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Badiou's Materialist Epistemology of Mathematics
Ray Brassier a
a
Centre for Research in Modern European Philosophy, Middlesex University, London N14 4YZ, UK
Online Publication Date: 01 August 2005
To cite this Article Brassier, Ray(2005)'Badiou's Materialist Epistemology of Mathematics',Angelaki,10:2,135 — 150
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ANGEL AK I
journal of the theoretical humanities
volume 10 number 2 august 2005
One establishes oneself within science from the
start. One does not reconstitute it from scratch.
One does not found it.
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Alain Badiou, Le Concept de modèle1
[T]here are no crises within science, nor can
there be, for science is the pure affirmation of
difference.
Alain Badiou, ‘‘Marque et manque’’ 2
i introduction
ray brassier
hroughout Badiou’s work, mathematics
enjoys a privileged status as paradigm of
science and of ‘‘scientificity’’ in general. This
has been a constant, from his first significant
philosophical intervention, the 1966 article ‘‘The
(Re)Commencement of Dialectical Materialism,’’3
notable for the way in which it already prefigures
his subsequent (career-spanning) preoccupation
with the relation between set-theory and categorytheory, to his most recent work, wherein Badiou
finally establishes a philosophical connection
between these two branches of mathematics
by arguing that the doctrine of being, laid out
via set-theory in Being and Event4 (1988), needs to
be supplemented by a doctrine of appearance that
mobilizes category-theory, as Badiou does in his
forthcoming Logics of Worlds.5 Two quotes,
separated by over thirty years, are indicative of
Badiou’s unwavering commitment to the paradigmatically scientific status of mathematics. The first
is from the aforementioned 1966 article:
‘‘[U]ltimately, in physics, fundamental biology,
etc., mathematics is not subordinated and expressive, but primary and productive.’’6 We shall try to
explain what this primacy of mathematical
‘‘productivity’’ entails for Badiou by examining
his early attempt to develop a ‘‘materialist
T
BADIOU’S
MATERIALIST
EPISTEMOLOGY OF
MATHEMATICS
epistemology’’ of mathematics in his first book,
The Concept of Model (1969). But first we shall
consider a second quote, from an interview with
Peter Hallward in 1998: ‘‘In the final analysis,
physics, which is to say the theory of matter, is
mathematical. It is mathematical because, as the
theory of the most objectified strata of the
presented as such, it necessarily catches hold of
being-as-being through its mathematicity.’’7 This
latter claim encapsulates an argument about the
relation between mathematical ontology and the
natural sciences implicit (though never explicitly
articulated) in Badiou’s most ambitious work to
date, Being and Event. Summarizing very briefly,
we can say the following: for Badiou axiomatic settheory is the science of being as sheer multiplicity,
the science of the presentation of presentation
ISSN 0969-725X print/ISSN 1469-2899 online/05/020135^16 ß 2005 Taylor & Francis and the Editors of Angelaki
DOI: 10.1080/09697250500417357
135
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badiou’s materialist epistemology
(rather than of what is presented); in other words,
the science that guarantees access to presented
reality.8 Thus not only does the Zermelo–Fraenkel
axiomatization of set-theory provide the fundamental infrastructure to which all of mathematics
can be reduced, it is also the guarantor for the
mathematical sciences’ access to reality. Consequently, the implication would seem to be that, for
Badiou, the ‘‘scientificity’’ of a given science is
directly proportional to its mathematization:
science is ‘‘scientific’’ precisely to the extent that
it is mathematical. By the same token, the less a
science depends upon mathematical formalization,
the less scientific it is. Hence Badiou’s deferential
nods towards physics and his notorious disdain for
biology, ‘‘that wild empiricism disguised as
science.’’9 But surely the claim that mathematics
provides the sciences with their ultimate horizon
of scientificity is damning evidence of Badiou’s
stubborn adherence to an unreconstructed (not to
say anachronistic) ‘‘Platonist’’ and ‘‘foundationalist’’ conception of science; one which privileges an
‘‘ideal’’ and ‘‘a priori’’ mathematical realm of
scientificity over science’s ‘‘empirical’’ and ‘‘material’’ dimensions, and attempts to ground the
latter’s access to reality in the former?10
Yet, as we shall see, it is precisely such
foundationalism, as well as all such distinctions
between a priori and a posteriori, ideal and real,
formal and material, that Badiou explicitly sets out
to undermine through the materialist epistemology of mathematics laid out in The Concept
of Model. Badiou’s materialist critique of the
ideological substructure which tacitly governs the
aforementioned continuum of distinctions
(formal/material, ideal/real, a priori/a posteriori)
common to empiricism and idealism (indeed, for
Badiou, idealism is merely a variant of empiricism) furnishes the key required in order to make
sense of his subsequent – and apparently startlingly eccentric – claims on behalf of the privileged status of mathematics in Being and Event.
If it is difficult to extract from Badiou’s work
a ‘‘philosophy of mathematics’’ conforming to the
norms of the academic sub-discipline of the same
name, or to render some of his philosophical
claims about science intelligible in terms of the
debates that define the field known as ‘‘philosophy
of science’’ (e.g., realism vs. instrumentalism,
analysis and reductionism, the nature of induction, the status of scientific law, inference to the
best explanation, etc.), this is not only because
mathematics functions as a synecdoche for science
in Badiou but also because his Platonist materialism challenges those empiricist doxas (principally the distinction between ‘‘formal’’ and
‘‘material’’ sciences) that, precisely because of
their uninterrogated but constitutive role in the
debates that characterize ‘‘philosophy of science,’’
have filtered down into the branch of the latter
known as ‘‘philosophy of mathematics.’’11 In this
regard, we shall find it instructive to contrast
Badiou’s materialist critique of what he regards
as the ideological distinction between ‘‘real’’ and
‘‘ideal,’’ common to empiricism and idealism,
with Quine’s celebrated intra-empiricist subversion of the analytic/synthetic distinction.
Early in The Concept of Model, Badiou,
alluding to Althusser, reminds us that ‘‘To talk
of science [la science] in the singular is an
ideological symptom.’’12 There are only sciences,
in the plural. Yet the synecdochal status of
mathematics vis-à-vis the sciences seems to engender a paradox whereby everything Badiou has
to say about ‘‘the sciences’’ is encoded in his
statements about one of them, mathematics. How
can Badiou reconcile his materialism, which
insists on the irreducible plurality of sciences,
with his Platonism, which ascribes a paradigmatic
status to mathematics? The answer lies in understanding how, for Badiou, Platonism is precisely
what recuses the empiricist distinction between
thought and object.13 Mathematics is neither
merely a formalist game, the arbitrary manipulation of intrinsically meaningless symbols, nor a
quasi-supernatural mystery presided over by a
select priesthood who enjoy a privileged vantage
onto a transcendent realm of eternal objects. What
singularizes mathematics as paradigm for science
is rather the exemplary nature of its autonomous
productivity. As we shall see, for Badiou, mathematical productivity (and a fortiori, scientific
productivity) consists in cutting or differentiating
the notational material upon which it operates.
Science is the production of stratified differences.
We shall dissolve the apparent contradiction
between Badiou’s ascription of a synecdochal
status to mathematics vis-à-vis the sciences, and
136
brassier
his insistence on the latter’s essential plurality, by
showing how for Badiou mathematics is not an
a priori formal science grounding the empirical
sciences’ access to reality but rather the
paradigmatic instance of a productive experimental praxis. This is the materialist dimension of
Badiou’s Platonism.
