Some thoughts on the subject in Badiou
Some thoughts on the subject in Badiou’s philosophy
[1] It has been a suspicion of mine for some time that Badiou’s philosophy, and especially his conception of the subject, is too closely tied to an ideal of political commitment. The need to keep alive the notion of meaningful political action directed toward a cause has resulted in a model of the subject cast in the restrictive mould of a perpetually active revolution. Though Badiou claims that this activity is creative, I have some problem in seeing it as anything other than a negative project aimed at overthrowing some concrete structural aspect of a situation. This suspicion rests with Badiou’s insistence on the model of set theoretical forcing as the archetypal example of an ‘event’.
[2] To understand my complaint we must turn to the mathematical significance of Cohen’s theory of forcing. The introduction of set theoretical forcing was required to prove that certain axioms of set theory were independent: two separate consistent models could be constructed, one in which the axiom held, and one in which it did not. The initial axioms that were of interest were the General Continuum Hypothesis (GCH) and the Axiom of Choice. What such proofs allow is the ability to clarify the necessity of certain axioms. Those axioms that turn out to be independent are not necessary to produce a consistent model of set theory. This emancipation from certain axioms allows set theory to explore a wider range of set theoretical models; even introducing new axioms which contradict those which have been proved to be independent. Most focus is put on GCH, partly due to Georg Cantor’s obsession with it, but mainly due to the fact that should it turn out to be the case that GCH is an integral part of set theory, the universe of sets would be a constructible and completely determined structure. The independence of GCH meant that set theory lost the ability to be completely determined but became a far richer and indeterminate structure as a consequence. The axiom of choice plays a significant role in Badiou’s philosophy and I will return to a discussion of it later.
[3] The question has to be asked: where are the creative and new aspects of set theory occurring? Is it in the process of forcing itself, or is it in the invention and application of new axioms? Badiou definitely locates the invention of the new in the process of forcing itself, in the construction of the generic sets integral to the procedure. But what is the process of forcing really doing? Its purpose, within mathematics at least, is to provide a partial proof of the independence of a given axiom; such proofs by forcing are called independence proofs. Hence, the process of forcing has as its aim a negative goal, to produce a model in which a given axiom fails, as part of a full proof to demonstrate the axiom’s independence. This proof has two possible uses; first, if an axiom is demonstrated to be independent it confirms that it is possible to use the axiom within a consistent system. Second, it provides an opportunity for invention: a new model can be constructed without the given axiom, even a new model using a new axiom. This second aspect is what occurred in the wake of the proof of GCH’s independence; it opened the door for all manner of Large (inaccessible) Cardinal Axioms. This greatly enlarged and complicated the field of set theory. To my mind it is in the invention and application of these new axioms that the creation of the new occurs. Badiou’s theory seems perpetually stuck in the overturning of a given axiom, obsessed in constructing a model in which an axiom, a structural component representing the status quo of a situation, fails.
[4] For me, the real process of invention/creation takes place after an axiom is proved to be independent. It is at this point that a new commitment is given, or a new selection made. It does not occur, as it does for Badiou, in the perpetual process of constructing a model in which a given axiom fails. What also puzzles me is Badiou’s introduction of the event as an unfounded set, this again seems an unnecessary complication required only in order to keep the subject bounded within finitude, and to prevent the possibility that an event may run its course and come to an end. There is a time to pledge commitment to a cause but there is equally a time to let things go, perhaps the real challenge of an event is to know when to finish. To give up on an event is, for Badiou, one of the forms of evil. Within the mathematical application of forcing the non-constructible sets which are appended to a model in order to produce a system in which a given axiom fails, are never presented as unfounded. An unfounded set being one that has no minimal element, such as the empty set, and can then belong to itself. The event, as an unfounded set, remains as a surplus providing the impetus for, and demanding the fidelity from, a finite subject.
