Of Mathematics and Radical Change: Alain Badiou’s Set-Theoretical Ontology

Ένα ενδιαφέρον εκλαικευτικό κείμενο, στο οποίο παρουσιάζονται με απλό κατανοητό τρόπο η Συνολοθεωρία, η Υπόθεση του Συνεχούς και η διαπραγμάτευση του Cohen


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Extending Badiou’s Mathematical Materialism to Account for Real Change: Beyond the Transcendence/Immanence Dichotomy .By Brian Smith

Ένα ενδιαφέρον κείμενο που παρέχει την πιο εκλαικευμένη παρουσίαση της διαδικασίας forcing, τόσο στη μαθηματική ( Cohen) όσο και φιλοσοφική πολιτική της αξία (Badiou)

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Extending Badiou’s Mathematical Materialism to Account for Real Change: Beyond the Transcendence/Immanence Dichotomy


Introduction

To begin with an oversimplification of the problem, I want to characterize continental philosophy’s obsession with immanence, over transcendence, as a subordination of conscious subjective thought to unconscious non-subjective thought, or processes. In French philosophy, at least, the figure of Sartre is central to this transition, representing the last bastion of a fully committed subjective position. The two volumes of the Critique of Dialectical Reason, which finally engage with the full complexities of material reality, appear to be hobbled by his fidelity to the absolute subjective freedom developed in Being and Nothingness. This fidelity appears anachronistic and conservative in the face of the growth and diversity of strucuralist and psychoanalytic accounts of reality.

The important question, both philosophically and politically, of how real change, creation and novelty are possible can now only be explained through the non-subjective processes of material reality or of unconscious desire. Consciousness and subjectivity are now seen as emergent phenomena, or interesting productions of an unconscious immanent field. The transformations and selections of this field cannot be explained in terms of conscious action or choice. The inexplicable occurrence of a selection, or event, at this level of pure immanence is the groundless moment that grounds the subsequent production of subjectivity and consciousness itself. The retention of anything that even slightly resembles a conscious subjective choice, is deemed a betrayal of immanence and the pure immanent moment of the event’s occurrence, and the unnecessary reintroduction of a transcendent moment.

Another way to characterize the problem is to take a brief overview of one popular area of materialist philosophy that favours immanence over transcendence: process philosophy. This area has received much attention over the last twenty years, due mainly to developments in computer science and the increasing use of computational and stochastic models in other areas of science such as: biology, psychology and neuroscience.

The basic structure of most process philosophies is to set up a simple division between a set of conditions and the matter that is processed by these conditions. The main aim is to show how complex phenomena can emerge through the application of simple rules of process on an essentially contingent, random or given material base. These are the two important moments of any process philosophy: the initial positing of the conditions and the designation of matter, and the subsequent processing of this matter, which is deemed successful to the extent to which interesting and complex structures emerge, especially structures associated with consciousness and subjectivity; though the term ‘interesting’ is rarely queried as much as it should be.

The difficult question that faces any such philosophy is to account for the relation between these two moments. How can the emergent structures produced through the processing of matter feedback and affect the conditions of the system itself? A thoroughgoing materialism would see both the conditions and that which is conditioned as both being matter, as more or less condensed or rigidified material structures. There is no intrinsic or essential difference between the conditions and the conditioned. Stable emergent phenomena can become conditions (habit becoming instinct at the level of the individual organism), or, in reverse, conditions may become dissolved into the flow of matter again (possibly the idea of cognitive plasticity popular in neuroscience). Hence all there is is matter, at the level of pure immanence there is nothing else, but it is never given in this flat way, it is always already folded in some way, such that this fold constitutes this division between conditions and the conditioned. To speak like Deleuze, the fold is that which constitutes a differential of speed between the slower conditions and the faster matter that is conditioned. This moment, or event, both grounds, initially, and disrupts, finally, any such process. In itself it must be seen as inexplicable, it cannot be a simple product of the process itself, as this would be to embed the process within another wider process and would produce a hierarchy of such processes becoming more and more general, repeating the pattern of a progressive, or Hegelian, dialectic.

