Math Matters – On Alain Badiou’s “The Concept of Model”
Following Ray Brassier’s (2005) short paper on Angelaki, let us assess how Badiou’s (1966) The Concept of Model advances towards a materialist epistemology of mathematics.
As is well known, Badiou’s work attributes to mathematics something of a paradigmatic position in determining “scientificity”. Already before the well known ‘mathematical turn’ in his thought, leading to the equation between mathematics and ontology in Being and Event, Badiou conceived of mathematics, since the very start of his philosophical oeuvre, as providing something of a standard for the rest of the scientific practices. The closer a theory/discipline is to inscription in the language of mathematics; the more scientific it can be esteemed to be. Mathematics is not subordinated to the scientific interests of the natural sciences, but in a certain way envelops their activity, giving them form and mobilizing their productivity: ‘‘[U]ltimately, in physics, fundamental biology, etc., mathematics is not subordinated and expressive, but primary and productive.’’ This means that mathematics is not a mere means for scientific production; but that its own development is what provides a standard and material support for scientific productivity as such.
We can anticipate here that Badiou’s conception of mathematics concerns challenging its purported status as a mere ‘formal’ discipline, somehow separated from the ‘empirical’ domains otherwise assigned to the natural sciences. By denouncing the ideological separation of science from the empiricist split between formal-empirical/scheme-content, and integrating scientific experimental practice into the experimental interplay within mathematics, Badiou thus seeks to provide a legitimate philosophical estimation of mathematics’ primordial productive function.
If after all, as Badiou came to think, the axiomatic of set-theory can provide the all-enveloping means such that it is equipped to describe a general theory of presentation (and thus of becoming the science which describes conditions of access to any presentable reality), then it is in and through mathematics that Badiou’s materialism must found itself in its theoretical and practical deployment. Thus while the Zermelo-Fraenkel axiomatization of set theory provides an all-encompassing domain for the operation of mathematics at large, all accessible reality must remain within the confines of mathematical susceptibility in its capacity to be underwritten by ontology’s generality, i.e. set-theory as the science of the pure multiple, without qualitative determinations. By the same token, to stand outside mathematics separates thought from the possibility to stratify differences, and so outside the material scientific domain of dynamic production and demonstration. Because of this, Badiou tends to favor physics over biology; insofar as the former mathematically provides the objectified strata of what it structures under its domain.
Ontology, on the other hand, conditions this possible reification insofar as it intrinsically forbids any objectification of the multiple; where the latter is revealed as the result of transitory operations; hence ontology’s capacity to embrace the multiple indistinctly of its domain of operation: it stratifies being by de-objectifying them insofar as it founds itself on denying a primal appearance, i.e. the existential negation of belonging by the axiom of void-set qua the non-being of the One.
Therefore, it seems natural for Badiou to say that the scientific esteem of science must be rooted in its mathematical inscription, since it must be ultimately subject to the sole ‘science of the real’ enveloped by mathematical set-theory, to use Lacan’s well-known phrase. Since, in general, the form of presentation is that of the One (as opposed to ontology), it is the scientific capacity to objectify and axiomatically deploy differences which assigns it to mathematical production.
The most general objection to this approach, Brassier notes, would be to accuse it of an archaic formalism, peculiar to Badiou’s own intransigent variety of discursive Platonism. Only under such a lens could Badiou’s idea that s mathematics could ‘found’ the rest of the sciences, since it is to imperatively disavow the empirical / material (experiential) side, subordinating their practice to the ideal/formal or ‘a priori’ dimensions. However, as we indicated in the beginning, Badiou is precisely looking to undermine the (empiricist, and idealist) distinction between the formal and empirical, to be in turn revealed in their ideologically unquestioned illegitimacy.
Badiou begins by challenging two dominant conceptions of mathematics, as:
1) A formalist game – An arbitrary manipulation of meaningless symbols.
2) A scientific practice – Granting access to a domain of transcendent objects.
Mathematics is described rather as a productive activity, which consists in the stratification and differentiation of its notional material. Or as Brassier puts it: “Science is the production of stratified differences.” So mathematics is not a formal a priori discipline guaranteeing access to empirical reality; but itself an experimental productive practice inseparable from reality, as in the purported duality of ‘bourgeois epistemology’ between form and content. This latter duality is thereby designated as an ideological notion. To this purpose, Badiou distinguishes between three registers:
1) Philosophical categories.
