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Robert Hughes's article offers an unexpected perspective on Alain Badiou's work and its
impact on the current intellectual and academic scene, a cliché-metaphor that
(along with its avatars, such as performance or performative, also a pertinent
theoretical 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).
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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 Félix 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.
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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 considerable 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, the 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 it 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
Gödel 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.
Gödel'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.
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In contrast to Badiou, the role of mathematics, most centrally topology
and geometry, in Deleuze or Deleuze and Guattari is less defined by mathematical
ontology, especially in the foundational, such as 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.
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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" ¶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
dimensions 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" ¶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.
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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'être et
l'événement), 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=mc2), 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, Levinas, 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.
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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 Levinas. (It seems to me that Hughes equates
"truth" and "event," too much ["Riven" ¶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" (¶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).
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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
Héloïse and Abélard, 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. -
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" ¶10). This is sensible, 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).
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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 naïve concept of set, naïve" 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 Gödel'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).
-
While Badiou, thus, establishes his ontology on the basis of more general
set-theoretical considerations, its crucial further dimensions are revealed by
Gödel'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. Gödel'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
Gödel (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
Gödel, that such is not the case. For Gödel 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 Gödel's "first incompleteness theorem." Gödel made
the life of mathematics even more difficult, and 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.
-
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 "consistent" in itself. Since,
however, Gödel'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 Gödel'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.
-
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 Gödel'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. -
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. -
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 Mallarmé 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)
-
Badiou's appeal to Mallarmé 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.
-
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 Romantics intuited and as Badiou's philosophy formulates
much more precisely, poetry and ethics, like poetry and truth, are not to be
disentangled" ("Riven" ¶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]" (¶21).
-
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" (¶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
Vérité," 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"
¶12).
-
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. (¶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" (¶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.
-
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 without-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" ¶12).
-
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" ¶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.
-
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? 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" (¶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 Héloïse and Abélard,
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.
-
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. It is, yet again,
literature that give us perhaps the best image of this multitude of the unique
multiples, that of a snowfall--a multitude of snowflakes, each with its unique
designs (its unique multiple) and unique trajectory of fall and eventual melting
down. 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. "All
the living and the dead"--past, present, and future.
Theory and Cultural Studies Program
Department of English
Purdue University
plotnit@purdue.edu
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Works Cited
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---. Positions. Trans. Alan Bass. Chicago: U of Chicago P, 1981.
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----. "Seminar on 'The Purloined Letter.'" Écrits. Trans. Bruce
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