The question of real change, creation, or novelty, can now be phrased in terms of the ideas introduced above. What I mean by real change is a change in the conditions of a system, the moment, or event, of a re-folding or re-configuration. One of the popular observations about process philosophy and the sciences that employ systems theory is that they are committed to a non-reductionist position. This is borne out by the notion of scale invariance that is at play in these theories: the interesting structures and processes are repeated at a many different levels. We find similar processes and emergent structures whether we consider cosmological, social, biological, atomic or sub-atomic processes. Therefore there will be no ultimate explanation in terms of some one fundamental material base; the question of real change remains an open and irreducible problem, concentrating more on the repetition of patterns, structures and information.

It is because the choice of this material base is arbitrary, the significance only being generated through the informational processing of this matter, that Badiou’s set theoretical ontology can equally be considered materialism. To choose sets as matter is as legitimate as to choose neurons, individuals or anything else. This is not Badiou’s own argument, as he wants to claim that sets provide the only possible basis for a pure materialism, following a somewhat reductive argument based on the use of mathematics to describe matter at the sub-atomic level in particle physics (Adrian Johnston’s critique in What Matter(s) in Ontology provides an adequate account of these problems). What is interesting about Badiou’s position is that he can give an account of real change that does not rely on an appeal to unconscious forces or desires. His model of subjectivity provides an account of the subject as a process of conscious commitment to an event; this process transforms the conditions of the system itself. The method of forcing is usually used to demonstrate the independence of certain axioms, through the production of a situation where the axiom in question fails. Badiou’s set theoretical materialism becomes significant not because it is the only one possible, but because it is one that demonstrates that certain productions are only possible through the notion of a subject consciously committed to a project/event. A materialist account of reality cannot be complete without a recognition that real change occurs through both conscious and unconscious processes.

What I want to show in this paper is how both positions are possible. Real change is possible as either a committed subjective and conscious project or as a non-subjective and unconscious process. Both positions are conceivable within a single systematic framework, with the conception of real change, or creation, remaining the same for both approaches. This single systematic framework is an extension of Badiou’s set theoretical ontology developed in Being and Event. The key point will be a critique of Badiou’s use of the Axiom of Choice. This axiom essentially embodies Sartre’s notion of freedom as a power of unconditional choice, but, unlike Sartre, it does not constitute subjectivity in itself. It is merely an immanent structural feature of any situation, or world, that has the potential for real change, or the creation of the new.

The Axiom of Choice is not a necessary feature of the standard Zermelo-Frankel (ZF) axiomatization of set theory, its use is always explicitly stated, and the abbreviation of ZFC denotes the addition of this axiom. Badiou’s claims regarding his ontology of set theory is based on ZFC, rather than the minimal axiomatization of ZF, without choice. The crude point that I wish to make, and one that I will not have time here to refine, is that the addition of the Axiom of Choice is essentially the addition of a transcendent and unnecessary axiom. The minimal purity of set theory, and Badiou’s claims relating to set theory as a general ontology as the presentation of being as multiple, are undermined by the extended inclusion of AC. A more thoroughly immanent, and Deleuzian, appropriation of set theory would drop AC, maybe moving all the way to a constructivist position where AC is seen as nothing more than a provable theorem within the constructible universe.

My extension of Badiou’s ontology is not an attempt to return to this constructivist level; one that I consider would reintroduce the spectre of determinism into the materialism debate. Rather, I want to propose that Deleuze can be used as a perverse Badiouian subject, and that the formalized axiom of freedom, AC, is not reduced to a constructivist model, but exceeded through an overproduction generic sets, or truth.

This approach will introduce a tension between Badiou and Deleuze that is not unlike the tension found in Sartre’s own conception of finality and counter-finality, developed mainly in volume two of the Critique of Dialectical Reason. This is the idea of how a group project, through its very process, can seem to generate finalities other than its own consciously posited project. How these counter-finalities affect the authentic functioning of a group is one of the fundamental concerns of accounting for real and significant change. It also addresses the problem of the perceived inflexibility of subjective fidelity in Badiou, a problem that leads to the possible fanatical appropriation of his philosophy.