2) Ideological notions.
3) Scientific concepts.
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). But, as we diagnosed above, these are ideological notions insofar as the differentiation of their terms is presupposed and not transparently accounted for:
‘‘[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.’’
So we must unveil this ideological envelopment in the case of bourgeois epistemology more thoroughly.
Carnap and Quine / Analycity and Synthesis, Form and Content
Brassier takes a brief detour through the history of logical empiricism to pave the way into Badiou’s critique of bourgeois epistemology. He begins by targeting Carnap’s project for a reductionist physicalism in order to neatly separate the physical empirical sciences corresponding to synthetic statements, and the artificial formal sciences corresponding to analytic statements.
But Quine (1951) famously subverts this possibility by attacking the distinction between synthetic/analytic statements, given the latter’s reliance on the notion of synonymy, which presupposes that meaning as intensionally transparent in propositionally/sentential form rather than extensionally defined. The latter is subject to logical transparency, the former presupposes that an entity attaches itself to a sign; in this case propositions are the correlates of sentential statements. But there is nothing ‘self-evident’ about the putative analytic statements such that their meaning could be rendered transparent by their correspondence to signs, so Quine concludes that these purported intensional propositions are noematic entities tethered to linguistic intentionality, which is “what [Aristotelian] essence becomes when it is divorced from the object of reference and wedded to the word.”
Quine doubts this dubious idea can found the notion of Analycity, just like the dogma of reductionism relies on the presupposition that one can neatly divide conceptual schemes between their formal and empirical aspects. With regards to this last dogma, Quine reminds us that theories are intrinsically holistic and so this separation is actually impossible: falsification in theory does not entail always particular statements of the theory to empirical data; considerations of revision evaluate the system as a whole, and there is no clearly demarcable procedure for this besides the evaluation of the pragmatic import of the theory at large, and of its specific premises. Quine thus recognizes that theories are underwritten by empirical evidence, since it is finally evidence which gives theory to revision. In this sense Quine refuses to let go of the last dogma of empiricism, as described by Davidson: the dualism of conceptual scheme and empirical content.
This way Quine refuses to distinguish between ontological speculation and scientific hypothesizing, which in addition to the indeterminacy of translation (the notion of reference is non-intensional, but always relative to a holistically articulated semantic structure) and ontological relativity (to be is to be the value of a variable) leads his subscription to the form/content dualism into an epistemological relativism. Brassier´s formulation is excellent: “the difference between Homeric gods and protons is merely one of degree rather than kind… Whatever superiority the myth of physical objects enjoys over that of the Homeric gods comes down to a question of usefulness.” So the empirical remains by necessity incommensurable to resources extrinsic to a properly defined semantics holistically sustained by the system as a whole: one cannot appeal to the intensional connection between meaning and sign to distinguish between analytic and empirical statements anymore than one can distinguish ontology from science by virtue of the latter’s reliance on the empirical and the former’s a prioricity. This way pragmatism reveals itself as the underlying doctrine behind Quine’s empiricism.
Because of this ultimate neutrality of the empirical, Quine cannot find room for philosophy to investigate how the theoretical ‘cultural posits’ relate to ‘empirical usefulness’, since there would be strictly no such mechanism for verification. This signals blindness to the underlying ideological distinction between form and fact which underwrites the logical empiricism commonly to Carnap and Quine, and the latter can thereby leave empiricism’s underlying condition unquestioned. Badiou says that for Quine it is finally indiscernible to say that the formal is a dimension of the empirical or the reverse. So it follows that philosophical inquiry into science would have to proceed from science’s intrinsic categories: since there is no higher ground of a prioricity discernible from its concrete semantic coordinate system such that it would be commensurable by a ‘transcendental enquiry’ of some sort: science is a cultural posit which presents its enquiry in terms immanent to itself. Only ‘empirical usefulness’ remains transcendent to the immanent interplay of the terms in the theory.
