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Badiou's Equations--and Inequalities: A Response to Robert Hughes's "Riven"
*Arkady Plotnitsky*
/ Purdue University/
plotnit@purdue.edu
(c) 2007 Arkady Plotnitsky.
All rights reserved.
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1. Robert Hughes's article offers an unexpected perspective on Alain
Badiou's work and its impact on the current intellectual and
academic scene, a clichÃ(c)-metaphor that (along with its avatars,
such as performance or performative, also a pertinent term) may be
especially fitting in this case, given that Badiou is not only a
philosopher but also a playwright. What makes Hughes's perspective
unexpected is its deployment of "trauma" as the main optics of
this perspective. While the subject and language of trauma have
been prominent in recent discussions, they are, as Hughes
acknowledges, not found in Badiou's writings nor, one might add,
in the (by now extensive) commentaries on Badiou. Hughes's reading
of Badiou in terms of trauma rearranges the "syntax" of Badiou's
concepts, as against other currently available readings of Badiou,
even if not against Badiou's own thinking, concerning which this
type of claim would be more difficult to make. In this respect
Badiou's thought is no different from that of anyone else. One can
only gauge it by a reading, at the very least a reading by Badiou
himself, for example, in Briefings on Existence (which I shall
primarily cite here for this reason and because it offers arguably
the best introduction to his philosophy).
2. Before proceeding to Hughes's argument, I sketch the conceptual
architecture of Badiou's philosophy from a perspective somewhat
different from but, I hope, complementing that offered by Hughes.
The language of conceptual architecture follows Gilles Deleuze and
FÃ(c)lix Guattari's view of philosophy, in What is Philosophy?, as
the invention, construction of new concepts (5). This definition
also entails a particular idea of the philosophical concept. Such
a philosophical concept is not an entity established by a
generalization from particulars or "any general or abstract idea"
(What is Philosophy? 11-12, 24). Instead, it is a conglomerative
phenomenon that has a complex architecture. As they state, "there
are no simple concepts. Every concept has components and is
defined by them . . . . It is a multiplicity" (16). Each concept
is a conglomerate of concepts (in their conventional sense),
figures, metaphors, particular elements, and so forth, that may or
may not form a unity; as such, it forms a singular, unique
configuration of thought. As a philosopher, Badiou is an inventor,
a builder of new concepts, just as Deleuze and Guattari are, even
when these concepts bear old names, as do some of Badiou's key
concepts, such as event, being, thought, and truth.
3. A distinctive, if not unique, feature of Badiou's philosophy, as
against that of most other recently prominent figures, is the
dominant role in it of mathematics, in particular of mathematical
ontology. It is true that major engagements with mathematics are
also found in Lacan and Deleuze, both of whom had a major
philosophical impact on Badiou's work. There are, however, also
significant differences among these thinkers in this respect, and
it is worth briefly commenting on these differences in order to
understand better Badiou's use of mathematics and mathematical
ontology. For Badiou, to use his "equation,"
"mathematics=ontology" (Briefings 59), and "ontology=mathematics."
(Badiou's first identity is not mathematical, and hence these two
identities are not automatically the same, but they appear to be
in Badiou.) Reciprocally, Badiou wants to give mathematics a
dimension of /thought/, specifically of ontological thought, which
he distinguishes from other, most especially logical, aspects of
mathematical thinking. "Mathematics is a thought," a thought
concerning Being, he argues in Briefings on Existence (45-62). It
is, accordingly, not surprising that Badiou is primarily
interested in foundational mathematical theories, that is, those
that aim at ontologically grounding and pre-comprehending all of
mathematics, such as set theory introduced by Georg Cantor in the
late nineteenth century, and in Badiou's more recent works, in
category and topos theories, as developed by Alexandre
Grothendieck in the 1960s. The latter offers a more fundamental
mathematical ontology, encompassing and pre-comprehending the one
defined by set theory. Indeed, it might be more accurate to
rewrite Badiou's equations just stated as "mathematical
ontology=ontology" and "ontology=mathematical ontology" (where
mathematical ontology could be either set-theoretical or
topos-theoretical). In other words, Badiou ultimately deals with
mathematical ontology rather than with mathematics, and deals with
it /philosophically/ and /not mathematically/, beginning with
defining as irreducibly multiple "the multiple without-One" (the
multiple minus One?), to be considered presently (Briefings 35).
The equations just stated are still produced by philosophy, and
not by mathematics, and the very argument that mathematics is a
thought is philosophy's task, even though and because mathematics
as ontology "functions as a [necessary] condition of philosophy"
(54). Necessary, but not sufficient! For, as will be seen, while
it is also "about identifying what real ontology is," philosophy,
especially as thinking of "event" or "truth," exceeds ontology,
since an event is always an event of "trans-Being," in part,
against Heidegger, at least early Heidegger (59). Both "event" and
"truth" are defined by Badiou in relation to what he calls
"situation," as something from within which, but also against
which or in discontinuously, an event emerges. It is this excess
that defines Badiou's inequality, which may be symbolically
written as "ontology=mathematical ontology <
trans-Being=philosophy." Making ontology mathematical and thus
also more Platonist is already a non-Heideggerian gesture,
deliberate on Badiou's part. Heidegger prefers pre-Socratic
thought, abandoned, albeit still concealed in Plato's philosophy
as well. One may thus write another pair of
equations--"philosophy=thinking the event" and "thinking the
event=philosophy"--which may, moreover, be accompanied by those
involving literature or poetry. This "thinking the event" might
also be equal to literature, at least a certain type of literature
or of literary thought. "Literature=philosophy," then, and, by the
same token, a certain redux of Heidegger in Badiou? As I discuss
below, and as Hughes's article suggests as well, to some degree
this may be the case. This relation between literature and
philosophy in Badiou's thinking the event may, however, also be
undecidable, in Jacques Derrida's sense rather than in that of
Kurt GÃ¶del in mathematical logic. Derrida's undecidability refers
to something that can "no longer be [unconditionally, once and for
all] included within philosophical (binary) opposition," in this
case, to our inability to decide unconditionally whether a given
thinking, such as that of the event in Badiou's sense, is either
literary or philosophical (Positions 43). It can be either, or
sometimes both simultaneously, in different circumstances, even
for the same thinker. GÃ¶del's undecidability refers to the
impossibility of proving certain propositions to be either true or
false by means of the systems within which they are formulated, a
concept that is crucial for Badiou in its own right.
