domingo, 6 de febrero de 2011

Matter Matters: In Dialog With Levi Bryant


- In Dialog With Levi Bryant -

   Levi wrote a wonderfully informative reply to my earlier, rather naïve inquiry into his philosophy. He begins by clarifying further his conception of knowledge in contrast to what he deems to be ‘an obsession with how we represent the world’. It is the latter, theoretically dubious obsession, which renders the demand of construing knowledge around the expectation of ‘adequation to the real’, well... inadequate.

 It does so for several reasons, in Levi’s account. Chief among these, he argues that:

a)      Such theories assume a representational framework in which propositions are made to correspond to states of affairs in the world.
b)      Such a propositional account places an emphasis on results, i.e. on establishing the conditions under which it can be said that a proposition p is true or false.

  Accordingly, the non-representationalist model advocated by Levi would be unassuming of these premises, and claim that, on the other hand:

c)      An alternative account of knowledge does not have to run the gauntlet of epistemological agnosticism about the real.
   The vision Levi wants to endorse is centered on experimentation understood as a productive practice. The real that such an account will enter in relation is construed in Levi’s philosophy in terms of the endogenous, virtual structure of the objects which populate the world, and their relations. Therefore, the earlier accusation that any linguistic category could coin the real virtual objects withdrawing from relations as apprehended in local manifestations can be assuaged if such an alternative account can specify knowledge about or of the real otherwise. Here is where Bhaskar and the question of scientific experimentation do some constructive work. Levi writes:

“I was very careful to emphasize experimentation, yet reference to experiment appears nowhere in Daniel’s post. Rather, all that we see in Daniel’s post is reference to propositions and questions of how they hook on to the world. And indeed, if we don’t examine actual engagement with the world, then the question of how propositions hook on to the world turns out, in many instances, to be deeply mysterious. It comes to seem as if any proposition is as good as another (that there’s no criteria for “degrees of adequation”).”

     It is true in that I was not sufficiently attentive in acknowledging the proposed distinction between an account of knowledge based on scientific practice and one based on a representational theory. I would only note, provisionally ,that I am not sure how Levi cashes out this distinction when saying that propositional accounts focus only on results, i.e. premise (b) above. It seems true enough that the idea of adequation suggests that there are discursive entities which express real properties.  Fine. But this does not mean, surely, that propositional accounts must be descriptively impervious to experimentation, to process and production. An account of representation need not preclude considerations about practice; or at least it is not obvious that or how they do, according to Levi. For we could be asking about the epistemic status of propositions which express a theory of knowledge centered on practice, such as Levi's or Bhaskar. Just like practice seemed missing from my account, Truth seems conspicuously missing from his. He speaks about 'the truth of a theory by putting it to work and seeing what it does'. But it remains ambiguous whether he means that truth is a property of theories, or a feature of the propositions of those theories. If there is a well defined sense in which we can speak of theories being true, then that seems just fine. But if truth remains circumscribed to the propositional ambit, then it seems perfectly fine to ask how propositional statements made by science relate to what they designate or refer to. If Levi wants to reduce truth to the 'conditions of verifiability of a proposition', cornering us into a representational account of science focused on results, I profoundly disagree. Set-theoretical ontology, for example, can perfectly accomodate Truth as a category apart from representation, while salvaging the latter and knowledge around the notion of veracity. It certainly does not follow that asking for conditions of situation and state-of-the-situation, presentation-representation correspondence, one necessarily obviates a theory of practice. On the contrary, representation works in terms of a situation and its State, presentation and its parts. Representation picks out those parts which may be separated by the axiom of separation in purely indistinct multiplicities, without any conceptual baggage bringing propositions into the story, nor those of a subject of representation. Indeed, like the structuralist tradition holds following Althusser, the individual is product of the structure; which in set theoretical ontology is described in terms of an individual being nothing more than a multiple of the situation. Representation is thus formally isolated from Truth qua production, as the latter is seen in the form of the construction of generic subsets, and the forcing of the situation's generic extension through various processes of verification in a trajectory with no representative name, i.e. the generic is indiscernible.

It seems that Levi thinks that asking about how we know the real conforms to descriptions is to be asking a question about propositions made from a 'linguistic bias'. But why should this be so? Even if practice is primary for Levi, one can still raise questions about the status of scientific statements within those theories. And as far as these statements purportedly describe or relate to relations to real objects in the world, it seems perfectly fine to ask how these are articulated in relation. This doesn't imply we must construe the problem around features of propositions and their external counterparts (as Badiou's account makes clear, for example) and it surely doesn't suffice to make discursive correspondence trivial. If the latter is cashed through the relation between a situation (set A) and its state (set S(A)), it can be perfectly specified without making any appeal to linguistic entities, or mysteriously pulsating external worlds. My question ultimately concerns for Levi the inscription of local manifestations when they are discursively formulated  and as they purport to be indexes of features about withdrawing virtual objects
The question would be only to try and get clear on what scientific experimentation is said to involve and how it allows this knowledge to obtain.