To accomplish this, logicist empiricism will replace the traditional representationalist account where formal theory models the world via representation, to a theory where the empirical models the formal. We will see this in detail as we progress. This allows Quine to relinquish the two dogmas while preserving empiricism. For this, naturalized epistemology it is scientific practice itself which models the epistemological theory of science: it constructs a scientific-model of science’s capacity to model. Mind you, having granted that the choice of theory is empirically pragmatic, epistemology is not set to inquire into the putative reasons for adopting the theory. It merely stipulates that the domain for interpretation of theory (its empirical aspect) already has chosen a domain of objects to be tested with respect to the theory. This way the empirical structure which models a theory of representation is the very realm of scientific theorization as such: a set of unspecified entities/objects. Note that this non-specificity is guaranteed by ontological relativity and, as we will see, delivers the epistemologically construction of the empirical domain in terms of sets. In particular, naturalized epistemology accomplishes this virtuous circularity by designating the domain of neurocomputational processes as the empirical model for scientific representation, while these processes are already a part of science:
“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]
Badiou claims in this regard that ‘‘If science is an imitative artifice [artisanat], the artificial imitation of this artifice is, in effect, Absolute Knowledge.’’ So, Brassier claims, the ideologically coated notion of model, moving from positivism to a pragmatist version of absolute idealism. But for Badiou this anchors Quine’s empiricism in the dogmatic irreflexivity obstructing materialism from properly putting science to the service of revolutionary motifs. The key lies in the concept of representation tout court, which idealizes science as an imitative theory instead of thinking the form of its production. This means we shouldn’t construct a dichotomy between formal theory and a pre-existing empirical reality (here, the pragmatist motivations behind naturalistic epistemology and its construction of scientific practice as a domain of interpretation empirically modeling theory). Rather, we must analyze the interplay of developing demonstrations and proofs which determine a precise historical material reality, whose structural specificity is the object of science [Pg: 139]. This way epistemology can be saved from the historicist variant of empiricism, according to Badiou. This pragmatist naturalization passes over the crucial difference between what Brassier distinguishes thus:
1) Cognitive production – Emergence of new theoretical practice through experimentation which likewise transforms the domain of interpretation. Domain interplay with axiomatic syntactic logical axioms transforming each other in an ongoing experimental practice which transforms the very reality it simultaneously describes.
2) Technical regulation – The use of models are subservient to strict conditions: Badiou offers the example of economics where the models are always in order to think the existing conditions of production rather than attempting to modify the very formal conditions for the conditions of production themselves. In this sense they merely regulate an existing practice through modelling rather than produce by way of an interplay between a domain of interpretation and a formal system (we will analyze this distinction in fuller detail on the next section).
So in distinction to vulgar empiricism’s blatant representational framework: Quine’s pragmatist version circulates around the closure of the gap between fact and form, both realms are reciprocally presupposed and are thus isomorphic in relation: “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.” [Pg: 140] We will then see how Badiou proposes to distance himself from naturalism while endorsing the rejection of any foundational ambitions by philosophy’s appeal to a realm of a prioricity. Badiou will reject integrating the sciences into the cognitivist evolutionary guise in which the neurocomputational processes model all scientific representation, and will insist on affirming the irreducible plurality of scientific practices and their discontinuous historicities. Subtracted from the guise of evolutionary history, Brassier nonetheless considers that this aversion to biology appears to overstep the properly Darwinian subversion of cleavage between natural and cultural history which Badiou seems resistant in accepting. But this shall not occupy us for now (we in fact know this is part of what motivates Brassier’s to challenge Badiou’s own subtractive ontology at the point where it lays credence to thought’s disruptive exclusive rights to evental change (arbitrarily allowing subjective decision to mediate between the grounded ontological presentation from the forced non-ontological presentation).
The Concept of Model
Let us proceed to examine Badiou’s reconstruction of the concept of model. For brevity’s sake, we will reconstruct the concept of model in its three essential components:
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.