4. In contrast to Badiou, Deleuze and Deleuze and Guattari define the
role of mathematics, most centrally topology and geometry, less by
mathematical ontology, especially in the foundational,
set-theoretical sense dominant in Badiou. Deleuze's or Deleuze and
Guattari's ontologies may be seen as conceptually /modeled/ on
certain mathematical objects, such as dynamical systems or
Riemannian spaces. (The actual definitions of these objects are
not important at the moment.) This /modeling/, resulting in a kind
of philosophical semblance or simulacrum of the mathematical
concept used is, however, different from Badiou's ontological
/equations./ In Lacan's case, it would be difficult to speak of
any /specifiable/ ontology, mathematical or other, at the ultimate
level, that of the Lacanian Real. In other words, in approaching
the Real, Lacan is attempting to think that which cannot be given
any /specifiable ontology./ The Real exists and has powerful
effects upon either the Imaginary or the Symbolic, but cannot be
represented or conceived of in specific terms. Lacan's use of
mathematical, such as topological, objects or concepts is defined
by this epistemology of the Real as irreducibly inaccessible, even
while it has powerful effects upon what can be accessed or
represented. For example, it may shape the topological
configuration of unconscious thought, as defined by the Symbolic.
But then this move beyond ontology is found in Badiou as well, and
even defines thinking what is ultimately most decisive: thinking
"thinking the event" and philosophy itself as nonontological, for
"it is also important for [philosophy] to be released from
ontology per se" (Briefings 59). Indeed, this "release" involves a
version of Lacan's Real, a major inspiration for Badiou's thought.
In Lacan and Badiou alike, the Real may be seen as /materially/
existing, in a form of nonontological ontology, as it were,
insofar as it cannot given a specifiable ontology, for example a
mathematical one. It is worth stressing that, in general, the
mathematical and infinitist grounding of his ontology
notwithstanding, Badiou's philosophy is materialist, just as is
that of Lacan, Deleuze, or Derrida. For all these thinkers
thought, including mathematics, is a product of the material
conditions and energies of our existence--from inanimate matter to
living matter to our bodies to the cultural, including political,
formations shaping our lives.
5. Hughes does not miss the /significance/ of Badiou's "reengagement
[as against the immediately preceding philosophical tradition,
especially as manifest in Hegel, Nietzsche, and Heidegger] of
philosophy with mathematics and set theory" ("Riven" Â¶1). He does
not, however, pursue, does not /engage/ with, this reengagement.
Such an engagement would allow Hughes to offer a richer and deeper
consideration of the problematics of Badiou's philosophy. On the
other hand, this non-mathematical restaging of Badiou's thought is
not altogether surprising and, to some degree, allows one to
perceive and articulate aspects of this thought sometimes hidden
behind the mathematical complexities of his argumentation.
Besides, one might also argue that, while certain /mathematical/
(set- or topos-theoretical) ontology essentially /grounds/
Badiou's thought and is helpful in understanding it more deeply,
his thought is ultimately not mathematical. It is fundamentally
/philosophical/, even, again, as concerns ontology, since equating
it with mathematical ontology is itself philosophical and not
mathematical. His thinking the event or truth is, as just
explained, philosophical on his own definition, by virtue of
exceeding ontology=mathematical ontology, without, however,
abandoning the latter. While justifiably attending to the
questions of event and truth, Hughes does not consider this
difference between ontology (as equal to mathematics or
mathematical ontology) and philosophy (as equal to thinking the
event). He is, however, right to place Badiou within a
philosophical tradition of thinking about art, ethics, and
politics defined, on the one hand, by the post-Heideggerian French
thought of Lacan, Levinas, Blanchot, and Derrida, and, on the
other by thinkers at the origin of Romantic thought ("Riven" Â¶3).
He is also right to relate this thinking to literature, at least
implicitly, and again specifically to Romantic literature, as well
as to Romantic philosophical thought. I return to these subjects
later. My point at the moment is the essentially philosophical
rather than mathematical nature of Badiou's thought (there is
hardly any /mathematics/ in the sense of its technical,
disciplinary practice in Badiou), and also the nature of
philosophical thought itself, as /against mathematics/, but not
/against mathematical thought/. If "mathematics is a thought,"
mathematics is also not only mathematics (in the sense of its
technical practice) but also a /philosophical/ thought, and hence
reaches beyond ontology on the philosophical side. This
philosophical dimension appears especially at the time of
"crises," defined by Badiou as largely synonymous with "events."
As Badiou writes, "mathematics thinks Being per se," or ontology,
"save for the rare moments of crisis" (Briefings 59). At such
moments it must move toward thinking the "event" which, according
to Badiou's definition, would make it philosophy or at least
philosophical. According to Badiou, "a 'crisis' in mathematics [as
ontology] arises when it is compelled to think its thought /as the
immanent multiplicity of its own unity/," and "it is at this
point, and only at this point, that mathematics--that is,
ontology--functions as a necessary [but, as I have said, not
sufficient] condition of philosophy" (Briefings 54). This
conjunction gives a crucial significance to the reciprocity
between mathematics, /as ontology/, and philosophy as thinking the
event or/as crisis, first, I argue, in mathematics itself,
understood in broader (rather than only ontological) terms at the
moment and, hence, beyond the purview of Badiou's argumentation
concerning mathematics. Secondly, given that at stake in Badiou's
statement is philosophy as such, this reciprocity between
mathematics as ontology of the irreducible multiple and philosophy
in thinking the event as crisis, and hence the discontinuity of
the event, is according to Badiou irreducible in philosophy and,
it follows, in Badiou's thought. This reciprocity between the
multiple and the discontinuous in Badiou is my main point here and
guides my argument from this point on.
6. As I see it, what defines Badiou's philosophy most essentially,
what he is most essentially a philosopher of, is above all a
philosophy of the multiple, specifically of the /ontologically/
multiple: ontology=the set-theoretical ontology=the irreducibly
multiple without-One. This is a particular view or interpretation
of mathematical ontology that is not necessarily shared by
mathematicians and philosophers working in foundations of
mathematics, even if it is ultimately true (in the sense of being
potentially irreducible). Indeed, Badiou would not speak of
interpretation here. According to him, "mathematics has the virtue
of not presenting any interpretations," and "the Real [understood
close but not identically to Lacan's] does not show itself as if
upon a relief of disparate interpretation." Instead, Badiou sees
the situation in terms of different "decisions of thought"
concerning what exists (e.g. made in the case of set-theory in
terms of constructible sets, large cardinals, or generic sets),
the decision through which thought "binds [one] to Being," "under
the imperative of an orientation [of thought]" (Briefings 56-57).