Put directly: what precise status these theories and models, their formulas, syntactical structures and theorems, bear on the phenomena they purport to describe? What is the relation between the production of domains of phenomena which is relative to scientific practice, and the phenomena themselves? What are these phenomena, which presume a kernel of givenness to an experimenting agent, supposed to be? Representations of a subject; a transparent influx of the world? The question of practice needs to be explained in terms of a relation between elements which enter into a distinctively ontologized role. What is finally the status of such scientific statements produced in this practice vis a vis the Real which they are said to model in domains if not analogy-resemblance-identity-contrariness, all of which articulate the fourfold of representation Levi abjures?   This will turn out to be a tricky issue, as we will see below.

    Levi has deepened his account, following Bhaskar and Pickering with care. The adherence to Bhaskar concerns the question of how we come to know the real as externally existing, and expresses several aspects of scientific experimentation in grounding externalism: out-of-phasing, differentiation, structure, stratification, etc. Pickering, on the other hand, explains how in broader terms scientific theory expands itself through practice by modeling new domains of phenomena, taking materially productive research into multiple and unforeseen directions.

    However, and most importantly, the actual development of this process is highly contingent on a set of experimental practices which are not reducible to what is merely expressed in the theory's results, i.e. in the actual statements or expressions presented through theories and models in their writing. The use, application and transformation of science implies, as  Pickering endorses, that  “…the construction of theoretical models is not sufficient to describe what takes place in the science and why what takes place takes place”.

   Thus, nothing tells us a priori how scientific theories will expand, or fail to do so. The material production of science is highly contingent in its becoming, and so the models produced obey no pre-established route,  no a priori principle to determine the course of their trajectory or successful application. In the context of laboratory settings, instrumental and engineering demands, and the laboring practice of scientists, we find nevertheless that the objects before us frequently surprise us or resist us. The becoming of a theoretical model is expressed thereby in terms of modeling vectors which index differential strata, and which in turn presuppose a process encountering an objective resistance from (external) material conditions. Resistance, therefore, becomes an index of the real,  as expressed in scientific experimentation in Levi’s words:

     “The claim that resistance is an index of the real is to be restricted to the epistemological register. In short, one of the ways in which we determine whether or not something is a being or entity is through encountering resistances.”

    This is in obvious connection to Bhaskar’s account, which provides a picture of scientific experimentation on the basis of how objects may be out-of-phase with their qualities, and in which we discover virtual powers by gradual, tentative approaches by acting upon the source material.  In fact, the good scientific theory, Levi claims, is one which can effectively produce these differences or resistances. That is, a good theory is one which finds new modeling vectors under contingent experimental conditions, which not just informs us about the conceptually expressible differences between and among the objects we’re dealing with. In addition to these, science also produces and stratifies the differences which form its material basis through its experimental practice. And I think Levi is right in expressing that scientific knowledge involves a productive aspect which involves a material transformation, and not just a representational theory concerned with proposition-world mappings, in the sense he earlier expressed.

   I think that at this juncture very interesting questions arise about the question of materialism in relation to the status of formalization, modeling, practice, knowledge, domains of phenomena, phenomena, and the Real. To be as brief as possible, I will outline where the core stakes lie as far as I can gauge:  

There is in both Pickering and Bhaskar’s accounts, as expressed by Levi, a distinction between theoretical model and material practice. This seems to be in accordance to the classic philosophical distinction between form and content which fuels the possible distinction between formal and empirical sciences. The role of scientific theories is to model the material  phenomena in domains, which are also produced in practice. So we have simultaneously the transforming of the phenomena itself into domains, and the production of the domains qua theoretical entities. This is a tricky relation which is not given proper explanation, for it is far from clear how the resistance of the Real in the transformation of manner relates to the theoretical domains in a non-representational way, i.e. outside the fourfold determinations outlined above. It seems tacitly assumed that the relations between the domains of phenomena qua theoretical postulates which reach over the phenomena, are in some way analogous or even isomorphic to the structure of Real phenomena themselves. But if this relation of isomorphy-analogy obtains, it must surely be said how the features of real resistance allows us to elucidate this relation, and thus to know of it. For to simply assume the theoretical domains, which are finally sets and thus mathematical entities, are transparently 'correspondent' to the Real is to smuggle in a gratuitous representational account between scientific discursive entities and the real world. How such an account is any less representational or presupposition-laden than accounts of linguistic representation, is beyond me.

 Models do not represent scientific practices themselves, but  purportedly apply to ever-expanding domains of phenomena conditioned by the practice of the scientist and his environment, including the laboratory instrumental conditions under which experiments occur. This is why I do not hesitate to call the accounts Levi follows as ultimately empiricist insofar as they accept the externality of matter in relation to what is expressed to the notional material of scientific theory. The latter only models the former through the production of domains of phenomena, while it is resistance qua inadequacy to theoretical postulates in the way of unpredictable experimental occurrences which become the index for the Real as non-produced instead. 