[5] The event as surplus ties in with Badiou’s interpretation of the Axiom of Choice, as essentially a faculty of freedom which belongs to a rational individual. This freedom is the capacity to become a subject, by being traversed by a truth and being taken up into it as an integral part of its infinite procedure. The subject is simply a finite portion, or segment, of a truth procedure. This fidelity to an event played out through the militant experimentation of the subject, who interrogates the situation with regards to an event. Now, within mathematics there is no need to actually carry out this labour; the labour of interrogating every element of a situation and asking whether it does, or does not, belong to an element of the event. All that is required is to know that such a procedure is possible. There is no interest in the actual content of this procedure, only in the consequence of it being carried out. The ‘evental’ set is forced to belong to the original model, and as a consequence a consistent model is built in which a specific axiom fails. But, for Badiou, the event itself is never forced to belong; it is always in the process of being forced. This process is carried out by subjects to the event. Therefore the process of building a model in which a specific axiom fails, or the status quo is undermined, is never complete, and is always in the process of happening.
[6] This need to keep the process of forcing as a never ending process, whilst maintaining the subject as finite, means that the event itself becomes a source of excess that provides the impetus for the continued action of the subject. Not directly, but through the faithful declaration of the subject, in his devotion to the cause. This means that the event itself must remain an unfounded set during this procedure: the subject’s experience of it being something akin to a pure experience, a pure sense, an intellectual intuition. Keeping the event as an unfounded set is a distinct deviation from how it is understood within its original mathematical context. The subject’s finite perspective, in the midst of carrying out a truth procedure, means that the event itself, which from the perspective of the situation remains un-founded (un-grounded), appears, through fidelity/faith, as self-founding (self-grounding). As was mentioned above, mathematically the process of forcing is carried out, and a model in which a specific axiom fails is produced. The ‘event’ does not remain as unfounded, only effectual in relation to a subject, but is forced to belong and in the process overturning an axiom. There is no interest in the actual content of this procedure. The interest for the mathematician comes after such a proof, in the potential opened up by a specific axiom becoming independent.
[7] This seems somewhat heavy going, and there is a fair degree of repetition. But in conclusion I want to draw attention to the point that I’ve been trying to make. Badiou’s use of set theory only goes so far before deviating from its standard use. Therefore, set theoretical forcing does not provide an archetypal model of the structure of the event. The model that Badiou uses deviates from its mathematical use by fixing the process of forcing as a never-ending process linked to the finite perspective of a subject. The subject is then fixed within very narrow limits; a subject committed to a cause. But this cause is never anything creative; it always seems reactive, perpetually caught within a process to undermine a specific aspect of the situation. A model that I find leans toward the political, despite Badiou’s four types of event.
[1] It has been a suspicion of mine for some time that Badiou’s philosophy, and especially his conception of the subject, is too closely tied to an ideal of political commitment. The need to keep alive the notion of meaningful political action directed toward a cause has resulted in a model of the subject cast in the restrictive mould of a perpetually active revolution. Though Badiou claims that this activity is creative, I have some problem in seeing it as anything other than a negative project aimed at overthrowing some concrete structural aspect of a situation. This suspicion rests with Badiou’s insistence on the model of set theoretical forcing as the archetypal example of an ‘event’.
[2] To understand my complaint we must turn to the mathematical significance of Cohen’s theory of forcing. The introduction of set theoretical forcing was required to prove that certain axioms of set theory were independent: two separate consistent models could be constructed, one in which the axiom held, and one in which it did not. The initial axioms that were of interest were the General Continuum Hypothesis (GCH) and the Axiom of Choice. What such proofs allow is the ability to clarify the necessity of certain axioms. Those axioms that turn out to be independent are not necessary to produce a consistent model of set theory. This emancipation from certain axioms allows set theory to explore a wider range of set theoretical models; even introducing new axioms which contradict those which have been proved to be independent. Most focus is put on GCH, partly due to Georg Cantor’s obsession with it, but mainly due to the fact that should it turn out to be the case that GCH is an integral part of set theory, the universe of sets would be a constructible and completely determined structure. The independence of GCH meant that set theory lost the ability to be completely determined but became a far richer and indeterminate structure as a consequence. The axiom of choice plays a significant role in Badiou’s philosophy and I will return to a discussion of it later.