The concept itself of "orientation of thought" becomes an
important part of Badiou's ontological thinking (53-54). That
Badiou's ontology of mathematics (which, again, equals ontology in
general) is that of the multiple without-One is, however, not in
doubt: this is his ontological and political orientation and his
decision of thought. He is a philosopher of the irreducibly
multiple. I would also add, however, that he is equally a
philosopher of the radical discontinuity, ultimately beyond
ontology (although it enters at the level of ontology as well) by
virtue of its connections, via the concept of event, to
"trans-Being" and hence to /philosophy/. It is true that, as the
title of Badiou's arguably most significant work, Being and Event
(L'Ã(tm)tre et l'Ã(c)vÃ(c)nement), would suggest, he can more properly be
defined as a philosopher of being and event, and of their
conjunction. It would be difficult to argue against this view. But
central as both of these concepts are to his thought, the
architecture of both is essentially defined by the concepts of
multiplicity and discontinuity. Indeed, we can describe these
concepts in parallel terms, or again by way of two equations,
being=the irreducibly multiple, the multiple-without One, and,
defined by Badiou as trans-Being, the event=the radical
discontinuity. In general in Badiou, as elsewhere in philosophy,
the architecture of each concept is also defined by its relation
to other concepts and, hence, by the entangled network of these
concepts. Not unlike those of modern physics (such as Einstein's
famous /E=mc^2 /), Badiou's equations encode the considerable
architectural complexity of the concepts involved or of their
relationships. Bringing the roles of multiplicity and
discontinuity in Badiou into a sharper focus allows one to
understand the architecture and mutual determination of his
concepts more deeply, in part through the connections, along both
lines (multiplicity and discontinuity), between Badiou's thought
and that of other key contemporary figures. Thus, while these
thinkers' thought is different, even as concerns multiplicity or
discontinuity, as a philosopher of the multiple Badiou is close
both to Deleuze and to Derrida, and as a philosopher of the
discontinuous he is closer to Lacan, de Man, LÃ(c)vinas, and, again,
Derrida--but not Deleuze. For even though Badiou draws inspiration
from Deleuze as a thinker of the multiple, one of his discontents
with Deleuze's "vitalist ontology," as he calls it, appears to be
the insufficient role of discontinuity there, as against, for
example, Immanuel Kant's "subtractive ontology." As a thinker of
discontinuity, Kant is also a precursor of Levinas, Lacan,
Derrida, and de Man.
7. Hughes's article correctly stresses the role of discontinuity and,
correlatively (they are not the same), singularity in Badiou's
thinking of event, truth, and ethics, and the Lacanian genealogy
(the Real) of this thinking, or its qualified connections to
LÃ(c)vinas. (It seems to me that Hughes /equates/ "truth" and
"event," too much ["Riven" Â¶5], which, indissociable as they may
be, they are not quite the same in Badiou.) Beginning with its
title ("Riven"), Hughes's article centers primarily on the role of
discontinuity in Badiou, especially that between "situation" and
"event," via what Badiou calls "the nothing of its all" or the
non-empty void linked to the Lacanian Real. Apart from a brief
discussion of "a set of component elements or terms" involved in
"the theatergoer's encounter with Hamlet" (Â¶9), the article gives
little, if any, attention to the role of multiplicity in Badiou,
at most mentioning it in passing, perhaps because the article's
argument bypasses the mathematical-ontological aspects of Badiou's
thought. Not coincidentally and, given its significance for
Badiou, even remarkably, the term "ontology" is never mentioned in
the article, except in a book title by Slavoj Zizek (who does not
miss it!) in the bibliography. However, Badiou's thought is
irreducibly defined by the ontology of the multiple and, again,
the equally irreducible combination of the multiple and the
discontinuous, in particular the trans-ontologically
discontinuous. The singularity of the event and the truth always
arises from and, through the non-empty void of the Real, grounds
the multiple (mathematical or other) of the situation and give
rise to a new multiple. This dynamics of the interplay between the
Real and the multiple gives the structure and the history of a
situation and an event (again, always exterior to the situation
from within which it appears) the architecture of "the multiple of
the multiple," a persistent locution in Badiou. According to
Badiou, given that "philosophy will always be split between
recognizing the /event/ as the One's supernumerary coming, and the
thought of its /being/ as a simple extension of the manifold,"
"the whole point is to contend, for as long as possible and under
the most innovative conditions for philosophy, the notion that the
truth itself is but a multiplicity: in the two senses of its
coming (a truth makes a typical /multiple/ or generic
/singularity/) and in the sense of its being (there is no /the/
Truth, there are only disparate and untotalizable truths that
cannot be totalized)" (Briefings 62; emphasis added). This
statement clearly reflects a crucial significance of the
relationships between the discontinuity of the event and the
multiple of being in Badiou. One cannot totalize all truths, and
each truth is itself an untotalizable multiple within the
singularity of its event. This situation (in either sense)
requires, for Badiou, "a radical gesture," which also manifests
philosophy's faithfulness to Lucretius, in whose (physical)
ontology of the multiple without-One "atoms, innumerable and
boundless,/ flutter about in eternal movement" (/De rerum natura/
II: 496; Briefings 62).
8. The multiple is everywhere manifest in Badiou's thought and is
expressly emphasized by Badiou as central to the ontology he aims
to establish. This view is confirmed by the proposition, cited
above, concerning mathematical thinking: at the time of a crisis,
"its thought /as the immanent multiplicity of its own unity/"
(Briefings 54). Given Badiou's view of mathematical ontology as
multiple-/without-One/, this can only be read in terms of the
ultimate /impossibility/ of this unity. (Otherwise the point would
merely restate Leibniz, who is among Badiou's precursors here, as
are Hegel and Kant [Briefings 141]). This multiplicity defines
Badiou's ontology and always enters an "event," by definition
always an event of crisis, "each time unique," or, in Derrida's
title phrase, "each time unique, the end of the world" (Chaque
fois unique, la fin du monde; published in English as The Work of
Mourning). In the ethical plane, this emphasis on the irreducibly
multiple serves both Badiou and Derrida (and in both cases, if
differently, against Levinas) to think the /events/ (plural) of
"evil," all evil. Some among such events are equally evil, but
each nevertheless is unique, as well as irreducibly multiple as
concerns the ontology involved, thus doubling the multiple.