     It must be said, however, that as far as Levi’s presentation is concerned, the account seems to presuppose a rather specific series of ontological commitments which make up the externality of this material world: that of practicing scientists, in laboratories, with equipment and instruments, and fabricating models. The material reality of scientific experimentation is underwritten by the agency of an anonymous scientific ‘practitioner’ within the strictures of intervention described in Pickering and Bhaskar in accordance to a very precise sense of what they take to be the work of scientists. 

     The problem as we have indicated is that it remains ambiguous in Levi’s explanation whether the
domains of phenomena onto which scientific theories are expanded are themselves material and part of the materially engaged practice, or rather part of the theoretical models which produce the scientific postulates, belonging to the sphere of mere ‘results’. It seems quite plausible to say that these domains are produced and material at once, since they constitute, alongside the apparatuses, laboratory conditions, and everything else, that which is being-worked-upon (the matter tested and transformed) and that which is being-modeled (the inscribed differentiation and stratification of produced differences in domains). By the same token Levi surely cannot mean that only scientists produce real phenomena or matter, since it is on the other hand matter which preconditions the existence of scientists and the possibility of any local manifestation. There must be thus a tacit distinction between domains of phenomena and the phenomena  themselves at work here, which is far more mysterious than it seems at first.

Levi's position is subtle: phenomena are of the class of local manifestations, manifestations of the virtual object. The latter never gives itself as such, but only locally and partially. Thus if the scientist may occasionally encounter resistance it is because the objects it deals with may be out-of-phase; not displaying a power(s) they have, and which somehow  prevents thorough and transparent
 isomorphy to the theoretical postulates. Yet the question between the domains of phenomena and phenomena remains somewhat ambiguous vis a vis the distinction between local manifestations and real object. Specifically, is there any sense of speaking of phenomena as already individuated, prior to the scientific production and modelling of domains? If the domains are not merely representing the phenomena, but also producing them, in what sense does one nevertheless claim that objects are individuated quite independently of the theoretico-practical resources of modelling which we use to test the powers of objects. 

One may tentatively assume the following: the ‘expansion’ of scientific theory consists in the production of new theoretical axioms-language along with the construction of new
domains which group phenomena, as much as a negotiation with the material conditions of production, i.e. the phenomena themselves. The domains of phenomena in question would thus be postulated sets of material beings/regions as sets which group local appearances which designate substances and their powers. These presuppose the phenomena which belong or are included in those sets, as tested by the possible resistance of a given powers in the object: the same object may or may not phenomenally be locally manifest in the appropriate domain. Thus water could be boiling or not, and this might allow us to stratify water across distinct differential domains for scientific production and testing. While the testable repetition of the same manifestations obtained under suitably comparable circumstance on the same material, guarantees the domains of phenomena pertain to virtual objects. These domains must thus be indexes of real powers in virtual objects, for Levi. It would make no sense for Levi to say thus that the local manifestation of an apple is simply a local manifestation of something which is of my own making, and that the object is in no relation to the statements-domains which scientist produce thus. That is, it wouldn't congruent with realism to say that the phenomena which appear and are modeled are power of the object as such. This would render the productivity of science impervious to the possible content of matter, irrespective of resistance, and of all the differentiations produced by the theory.

However, for this account to be rendered coherent, Levi must imply that the
domains do not themselves produce the phenomena which belong to it; since that would mean that all phenomena are produced as individuated in the development of scientific practice and domains, something which would be either correlationist or Idealist (we will see why later). Finally, how are the phenomena in question, assumed as already individuated, known-as pertaining to the real object apart from any domains under which they are grouped in scientific statements? If the groupings which give know-that through know-how are always associated with a produced expressible domain, that is as a set, how is one to understand phenomena as non-produced, the phenomena themselves, as anchored in a subsistent and structured real already individuated into parts and powers, and not merely relative to our knowledge. Resistance here does not help in defining the relation between the phenomena manifested and as compiled in domains on the one hand, and the powers themselves as hosted by a virtual structure on the other. The answer may be anticipated from the following. Levi’s criteria, following Bhaskar and Pickering, requires that: “There are degrees of adequation based on the differences that the model has consistently been able to produce in practice.
   This all seems to suggest that knowledge-that or adequation is measured against the standard of produced differences, the domains of phenomena which stratify and differentiate distinct phenomena.  The question here becomes again one of individuation: structure and difference are known in the production of differences which adequately model the real, but the real as such is known only in its resistance to full  integration, or structural isomorphy, i.e to the produced theory and expressed domains.   Seeing that Levi expressly rejects representationalist accounts, he surely does not mean to say that domains of phenomena represent phenomena, or things themselves. These domains are the material objects themselves, and not just the expressions which we give of them; the two are distinct but bear structural relation. Yet how this account is not representational is completely beyond me. Perhaps I am wrong, but correspondence theories coded in mapping mathematical structures to entities in the world are no less problematic epistemologically than those between words and world. If Levi's account is so thoroughly anti-representational, how are the domains produced made to express properties of objects themselves, rather than of mere appearances ala Kant relative to transcendentally individuated phenomena, or by virtue of the very domains constructed in formalization, which makes them formally individuated? Consistency only says that a  phenomenon will appear thus and may be modeled thus consistently, but it doesn't tell me how this phenomenon and its allotment into domains obtains for being-it-itself. This cannot be left in suspense. For how do we gauge not just that there is a real resistance outside our domains of difference, but realities which are numerically correspondent to our singularly produced domains as expressed by sets.

