The Concept of Model is the first book Badiou published in philosophy, and in it he initiates a lifelong concern not only with mathematics and mathematical logic, but also with the ways in which philosophy can receive these disciplines as a condition for the philosophical thinking of truth and change. The concept of model, itself, will go on to occupy a pivotal position in Badiou’s work, orienting in productive and problematic ways his later use of mathematical set theory, and, as Oliver Feltham has argued, giving him an apparatus by which to think the compossibilization and interaction between various truth procedures in addition to mathematics. But what is a model?
The simplest, and least adequate, answer is that a model is a pair, consisting of (1) a structure that a given formal theory can be taken to be theory ‘about’, and (2) an interpretation that systematically, and functionally, links the terms of the theory to the structure in question, in such a way that we can say that the axioms of the theory are ‘true’ or ‘valid’ for the model, and in such a way that the rules by which the theory transforms its axioms into theorems ‘preserve truth’. This simple idea can give rise to numerous misapprehensions, so it is best to go over things more carefully.
First off, we should resist any temptation to view the model/theory distinction as the distinction between an object and its discursive representation. This, by Badiou’s lights, is the error of the empiricist epistemology of models (CM 18-22). It is inadequate on two counts: To begin with, the model/theory distinction is, strictly speaking, internal to mathematical practice: both a formal theory and its models are mathematical constructions, and no structure can ‘deploy a domain of interpretation’ for a mathematical theory if it is not already embedded ‘within a mathematical envelopment, which preordains the former to the latter’ (CM 42). The point of interpreting a structure as a model for a theory (or interpreting a theory as the theory of a structure) is not to mathematically represent something already given outside of mathematics, but to generate a productive interaction between already-mathematical constructions, opening each to new, essentially experimental techniques of verification and variation: determining the relative intrications and independences among concepts, establishing the extent of a concept’s mobility and applicability, sounding out unseen harmonies between apparently heterogeneous domains, and exposing what the logician Girard has called ‘disturbances’ and points of ‘leakage’, the ‘cracks in the building’ which ‘indicate what to search and what to modify.’[i] Freeing it from the doublet that binds representations to their objects, Badiou proposes
to call model the ordinance that, in the historical process of a science, retrospectively assigns to the science’s previous practical instances their experimental transformation by a definite formal apparatus. […] The problem is not, and cannot be, that of the representational relations between the model and the concrete, or between the formal and the models. The problem is that of the history of formalization. ‘Model’ designates the network traversed by the retroactions and anticipations that weave this history: whether it be designated, in anticipation, as break, or in retrospect, as reforging. (CM 54-5; trans. modified)
That it is indeed a network of relations that are at stake in the concept of model, and not the bilateral mirror-play of object and representation, is pressed on us by the fact that, in general, no privileged relation obtains between a syntactically formulated theory and a structure interpreted as its model: more often than not, a theory admits of a vast multiplicity of models, which only in the rarest of cases map on to one another in any strict sense (where a strict mapping—or, precisely, an isomorphism—exists between all the models of a theory, that theory is said to be categorical, but this is quite uncommon); similarly, a given structure can in most cases be equipped with distinct interpretations, each of which making of it a model for quite different formal theories. It is even possible, with a bit of tinkering, to interpret the literal structure of a formal theory as a model for the theory itself—a technique which often proves useful in logic (an example of this technique is given in the Appendix to The Concept of Model).[ii]
The (‘ideologically’ motivated) intuitions that push us to see the mirror-play of object and representation in the model/theory distinction are strong ones. It is instructive to learn that even Paul Cohen—to whom we owe some of the most significant proofs that have ever been written regarding the relation between Zermelo-Fraenkel Set Theory and its models, including his proof of the independence of the continuum hypothesis, in which the concepts of forcing and the generic, so decisive for Badiou’s philosophy, first see the light of day—would confess that
The existence of many possible models of mathematics is difficult to accept upon first encounter […]. I can assure you that, in my own work, one of the most difficult parts of proving independence results was to overcome the psychological fear of thinking about the existence of various models of set theory as being natural objects in mathematics about which one could use natural mathematical intuition.[iii]