Indeed, as will be seen, as a multiple without-One, this
ontological multiple is already doubled, is already the multiple
of the multiple, which gives Badiou's overall situational ontology
the form of the multiple of the multiple of the multiple. It
follows that there is no single or absolute, absolutely radical
evil, no matter how horrific or difficult to confront or even to
imagine evil events may be and how much we try, as we must, to
prevent their occurrence; there are other evil events and forms of
evil that are comparable (in their evilness), but again each is
different (Ethics 61-67). The (mathematical) ontological
multiplicity found within an event or the situation that brings
the event about always makes it political, as against Levinas's
thinking of the ethical. Badiou elegantly reads certain specific
actual political and mathematical "orientations" in terms of each
other, in particular by mutually mapping the theory of generic
sets and what he calls "generic politics" as "something groping
forward to declare itself," both defined by the multiple
without-One (Briefings 55-56). The same political complexity also
defines other (more "positive") "events," political, cultural,
mathematical, scientific, aesthetic, erotic, and so forth:
the French revolution in 1792, the meeting of HÃ(c)loÃ¯se and
AbÃ(c)lard, Galileo's creation of physics, Haydn's invention of
the classical musical style [vis-vis the Baroque] . . . But
also: the cultural revolution in China (1965-67), a personal
amorous passion, the creation of Topos theory by the
mathematician Grothendieck, the invention of the twelve-tone
scale by Schoenberg." (Ethics 41)
Badiou's concept of the "situation," defined by the analogous
complexity of the interaction between the singular and the
multiple (e.g. Ethics 16, 129), always entails the possibility of
an /eruptive/ event, such as those just mentioned, which,
reciprocally, can only emerge in relation to a situation. This
eruptive singularity of the event can, accordingly, only be
comprehended philosophically, by the (trans-Being) thought of
philosophy and not by (mathematically) ontological thought, but it
cannot be considered apart from ontology and its irreducible
multiple.
9. The conceptual architecture just outlined makes Badiou a
philosopher both of the irreducibly multiple and of the
irreducibly discontinuous. Badiou rarely invokes the /term/
"discontinuity" itself. The /concept/ is, however, clearly central
to his thought, in particular as concerns "event," as each event
is, again, always defined by the radical, unbridgeable,
end-of-the-world-like discontinuity of a crisis (but is always an
opening to the irreducibly multiple). Invocations of discontinuity
are found throughout Badiou's writing, and Hughes lists quite a
few of them, "riven, punctured, ruptured, severed, broken, and
annulled," which he links to trauma ("Riven" Â¶10). This is
correct, because trauma always entails discontinuity. On the other
hand, the latter is a more general concept, as is "event," and
Badiou uses both concepts more generally as well, in part by
linking them to multiplicity. In order to understand how the
multiplicity of ontology and the discontinuity of event work
together in Badiou, I turn to set theory, as opening "the very
space of the mathematically thinkable" (Briefings 42).
10. As we have seen, according to Badiou, "ontology" [including in its
irreducible multiplicitous form] is nothing other than mathematics
[of set theory or, later, topos theory] itself," or again,
mathematical ontology=ontology, ontology=mathematical ontology
(40). Given my limits here, I confine this discussion mostly to a
/naÃ¯ve/ concept of set, naÃ¯ve" being an accepted term in
mathematics in this context. A set is a collection of certain
usually abstract objects called elements of the set, such as, say,
the numbers between 1 and 10, which is a finite set, or of all
natural numbers (1, 2, 3, 4, etc.), which is an infinite set, a
countable infinite set, as it is called. There are also greater
infinities, such as that of the continuum, represented by the
numbers of points in the straight line. The resulting ontological
multiplicity or manifold is, Badiou argues, unavailable to
unification, to the One, and, as will be seen presently, this
multiplicity is also inconsistent, while nonetheless enabling a
set-theoretical ontology. While, on the one hand, "the /set/ has
no other essence than to be a manifold" and while, with Cantor, we
recognize "not only the existence of infinite sets, but also the
existence of infinitely many such sets," that is, sets possessing
different magnitudes of infinity, "this infinity itself is
absolutely open ended" (41). In particular, it cannot itself ever
be contained in a set. There is no "the One" of set theory,
because the set of all sets does not exist or at least cannot be
consistently defined, in view of the well-known paradox related to
the question of whether this set does or does not contain itself
as an element. (It is immediately shown not to be the set of all
sets in either case.) Set-theoretical multiplicity is both
ultimately uncontainable by a single entity or concept and is
inconsistent, because of the impossibility of giving it the
overall cohesion of a whole and because, as will be seen
presently, it contains systems that are expressly inconsistent
with each other in view of GÃ¶del's incompleteness theorems.
According to Badiou: "Ontology, if it exists has to be the figure
of inconsistent multiplicities as such. This means that what lends
itself to the thought of ontology is a manifold without a
predicate other than its own multiplicity. It has no concept other
than itself, and nothing ensures its consistency. . . . Ontology
is the thought of the inconsistent manifold, that is, of what is
reduced without an immanent unification to the sole predicate of
multiplicity" (36, 40). Accordingly, "ontology, or the thinking of
the inconsistent pure multiple, cannot be guaranteed by any
principle" (39). In view of these considerations, Badiou's
"initial [philosophical] decision was to contend that what can be
thought of Being per se is found in the radical manifold or a
multiple that is not under the power of the One, . . . [in] a
'multiple without-One.' . . . The multiple is radically
without-One in that it itself consists only of multiples. What
there is, or the exposure to the thinkable of what there is under
the sole requirement of the 'there is,' are multiples of
multiples" (35, 40). Or via Plato's Parmenides, which already
grapples with this situation and its "inconsistent multiplicity,"
what we encounter here is "an absolutely pure manifold, a complete
dissemination of itself" (46).