The answer here is that we can't, ever, for sure! But the question is more serious than that. Not being a materialism which favors scientific differentiation in production, Levi's own irreductionist account must tweak the criteria for knowledge about the real apart from conditions of pure differential production within scientific understanding. But the nearness to the real in terms of the withdrawal of objects appears difficult to discern in epistemological grounds: domains of phenomena are produced, as sets of differentiated individuated local manifestations expressed in sets. These sets are associated with powers in some unique object. The former can be out-of-phase with the latter. This tells us that domains model groups of local manifestation. But we do not only isolate arbirary local manifestations of powers, but the objects that presumably bear these powers. The variety of postulated objects is simply contingent on the stratification of domains, and these all fold back to the real as resistance. How are we to understand the individuality of such a non-produced phenomenon, then, lest we describe how the domains of phenomena under which they are individuated, are in some way not merely produced, but also correspondent to a real object/part? How is this correspondence understood or how can it can potentially fail? The 'out-of-phasing' of an object with respect to its properties, and the other criteria Bhaskar proposes just begs the question, since it continues to assume correspondence between the structure of produced domains which stratify differences in phenomenal, locally actualized appearance, and that of the virtual powers of objects themselves. But this correspondence seems wanting and threatening to an account that has sworn to abjure representation. 

As I indicated earlier, there are ways to construct an account of representation without falling into a representationalist vision of the sort Levi has in mind. Badiou’s distinction between parts and elements codes representation in terms of how the State doubles the count of a situation by counting its discernible parts (subsets). The twofold count is what conforms to the situation's structure. But in Badiou’s account there is no agency of the ‘experimental subject’ modeling a phenomenal field or doing the 'counting', since there is no subject of science, no ‘practicioner’ underlying its mathematized kernel. In Levi’s account, however, it becomes either necessary to specify how practical scientific knowledge models material phenomena in a way in which the latter can be said to be knowledge of the in-itself, and not just knowledge of the for-us; not just relatively.

For if we must say that all we know of the Real is that it resists our construal of domains in practice through which we differentiate and stratify, then the Real becomes, in a manner similar to Laruelle, in principle refractory from any ontologizing/formalizing structure which might individuate it in continuity with our fabricated domains. That is, lest we surreptitiously claim that domains are not really just produced, but representational or formalist, i.e. that the domains bear analogic resemblance or structural isomorphism to the structure of phenomena themselves. In the latter case, phenomena themselves would be theoretical postulates, and it is doubtful we can call such a theory a realism. Since in Pickering's account one can only survey the proximity of knowledge to phenomena themselves by virtue of the differences the model(s)  produces,  one must explain how these differences are not just indexes for the Real qua unspecified being in withdrawal (both quantitatively and qualitatively), but to the Real as structurally adequate to these differences or continuous with them. And indeed Levi acknowledges that we have not done with adequation, but merely with 'representational' accounts. Without criterion for adequacy, the domains couldn’t be of phenomena qua local manifestations of virtual structures. Since the question is now not only which domains we may produce in scientific practice but how these domains stand for features of the real, the question seems pending. Put more provocatively: how do we know the structure of what withdraws if what manifests itself is always known under conditions of production which remain specific to these conditions (the domain of phenomena)? 

In spite of all the insightful precautions, the demand to clarify the nature of proximity between production-dependent domains and production-independent phenomenal appearances qua indexes of real objects raises interesting questions. It seems that Levi’s subscription to Pickering oscillates between a representational empiricist account where the domains of phenomena merely designate/organize/produce phenomena which are individuated prior to the production of domains through analogic resemblance,  and the Idealist/formalist thesis according to which individuation obtains intrinsically as a function of scientific production, and the latter to the construction of difference, i.e. the domains of phenomena.

The former option cannot, however, appeal to resemblance and at the same time attempt to dodge representation without flirting with the status of scientific statements and questions about conditions for verification. The latter may provide one with isomorphy between domain and phenomena, but the externality of matter as resistence becomes either annulled as structured, or else an intractable index discontinuous with the domains produced: mysticism. That the latter option can be reconciled with a realism about scientific differentiation is dubious, since it seems to entail an unbridgeable qualitative gap between domains of phenomena on the one hand, which bear isomorphic relations to phenomena; and the Real 'as such' which remains known by a resistance in the order of given experience. This last option seems to me too close to the mystical option, and I doubt Levi would want to endorse it. The possible idea of a given without givenness, on the other hand, a non-individuated index of withdrawal, seems too close to Laruelle's practicism.