From the syntactical giveness of axioms and rules of deduction one can derive theorems, but it is a requirement that there be at least one expression in the system which is not a theorem (directly deducible from the axioms), to avoid the redundancy of the rules of deduction and ensure the consistency of the system. Next, to establish that the formal system gives a specific deductive structure the rules of correspondence map expressions of the system to expressions which belong to the well defined domain for interpretation. Because semantics deals with these rules, it is defined extensionally. With this in place, the basic requirement for the construction of the concept of model can be given: a model is a formal system for which every deducible expression (theorem) corresponds to a ‘true’ statement in the domain of interpretation. Recall that the label ‘true’ is intended to merely mean ‘demonstrable’, and so it doesn’t carry any additional baggage: it is a mere functional operation of the system. Thus, if all deducible expressions in the system correspond to a true statement on the domain of interpretation the latter is a model for the system. [Pg: 141]
Scientific theory demands consistency, its experimental aspect demands the examination and building of concrete models. So scientific apparatuses are tools for such model building, the formal system is constrained by the model it uses to test the rules of correspondence (on which furthermore the consistency of the system depends) and the syntactical rules, and their interplay is the scientific practice as such. On the tripartite structure of formal systems, Brassier makes three remarks:
1) The specificity of the domain of objects must be defined in a concrete set of objects, which establishes it within mathematics, and thus renders possible the scientific eligibility of semantics.
2) The requirement that the number of symbols be finite means they are denumerable using whole numbers. Every well formed expression needs to have a denumerable number of indecomposable terms. Here the reclusiveness is not to set-theory, but to arithmetic: ’One establishes oneself in science from the start. One does not reconstitute it from scratch. One does not found it.’’
3) There is a crucial distinction between:
-Logical axioms – depending uniquely on the logical connectives and unaffected by the substitution of fixes constants in it.
-Mathematical axioms – Singularizes at least one of the fixed constants by separating it from at least one other. It is thus sensitive to substitution.
We can thus explain the construction of a model as follows: by the set-theoretical of the structure/domain of interpretation and the correspondence function F, one defines the validity and invalidity of a well formed expression of the system relative to the structure. Next, one specifies under what conditions a structure is a model for the system by establishing a correspondence between syntactic deducibility (an expression A is a theorem of the system) and semantic validity (that A is valid for a, the, or every structure). Then we can define a closed instance of expression A when all of its free variables have been replaced by fixed constants: A (a/x), (b/y), (c/z). [Pg: 142-143]
In turn, Validity is defined thus: expression A is valid for a structure if for every closed instance A’ of A, one gets A’ = True. If the axioms of the system are valid for a structure, it will follow the theorems are likewise valid. So via the correspondence function F we can go from deducible theorem to the idea of validity for structure. Finally the concept of model is defined as follows: “A structure is a model for a formal theory if all the axioms of that theory are valid for that structure.” [Pg: 143]
Logical axioms are valid for every structure, mathematical axioms only for particular ones. Logic is thus equivalent for the structural operational consistency, and mathematics differentiates between types of structure. Logic itself is doubly articulated between the syntactic and semantic elements of the system: the axioms are modified for structures and the opposite as well in the experimental construction of logical systems. It follows that a purely logical system has no semantic indication of its models, since it can singularize any as validating it in distinction from other structures and becomes thus equivalent to structure as such. Therefore, mathematical axioms provide the differentiating power in a formal theory for individuation wherein the concept of model becomes relevant or indeed possible, since it is only when a structure is a model but another isn’t that a gap between the syntactic logical machinery and the structural domain can be cashed out in terms of which axioms are logical (purely syntactic) and which are on the other hand mathematical (sensitive to substitution); the former index the unity of the system and structure, the latter their difference. Models are thus centrally constructed around the “…differentiating power of logico-mathematical system.”
“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.” [Ibid]
Experimentation and demonstration
Because the nature of the scientific activity finds itself split within mathematics between experimentation (building systems and models) and demonstration (testing structural validity for systems) in the dual interplay between set-theory and recursive arithmetic, it is misleading to think that the concept of model implies a relation between thought and its empirical exteriority. Structures are only domains for interpretation for systems in accordance to the syntactical axiomatic (syntactic) and the set-theoretical definition of rules for correspondence (semantics) which remain intrinsic to mathematics; arithmetic and set-theory in an endless interplay which is experimentation.