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ii the formal and the empirical
In The Concept of Model, Badiou is operating
under the aegis of two fundamental distinctions:
1. The Althusserian distinction between
‘‘historical materialism,’’ understood as the
Marxist science of history, and ‘‘dialectical materialism,’’ understood as the latter’s philosophical
counterpart.14
2. Badiou’s own distinction between ideological notions, philosophical categories, and scientific concepts.
Philosophy, constituted by its reactive and/or
parasitic relation to scientific innovation on the
one hand, and its subservience to dominant
ideological interests on the other, is defined as
the practice of an ‘‘impossible relation’’ between
science and ideology.15 For the most part, philosophy consists in the ideological envelopment of
science: philosophical categories denote ‘‘inexistent’’ objects wherein concepts and notions are
variously combined.16 Informed by the Marxist
science of history, the task of a materialist
philosophy (as opposed to a ‘‘philosophy of
matter,’’ which merely synthesizes an inexistent
category ‘‘matter’’ through the notional envelopment of physico-biological concepts) is to expose
and critique the reactionary ideologies encoded in
various ‘‘philosophizations’’ of science and to
supplant them by materialist categories capable
of being deployed in the service of revolutionary
ideology (philosophy, according to a famous
Althusserian slogan, being ‘‘the class struggle in
theory’’).17 Badiou’s aim in The Concept of
Model is to isolate the scientific – i.e., logicomathematical – concept of model from its notional
envelopment by the categories of bourgeois
epistemology – central to which is the distinction
between the ‘‘formal’’ and the ‘‘empirical’’ – and
to construct a category of model consonant with a
materialist history of the sciences.
137
We shall begin by recapitulating Badiou’s
critique of bourgeois epistemology. Ideological
formations are structured as continuous combinations of variation on a difference whose principle is
presupposed but never given in the series which it
governs.18 It is a characteristic of such formations
that their notional variants are incapable of
examining or legitimating their own underlying
principle. The unthematized variational principle
governing bourgeois epistemology is the notional
difference between theoretical form and empirical
reality: science is a formal representation of its
object, whether the representation be characterized
in terms of the effective ‘‘presence’’19 of the object,
as is the case with empiricism, or in terms of the
anteriority of a formal apparatus, i.e., of the
mathematical code whereby the object is represented, as is the case with formalism (Badiou seems
to have structuralism specifically in mind here).
But in either case what must be borne in mind,
Badiou insists, is that ‘‘[E]mpiricism and formalism have no other function here besides that of
being the terms of the couple they form. What
constitutes bourgeois epistemology is neither
empiricism, nor formalism, but rather the set of
notions through which one designates first their
difference, then their correlation.’’20 Thus the
materialist critique of bourgeois epistemology
must first identify the hidden theme, whose
characteristic structure is that of a differential
correlation between two opposed terms, governing
the ideological continuum of notional variants.
Badiou identifies a canonical variant of this theme
in the opposition between Carnap and Quine
regarding the status of the distinction between
formal and empirical sciences. In ‘‘The Logical
Foundations of the Unity of Science,’’21 Carnap
begins by positing the difference between formal
and empirical sciences before proceeding to seek
rules of reduction governing the conversion of the
terms of one empirical science into another.
Carnap argues that biological terms are convertible
into physical terms: Physics provides a sufficient
basis for the reduction of biology.22 Thus the
language of science can be unified in so far as a
‘‘physicalist’’ language provides a universal basis
for reduction for all the empirical sciences. Finally,
Carnap’s project – and more generally, the logical
empiricist approach to the issue of the unity of
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badiou’s materialist epistemology
science – culminates in the question of the relation
between the fundamentally ‘‘physicalist’’ language
of empirical science and the ‘‘artificial’’ languages
of the formal sciences; in other words, the relation
between the synthetic statements of the former and
the analytical statements of the latter. But this is, of
course, precisely the distinction that Quine calls
into question in his celebrated 1951 article ‘‘Two
Dogmas of Empiricism.’’23 According to Quine,
the notion of ‘‘analyticity’’ cannot withstand
critical scrutiny: it relies on a notion of ‘‘synonymy,’’ i.e., ‘‘sameness of meaning,’’ which in turn
presupposes a theoretically transparent account of
the intensional dimension of ‘‘meaning.’’ Quine
challenges the intelligibility of the former and
the possibility of the latter. While the notion of
linguistic ‘‘extension’’ or reference can be
rendered logically transparent, ‘‘intension’’ understood as a noematic entity tethered to the linguistic
sign by noetic intention is simply ‘‘what
[Aristotelian] essence becomes when it is divorced
from the object of reference and wedded to the
word.’’24 Such a dubious metaphysical doctrine
(‘‘the ‘idea’ idea’’) cannot provide a reliable
warrant for the notion of analyticity. Moreover,
Quine goes on, the dogma of the analytic/synthetic
distinction is indissociable from another empiricist
dogma, the dogma of reductionism, manifested in
Carnap’s belief in the possibility of decomposing
the truth of scientific statements into a formal or
linguistic component on the one hand, and a
factual or empirical one on the other. Pace
Carnap, Quine insists on the intrinsically holistic
character of the conceptual scheme called
‘‘science’’ and maintains that it is impossible to
separate in it the contribution of language (i.e.,
conceptual convention) from the contribution of
experience (empirical data):
The totality of our so-called knowledge or
beliefs, from the most casual matter of geography and history to the profoundest laws of
atomic physics and even of pure mathematics
and logic, is a man-made fabric which impinges
upon experience only along the edges [. . .] But
the total field is so underdetermined by its
boundary conditions, experience, that there is
much latitude of choice as to what statements
to re-evaluate in the light of any single contrary
experience. No particular experiences are
linked with any particular statements in the
interior of the field, except indirectly through
considerations of the equilibrium affecting the
field as a whole.25
Thus although acknowledging the underdetermination of scientific theory by empirical
evidence, Quine refuses to abjure what Donald
Davidson subsequently criticized as the ‘‘third’’
and ultimate dogma of empiricism: the dualism of
conceptual scheme and empirical content.26 In
demolishing the analytic/synthetic distinction on
empiricist grounds, Quine rejects Carnap’s positivist ‘‘double standard’’ in the treatment of
scientific hypotheses on one hand, and ontological
questions on the other.27 No hard-and-fast dividing line can be drawn to demarcate scientific
hypothesizing from ontological speculation.
Coupled with the Quinean doctrines of the
indeterminacy of translation, which claims that
reference is inscrutable unless relativized to a
specific semantic coordinate system, and ontological relativity, which insists that ‘‘to be is to be the
value of a variable,’’ this scheme/content dualism
leads Quine to embrace an epistemological relativism according to which the difference between
Homeric gods and protons is merely one of degree
rather than kind: ‘‘Both sorts of entities enter our
conception only as cultural posits.’’28 Whatever
superiority the myth of physical objects enjoys
over that of the Homeric gods comes down to a
question of usefulness:
As an empiricist, I continue to think of the
conceptual scheme of science as a tool, ultimately, for predicting future experience in the
light of past experience [. . .] The myth of
physical objects is epistemologically superior to
most in that it has proved more efficacious than
other myths as a device for working a manageable structure into the flux of experience.29
Thus pragmatism is revealed as the truth of
Quinean empiricism. Quine proposes to supplant
Carnap’s logical empiricism with a pragmatism
which plainly exposes the point at which
empiricism cannot but concede its own constitutive subordination to ideological imperatives
whose status it is incapable of problematizing.