Rethinking Materiality - Mathematics and Matter

At this juncture, we can point out how naturalized epistemology is already denounced by Badiou for its apparent circularity: it attempts to clarify how productivity anchors itself on a resistant material basis, while at the same time explain the latter by the production of domains which are not merely heuristic with respect to the phenomena it analyses, but correspondent or adequate to it according to an extrinsic relation of analogical resemblance (albeit it remains unstated and unconceptualized in Levi’s introductory presentation):

“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 [a theory of how science records ‘facts’].” [Pg. 139]

 As I see it, this is the result of the empiricist adherence to the form-content distinction which underwrites both Bhaskar and Pickering’s account.  Theories produce models, and the latter map into external material reality. This is facilitated by the theory of praxis underlying these accounts, the scientific standard for an agency which knows. I would love to know what Levi thinks of Badiou’s criticism to this account in his first book The Concept of Model.
The core issue for Badiou is how to think of a robust materialism which accounts for its immanent historicity. Thus the latter requirement seeks to think of matter dialectically, and yet not as transcendent object for representational reification, or constituted by a subjective agency in Idealist manner. Matter is immanent to itself, and its becoming is dialectical and yet not anchored on any external guarantor. Thus Badiou will reject any accounts which locate at the basis of scientific phenomena the agency of a subject or individual, or community, which is tethered to inherit the weight of scientific productivity. Against Heidegger, who believes that science is to face a crisis at a loss for a metaphysical grounding, Badiou explains that there is no subject of science; and that therefore there are no crises in science. Science is its own subject:

“Such a state of affairs is symptomatic of contemporary science and represents its innermost danger… science will not allow itself to enter such a crisis because it is already much too preoccupied with the realm of serviceability” (Heidegger, The Fundamental Concepts of Metaphysics, Pg. 189

“[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, 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…There are no crises within science.” (Badiou: ‘Mark and Lack’)

   Badiou’s way to do this cuts right into the central question about what constitutes scientificity. For Badiou, the empiricist model which distinguishes between form and content rests still on a transcendental conception, i.e. every empiricism is already in itself an Idealism. It presupposes a split between theory and practice in terms of formal systems and the material realm which provides the ‘content’ for the verification of the experiments of these systems. Therefore, the theoretical framework models materiality as such. This empiricist bias acknowledges that there is an externality of matter to its knowing, where the latter must situate itself as a ‘blind spot’. This is precisely the status of the scientific practice favored by Levi described by Pickering: the material activity of the scientific community producing models after a resisting reality. Thus against the hackneyed purported split between the formal and empirical sciences, the natural scientist can avow that only the latter deserve to be called scientific, an enquiry into the real, since they only interact with the material real, external to theories and their models. Theory models reality, reality itself withdraws from theory.

   Yet for Badiou precisely what distinguishes scientific production is its thorough mathematicity. Of course, as is well known, this underwrites his favoring of physics and his questionable dismissal of biology (that ‘wild empiricism’, as he calls it). For Badiou, even Quine’s naturalism cannot escape pragmatism as long as it locales the formality of scientific productivity apart from materiality itself. Thus Badiou’s extant mathematical Platonism: it is at once a theory of the most formal as the only material. Mathematics says what is explainable of being qua being. There is no ‘externality’ for practice to anchor itself and for its theories to model. Strictly speaking, what we have is always two thoroughly mathematical structures:

1)      An axiomatic framework.
2)      A domain of Interpretation.

The history of the scientific becoming concerns how it continually differentiates its own notional material in order to escape its appropriation by bourgeois epistemology. Scientific concepts stratify notional difference, while ideological notions usually envelop these, i.e. the form-concept distinction is one of such notion. Philosophy thereby wanders between a reactionary ideological envelopment of science, and a potential unveiling of these envelopments deploying truly materialist categories.  The obvious political flair of the last sentence shouldn’t distract us to what is being implied here. What science does is to immanently sublate the subordination of scientific concepts to ideological and notions, by re-stratifying that which philosophy de-stratifies. The appeal to an ‘external category called matter falls to a mere philosophy of matter, while a true materialist philosophy is immanently materialistic. In this regard, Badiou agrees with Levi in that: “The reality of the epistemological materialism which I am trying to introduce […] is indissociable from an effective practice of science”. However, this practice of science cannot be explained by the activity of an agent building theories to model material reality. Rather, the process whereby scientific production stratifies differences immanently is by constantly splitting its notional material in order to produce concepts for which no category/notion exists yet. And it is in this challenge that precisely we see how Badiou seeks to reject both empiricism and Idealism in the same stroke:

    “The broader epistemological import of this project is to develop a materialist account of the nature of scientific theory that challenges both the empiricist assumption that scientific theories merely model empirical reality, and the idealist claim that reality is nothing but an inert support for scientific theory.” (Brassier, 2010)

    The productivity of science is anchored on account of how it operates over its mathematical notation, in which it is progressively formulated: ““[U]ltimately, in physics, fundamental biology, etc., mathematics is not subordinated and expressive, but primary and productive”. This primacy and productivity must be explained, however, in its three fundamental aspects: stratification, differentiation, and how it is an index of materiality itself. And this must be so without any anchoring of it externally on a notion of subjective intervention. We could provisionally remark that this early conception of mathematical production as immanent, being its own subject, still coincides with the latter ontological formulation from Being and Event according to which mathematics speaks resolutely what can be known of being qua being. In any case, Badiou’s account seems very close to Bhaskar in what it needs to explain, and indeed it all turns on whether Badiou can effectively produce a theory of immanent material productivity through mathematics without resorting to any externality or endorsing a Pythagoreanism about number, i.e. stratified difference must index materiality itself. But if mathematicity is said to cut its own notional material, this must be a way which does not imply some external agency which ‘does the cutting’; it must be intrinsic to its own operations.  

