which the Nazis built their ideology. The issue of gay and lesbian rights today has shifted from biological considerations to civil/cultural considerations (the right to legal marriages, for example). The policy of lobby groups such as the NGLTF (National Gay and Lesbian Task Force) is to define gay/lesbian in civil terms as a means of dissociation from the biological practices of the Nazis, which were never wholly distinct from ethnical and racial categories. [^]

[34] Written from the perspective of her eight-year-old self before deportation to Theresienstadt concentration camp in 1942, “I am a Star” is the opening poem of Inge Auerbacher’s [i]I am a Star: Child of the Holocaust (1986). [^]