If Quine sees no need for philosophy to investigate
the mechanism of conceptual correlation whereby
138
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brassier
‘‘cultural positing’’ is supposedly adjusted
to ‘‘empirical usefulness,’’ this is because the
putative transparency of that mechanism reveals
how empiricism remains conditioned by a set of
ideological norms whose structure it cannot
perceive. Thus the opposition between Carnap
and Quine remains internal to an empiricist
problematic structured around a difference
between fact and form which neither can afford
to question. In this regard, Quine’s audacious
subversion of the analytic/synthetic distinction
obscures and perpetuates the more fundamental
difference between formal and empirical which
underlies it. Quine merely negates a distinction
which Carnap seeks to reduce: ‘‘Whereas that
reduction is essential to Carnap’s discourse, all
that matters in Quine’s is the justification of the
claim that it is not necessary to reduce what can
be conveniently denied.’’30 Quine’s negation is
convenient because it leaves empiricism’s own
ultimately empirical yet empirically imperceptible
condition of possibility untouched. For Quine, as
Badiou points out, it comes to the same thing
whether one says that the empirical is a dimension
of the formal, or the formal a dimension of the
empirical. Quine’s naturalization of epistemology
and his doctrine of the reciprocal containment of
epistemology and ontology entail that the philosophical investigation into the scientific representation of the world be carried out from within the
ontological framework provided by science
itself. As we shall see in section iii, logical
empiricism appropriates the scientific (i.e., logicomathematical) concept of model but overcodes it
in terms of the formal/empirical distinction in
such a way that it is no longer the formal that
models the empirical (as in vulgar empiricism)
but rather the empirical that models the formal
(as it does for Carnap). Quine’s naturalization of
epistemology and subversion of the analytic/
synthetic distinction reveals one way of overcoming this formal/empirical dichotomy without
relinquishing empiricism: by grounding the
formal modelling of the empirical in an empirical
modelling of the formal. In other words, by
proposing as ultimate horizon for naturalized
epistemology the construction of a scientific
model of science’s model-constructing capacity
in general. For Badiou in Concept of Model,
139
writing in the late 1960s, this strategy is exemplified by the conjunction between cybernetics
and empiricism, and, more specifically, by the
integration of AI research into the programme of
naturalized epistemology spawned by Quine.31
The goal of the latter consists in explaining the
congruence between the world and its scientific
representation by elaborating a scientific theory of
representation. The scientific modelling of reality
is explained in terms of neurocomputational
processes which are themselves part of science:
scientific representation is integrated into
the science of representation (which is itself a
representation of science).32 Thus in a surprising
empiricist mimesis of the serpent of absolute
knowledge swallowing its own tail, naturalized
epistemology seeks to construct a virtuous circle
wherein the congruence between fact and form is
explained through the loop whereby representation is grounded in fact and fact is accounted for
by representation. As Badiou puts it: ‘‘If science
is an imitative artifice [artisanat], the artificial
imitation of this artifice is, in effect, Absolute
Knowledge.’’33 Thus the ideological deployment
of the category of model allows empiricism to
progress from positivism to a pragmatist variety of
absolute idealism.
But for Badiou this pragmatist idealism and the
empiricist representation of representation concomitant with it remain beholden to ideological
doxas which, despite the latter’s reflexivity, they
are incapable of registering. The idealization of
science as imitative artifice occludes its reality
as process of cognitive production, which Badiou
maintains should not be understood in terms of
a confrontation between formal operations and
a pre-existing empirical reality but rather as a
theoretical practice developing demonstrations
and proofs within a determinate historical materiality whose structural specificity is itself the object
of a science: historical materialism. Only in light of
the latter does epistemology become sensitive to
the relation between science and its ineliminable
ideological representation without relapsing into
historicism (which is itself a variant of empiricism
for Badiou). Moreover, the pragmatist usage of the
category of model elides the distinction between
cognitive production and the technical regulation
of concrete processes. This latter elision is
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badiou’s materialist epistemology
exemplified by economics, where the use of
models consists in passing off the discipline’s
own technical subservience to conditions of
production as the timeless necessity of a specific
type of economy, whose benefits the models
exemplify. Here again, Badiou insists, the task of
materialist epistemology consists in exposing the
representational idealization of science as imitative
artifice by providing an account of the autonomy
of scientific practice vis-à-vis its ideological representation while acknowledging its constitutive yet
non-empirical historicity.
Ultimately, every variant on the fundamental
theme of empiricism, which consists in the
difference between fact and form, is faced with
the problem of how to articulate the unity of that
difference. For vulgar empiricism, the unity of the
duality of fact and form is posed in terms of the
question of the model’s reproduction or functional
simulation of reality. Thus an extrinsic relation
of analogical resemblance is invoked in order to
bridge the gap between the supposedly inert
opacity of empirical fact on the one hand, and
the active construction of theoretical form on
the other. Here, of course, the precise nature of the
desired ‘‘resemblance,’’ ‘‘simulation,’’ or ‘‘reproduction’’ remains vague and ambiguous. For the
brand of pragmatist idealism spearheaded by
Quine, however, the unity of the difference can
be unearthed by sealing the gap, by replacing
congruence with reciprocal presupposition,
by supplanting ‘‘resemblance’’ and ‘‘simulation’’
with isomorphy, and by ensuring the double
articulation of fact and form. No longer inert
and passive, the structure of the empirical itself
generates the form of representation that will
account for it. Here, evolutionary epistemology
and ultimately natural history provide the
explanatory fulcrum for explaining the relation
between empirical fact and theoretical form.
As we shall see in section iv, Badiou’s own
stance in Concept of Model exhibits both surprising parallels and profound divergences with
Quine’s. Like Quine, Badiou insists on philosophy’s dependence upon science and on the
immanent autonomy of scientific thought. Like
Quine, he refuses any recourse to a sciencetranscendent philosophical foundationalism.
But unlike Quine, Badiou will have no truck
with naturalism and hence refuses to reintegrate
the sciences (i.e., mathematics) into a broader
evolutionary and ultimately biological narrative
about the development of human cognitive
prowesses. For Badiou, the irreducible variety of
scientific practices each harbour discontinuous
historicities that remain internal and immanent
to each of them; historicities which cannot be
reabsorbed into an all-encompassing bioevolutionary narrative about the human organism’s ‘‘science-forming’’ faculties. In this regard,
notwithstanding a justifiable aversion to spuriously ‘‘totalizing’’ evolutionary narratives, and
even though viable evolutionary accounts of the
mathematical sciences (arguably the most spectacular manifestation of human cognitive prowess)
remain a very distant prospect, Badiou’s apparent
refusal to countenance any mediating nexus
between natural history and the science of history
betrays an all-too-ideological antipathy to biology
– as though in spite of Darwin, biology still
harboured too many residues of its Aristotelian
inception for even a heterodox Platonist to
stomach. But however unappetizing the prospect
of a naturalized epistemology may be to Badiou in
its pragmatist idealist guise, once one has
discounted transcendentalism, as Badiou has, it
becomes difficult to reconcile insistence on the
autonomy of the sciences as discrete registers of
cognitive production with an unqualified disdain
for the one scientific discourse that is in a position
to mediate between natural and cognitive production, or phusys and praxis. For is it not precisely
the appeal to an absolute (theological) cleavage
between two fundamentally different kinds
of history, natural history and cultural history,
or hyletic history and noetic history, that Darwin
revoked?
iii the concept of model
Throughout this section, I will simply be recapitulating the main features of Badiou’s own
meticulous reconstruction of the mathematical
concept of model. Since I am not qualified to
judge the exactitude of Badiou’s handling of
the mathematical details, I shall confine myself
to summary and paraphrase and reserve my
own appraisal of Badiou’s philosophical claims
140
brassier
to section iv, in which I shall consider the
account of the historical materiality of mathematical productivity which Badiou proposes in
light of the scientific theory of modelling outlined
below.