Badiou thereby proposes to assess the scientific status of the concept of model to purify it from its notional (ideological) baggage. One can attribute this ideological notional coating of the concept of model present in bourgeois epistemology by diagnosing the latter as being structured around an unexplained differentiation – based on an unquestioned assumption and perfectly incapable of examining their underlying principle. In the case of bourgeois epistemology, this is the distinction between:

a) Theoretical form – The formal theory (mathematics) which has a function.
b) Empirical reality - Which lies independently of formalizations.

The articulation between the two has in turn (at least) two variants: the representationalist idea of theory mirroring pre-given / presented objective reality, or the structuralist thesis governing the idea of an anteriority of a formal apparatus where the theory provides the form of representation for access to reality (no reality without theory).

Let us attempt to understand each of the stages involved in Badiou’s presentation, for the sake of clarity. Following Brassier’s (2010) excellent presentation, we can begin by distinguishing:

1) Formal system (syntactical aspect):
- Finite set of symbols
- Logical operators (negation and implication)
- Individual constants (a, b, c)
- Predicates (P, Q, R)
- Variables (x, y, z…)
- Quantifiers: (Universal, Existential)
- Rules of formation
- Rules of deduction (generalization, separation)
- A list of axioms

2) Structure
- Domain for interpretation (non-empty set V)
- Marks (true/demonstrable, false/non-demonstrable)
3) Rules of correspondence (semantic aspect)
- Correspondence function F mapping individual constants to some element of V, and predicative constants, to some subset of V.

    This general layout expresses how scientific activity operated by the elementary mechanisms of concactenation, formation and derivation. The stratification of symbols in strings become solely the input to produce differentiations at distinct levels through deductions which allow for operations of generalization or separation, in accordance with an axiomatic framework. The intelligibility of statements in the theory are determined on the basis of its internally regulated production of differences (the deductions themselves).

Next, we explain how scientific concepts are enveloped in a historical development along categories and notions. This happens essentially in three steps: first science breaks away from ideological capture in also breaking away with its philosophical categorical representation; then this break is recaptured by philosophical categories (which can be the scientist’s own interpretation); science breaks with this second representation by reconfiguring around new limits independently and inherently established through its own formalization.  This is what renders the material becoming of science, as tethered to mathematicity, also dialectically engrossed in the tripartite constellation of concepts, notions and categories; science, ideology and philosophy.

Brassier’s tracing of this movement here is helpful and can be quoted in full:

     “The first break is the point at which scientific discourse subverts its current ideological representation by producing a difference that cannot be subsumed by extant philosophical categories. Philosophy then represents this break by de-stratifying (re-categorizing) this stratified difference, thereby re-enveloping the new concepts produced by science within extant ideological notions (these will be novel philosophical categories). Lastly, the moment of reconfiguration is the moment of the second break understood as a re-stratification of what philosophy has de-stratified; a re-stratification wherein the parameters of scientific discourse are re-established independently of science’s current philosophical representation.”

    Badiou himself traces the historical becoming of these developments, from Pythagoras to Godel, in seven moments which serve to articulate this movement:

I) The existence of a historical mathematics (namely ‘intuitive’ arithmetic); one that is open in principle (indefinitely stratified signifier).

II a) The ideological re-presentation of this existence as the trans-mathematical norm of completely controllable rationality (the ideological destratification of the mathematical signifier).

II b) The posing of a question to mathematics about their conformity to this ideological norm, in the form of the axiomatic and formalist intention, whose goal is to display a founded transparency (this is the ideological motivation of Frege and Russell).

III) Break: the mathematical treatment of this ideological re-presentation of mathematics via the actual construction of formal systems that ‘represent’ historical arithmetic (Principia Mathematica).

IV a) The ideological re-presentation of this break: formal systems conceived as trans-mathematical norms of rational closure via the idea of a nomological system (Husserl).

IV b) The posing of a question to mathematics about their absolute conformity to the ideological norm of closure: this is the metamathematical intention, relative to the internal demonstration of a system’s consistency (Hilbert).

V) Break: the mathematical treatment of ideological re-presentation via the actual construction of a mathematical metamathematics (the arithmetization of syntax).
Gödel’s Theorem: the structural stratification of the mathematical signifier does not answer the ‘question’ of closure.

VI) Ideological re-presentation of this break: Gödel’s Theorem is experienced as a limitation relative to the normative expectation.
            Ideological exegesis of this ‘limitation’ as:
-         openness of speech and concealment of being (Ladrière);
-         finitude;
-         splitting, suture;
-         ...