Due to the limited time that I have available to me, I want to concentrate on one singular aspect of this possible connection between Badiou and Deleuze. I will first present two definitions of freedom that I associate with Badiou and Deleuze respectively. Before examining the general structure of Badiou’s use of forcing, used to describe the subjective procedure of making the consequences of an event consist in a situation, thus transforming it. I will then look at the formal differences involved in the proof of the independence of AC, which also utilizes Paul Cohen’s theory of forcing, though in a style very different to that of proving the independence of the Continuum Hypothesis. Finally I will associate this type of procedure with, at least, the spirit or style of Deleuze’s philosophy.


Part One: Freedom, Event and Subjectivity

The central term in establishing the movement, between Badiou and Deleuze, is freedom. To begin with I will present a characterization of their respective positions in terms of two definitions, which will then provide a useful conceptual point of reference.

Definition One: Badiou

Freedom is the capacity to affirm an event. The event occurs in a subjectively consistent situation.

Definition Two: Deleuze

Freedom is the affirmation of an event. The event occurs in a subjectively inconsistent situation.

The important distinction between these two definitions is the difference between an event and its affirmation. In the first definition there is a distinct separation, or gap, between the event and its affirmation, whilst in the second definition the difference is indiscernible; the event and its affirmation become inseparable. For Badiou, a subject affirms, or, more precisely, a subject is the process of the affirmation of, an event. While, for Deleuze, an event is its own affirmation; if the event and its affirmation are separated, the original intensity of the event is lost and covered over, especially if it is taken up by a subject. Whenever Deleuze mentions freedom it is always exercised immanently by an event, idea, concept, or another similar Deleuzian term, whilst the subject is always explicitly rejected or overcome. In the eternal return, or event, neither the agent nor the condition returns.

For Deleuze, and the second definition of freedom, the subject is not capable of affirming the event, such that the immanent affirmation of the event always appears as inconsistent and paradoxical to a subject. In affirming this paradoxical moment subjectivity is dissolved, allowing the event to express itself. The situation in which the event expresses and affirms itself is only ever paradoxical relative to a subject, in itself this situation has its own consistency, it is never pure inconsistency or chaos.

At this stage, in order to bridge the gap between Badiou and Deleuze we can posit the difference in terms of this unconditional moment. For Deleuze, the unconditional trace, or singularity, of an event is always subjectively inconsistent, and subjectively inseparable from the event itself. For Badiou, the unconditional trace is always subjectively consistent and separable from the event itself. What is required is a model where the event is sometimes subjectively consistent, and sometimes inconsistent, such a model could support both positions. This would be to introduce the idea of the degree of a singularity’s unconditioned nature: unconditioned to a degree such that a subject can consistently affirm it, or surpassing this degree, such that the subject is ungrounded and becomes inconsistent. This distinction will only be possible through an analysis of the event as a contingent or chance occurrence. Only be focusing on what is meant by chance or contingency in the works of Badiou and Deleuze will it be possible to offer a model that bridges the above gap.


Part Two: Badiou’s Use of Forcing

Badiou’s use of forcing as a model of subjective endeavour is limited to the standard model put forward by Paul Cohen for proving the independence of the Continuum Hypothesis. As Badiou states in The Clamour of Being: ‘the form of all events is the same’. This method proceeds by taking a ground model of set theory, M, and producing a generic extension, M[G], by adding the generic sets G, such that G are non-constructible sets and GÏM, via the method of forcing. The Continuum Hypothesis fails in this new model; in the ground model extended by the addition of generic sets G. The subject has as its finality, or project, the construction of this generic extension, where the presupposed condition fails; in this cases the Continuum Hypothesis. Badiou defines truth as these generic sets, and a subject as the localised process of making these sets consist in the extended situation. The extended situation is, for Badiou, a space in which new knowledge is possible, on the basis that things that were undecidable become decidable due to the supplementation of a truth. As Badiou states:

Thought in its novelty, the situation to-come presents everything that the current situation presents, but in addition, it presents a truth. By consequence, it presents innumerable new multiples.