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syntax and semantics
Badiou’s account of the difference between the
category and the concept of model provides the
key to understanding how logical empiricism
effectively recodes the distinction between the
syntactical and semantic dimensions of logicomathematical systems in terms of the distinction
between formal and empirical science. In a given
formal system, the set of rules specifying the
difference between well-formed and illicit combinations of symbols, how expressions are to be
formed and connected to or derived from one
another, defines the system’s syntactical aspect.
The rules of deduction or syntax of a formal
system allow one to derive theorems from an initial
set of axioms. But not all the well-formed expressions in the system can be theorems, otherwise
every expression would be legitimate and the rules
of deduction would be redundant. Thus there must
be at least one theorem which cannot be derived
from the axioms by way of the rules of deduction.
This is a formal requirement necessary in order to
ensure the consistency of the system.34 Moreover,
in order to verify that a syntactical construction is
not entirely arbitrary and that a formal system
actually expresses a specific deductive structure, it
is necessary to establish a relation of correspondence between expressions of the system and
expressions belonging to a well-defined domain
of ‘‘objects.’’ Obviously, neither analogy nor
resemblance suffices when it comes to defining
this relation; what are required are well-defined
rules of correspondence. Everything pertaining
to these correspondence rules will relate to the
semantics or the interpretation of the system.
Given this characterization of semantics, ‘‘meaning’’ has a purely extensional character: to talk of
the semantics or meaning of a system is to talk of
its various interpretations as governed by these
rules of correspondence. Once one has defined
the rules of semantic correspondence for a
system, one possesses the basic requirement for
141
constructing the concept of model. This is that
every deducible expression (or theorem) of the
system be linked to a ‘‘true’’ expression in the
structure that serves as its domain of interpretation. Badiou emphasizes that the use of the term
‘‘true’’ in this logico-mathematical context is not
intended to carry any ideological-philosophical
baggage; it is simply defined in terms of a
functional division enforced by a formal mechanism that invariably distinguishes between two
classes of expression: ‘‘true’’ (or ‘‘demonstrable’’
or any other equivalent scientific valence) statements on the one hand; ‘‘false’’ (or ‘‘indemonstrable’’ etc.) statements on the other. If every
deducible expression in the system can be made to
correspond to a ‘‘true’’ statement in the domain of
interpretation, then the latter is effectively a model
of the formal system. The reciprocal of this claim
is stronger: if for every true statement in the model
there corresponds a deducible expression in the
system, then the system is said to be ‘‘complete’’
for this particular model.35 There is, in effect, a
whole gamut of semantic properties which can be
studied using mathematics: in so doing one
effectively catalogues the properties of the scientific concept of model.36
Logical empiricism exports this concept into
epistemology by characterizing science’s purely
mathematical or ‘‘formal’’ dimension as its syntactical aspect, and by designating its experimental or
‘‘material’’ aspect as a semantic interpretation of
its formal dimension. Whereas science’s formal/
theoretical dimension is said to be governed by the
demands of consistency, its empirical/experimental aspect is said to necessitate an examination of
concrete models. Experimental apparatuses are at
once instruments for constructing such models
and the realm within which to deploy the rules of
correspondence between formal calculation and
concrete measurement.37 The choice of scientific
theory is constrained by the experimental model
and correspondence rules on the one hand, and by
the system and its syntactical rules on the other.
Thus, in Carnap’s Meaning and Necessity,38 for
example, science is structured by the interplay
between the constraint of syntactic deduction
and the exactitude of semantic interpretation.
But for Carnap, unlike vulgar epistemology, it is
the empirical that functions as a model for
badiou’s materialist epistemology
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syntactical artifice, rather than the reverse.
Whereas the vulgar epistemological category of
model is forced to rely on para-theoretical notions
of ‘‘resemblance,’’ logical empiricism is able to
characterize the appropriate epistemological
criteria governing the relation between science’s
formal and empirical aspects in terms of demonstrable theoretical properties such as that of
syntax’s completeness relative to a given model.
Nevertheless, we shall see below why this is still
not enough to legitimate its appropriation of the
concept of model.
system and structure
The three prerequisites for the concept of model
are: (1) a formal system comprising a (finite) set of
symbols (logical operators, individual constants
[a, b, c . . .], predicates [P, Q, R . . .], individual
variables [x, y, z . . .]) þ rules of formation þ rules
of deduction þ a list of axioms; (2) a structure
which provides the domain of interpretation for
the system, and which is defined as a non-empty
set V comprising a list of individual elements
and a list of subsets þ two supplementary
marks ‘‘True’’ and ‘‘False’’; (3) a correspondence
function mapping individual constants of the
system to some element of V, and predicative
constants of the system to some subset of V.
With regard to the above, there are three
important remarks to be made. First, the use of
the concept of set is absolutely decisive. The vague
notion of a ‘‘domain of objects’’ is dangerously
equivocal and tainted with empiricism. Only if the
notion of interpretative domain is cashed out in
terms of the mathematical concept of set can the
concept of model be scientifically articulated.
The scientific status of ‘‘semantics,’’ and hence
of the concept of model, depends upon the former
being established within an existing branch of
mathematics, so that the rules governing the
interpretation of a formal mathematical system
are formulated within a (non-formal) branch of
mathematics itself.
Second, the stipulation that the list of formal
symbols comprised by the system be finite means
that they should be denumerable using natural
whole numbers. This is another indispensable
requirement for the construction of the concept
of model. Every well-formed expression of the
system should consist of a denumerable or, in the
case of most systems, finite series of indecomposable symbols. It is a condition for the scientific
theory of models that the formal language of the
system not be continuous. Just as the recourse
to set-theory proved necessary for the scientific
characterization of semantics, there is an unavoidable recourse to the mathematics of whole
numbers (and to recursive arithmetic in particular) in the conception of syntax. The deployment
of a scientific concept of model necessarily
presupposes the existence (or ‘‘validity’’) of
these mathematical practices. This explains
the remark which served as the first of our
two epigraphs by Badiou: ‘‘One establishes oneself
in science from the start. One does not reconstitute
it from scratch. One does not found it.’’
Third, it is important to note the intrasyntactical distinction between the system’s logical
and mathematical axioms. A logical axiom is one
whose functioning depends solely on the logical
connectives which figure in it and remains
unaffected by substitution of the fixed constants
(individual or predicative) contained in it. A
mathematical axiom is one which singularizes at
least one of the fixed constants that figure in it by
separating it from at least one other; thus it is
sensitive to their substitution.
Given these prerequisites, the construction of
the concept of model proceeds in the following
way. Using the set-theoretical resources of the
structure and the correspondence function , one
defines the validity and invalidity of a well-formed
expression of the system relative to the structure.