VII) Break: the general theory of the limitation-effect, positively conceived as a structural instance of certain mathematical objects (Smullyan’s epistemological truth).
The epistemological upshot of this convoluted adventure reminds us that mathematics operates upon its own existence such as it is designated in ideology; but that this operation, conforming to the specific constraints of a science, takes the form of a break, such that the (ideological) questions which make up the material upon which mathematics carries out its working reprise, find no answer in the latter.
(Badiou, ‘Mark and Lack’, 172)

      Philosophy de-stratifies science insofar as it renders its pure notional differentiation subject to some categorical operator which subordinates/sublates this. A good example here would be, as Deleuze has developed, how Hegel or Leibniz seize infinity into the dialectic of identity. The former subordinates infinite differentiation to the moment of essence in the infinitely large, as the historical movement of determinate negation which renders lays the ground on contradiction. For the latter, the vicediction of the infinitely small indistinguishes the difference (the mirroring of the monads), which it excludes in essence (Difference and Repetition, Chapter II, Pgs 42-46). The post-Cantorian reconceptualization of infinity, however, allows us to differentiate between various infinities of variable cardinal order, which devolves in the sequential expansion of ever growing, perfectly determinate sets (the series of alephs; see Being and Event, Meditations 14, 26-28). By edifying a new layer of stratification mathematicity produces a difference for which no category yet exists. This ideological re-presentation which is curiously also a philosophical operation (though not, it must be stressed, one made exclusively by philosophers) is constantly shed off by formalization.

   Yet what does it mean to say that this production is, strictly speaking, material? It does not, Badiou emphasize, pertain to the spurious category of ‘matter’ invoked as the kernel of content or sensible material upon which formalization operators. Just as radically anti-phenomenological as it is anti-empiricist, set-theoretical ontology abhors construing matter around any form of panthematic presentation, the non-identical sentiendum of the aesthetic, even in its contrived naturalistic guises. Matter is a direct index of the scriptural materiality wherein difference is produced. But, as Brassier credits Zachary Luke Fraser in pointing out, this cannot pertain to the substance of graphic marks in paper, ink, and so on. This materiality is incorporeal, and here lies the real break with OOO; it does not pertain to any form of circumscribed substantiality. Since productivity is the index for materiality, Badiou emphasizes this Marxist link and insists that it is the formal stratification which produces difference and which produces intelligible differences apart from any category of representation, so that only then “ signifying order can envelop the strata of its discourse”.  The domains of interpretation which serve as the structural support for the theory are thoroughly mathematized, and obey perfectly the extensional order of sets, without any intensional determinant playing the role of specifying the ‘nature’ of its contents. As Brassier puts it:

    “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.” [Ibid: Pg, 144]”

The ‘rule governed transparency’ of the mathematical machinery which articulates scientific productivity make the dualism of subject and object, and the idea of an experimental practice as anchored on a mediating agent, a residue of the bourgeois empiricism which remains ideologically enveloped. The means of mathematical production are likewise produced mathematically on their own account “there would be no formal systems without recursive arithmetic, and no rigorous experimental protocols for such systems without set-theory.” And it is in this element that scientific production finds its historicity, which becomes then nothing but the history of materialism as it negotiates its field with ideology: the problem is finally the history of formalization. Discontinuity is hardly a problem for science which needs to be accounted for irrespective of its immanent resources by appeals to the practicing scientist, laboratory conditions, and the externality of produced matter over modeling formal systems. Rather, discontinuity is immanent to its productive method, cutting itself loose perpetually. As Brassier concludes:

“Badiou’s account is one in which there is nothing that science cannot know because, dispensing with every vestige of substance, Badiou’s formalist ontology leaves no room for the inconceivable or unconceptualizable. Truth is the sole exception to the order of being, but science is one of the harbingers of truth.”  

I should say that I am aware that this position has been criticized for different reasons. Brassier’s Sellars-inspired account insists that formalization remains helplessly abstract if not wed to a perceptual realism, and therefore proposes to rehabilitate the cog of representation altogether for epistemology. Daniel W Smith insists, contra Badiou, that the field of axiomatics that Badiou’s set-theoretical approach favors obviates the intrinsic role problematics play in scientific production, leading the way to rehabilitate the Deleuzean conception of intensive multiplicities which morphogenetically become the source for material dynamisms, and not just the bare extensional stasis of the axiom.