The subject, as finite and localized, is focused on the generic extension as a new situation, a situation to come. But truth, in the form of these generic sets, only has value to the extent that new thoughts, knowledge, or multiples, are made accessible and decidable in this situation to come. This happens in the extension for the standard approach because the condition, or hypothesis, whose independence is sought, fails in the extension. The presentation of the generic sets in the generic extension makes the Continuum Hypothesis fail.


Part Three: The Axiom of Choice, and Its Style of Forcing

The most immediate difference in the proof of the independence of the Axiom of Choice is that the axiom does not fail in the generic extension. In Thomas Jech’s classic textbook, Set Theory, we find the following clear description:

If the ground model satisfies the axiom of choice, then so does the generic extension. However, we can still use the method of forcing to construct a model in which AC fails; namely, we find a suitable submodel of the generic model, a model N such that MÍNÍM[G].

The first point to note is that the mere presentation of these generic sets is not sufficient to make the Axiom of Choice fail. It is important how and where these generic sets are presented. The ‘suitable submodel’, that Jech posits, utilizes the generic sets of the extension but extracts them from the extension to present them in a distilled or concentrated manner in the submodel. The model in which the Axiom of Choice fails is between the ground model and the extension, it does not fail in the extension. This is a point that Badiou overlooks in Being and Event, in questioning the validity of the axioms of Zermelo Frankel set theory he remains fixated on the generic extension: ‘They [the axioms of set theory] are… veridical in any generic extension’. But the Axiom of Choice does not fail in the extension, but in a submodel between this extension and the ground model.

Even this small structural change raises two important questions with regards to Badiou’s philosophy. One, are generic sets truths in essence, or are they only truths when their presentation makes some axiom or hypothesis of the ground model fail? Two, how does this approach to forcing change the intention of the subject? Is the subjective finality directed toward the situation to come, of the generic extension, or is it directed toward this submodel that is between the extension and the ground model, where the Axiom of Choice fails? Or could we see a division between an unconscious drive, and desire, toward the submodel, in opposition to the proclaimed conscious subjective intention, directed toward the generic extension?

The question of how a Badiouian subject committed to this style of forcing, whether of AC, or some other independence proof, would differ from the standard model presented in Being and Event, and, for the most part, held to in Logics of Worlds, is not easy to answer. I offer Deleuze, or the constant Deleuzian project, of pushing subjectivity to its limit, such that it is overcome or dissolves in the moment of the eternal return, or third synthesis of time as a possible example of this type of subject. If the form of this proof is examined in closer details the Deleuzian aspects become more pronounced, especially with respect to Badiou’s own criticisms of Deleuze. The accusation of monotony in Deleuze, to take one brief example, can be explained by the fact that the result in the proof of the independence of AC is achieved through a massive over production of generic sets, and that the result is not obtained in the generic extension. For Badiou, Deleuze’s approach results in nothing, no change, because of Badiou’s fixation on the generic extension, which, here, is not the place of real change.

Conclusion

The main concluding points that I want to make are that Badiou’s philosophy allows us to reinvent the concept of the subject, and committed conscious action. This is probably his greatest appeal as a philosopher, especially for political philosophy. It is a breath of fresh air to be able to recognize that a subject is capable of being committed to the cause, or project, that they claim to be. There is no subversive or unconscious real aim, as opposed to the consciously claimed apparent aim. Badiou’s appeal to the formalism of set theory allows him to make this claim, but, at the same time, it opens the possibility of an alternative subject, one that could be unaware of their real purpose. Badiou cannot escape from the dilemma of authenticity; the figures of reactive and obscure subjectivity, introduced in Logics of Worlds, essentially break with the correct formal structure of the faithful subject, but the alternative form of subjectivity I offer here keeps within the formalism of set theoretical forcing. The subject must really question the finality toward which they are working; it is not necessarily the consciously posited world-to-come of the generic extension.