One then specifies the conditions under which a
particular structure is a model for the system by
establishing a relation between syntactic deducibility (i.e., the fact that an expression A is a
theorem of the system) and semantic validity (i.e.,
the fact that A is valid for a structure, or a
particular type of structure, or even any structure
whatsoever). A ‘‘closed instance’’ of the expression
A is an expression of the type A(a/x) (b/y) (c/z),
wherein all of A’s free variables have been replaced
by constants. The following definition of ‘‘validity’’ can then be proposed: an expression A in the
system is valid for a structure if, for every closed
instance A0 of A, one obtains A0 ¼ True for that
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structure. More particularly, a closed expression A
is valid if A ¼ True since it has no other closed
instances apart from itself (nothing in it is
substitutable).39 One can then use this definition
of validity to demonstrate that if the axioms of a
system are valid, all the theorems of that system
are also valid. In effect, since a deduction begins
with an axiom, and subsequently comprises
nothing but axioms or expressions derived from
previous expressions via the application of the
rules of deduction, then, if the axioms are valid,
every expression used in the deduction is also
valid. Thus the correspondence function which
sustains the procedures of evaluation allows us
to infer from the syntactic concept of deducible statement (theorem) the semantic concept
of statement-valid-for-a-structure. This allows
Badiou to propose the following definition of
model: ‘‘A structure is the model of a formal
theory if all the axioms of that theory are valid for
that structure.’’40
the logical and the mathematical
The distinction between logical and mathematical
axioms can be characterized semantically in terms
of the scope of their respective validities: whereas
logical axioms are valid for every structure, a
mathematical axiom is valid only for particular
structures. Thus, from a semantic point of view,
logic is equivalent to the ‘‘systematicity’’ of
structure as such, whereas mathematics is equivalent to the theory of the types of structure.41 But
this is not to say that logic enjoys some putatively
‘‘trans-historical’’ status as condition of possibility
for mathematical rationality as such; or that it must
always already be there as the condition for, rather
than the result of, the history of reason. In order to
overcome this dichotomy between logicist transcendentalism and historicist relativism, Badiou
suggests that logic itself be conceived as doubly
articulated between syntactic system and semantic
structure. The opposition between history and a
priori is circumvented by the relation of reciprocal
presupposition between the logical practice inherent in every semantic demonstration and the
experimental construction of particular logical
systems. Thus the ‘‘trans-historicity’’ of logic can
be scientifically accounted for in terms of the
143
experimental property whereby a purely logical
system, all of whose axioms are logical, contains no
semantic indication of its models. Since every
structure is a model for such a system, the concept
of model is not logically discernible from that of
structure for it. Accordingly, it is mathematical
axioms that suspend this semantic indiscernibility
of logical structure by effectuating the inscription
of a structural gap between syntactic system and
semantic structure; a gap within which the concept
of model comes into play. Thus the concept of logic
neither transcends nor subsumes mathematics; it
remains inseparable from the couple which it
forms with the latter. The contrast between the
logical and the mathematical is a syntactical
redoubling of the semantic distinction between
structure and model. Given two structures whose
difference is indexed by the fact that one of them is
a model for a given formal system while the other is
not, it becomes possible to classify the axioms of
the system into those that are logical and those that
are mathematical. The former index the unity of
system and structure while the latter index their
difference.42 Thus ‘‘[a] model is the mathematically constructible concept of the differentiating
power of a logico-mathematical system.’’43
experimentation and demonstration
The double occurrence of the term ‘‘mathematics’’
in the formulation quoted above indexes the way
in which the means of mathematical production,
namely the conjunction of experimentation and
demonstration, are themselves mathematically
produced. This double articulation cannot be
transplanted outside of mathematics or duplicated
in the relation between a supposedly formal theory
and its putatively material instantiation. Thus,
given the manner in which the construction of
the concept of model depends at every step on
the double articulation between two particular
branches of mathematics, namely recursive
arithmetic and set-theory, it is profoundly
misleading to claim, as some philosophers do,
that the concept of model indexes formal thought’s
relation to its (empirical/material) ‘‘exterior.’’
Extra-systemic (structural) inscriptions are only
capable of providing a domain of interpretation for
those of the system according to a mathematical
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badiou’s materialist epistemology
envelopment which subordinates the former to the
latter. Syntax is an arithmetical discipline; semantics a set-theoretical one.44 The theory of the
apparatuses of inscription conceived as mathematical objects is arithmetical. It allows one to engage
in an ordering and inductive numbering of the
experimental set-up; to evaluate its power and
complexity through reasonings bearing on the
structure of inscription which the system either
allows or prohibits. By way of contrast, the theory
of the usage of these apparatuses, conceived as
experimental operations, classifies the regions of
the mathematical material which is to be processed
by the apparatus; this is the aim of the concept of
structure, which is itself produced by the most
general, most all-enveloping mathematical theory
produced thus far: set-theory.45
The sole basis for the unity-in-difference
of syntax and semantics lies in the intramathematical relation between the arithmetical
material and the set-theoretical material. Any
attempt to export the concept of model outside
the mathematical realm violates this necessarily
intra-mathematical relation and is thus illegitimate. In this regard, logical empiricism’s attempt
to export the mathematical concept of model into
epistemology is doubly illegitimate. First, because
it tries to theorize science in general on the basis of
a difference between syntax and semantics that is
merely an ideological distortion of the regional and
intra-mathematical distinction between recursive
arithmetic and set-theory. Second, because its
conception of empirical facts as models for
formal theories is merely an analogon for the
intra-mathematical correspondence between
system and structure and fails to register the
crucial respect in which the modelling of a formal
system constitutes a means for experimenting
upon and ultimately transforming the rigour or
generality of that system. This perpetual labour
of intra-mathematical transformation becomes
inoperative in cases where the putative domain
of interpretation is not already mathematical
and hence semantically assignable as capable of
corresponding to a syntactical apparatus.
Thus, contrary to what the ideological appropriation of the theory of modelling suggests,
scientific practice does not proceed from formal
theory to concrete model, or from system to
structure, but from structure to system, from
model to theory and back again in ceaseless
dialectical interplay. The historical reality of
mathematical production does not confront us
with the challenge of testing the theory through
the model, but of testing the model by means
of the theory. The problem faced by actually
existing mathematical practice, as opposed to its
epistemological idealization, consists in specifying
the formal theory modelled by the historical
plurality of structures, in identifying the appropriate syntactic signature – i.e., the formal theory –
for a given type of structure. This is the problem of
mathematical formalization. If ‘‘[s]emantics is an
experimental protocol,’’46 this is not, Badiou
insists, because a model is an experimental realization of a formal system, but because, on the
contrary, it is a structure embodying a conceptual
demonstration whose experimental verification is
carried out by means of inscription in a formal
syntax. The formal system is retrospectively
constituted as the experimental moment, the
material linkage of verification, in the wake of the
conceptual demonstration articulated in the model.
Thus formalized syntaxes are materialized theories:
means of mathematical production just like the
vacuum tube or particle accelerator for physics: ‘‘It
is precisely because it is itself a materialized theory,
a mathematical result, that the formal apparatus is
capable of entering into the process of the production of mathematical knowledge; a process in which
the concept of model does not indicate an exterior
to be formalized but a mathematical material to
be tested.’’47 It is the system that is formalized
by means of the demonstration provided by
the model. Formalization proceeds through
the experimental verification of the conceptual
demonstration provided by the model. The latter
provides the material to be tested by its inscription
in a formal system:
[T]he philosophical category of effective procedure, of what is explicitly calculable through
a series of unambiguous scriptural manipulations, lies at the heart of all mathematical
epistemology. This is because this category
distils the properly experimental aspect of
mathematics, that is to say, the materiality of
its inscriptions, the montage of notations
[. . .] Mathematical demonstration is tested
144
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[s’éprouve] through the rule-governed transparency of inscriptions. In mathematics, inscription represents the moment of verification.48
Thus the formal system provides the instrument
of experimental inscription required in order to
verify the conceptual demonstration deployed
in the structure of the model; but this verification
in turn becomes a means of formalization.