The morale of the story is finally that as long as epistemology is wed to an external concept of matter, even if it is wed to experimental production, remains subject to the representational exigency (analogical resemblance) or the formalist coup (isomorphy). The former rehabilitates the representational cog Levi has sworn to abjure, while the latter vitiates resistance qua index for reality; insofar as it must admit of a structural continuity in the isomorphy between model and matter, form and content, produced domain of phenomena and the phenomena themselves. For Levi cannot settle with the thesis that phenomena only exist as produced or that they are individuated only in scientific production, without surreptitiously re-anthropomorphizing his account, and inflecting OOO into the proto-Idealist fable in which only knowing systems individuate objects, or the proto-Deleuzean notion that pre-individual singularities are non-individuated prior to psychic agency. The neat separation between epistemology and ontology, and circumscribing the question of the scientific determination of the real to the former, is thereby threatened. The ontological status of matter remains fatally ambiguous between the thesis according to which it remains in utter externality to thought (known only in passive resistance), and the thesis according to which it isomorphic to the structure of knowledge as individuated by scientific domains and theories (known by the active production of domains of difference). Since the phenomena which belong to these domains cannot be functions of scientific practice for OOO, they must be individuated isomorphically to at least certain formal structures or models at time, or else remain utterly qualitatively distinct and thus unknowable wholesale. But isomorphy only dodges representation by rendering the gap between form and content effectively null. Such an option, does not permit the external conception of matter as substantially resistant, but makes it rather formally accessible. Levi would need the latter to encode the difference between withdrawing virtual being, and local manifestations. Isomorphy can only maintain a concept of a resisting real in the form of a remainder, the matter external to the domains and which becomes palpable in practice by experiencing out-of-phasing. But if this resistance resists the transparent isomorphy of the scientific theory, then it must be because what doesn’t withdraw can never be isomorphic to the real, i.e. the latter is precisely what always remains outside. But then the entire production of difference which is supposed to tell us more about the real ends up being ideologized as a monotonous stupidity where stratification can’t be measured in its proximity (or lack thereof) to the real. Formalization operates by dissecting itself on the basis of a withdrawal of the real, but this renders impossible to verify that real can ever be isomorphic with any formal structure. This gap remaining open, the domains of difference made up by phenomena remain just that, phenomena, local manifestations seized into domains by production in our dealings, but never in principle knowledge of the real. At best, the practitioner is laid with the epiphanic encounter with the Real’s slippery asymmetry against its modelling attempts; the excess of a first mover, while the scientist in turn embodies the relentlessness of a dog trying to bite its own tail. The laboratorial considerations of scientific practice remain, it must be said, no less invested by ideological presuppositions than the representationalist prejudice Levi wishes to denounce. In the intractable alienation of the Real, one cannot but wonder why the production of difference constitutes anything like an approximation? That seems to gauge the intelligence of the dog in direct proportion to how many loops it has ran chasing its own tail.

In the end, either the real remains totally knowable and we have formalist isomorphy, or we know it progressively and only partially, in which case we have representationalist analogical resemblance, or we make stratification thoroughly knowledge-relative and make the real totally unknowable and we oscillate between between correlationism and Idealism. In any case, it is the externality of matter, the withdrawal of its substantive being, which resurrects the epistemological juncture with metaphysics anew, in what was to be a neat separation of their position. 

The resolutely pragmatic considerations underwritten in Levi’s realism through Pickering and Bhaskar’s account seem under that light, produced by an adherence to the form-content distinction. I think Badiou’s epistemological insights might clarify where certain core issues could be, even if I, for one, remain highly uncertain. In particular, since Badiou ends up bringing up the agency of a post-evental subject which can only work through the meta-ontological role of philosophy vis a vis the primary ontological role attributed to set-theory qua a theory of presentation. The entire dialectics of the event, even in its reworked form in Logics of Worlds remains to me wed to a dubious emergence of a ‘singularity’ which however audaciously smuggled, remains gratuitously in exception to the axiom of foundation. Nothing in the materialist dialectic of objects helps to account for this emergence, since ontology knows nothing of the event, even if it can specify the post-evental Truth in terms of the generic procedure. Without the act, however, Badiou’s ontology remains static, and open to the Deleuzean bite in favor of problematics. But the latter likewise can only proceed by reproducing, in Badiou’s estimation, a dualism of Idea and matter, virtual and actual, at the expense of univocity. Finally, if the materialism purported by Badiou is not an Idealism, in spite of his endorsement of the Parmenidean identification of being and thought, it is because the productivity of matter is immanent to itself and its own notional material. But an ontology identified with mathematics can have no truck with the spuriously external notions of ideology or philosophy any more than it can have concern with ‘matter’ or laboratory instrumentation. This is why the process of novelty occurs through four domains and is ontologically completely generic in its being. The philosophical meta-ontological intervention in Badiou’s discourse operates precisely as a point of exception where presentation  now solicits ontology to become a theory for its conditions  to render its models in art, science, love and politics. All of these are mere historical configurations of the base generic material being expressed through mathematics. The agency of a ‘pure subject’, then, comes to intervene in the emergence of a ‘singularity’ in the world, regulating the passage of the inexistent of the world into maximal existence, in disconnected trajectories. This meta-ontological exception is however, a lingering point in Badiou’s account for reasons Brassier (2007) has already pointed out, and which remain pressing in my estimation as late as LOW. But enough for now.

  I am very glad, however, to see Levi’s account illuminated by his references to Pickering and Bhaskar, and hopefully points towards a dialog surrounding these fascinating issues. I want to thank him again for his time and insightful production.

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