11. While Badiou, thus, establishes his ontology on the basis of more
general set-theoretical considerations, its crucial further
dimensions are revealed by GÃ¶del's famous discovery of the
existence of undecidable propositions in mathematics and by Paul
Cohen's findings (along the lines of undecidability) concerning
the mathematical continuum. The latter is, as I said, defined by
the order of the infinite larger than the countable infinity, 1,
2, 3, . . . etc., of natural numbers, but in view of Cohen's
theorem it is ultimately undecidable whether there is something in
between. GÃ¶del's concept of an undecidable proposition is arguably
his greatest conceptual contribution. An undecidable proposition
is a proposition whose truth or falsity cannot, in principle, be
established by means of the system (defined by a given set of
axioms and rules of procedure) in which it is formulated. The
discovery of such propositions by GÃ¶del (in 1931) was
extraordinary. It undermined the thinking of the whole preceding
history of mathematics (from the pre-Socratics on), defined by the
reasonable idea that any given mathematical proposition can, at
least in principle, be shown to be either true or false. We now
know, thanks to GÃ¶del, that such is not the case. For GÃ¶del
proved--rigorously, /mathematically/--that any system sufficiently
rich to contain arithmetic (otherwise the theorem is not true)
would contain at least one undecidable proposition. This is
GÃ¶del's "first incompleteness theorem." GÃ¶del made the life of
mathematics even more interesting with his "second incompleteness
theorem" by proving that the proposition that such a system, say,
classical arithmetic, is consistent, is itself undecidable. In
other words, the consistency of the system and, hence, of most of
the mathematics we use cannot be proven, although the possibility
that the system and with it mathematics may be shown to be
inconsistent remains open.
12. Given the undecidablity of certain propositions inevitably found
in any sufficiently rich axiomatic system, one can in principle
extend the system in two incompatible ways by accepting /by a
decision of thought/ such a proposition as either true or false.
This allows one to have two different systems--incompatible with
each other, but each perfectly consistent in itself. Since,
however, GÃ¶del's first theorem would still apply to each system,
new undecidable propositions will inevitably be found in each.
This makes the process in principle infinite, that is, potentially
leading to the infinite multiplicity of mutually inconsistent
systems, each of which, moreover, can never be proven to be
consistent in view of GÃ¶del's second theorem. This situation
becomes especially dramatic in the case of Cantor's famous
continuum hypothesis, which deals with the question: How many
points are there in the straight line? It states, roughly, that
there is no infinity larger than that of a countable set (such as
that of natural numbers: 1, 2, 3, etc.) and smaller than that of
the continuum (as represented by the number of points on the
straight line). The answer to this question is crucial if one
wants to maintain Cantor's hierarchical order of (different)
infinities, and hence for the whole edifice of set theory. The
hypothesis was, however, proven undecidable by Cohen in 1963.
Accordingly, one can extend classical arithmetic in two ways by
considering Cantor's hypothesis as either true or false, that is,
by assuming either that there is no such intermediate infinity or
that there is. This allows one, by decisions of thought, to extend
the system of numbers, arithmetic, into mutually incompatible
systems, in principle, infinitely many such systems--a difficult
and for some an intolerable situation. The question of how many
points are on the straight line cannot be determinately answered.
13. Instead of seeing this situation as difficult and even
intolerable, Badiou finds in it both a support for his program and
a special appeal or even beauty. As he writes:
As we have known since Paul Cohen's theorem, the Continuum
[h]ypothesis is intrinsically undecidable. Many believe
Cohen's discovery has driven the set-theoretic project into
ruin. Or at least it has "pluralized" what was once presented
as a unified construct. I have discussed this enough elsewhere
for my point of view on this matter to be understood as the
opposite. What the undecidability of the Continuum hypothesis
does is complete Set Theory as a Platonist orientation [in
Badiou's sense]. It indicates its line of flight, the aporia
of immanent wandering in which thought experiences itself as
an unfounded confrontation with the undecidable. Or, to use
GÃ¶del's lexicon: as a continuous recourse to intuition, that
is, to decision. (99)
The appeal to Deleuze's concept of "line of flight" is worth
noting. This view of the situation also shapes Badiou's
understanding of Plato's thought, juxtaposed by him to
conventional, especially conventional mathematical, Platonism.
Plato's /thought/, Badiou argues, is interested primarily in "the
movement of thought," and "the undecidable commands the perplexing
aporetic style of the dialogues. This course leads to the point of
the undecidable so as to show that thought precisely ought to
decide upon the event of Being: that thought is not foremost a
description or construction, but a break (with opinion and/or with
experience) and, therefore, a decision" (90). For Plato, as for
the /truly/ Platonist set-theoretical thinking, and for Badiou,
"it is when you decide upon what exists that you bind your thought
to being" (57). These statements bring together, in a firm
conceptual architecture, Badiou's key concepts, invoked here, from
the ontology of the multiple to thought to Being to event and
through it, the interconnective discontinuity between ontology and
philosophy.
14. This conceptual and epistemological architecture is set up in
Being and Event. Badiou retraces it (with some new inflections,
especially along the lines of topos theory) in Briefings on
Existence: A Short Treatise on Transitory Ontology (published in
French in 1998). The book is not cited by Hughes, perhaps because
it largely reprises Being and Event, apart from its discussion of
topos theory. It is arguably the best available condensation of
Badiou's philosophy, apart from the ethical problematic to which
Badiou turns in Ethics, which also appeared in French in 1998.
Briefings on Existence establishes the architecture just sketched
by starting, in the "Prologue: God is Dead," with the radically
materialist grounding of our being and thought, that of
mathematics and set theory, or of the infinite, included. Chapter
1, "The Question of Being Today," situates the question of Being
in this framework of materiality and infinity, but now in relation
to the ontology of the irreducible and inconsistent manifold or
multiple, the multiple without-One, grounded in the axiomatics of
set-theory, which is considered, as "a thought," in Chapter 2,
"Mathematics as a Thought." With this argument in hand, Badiou is
ready to define "the event as trans-Being" in Chapter 3. This
definition establishes the crucial difference between mathematics
as ontology (but again, their equation is stated and legitimated
by philosophy), and philosophy as that which, while also "all
about identifying what real ontology is," is ultimately released
from ontology. Indeed, philosophy is a theory of what is "strictly
impossible for mathematics" and "a theory of /event/ aimed at
determining a trans-being." As I said, however, one might want to
replace mathematics with mathematical ontology here, thus leading
to an inequality defined earlier: "ontology=mathematical ontology
< trans-Being=philosophy." According to Badiou, thus also defining
"/event/" (as the concept is conceived in Being and Event):
On the other hand [as against the ontological determination
defined by set or topos theory], a vast question opens up
regarding what is subtracted from ontological determination.