Demonstration and formalization are bound
together in a material dialectic of reciprocal
presupposition. It is this dialectic, and the
mechanism of inscription upon which it depends,
that lie at the heart of Badiou’s claims about
the historical materiality of mathematical
production.
iv the historicity of mathematical
production
Badiou’s reconstruction of the concept of model
provides the basis for the construction of a
category of model which is to be mobilized
within a dialectical-materialist account of the
historicity of scientific practice. Badiou is explicit
about the structure of his argument: there can be
no question of using the concept of model as the
basis for a theory of mathematical historicity,
which would amount to a transparently ideological
misappropriation of a scientific concept; rather, it
is a question of deploying an explicitly materialist
category of model on the basis of a theory of
the historicity of mathematical science already
implicit in Badiou’s preliminary critiques of the
notional uses and abuses of the concept of model.
The ‘‘materiality’’ of mathematical practice is
not to be understood as an analogue of the
inexistent philosophical category ‘‘matter,’’ but
rather as an index of the scriptural production of
difference. This account of scriptural materiality
is, so to speak, the esoteric subtext of Badiou’s
materialist epistemology of science. It has to be
reconstructed on the basis of various suggestive
but elliptical hints scattered throughout Concept
of Model and another roughly contemporaneous
(i.e., 1967–69) text, the extraordinary ‘‘Mark and
Lack: About Zero.’’49 At the conclusion of the
latter, Badiou writes: ‘‘Science is the veritable
archi-theatre of writing: traces, crossed out traces,
traces of traces; movement wherein there is not the
145
slightest prospect of recentering the detestable
figure of Man: the sign of the nothing.’’50
Badiou’s recurrent emphasis on the materiality
of logico-mathematical inscription in these early
epistemological writings seems to effect a critical
conjunction of Lacan’s theses about the agency of
the letter and Derrida’s claims about the disseminatory force of archi-writing. Logico-mathematical
inscription circumvents the metaphysical primacy
of the linguistic signifier via a ‘‘stratified multiplicity’’51 of differential traces which ‘‘no signifying order can envelop’’;52 one which pulverizes the
presence of the object – ‘‘Neither the thing nor the
object has any more existence here than has their
traceless exclusion’’53 – and dissolves the unity of
the subject:
[T]here is no subject of science. Infinitely
stratified, adjusting its transitions, science is a
pure space, without a reverse or mark or place
of what it excludes. It is foreclosure, but
foreclosure of nothing, and so can be called
the psychosis of no subject,54 hence of all; fully
universal, shared delirium, one only has to
install oneself within it to become no-one,
anonymously dispersed in the hierarchy
of orders. Science is an Outside without a
blind-spot.55
Here, we encounter all the essential features of
Badiou’s Platonist materialism: scientific thought
is outside, beyond the enclosure of ideological
representation; not because the subject of science
enjoys intuitive access to a realm of transcendent
objects, but on the contrary, because the remorselessly mechanical ‘‘rule governed transparency’’
of logico-mathematical inscription dissolves the
consistency of the object and the coherence of the
subject in the infinitely stratified multiplicity of
scientific discourse. The immanent autonomy of
scientific discourse, its non-representational character, is a consequence of its machinic nature,
since ‘‘[a] formal system is a mathematical
machine, a machine for mathematical production,
positioned within that production.’’56 But
the means of mathematical production are themselves produced; the mathematical machine
or instrument is also a mathematical product,
a result: there would be no formal systems
without recursive arithmetic, and no rigorous
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badiou’s materialist epistemology
experimental protocols for such systems without
set-theory.
Yet it is because this scientific reproduction of
the means of production harbours a constitutive
historicity that science’s self-reproduction is inherently differential. Or rather, it is the inherently
differential (dialectical) nature of scientific
reproduction that generates its historicity. The
perpetual dialectic of demonstration and experimentation is the motor of scientific history.
Scientific re-production is self-differentiating
because of the way in which science itself intervenes within a determinate epistemological
conjuncture by means of formal experimentation.
Thus, for example, by proving the consistency of
a model of axiomatic set-theory with the Axiom
of Choice and the Continuum Hypothesis, Gödel
demonstrates that these two axioms can be
integrated into the formal theory without compromising its coherence. He thereby provides a
conceptual sanction for mathematical practice:
‘‘In doing so, [Gödel’s experimentation] transforms, not the theory, but the status of the theory
within the historical process of the production of
knowledges.’’57 Given a mathematical configuration inscribed within the history of that science, to
treat it as a model of a formal system is to situate
its specificity by transposing it beyond the narrow
ambit of the spontaneous illusions engendered by
its singular production and into the wider mathematical space constituted by the various models of
the system. Consequently, the experimental apparatus is a nexus of practices. The double articulation of formal experimentation and conceptual
demonstration becomes the driving force for
science’s own epistemic interventions within
determinate historical configurations. In the
history of a science, the experimental transformation of practice via a determinate formal apparatus
retrospectively assigns the status of model to those
antecedent instances of practice. Conversely,
conceptual historicity, which is to say the ‘‘productive’’ value of formalism, derives both from its
theoretical dependency as an instrument and from
the fact that it possesses models, i.e., that
it is integrated into the conditions of the production and reproduction of knowledge: as Badiou
states: ‘‘[s]uch is the practical guarantee of formal
set-ups.’’58
For Badiou, then, the materialist category of
model designates formalism’s own retrospective
causality upon its own scientific history, which
conjoins an object (a model) and a usage (a
system). The historicity of formalism consists in
the ‘‘anticipatory intelligibility’’ of what it retrospectively constitutes as its own model.59
Ultimately, the fundamental epistemological
problem is not that of the nature of the representative relation between the model and the
concrete, or between the formal and the model;
rather, ‘‘[t]he problem is that of the history of
formalization.’’60 The materialist category of
model proposed by Badiou designates the meshwork of retroactions and anticipations from which
the history of formalization is woven; the history
of its anticipatory cuts and its retrospective
reconfigurations.61 The historicity of scientific
(re-)production is constituted by this differential
meshwork of epistemic cuts and reconfigurations.
Thus there is no need to invoke empirically
arbitrary, para-theoretical ‘‘paradigm shifts’’ to
account for the structural discontinuities that
punctuate scientific history. Discontinuity is
already inherent in the immanent conceptual
mechanisms of scientific practice, for ‘‘science is
precisely that which is ceaselessly cutting itself
loose from its own indication in re-presentational
space [i.e., ideology].’’62 This is the key to understanding the second of our introductory epigraphs:
‘‘[T]here are no crises within science, nor can
there be, for science is the pure affirmation of
difference.’’
Contra naturalism, the ‘‘science’’ within
which Badiou recommends thought establish
itself in spite of the otiose prevarications of
transcendentalism cannot be mistaken for its
empiricist representation or conflated with an
ambient scientific worldview, a diffuse ideological distillate synthesized from various
scientific disciplines (such is the composition
of Neurath’s boat); rather, it is an entirely
autonomous,
ceaselessly
self-differentiating
mode of theoretical practice
invariably defined by a
specific historical conjunction
between conceptual demonstration and formal experimentation.