The arithmetic inscription into natural whole numbers stratifies the differences for the experimental practice, allows order and inductive numbering within the parameters for validity established within the system’s syntactic stricture. The concept of structure on the other hand regulates the usage of the experimental operations as produced by set-theory, concretely classifying the mathematical material according to rules of correspondence. It is thus the unified interplay between arithmetic and set theory which regulates the interplay of syntactic and semantic strictly within the mathematical discourse. From this, Badiou concludes the double illegitimacy of logical empiricism [Pg: 144]:
1) The notional distinction between formal syntactic and formal semantics ideologically coats the fundamental interplay of arithmetic and set-theoretical material which is strictly intra-mathematical.
2) The notion encoding the relation model to system as that between empirical fact and formal theory fails to register how the interplay between the modelling of structure serves also as an experimental basis to test/revise the formal system itself. It is thus incapable of distinguishing, as we saw, between the dual interplay which leads to cognitive production from merely technical regulation, where models must ‘conform’ to the underlying principles of the theory. This is especially true in cases where the domain of interpretation serving as structure is not yet entirely formalized in mathematics and so not prone to semantic correspondence by a syntactical system, which leads to transformations within the system.
Thus, against its standard ideological envelopment of scientific modelling, science does not proceed from theory to model or system to structure, at least not necessarily. The directionality is reversible and constitutes the constant interplay of scientific experimentation and production. The historical becoming of production in mathematics is thus not the testing of a theory for a model, but the specification of theories to account for the existing plurality of structures by devising an appropriate formal/syntactic means of inscription. The conceptual demonstrations in a given structure are thus to be inscribed formally into an appropriate syntax; which renders the idea that the model is a mere means for confirmation of the theory obsolete:
‘‘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.” [Pg: 144]
So the system is formalized by the syntactical inscription of the model: semantics does not concern the modelling of the empirical by the formal, but the productive inscription of structural conceptuality into the axiomatic of a formal system, a procedure which amounts to experimental verification of the demonstrative model:
“[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 [s’e´prouve] through the rule-governed transparency of inscriptions. In mathematics, inscription represents the moment of verification.” [Ibid]
Verification is a means of formalization: demonstration and formalization are knit in their mutual implication.
The Historicity of Mathematical Production
Badiou seeks to construct a category of model which can describe within a dialectical-materialist framework the historicity of scientific practice. Materialism, Brassier underlines, is not to be understood in the vulgar sense of an appeal to substantial ‘matter’, but simply as an index of differential stratification of the pure multiple in objectified domains within a formal system. This is what, for Badiou, amounts to a sort of ‘discursive materiality’ or ‘scriptural materiality’, as Brassier calls it. These are thus irreducible to one scriptural domain, or to the order of a signifying operation; it attests to the formal differentiating capacity of logico-mathematical scripture which transforms itself within mathematical productivity, thereby eradicating the primacy of the object or the world proper to empiricism:
“[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.”
So science can stand outside the domain of ideology not because its subject penetrates into the hard core of reality or a realm of transcendent objects, but simply because of the axiomatic differential stricture of mathematics which proceeds without concern for its object, which stratifies pure multiplicities and for which the procedure of experimentation is precisely mobilized insofar as it subverts the very formal stricture in which conceptually determined structures are modeled and inscribed. There are no formal systems without recursive arithmetic and no rigor for experimental protocols without set-theory. This productivity renders science an essentially differential, rather than primordially representational activity, in the dialectic of demonstration and experimentation. A great example here is Gödel’s proof of the consistency of a model for axiomatic set-theory along the coherent integration Axiom of Choice and the Continuum Hypothesis, which does not transform the theory as much as the status of the theory in scientific historical production. So the constant sublation of a system presenting it retroactively as a model for a new experimental production organizes the dialectical movement of science:
“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]
So finally, we can say that Badiou thinks of the materialist category of model as designating the retrospective causality in formalism, in the connection between an object (model) and a usage (system). It modifies its internal systemic conditions to then be capable of registering a structure as being a model for the system. This perpetual movement of science along with its structural breaks and discontinuities can be rendered transparent without recourse to para-theoretical notions of ‘paradigm-shifts’ or anything of that sort: ‘‘Science is precisely that which is ceaselessly cutting itself loose from its own indication in re-presentational space [i.e., ideology]…. ‘‘[T]here are no crises within science, nor can there be, for science is the pure affirmation of difference.’’’’.
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