This is the question of confronting what is not Being /qua/
Being. For the subtractive law is implacable: if real ontology
is set up as mathematics [mathematical ontology] by evading
the norm of the One, unless this norm is reestablished
globally there also ought to be a point wherein the
ontological, hence mathematical, field is de-totalized or
remains at a dead end. I have named [in Being and Event] this
point the "/event/." While philosophy is all about identifying
what real ontology is in an endlessly reviewed process [such
as from set to topos theoretical ontology], it is also the
general theory of the event--and it is no doubt the special
theory, too. In other words, it is the theory of what is
subtracted from ontological subtraction [such as that found in
Kant's subtractive ontology]. Philosophy is the theory of what
is strictly impossible for mathematics. (60)
This is a crucial point, a crucial /thought/. The distinction
between the special and general theory of the event may be best
understood, in part via Bataille (restricted vs. general economy),
as that between the representational theory of the event and the
theory that reveals something in the event, or the truth, that
exceeds, irreducibly, any representation or any specific ontology.
This thought governs the remainder of Badiou's argument in the
book, developed via analyses of philosophical frameworks (Plato,
Aristotle, Spinoza, Kant, and Deleuze), and set- and topos
theoretical mathematics.
15. It is also in this "space" of the relationships between Being and
trans-Being, the space defined by the non-empty void of the Real
(in Lacan's sense, extended by Badiou), that Badiou, in closing
the book, also re-establishes the relationships between Being and
appearing (153-68). In Lacan and Badiou alike, the trans-Being of
the Real is conceived as the efficacity of both Being and
appearing, or even of trans-Being of the event. Badiou, thus, also
brings together his faithfulness to (the true) Platonism and his
faithfulness to a reversal of Platonism, and thus to modern
philosophy (163), in order to reveal "Being itself in its
redoubtable and creative inconsistency. It is Being in its void,
which is the non-place of every place" (169). Given this relation
to appearing, Being comprises, from its two different sides, both
the multiple without-One and the void, and in both of these
aspects it is produced, as an effect or set of effects, by the
efficacy of the Real. The /appearance/ of Being is traced through
an important discussion of the relationships between mathematical
ontology and mathematical logic, via topos theory. These
relationships and Badiou's view of mathematics as a thought and
ontology, as against logic, are important in this context and for
Badiou's thought in general. Here suffice it to say that logic is
linked to appearance, thus also giving the double genitive to the
phrase "logic of appearance," and mathematical ontology is related
to Being, /is/ Being. The configuration of Being and appearing,
just defined, is that of "event," which Badiou's use of
mathematical ontology helps him establish in order to move beyond
ontology to philosophy. As Badiou writes:
This [the configuration just described] is what I call an
"event." All in all, it lies for thought at the inner juncture
of mathematics [as ontology] and mathematical logic. The event
occurs when the logic of appearing [the double genitive sense]
is no longer apt to localize the manifold-being of which it is
in possession. As MallarmÃ(c) would say, at that point one is
then in the waters of the wave in which reality as a whole
dissolves. Yet one also finds oneself where there is a chance
for something to emerge, as far away as where a place might
fuse with the beyond, that is, in the advent of another
logical place, one both bright and cold, a Constellation. (168)
16. Badiou's appeal to MallarmÃ(c) signals that, along with being the
space of philosophy (the space with which it is concerned and
which it also occupies), this space also appears to be, in
Blanchot's title phrase, the "space of literature" (the same
parenthesis applies). It is the space from which, in a
non-reversing reversal of Plato, mathematics, which only relates
to Being or ontology, is exiled or rather into which it is only
partially allowed as a tenant, as against poetry, which inhabits
this space as a resident alongside philosophy. This orientation
and decision of thought also brings Badiou closer to Heidegger,
especially the later Heidegger, when the ontological projects of
Being and Time (and several projects following it) are replaced
with a certain conjunction of thinking and being, via Parmenides's
fragment, "The Same is Both Thought and Being," which Badiou
invokes (52). (The English translation is that of Badiou's French
translation, different from Heidegger, and indeed the translation
of this statement is itself a decision of thought.) At this stage
of his thought, the true "thinking" [/denken/] is fundamentally
linked by Heidegger to the thinking and language of poetry. The
proximity between Badiou and Heidegger thus reemerging is tempered
by differences (no equation here), most essentially because
mathematical ontology and the equation "mathematical
ontology=ontology=the multiple without-One" is retained by Badiou,
as against Heidegger (on both counts: mathematics and
multiplicity). This equation remains crucial, even if one can now
add an inequality ontology< poetry=philosophy (as thinking
"event"), or perhaps with poetry and philosophy in the undecidable
relationships (in Derrida's sense) to each other. Either way,
mathematics, poetry, and philosophy are brought together, in a
Constellation.
17. A reader of Badiou, or of Hughes's article, would not be surprised
by the presence of literature in this space, any more than by the
presence of this space in literature, by its becoming, in
Blanchot's title phrase, the space of literature, defined by
Blanchot along similar lines (of the discontinuity of the event).
Hughes's article, to which I am now ready to return, deserves
major credit for its exploration of the role of literature in
Badiou's thinking of the event (ethical, aesthetic, or other), and
additional credit for relating the situation and the event of
Badiou's thought to the Romantic tradition, indeed to many
Romantic traditions, which also form a multiple without-One.
Hughes is also right to bring these aspects of Badiou's thought to
bear on Badiou's thinking (of) subjectivity and the ethical, and
the connections (proximities and differences) between this
thinking (or Badiou's thought in general) and that of Lacan and
Levinas. Hughes's "ventur[ing]" a (re)formulation of Badiou's
ethical maxim as "/one must poeticize/" is compelling, especially
given that the true ethical imperative (under the full force of
which we come rarely, according to Badiou) is by event and truth.
As Hughes says: "One might venture it as a new formulation of
Badiou's ethical maxim: /One must poeticize/. That is, one must
exceed one's situation and assume an ethical relation to the event
by striving to name it through poetry. As the Romantic intuited
and as Badiou's philosophy formulates much more precisely, poetry
and ethics, like poetry and truth, are not to be disentangled"
("Riven" Â¶21). This is, I think, quite true, as is the more
general claim that Badiou "is suggesting a special role for poetry
in the elaboration . . . of truth [in his sense]" (Â¶21).