146
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notes
1 Le Concept de modèle. Introduction à une
épistémologie matérialiste des mathématiques
[The Concept of Model: Introduction to a
Materialist Epistemology of Mathematics] (Paris:
Maspero, 1969) 42. It is important to bear in mind
that, in the context in which Badiou is writing here
(i.e., the context defined by the work of Althusser,
Bachelard, and Canguilhem),‘‘epistemology’’ refers
to the ‘‘theory of science,’’ and not the ‘‘theory of
knowledge’’ as commonly understood in AngloAmerican academic philosophy. According to the
latter, epistemology is concerned principally with
problems concerning the nature of rationality,
belief, truth, scepticism, etc. ^ problems whose
philosophical scope far exceeds that of ‘‘philosophy
of science’’ proper. Thus the question about ‘‘how
science represents the world’’ is subsumed by the
larger issue concerning the precise nature of the
epistemic relation between mind and world. But
for Badiou, as well as for Althusser, Bachelard,
and Canguilhem, such questions cannot be the
concern of epistemology proper since they
remain fatally enmeshed in the empiricist prejudices of representationalism, which can only
obstruct proper philosophical understanding of
scientific theory and practice. One suspects that
further investigation into the deeper conceptual
ramifications harboured by this seemingly trivial
nominal difference would go a long way towards
explaining the fundamental philosophical divergence between the concerns of post-Bachelardian
‘‘epistemology’’ in France and those of AngloAmerican ‘‘philosophy of science.’’
2 ‘‘Marque et manque: A propos de ze¤ro’’ [Mark
and Lack: About Zero], Cahier pour l’analyse no.
10 (1969) 165. Badiou’s article is actually dated
‘‘January 1967,’’ though it was not published until
1969.
3 ‘‘Le (Re)Commencement du mate¤rialisme dialectique,’’ Critique no. 240 (1966): 438 ^ 67.
Ostensibly a review article focusing on three
works (Althusser’s Pour Marx (Paris: Maspero,
1965) and ‘‘Mate¤rialisme dialectique et mate¤rialisme historique,’’ Cahiers Marxistes-Léninistes no.
11 (Apr. 1966); and Lire Le Capital by Althusser,
Balibar, Establet, Macherey, and Rancie¤re, 2 vols.
(Paris: Maspero, 1965)), this early piece not only
provides a magisterial critical overview of the
Althusserian project but is also a powerfully original philosophical intervention in its own right.
147
4 L’Être et l’événement (Paris: Seuil, 1988). An
English translation by Oliver Feltham is due to be
published by Continuum in 2006.
5 Logiques des mondes (Paris: Seuil, 2005). For an
introduction to the latter, see the ‘‘Logics of
Appearance’’ section in Badiou’s Theoretical
Writings, eds. R. Brassier and A. Toscano
(London: Continuum, 2004) 163^231.
6 Badiou, ‘‘Le (Re)Commencement du materialisme dialectique’’ 464.
7 ‘‘Politics and Philosophy: An Interview with
Alain Badiou,’’ Angelaki 3.3 (1998) 127.
8 Cf. Being and Event, Introduction (Paris: Seuil,
1988) 7^27.
9 Alain Badiou,‘‘Mathematics and Philosophy: The
Grand Style and the Little Style’’ in Theoretical
Writings, eds. R. Brassier and A.Toscano (London:
Continuum, 2004) 16.
10 Daniel Smith provides a sophisticated variant
of this particular criticism whilst delivering a
sharp rejoinder to Badiou’s reading of Deleuze
in ‘‘Badiou and Deleuze on the Ontology of
Mathematics’’ in Think Again: Alain Badiou and
the Future of Philosophy, ed. P. Hallward (London:
Continuum, 2004) 77^93. Smith elaborates on the
Deleuzian distinction between ‘‘royal’’ and ‘‘minor’’
(or ‘‘nomad’’) science by developing a highly illuminating contrast between the ‘‘axiomatizing’’ and
‘‘problematizing’’ tendencies in mathematical practice. He then uses this distinction as an interpretative prism through which to correct what he takes
to be Badiou’s misreading of Deleuze, and criticizes
the former for his narrowly ‘‘royalist’’ conception
of science and his exclusively ‘‘axiomatic’’ characterization of mathematics. Yet although this
distinction between axiomatics and problematics
^ partially rooted in the work of Albert Lautman
^ is undoubtedly of considerable philosophical
significance, it is doubtful that it can be used to
defend Deleuze against Badiou in the way that
Smith attempts to do here. One cannot help
suspecting that the distinction between ‘‘problematics’’ and ‘‘axiomatics,’’ like that between
‘‘minor’’ and ‘‘royal’’ science upon which it is based,
is merely a reiteration (rather than an independent
conceptual legitimation) of the Deleuzian distinction between intensive and extensive multiplicities
(or open and closed, smooth and striated, virtual
and actual, etc.); the fundamental distinction
around which Deleuze’s entire philosophy is
badiou’s materialist epistemology
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coordinated but whose necessity Badiou ^ regardless of the undeniable infelicities in his reading
of Deleuze ^ is surely entitled to call into question. Invoking the rights of problematics against
axiomatics in order to defend the necessity of the
distinction between intensive and extensive multiplicities will prove to be of no avail if the former
distinction turns out to be a restatement of the
latter.
11 Indeed, there seem to be legitimate grounds for
claiming that ‘‘philosophy of science’’ as we know it
was spawned by a particular philosophical doctrine: logical empiricism. Thus in his contribution
to the MIT anthology The Philosophy of Science,
co-editor Richard Boyd declares:
Almost all work, foundational or applied, in
English-language philosophy of science
during the present [twentieth] century has
either been produced within the tradition of
logical empiricism or has been written
in response to it. Indeed it is arguable
that philosophy of science as an academic
discipline is essentially a creation of logical
empiricists and (derivatively) of the philosophical controversies it sparked. (Richard
Boyd, ‘‘Conf|rmation, Semantics, and the
Interpretation of Scientif|c Theories’’ in
The Philosophy of Science (Cambridge, MA:
MIT P,1991) 3)
12 Badiou, Le Concept de modèle 12. The Concept
of Model originated as Badiou’s contribution to
a seminar series set up by Althusser entitled
‘‘Cours de philosophie pour scientifiques’’
[Course in Philosophy for Scientists] at the E¤cole
Normale Supe¤rieure during the 1967^ 68 academic
year. Althusser’s first four lectures in the series,
to which Badiou is alluding here, were entitled
‘‘Philosophy and the Spontaneous Philosophy of
the Scientists’’ and can be found in his Philosophy
and the Spontaneous Philosophy of the Scientists
and Other Essays (London: Verso, 1990) 69^144.
Althusser’s fifth lecture in the same series was
entitled ‘‘Du cote¤e de la philosophie’’ [On the Side
of Philosophy] and published posthumously in
vol. II of his Écrits philosophiques et politiques
[Philosophical and Political Writings] (Paris: Stock/
IMEC, 1997) 265^308. Badiou may also be alluding
to a passage here that occurs on page 277.
13 Here we have an example of how a thesis which
has been central to Badiou’s work from the beginning is not stated explicitly until much later.Thus it
is not until 1998’s ’’Platonism and Mathematical
Ontology’’ (in Theoretical Writings 49^58) that
Badiou explains how his conception of Platonism
subverts the basic distinction between thought
and object, which in Husserlian phenomenology
is given a more subtle characterization in terms
of the correlation between ‘‘noesis’’ and ‘‘noema.’’
14 The characterization of historical materialism
as a ‘‘science’’ of history is obviously contentious,
particularly in light of Badiou’s implicit identification of mathematics with scientificity; moreover,
it is one to which I believe he no longer subscribes,
but I shall not question it here. Be that as it may,
Badiou’s emphasis on the specific mode of conceptual ‘‘productivity’’ which he discerns in mathematical practice is obviously tied to the central
role of productivity in historical materialism.