An avatar of this prejudice—which Cohen magnificently overcame—is the distinction between standard and non-standard models, which is even today commonplace in mathematical literature. The ‘standard’ model of a theory, in a nutshell, is simply the structure that the theory is intended to describe, together with an interpretation that puts things together in the expected manner. A ‘non-standard’ model is a structure and interpretation that deviates, often wildly, from these educated expectations. (To put it another way, a ‘standard interpretation’ obeys the spirit of the law; a ‘non-standard’ one adheres only to its letter.) Though he rarely addresses this distinction head-on, ever since his remarkable study of Abraham Robinson’s non-standard analysis (the non-standard model that Robinson constructed for the infinitesimal calculus), Badiou has engaged with mathematics in such a way that the distinction between the standard and the non-standard can confront his readers only as an obstacle to understanding. Nowhere is this distinction less pertinent than in set theory, and no single insight does a better job of linking Badiou’s ontological use of set theory with his pronouncement that ‘the One is not’ than the realization that, in all rigour, a standard model for set theory does not exist. If there is anything that set theory is expected to be a theory about, it is the ‘universe of all sets’, but it was a theorem already known to (and considered to be of tremendous importance by) Georg Cantor that the set of all sets cannot exist, on pain of inconsistency.[iv]
If set theory is ontology, but an ontology which, ungrounded by the annulment of the One, has no standard model, then there is every reason to expect that the rules for its interpretation cannot be given in one stroke—a fact which has caused no end of frustration for Badiou’s exegetes—and that they must (within strict but underdetermining constraints) be reinvented situation by situation. The difficulty that remains, of course, is that of escaping the iron strictures of The Concept of Model, which forcefully argues that only an already-mathematical structure can model a mathematical theory. This may be true for mathematics qua mathematics, but it cannot (on pain of a philosophical suture) be maintained for mathematics qua condition for philosophy. The philosophical category of model, conditioned by the mathematical concept, can not remain (as it does in ’68), a purely epistemological category. What is needed, as Oliver Feltham has forcefully argued, is a category of ‘modelling’ that
is the inverse of the procedure of conditioning. In modeling the syntax is constructed in philosophy and then tested in diverse semantic fields such as revolutionary politics or Mallarmé’s poetry. In contrast, with conditioning it is a particular generic procedure such as set theory that provides the syntax and philosophy provides the semantic domain: hence ‘metaontology’ is a model of set theory. (Feltham, 132)
This inverse operation is not contrary to, but demanded by philosophy’s mandate to compossibilize radically heterogeneous conditions, for
if it must circulate between a multiplicity of artistic, scientific, political and amorous conditions, [philosophy] can never be perfectly faithful to one truth procedure alone. Thus, with regard to the comparison between modeling and conditioning, one cannot simpy assert that it is always a truth procedure alone that furnishes the syntax for the model; sometimes it is also philosophy that provides part of the syntax, based on its encounters with other conditions. (Feltham, 132)
It is in this light that we should see in the concept of model the first condition, issuing from the truth procedure of mathematics, of Badiou’s philosophy, the philosophical effects of which make it possible for Badiou, many years later, to put into practice a full and unsutured philosophy under conditions.
See also: Conditions, Suture, Forcing, Generic, Ontology, Mathematics, Set Theory, Ideology.
[i] See Jean-Yves Girard, ‘Linear Logic,’ in Theoretical Computer Science, vol. 50 (1987), p.14, and ‘Locus Solum,’ in Mathematical Structures in Computer Science, vol. 11 (2001), pp. 441, 485.
[ii] This is not to suggest that the relations between theories and models are so loose and wooly that they tell us nothing of interest about either structure. Quite the contrary: since Gödel’s famous completeness theorem (which is a bit less famous, perhaps, than his celebrated incompleteness theorems, at least outside of mathematics), we have known that there is a strict equivalence between saying that a theory has a model (that it is ‘true’ of a certain structure) and that the theory is consistent (that it doesn’t prove everything).
[iii] Paul J. Cohen, ‘The Discovery of Forcing,’ Rocky Mountain Journal of Mathematics, Vol. 32, Nº 4 (Winter 2002), p.1072.
[iv] Negligence of this fact, I believe, is responsible for the widespread impression that Cohen’s proof is something altogether ‘artificial’—impressions which draw their apparent strength from the fact that Cohen’s key procedure—the forcing of a generic extension to a model—depends on his decision to take as his starting point a countable model of set theory. ‘Countable’ here means that the elements of the model-structure can, if immersed in a sufficiently large super-structure, be shown to be in a one-to-one correspondence with the set of natural numbers—but which, in itself, lacks all the ties that would bind its terms to such tiny infinities, and so is not ‘countable for-itself’ and capable of harbouring interpretations of any theorem about transfinite sets that the theory can throw at it. The astonishing fact that such a creature could be a model for a theory of sets of infinitely many different, and ascending, orders of infinity is what was shown to be true by the Löwenheim-Skolem theorem—whose authors, it should be noted, saw their result as something approaching a reduction to the absurd of the formalistic and axiomatic methods of set theory’s pioneers, and as the eternal inadequacy of the letter to the spirit. History would retain the theorem, but invert its moral, winning for formalistic and model-theoretic methods an unprecedented array of freedoms.
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