18. It appears to me, however, that Hughes disentangles too much both
from mathematics and from mathematical ontology, as the ontology
of the multiple and the political without-One, and from ontology
in general in Badiou. Hughes's invocation of Poe's "The Purloined
Letter" on his way to his conclusion just cited is apposite here:
"We might think of this [this special role of poetry] as somewhat
akin to the insight of Poe's Dupin, who says, referring to the
Minister [a poet and a mathematician] who has purloined the royal
letter, that "as a poet and mathematician, he would reason well;
as mere mathematician, he could not have reasoned at all" (Â¶22).
We might recall that Poe's "The Purloined Letter" and Lacan's
reading of it in his "Seminar on 'The Purloined Letter'" engage
the question of the relationships between poetry and mathematics.
However, could the Minister think as a mere poet, at least as
sharply as Dupin, who out-thinks him? Perhaps he could not, at
least if we read the story through Badiou's optics, where the
thinking of the Real is at stake, and Poe does not say that the
Minister could either. In fact, both Lacan's and Derrida's
readings (in "Le Facteur de la VÃ(c)ritÃ(c)," in The Post Card) place
Dupin in a position that is more akin to that of a philosopher in
Badiou, as both a poet and a mathematician. Part of Derrida's
critique of Lacan is that, unlike Dupin/Poe, Lacan does not think
the multiplicity and dissemination of writing in his reading. One
might add that Lacan also places the whole case too much in the
Symbolic register, thus both reducing the multiple to the Oedipal
and, as it were, forgetting the Real. In any event, in my view
Badiou's ontology of the multiple without-One and its political
underpinnings and implications could have sharpened and enriched
Hughes's analysis of the ethical and/as literary problematics in
Badiou. Hughes invokes the multiple only briefly in his discussion
of a theatergoer's encounter with Hamlet in Being and Event
("Riven" Â¶12).
19. Consider, for example, how the connections between Badiou and
Levinas appear from the perspective of Badiou's mathematical
ontology of the multiple. Hughes does note the potential role of
the mathematical considerations for Badiou, including as concerns
the difference between him and Levinas. Thus, he says:
Badiou's mathematical grounding and conceptualization of
alterity, his "numericalities" of solipsism and the Infinite,
/his set-theoretical elaboration of the event/, and his
insistent recourse to the category of /truth/ as the grounds
for the specifically /ethical/ force of alterity and the
infinite--all this is quite foreign to Levinas's sensibility .
. . This [along with other factors that I omit for the moment,
given my context] also gives Badiou a broader scope for
thinking the ethical in places--art, science,
/politics/--where Levinas's writings do not often venture.
(Â¶22; some emphasis added)
Hughes, however, does not take advantage of the mathematical
aspects of Badiou's thought, in particular "/his set-theoretical
elaboration of the event/," which entails and enacts the multiple
without-One and, within it, the political, as considered here. As
a result, Hughes's analysis ultimately leaves Badiou's thought
within the domain of rupture, discontinuity, inscribed "through
tropes of /trauma/" (Â¶23). The Levinasian ethical /situation/ (the
term can be given Badiou's sense as well) is defined by an
encounter with the radical, irreducible alterity of the Other
(/Autrui/), which should not be simply identified with a person or
a subject. (This alterity is not unlike that of Lacan's Real in
epistemological terms, but is different in ethical terms, is
ethical.) It may be noted that Badiou, and some of his followers,
tend to over-theologize Levinas's thinking on this point. Contrary
to Badiou's argument in Ethics, while Levinas's thought has
significant theological dimensions, the Other as /Autrui/ is not
theological, even if it is /modeled/ on theology, and as such may
be better termed, via Heidegger and Derrida, "ontotheological."
For the moment, the appearance of the Other is the /event/ that
transforms the /situation/ (again, in Badiou's sense) in which
each of us finds oneself when the Other appears. This appearance
(including in the sense of phenomenon) redefines our world, or
home, since, according to Levinas, we must welcome the Other with
hospitality (Totality and Infinity 27). Levinas's conception is
more complex because the event of the appearance of the Other has
always already occurred, thus making /ethics/ and its /infinity/
precede /totality/, which Levinas often sees as defining
/philosophical/ thought. These complexities do not, however,
affect my argument here.
20. In contrast to Levinas, for Badiou any ethical event, good or
evil, or beyond good and evil, while it may involve an encounter
with the other (no capital), cannot be defined by the alterity of
the Other as /Autrui/ (with capital O or A) in Levinas's sense.
Any situation or any event is defined by and defines (but cannot
be contained by) the mathematical and, correlatively, political
ontology of the multiple without-One, by /Badiou's infinite./ As
such, it is not only without totality, but also without /Levinas's
infinity/, which appears as a form of totality from where Badiou
stands, since it is defined by the One (as the Other), rather than
by an infinitely multiple without-One. Accordingly, in an ethical
event, as in any other event, we always confront the Real and its
alterity through the manifold or the multiple, whether we do it
together with others or/as in encountering an other (and hence, as
against Levinas, still always together and never apart within the
multiple). Subjectivity, it follows, is this political
multiplicity as well, and hence every subject is a multiple
without-One. By the same token, "the truth itself is but a
multiplicity: in the two senses of its coming (a truth makes a
typical /multiple/ or generic /singularity/) and in the sense of
its being (there is no /the/ Truth, there are only disparate and
untotalizable truths that cannot be totalized)" (Briefings 62;
emphasis added). An ethical or any other situation or any event is
always political, infinite yet multiple, multiple without0 One,
without any possibility of unity or totality. Because of the role
of the political multiple-without One of Being, always involved in
an event, the multiple is irreducible in the trans-Being of the
event as well. There is no event, no encounter, ethical or other,
that can ever be ontologically single; it can only be singular in
the sense of its uniqueness or discontinuity relative to its
situation, on the one hand, and to other situations and events, on
the other. Hence the political is irreducible in and defines the
ethical, rather than being grounded in the Levinasian ethical
Other. In his Adieu to Emmanuel Levinas, Derrida offers a
respectful and subtle, but firm, critique of Levinas along similar
lines, although there are also differences between Badiou and
Derrida, specifically insofar as there is no ontological infinite
(in Badiou's sense) in Derrida. It also follows that, while Hughes
is right to stress the singularity of the event and its alterity
or exteriority to the situation, it is not possible to speak as
Hughes does along more Leibnizean lines of the Oneness of the
situation in Badiou or to read "its /all/" "as the Oneness of
one's multifarious elements" ("Riven" Â¶12).