15 Badiou, Le Concept de modèle 62.
16 Badiou gives three examples: the Platonic
category of ‘‘ideal number’’ denotes an inexistent
‘‘adjustment’’ between arithmetical concepts and
hierarchical moral-political notions; the Kantian
categories of ‘‘space’’ and ‘‘time’’ combine
Newtonian concepts with notions that are relative
to human faculties; and the Sartrean category of
‘‘History’’ combines Marxist concepts with
metaphysico-moral notions such as temporality,
freedom, etc. Regarding the second of these
examples, it should go without saying that Badiou
is using the term ‘‘category’’ here in a sense
entirely distinct from Kant’s and is perfectly well
aware that for Kant space and time are ‘‘forms of
intuition’’ rather than ‘‘categories.’’
17 As we shall see, the true philosophical index of
‘‘materiality’’ in this conjunction between historical and dialectical materialism is that of the
‘‘productivity’’ of a given theoretical practice. As
Badiou states: ‘‘The reality of the epistemological
materialism which I am trying to introduce here is
indissociable from an effective practice of science’’
(Le Concept de modèle 29).
18 Badiou, Le Concept de modèle 12.
19 The use of the term ‘‘presence’’ here is perhaps
intended as an allusion to Derrida’s work, with
which Badiou was certainly already familiar (cf.,
for instance, Badiou, ‘‘Le (Re)Commencement du
mate¤rialisme dialectique’’ 445); one possible implication being that the deconstruction of logocentrism can be enlisted as part of the critique of
empiricist epistemology.
148
brassier
20 Badiou, Le Concept de modèle 9.
28 Ibid. 44.
21 Reprinted in The Philosophy of Science, eds.
R. Boyd, P. Gasper and J.D.Trout (Cambridge, MA:
MIT P,1991) 393^ 404.
29 Ibid. 44.
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22 Carnap is careful to distinguish between unification couched in terms of the logical reducibility
of terms, which he endorses, and unification
understood as the derivation of the laws of
one science (e.g., biology) from those of
another (e.g., physics), about which he expresses
reservations:
[T]here is a common language to which both
the biological and the physical laws belong
so that they can be logically compared and
connected. We can ask whether or not a
certain biological law is compatible with the
system of physical laws, and whether or not
it is derivable from them. But the answer to
these questions cannot be inferred from the
reducibility of the terms. At the present
state of the development of science, it is
certainly not possible to derive the biological
laws from the physical ones. Some philosophers believe that such a derivation is forever
impossible because of the very nature of
the two f|elds. But the proofs attempted so
far for this thesis are certainly insuff|cient.
(Carnap, ‘‘Logical Foundations of the Unity
of Science’’ in The Philosophy of Science,
eds. R. Boyd, P. Gasper, and J.D. Trout
(Cambridge, MA: MIT P,1991) 403)
23 In From a Logical Point of View, 2nd ed.
(Cambridge, MA: Harvard UP,1980) 20 ^ 46.
24 Willard Van Orman Quine, ‘‘Two Dogmas of
Empiricism’’ in ibid. 22.
25 Ibid. 43.
26 Donald Davidson, ‘‘On the Very Idea of a
Conceptual Scheme’’ in Enquiries into Truth and
Interpretation (Oxford: Clarendon, 1984) 183^98.
Although I cannot do so here, it would be instructive to compare and contrast Davidson’s critique
of Quine’s scheme/content dualism with Badiou’s
critique of empiricism. Despite the semblance of a
shared antipathy to empiricism, Badiou’s definition
of the latter is wider ranging than Davidson’s and
I suspect the latter’s work would still seem all
too empiricist to Badiou.
27 Quine,‘‘Two Dogmas of Empiricism’’ 45^ 46.
149
30 Badiou, Le Concept de modèle 11.
31 A conjunction which birthed cognitive science,
and whose most distinguished contemporary
representative is arguably Daniel Dennett.
32 See, for example, Paul Churchland, A
Neurocomputational Perspective: The Nature of
Mind and the Structure of Science (Cambridge,
MA: MIT P, 1989), or more recently, Jean-Pierre
Changeux, The Physiology of Truth: Neuroscience
and Human Knowledge (Cambridge, MA: Harvard
UP, 2004).
33 Badiou, Le Concept de modèle 21.
34 Badiou examines this fundamental formal
requirement at length in the appendix to Concept
of Model 69^90.
35 In his famous theorem of 1931, Go«del demonstrated the incompleteness of the formal system
of arithmetic, i.e., of a formal system capable
of being modelled by ‘‘classical’’ or recursive
arithmetic, by showing how its model contains a
true statement for which there is no corresponding deducible theorem in the system. Thus a
system may be consistent but incomplete, or
complete but inconsistent, but it cannot be both
consistent and complete. Cf. ‘‘On Formally
Undecidable
Propositions
of
Principia
Mathematica and Related Systems’’ reprinted in
Frege and Gödel: Two Fundamental Texts in
Mathematical Logic, ed. Jean Van Heijenoort
(Cambridge, MA: Harvard UP,1970) 87^107.
36 Cf. Badiou, Le Concept de modèle 24 ^25.
37 Cf. ibid. 25.
38 Rudolf Carnap, Meaning
(Chicago: U of Chicago P,1956).
and
Necessity
39 Cf. Badiou, Le Concept de modèle 41.
It should be noted that this procedure is
constructed by means of recurrence over
the ‘‘length’’ of expressions, i.e. over the
number of symbols which constitute them.
One begins with elementary expressions of
the type P(a), which are directly evaluated
in the structure, by examining the eventual
belonging of a’s semantic ‘‘representative’’ to
the subset of the universe represented by
P. One then adjusts the procedure which
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badiou’s materialist epistemology
allows one to evaluate an expression A on
the basis of the (supposedly acquired) evaluation of the shorter expressions contained in
A, or contained in its closed instances. Thus
the evaluation of B is carried out on the
basis of the evaluation of B, while that
of (Ex)B is carried out on the basis of
B(a/x), etc. The conviction that these rules
guarantee the existence of an evaluation for
an expression of any length whatsoever
amounts to admitting the legitimacy of
reasoning by recurrence over whole
numbers (in this case, over the number of
symbols that enter into the composition of
an expression). (Badiou, Le Concept de
modèle 42)
56 Badiou, Le Concept de modèle 54.
57 Ibid. 64.
58 Ibid. 67.
59 Ibid. 67.
60 Ibid. 68.
61 Ibid. 68.
62 Badiou,‘‘Marque et manque’’ 165.
40 Ibid. 44.
41 Ibid. 44 ^ 45.
42 Ibid. 47^ 48.
43 Ibid. 52.
44 Ibid. 55.
45 Ibid. 55^56. Significantly, Badiou also mentions
category-theory here as a potential rival to settheory in terms of all-enveloping generality.
46 Badiou, Le Concept de modèle 48.
47 Ibid. 58.
48 Ibid. 34.
49 Cf. n. 2 above. Not least among this text’s
many extraordinary features is its remarkably illuminating analysis of Go«del’s famous incompleteness theorem, and its penetrating critique of
certain popular philosophical misinterpretations
of Go«del’s work.
50 Badiou,‘‘Marque et manque’’ 164.
51 Cf. ibid.
52 Ibid.163.
53 Ibid.156.
54 An allusion to Jacques-Alain Miller’s claim that
‘‘[E]very science is structured like a psychosis’’ in
‘‘L’Action de la structure’’ [The Action of
Structure], Cahiers pour l’analyse no. 9 (1968);
reprinted in Miller’s Un début dans la vie (Paris:
Le Promeneur, 2001) 57^79.
55 Badiou,‘‘Marque et manque’’ 162.
Ray Brassier
Centre for Research in Modern European
Philosophy
Middlesex University
Trent Park
Bramley Road
London N14 4YZ
UK
E-mail: ray.brassier@btopenworld.com