[3] The question has to be asked: where are the creative and new aspects of set theory occurring? Is it in the process of forcing itself, or is it in the invention and application of new axioms? Badiou definitely locates the invention of the new in the process of forcing itself, in the construction of the generic sets integral to the procedure. But what is the process of forcing really doing? Its purpose, within mathematics at least, is to provide a partial proof of the independence of a given axiom; such proofs by forcing are called independence proofs. Hence, the process of forcing has as its aim a negative goal, to produce a model in which a given axiom fails, as part of a full proof to demonstrate the axiom’s independence. This proof has two possible uses; first, if an axiom is demonstrated to be independent it confirms that it is possible to use the axiom within a consistent system. Second, it provides an opportunity for invention: a new model can be constructed without the given axiom, even a new model using a new axiom. This second aspect is what occurred in the wake of the proof of GCH’s independence; it opened the door for all manner of Large (inaccessible) Cardinal Axioms. This greatly enlarged and complicated the field of set theory. To my mind it is in the invention and application of these new axioms that the creation of the new occurs. Badiou’s theory seems perpetually stuck in the overturning of a given axiom, obsessed in constructing a model in which an axiom, a structural component representing the status quo of a situation, fails.
[4] For me, the real process of invention/creation takes place after an axiom is proved to be independent. It is at this point that a new commitment is given, or a new selection made. It does not occur, as it does for Badiou, in the perpetual process of constructing a model in which a given axiom fails. What also puzzles me is Badiou’s introduction of the event as an unfounded set, this again seems an unnecessary complication required only in order to keep the subject bounded within finitude, and to prevent the possibility that an event may run its course and come to an end. There is a time to pledge commitment to a cause but there is equally a time to let things go, perhaps the real challenge of an event is to know when to finish. To give up on an event is, for Badiou, one of the forms of evil. Within the mathematical application of forcing the non-constructible sets which are appended to a model in order to produce a system in which a given axiom fails, are never presented as unfounded. An unfounded set being one that has no minimal element, such as the empty set, and can then belong to itself. The event, as an unfounded set, remains as a surplus providing the impetus for, and demanding the fidelity from, a finite subject.
[5] The event as surplus ties in with Badiou’s interpretation of the Axiom of Choice, as essentially a faculty of freedom which belongs to a rational individual. This freedom is the capacity to become a subject, by being traversed by a truth and being taken up into it as an integral part of its infinite procedure. The subject is simply a finite portion, or segment, of a truth procedure. This fidelity to an event played out through the militant experimentation of the subject, who interrogates the situation with regards to an event. Now, within mathematics there is no need to actually carry out this labour; the labour of interrogating every element of a situation and asking whether it does, or does not, belong to an element of the event. All that is required is to know that such a procedure is possible. There is no interest in the actual content of this procedure, only in the consequence of it being carried out. The ‘evental’ set is forced to belong to the original model, and as a consequence a consistent model is built in which a specific axiom fails. But, for Badiou, the event itself is never forced to belong; it is always in the process of being forced. This process is carried out by subjects to the event. Therefore the process of building a model in which a specific axiom fails, or the status quo is undermined, is never complete, and is always in the process of happening.
[6] This need to keep the process of forcing as a never ending process, whilst maintaining the subject as finite, means that the event itself becomes a source of excess that provides the impetus for the continued action of the subject. Not directly, but through the faithful declaration of the subject, in his devotion to the cause. This means that the event itself must remain an unfounded set during this procedure: the subject’s experience of it being something akin to a pure experience, a pure sense, an intellectual intuition. Keeping the event as an unfounded set is a distinct deviation from how it is understood within its original mathematical context. The subject’s finite perspective, in the midst of carrying out a truth procedure, means that the event itself, which from the perspective of the situation remains un-founded (un-grounded), appears, through fidelity/faith, as self-founding (self-grounding). As was mentioned above, mathematically the process of forcing is carried out, and a model in which a specific axiom fails is produced. The ‘event’ does not remain as unfounded, only effectual in relation to a subject, but is forced to belong and in the process overturning an axiom. There is no interest in the actual content of this procedure. The interest for the mathematician comes after such a proof, in the potential opened up by a specific axiom becoming independent.
[7] This seems somewhat heavy going, and there is a fair degree of repetition. But in conclusion I want to draw attention to the point that I’ve been trying to make. Badiou’s use of set theory only goes so far before deviating from its standard use. Therefore, set theoretical forcing does not provide an archetypal model of the structure of the event. The model that Badiou uses deviates from its mathematical use by fixing the process of forcing as a never-ending process linked to the finite perspective of a subject. The subject is then fixed within very narrow limits; a subject committed to a cause. But this cause is never anything creative; it always seems reactive, perpetually caught within a process to undermine a specific aspect of the situation. A model that I find leans toward the political, despite Badiou’s four types of event.
posted by Brian at 11:33 am
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