21. To some degree, the argument just given also applies to Lacan's
use of the Real, in part in juxtaposition to Badiou's concept of
the Real or how this concept can be used and developed, and has
been used and developed by Badiou. That is, Lacan's use of the
Real may also be seen as to some degree bypassing the multiple and
the political, and centering primarily on individual subjectivity
or intersubjectivity and on the ethical, as innovative and radical
a move as the introduction of ethics into psychoanalysis might
have been. Apart from other key differences (such as those those
having to do with the role and architecture of language,
signification, desire, the Imaginary and the Symbolic), Lacan's
claims concerning the ethical are not as strong as those of
Levinas. Indeed, by being placed within the triangularity of the
Oedipal (transformed, as against, Freud, via the economy of the
signifier) and hence within a certain Oedipal politics, Lacan's
ethical order or subjectivity is at least implicitly political.
Nevertheless, one can speak of a certain curtailment of the
multiple and/as the political, and Deleuze and Guattari have
criticized Lacan along these lines in Anti-Oedipus. It appears to
me that on this point, too, Hughes's analysis can be deepened and
must, to some degree, be adjusted. Let me reiterate that, even
apart from being an extraordinarily powerful concept in its own
right, Lacan's Real is crucial for Badiou and even irreducible in
his philosophy (including in his sense, as discussed above).
Hughes, accordingly, is correct to give major attention to this
significance, specifically in the context of Badiou's ethical
thinking, and to link Lacan's Real and its connection to language
to the problematics of literature and Romanticism, and to the way
both think "the void of the real" ("Riven" Â¶13). The concept is
indeed especially significant, as Hughes argues, for Badiou's
concept of event, as always the event of trans-Being: the "trans"
of this "trans-Being" is essentially linked to the Real in Lacan's
sense, or rather--and this is my point here--in Badiou's sense.
For it seems to me that Badiou's deployment of Lacan's Real is
assimilated into his thinking though the multiple and the
political without-One, or it follows without-Three (the Oedipal
three), as much as without-Two, the ethical Two of Levinas, which
is still ultimately the One (Lacan's Three is three). The Real
cannot by definition be reduced to ontology--any ontology but
especially Badiou's mathematical ontology--any more than can an
event and its trans-being, grounded in (as arising from) the Real.
I would argue, however, that this disruptive work of the Real, as
understood by Badiou, cannot be dissociated from the multiple
without-One. The Real acts upon this ontology and disruptively
transforms its multiplicity by giving rise to events, but only
into another multiplicity.
22. Now, is this transformation /traumatic/? Or, more generally and
more pertinently to Hughes's argument, how does Badiou's concept
of event relate to that of trauma, especially as considered by
Lacan, via the Real? As Hughes says, Badiou does not appeal to and
does not primarily, if at all, think the event as trauma. As
Hughes states, "to be clear, 'trauma' is not a word Badiou himself
employs . . . he uses an array of others to describe his
subjects--riven, punctured, ruptured, severed, broken, annulled,
and so forth" (Â¶10). Hughes gives his reasons for his appeal to
trauma. One might ask, however (and this question appears to be
missing in Hughes's article): Why does Badiou not appeal to
trauma? Although Badiou's ontology of the multiple could be
brought to bear on this question as well, the main reason for
Badiou's avoidance of trauma, I contend, lies in the nature of
trauma as being primarily, fundamentally about the past event and
its primarily negative, traumatic impact on the (post-event)
future. By contrast, even though he grounds his concept of event
in a Lacanian concept of the Real, Badiou appears to be primarily
concerned with the future, and moreover with the positive,
transformative future of events (which may of course have occurred
in the past). This futurity is part of the architecture of
Badiou's concept of event, and defines actual events--of whatever
kind and whenever they occur--as futural events. Let us revisit
Badiou's list of events cited earlier: "The French revolution in
1792, the meeting of HÃ(c)loÃ¯se and AbÃ(c)lard, Galileo's creation of
physics, Haydn's invention of the classical musical style, . . .
the cultural revolution in China (1965-67), a personal amorous
passion, the creation of Topos theory by the mathematician
Grothendieck, the invention of the twelve-tone scale by
Schoenberg" (Ethics 41). These events may have been traumatic and
may have left their traumatic effects or traces, but it is their
futural impact, as creating new situations, that is above all at
stake for Badiou, even in the case of a personal amorous passion,
or love. Indeed, there is no need to say "even," for passion and
love are about the present and future, even though they can and
sometimes do have traumatic effects. Lacan's concept of the Real
easily allows Badiou to give it this futural dimension because,
apart from Badiou's mathematical (ontological) and political
extension of the Real, it is indeed a more general concept rather
than something that isirreducibly connected to trauma. It is true
that, according to Lacan, "the function . . . of the real as
encounter . . . first presented itself in the history of
psycho-analysis in a form that was in itself already enough to
arouse our attention, that of trauma" (Four Fundamental 55). That,
however, need not mean and, I would argue, does not mean that the
function of the Real is limited to trauma, even in Lacan or in
psychoanalysis; quite the contrary, and Badiou is right to take
advantage of the broader sense of Lacan's extraordinary concept in
defining his conception of the event.
23. The futural orientation (also in Badiou's sense) of his thought of
the event is, however, a more complex matter. For this orientation
not only poses a question for Hughes about his reading of this
concept in terms of or via tropes of trauma, it also poses a
question for Badiou from the traumatic side of the Real. The
significance of this futural orientation of thinking the event is
undeniable, including in our understanding of history, and hence
the past, as shaping our present and future. But the past, the
ghosts of the past inevitably haunt us, many ghosts of many pasts,
for this ontology is multiple without-One, too--that of the
multitudes of the living and the dead, each with its own end of
the world, unique each time, both multiple and unique, like the
ontology of snowflakes. It is, yet again, literature that brings
together both these multiples without-One, that of the living and
the dead and that of the falling snow. It does so in Joyce's
thinking an event of trans-Being, human (the passion of love),
literary, and political, in ending, /uniquely and multiply
ending/, "The Dead" and Dubliners: "the snow falling and faintly
falling, like the descent of their last end, upon all the living
and the dead," of Ireland and of the world. "/All/ the living and
the dead," each unique and multiple, as snowflakes in a
snowfall--past, present, and future.
/ Theory and Cultural Studies Program
Department of English
Purdue University
plotnit@purdue.edu /
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