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
3 Comments:
I agree with a lot of what you say here Brian. I am very wary myself of criticising Badiou when it comes to his ontology because set theory is really not my strong point. You of course are not troubled by this limitation.
Political can become a restrictive notion, I agree very much. That what Badiou sees as creative is in fact negative seems a very strong point. I would certainly see in the absense of a virtual a negativity. Badiou says that the virtual-actual couple in Deleuze introduces transcendence but if its movement in-between is considered it becomes possible to argue against him that this is the apparatus of creativity. I can see that the process of undermining the situation may be negative because it isn't able to be a 'creative drestruction.' There has to be a creativity and not simply an undermining. The movement that is forced and violent cannot be disconnected from a creative ontology. However, I can't really comment confidently upon the mathematics.
I will be looking at differentuial calculus over the next term and perhaps this would be a chance for a discussion on maths and philosophy. An area where I would benefit from Brian's expertise.
It is interesting to witness the critical developments, nay scepticism, in your thoughts regarding Badiou Brian thank you. I confess I was surprised at first that you cast Badiou in the pose of the reactive, negative figure – though if you’re right his mathematical ontological presentation is in effect a shield for his politics. In the context of having called Nietzsche the prince of sophists, this is quite a ruse. I suspect that many Badiouians would like to keep the politics and forget (or easily accept) the mathematics. What makes your position interesting is that you seem to want the opposite – some form of set theoretical groundwork for philosophy, but without the long march (or militant experimentation). It appears rather than call Badiou himself the sophist you’d rather affirm that he’s tapped into something important, but surrendered it to a reactive politics – though one might ask in this context, was the French revolution reactive? (the triumph of reason or the reaction of the peasants?)
I’m interested in hearing more about this notion of invention you’re setting up. Should we think mathematicians the true axiomatic pioneers? Or rather could we for instance think of the empiricist setting up experiments in such terms? The ability to delimit one’s commitment to a given axiom sounds interesting but it appears to shoot holes in the existential fidelity of such a given commitment. I wonder whether to regard a commitment with respect of a limit – a finite value – is already to render it ontologically vacant. It appears that in favour of the struggle of the finite subject taken up into an infinite movement you have a metaphysician constructing abstract axiomatic models – commitment in this context loses its “devotional” Kierkegaardian/Sartrean flavour and becomes what exactly? I wonder which direction one is going with the strict affirmation of mathematics as a philosophical cornerstone – back into idealism? You say that the subject in the context of its task in Badiou is set within narrow limits. This seems to me a little unfair – since it is the cause itself in which the subject grasps its own existence as a possibility. Or to pose it another way: How else might we consider the constitution of the subject?
I'm really enjoying your discussion of Badiou, but I wonder whether your remarks here are entirely fair. Perhaps it is better to focus on art, science, and love as paradigms of truth-procedures, rather than politics, as these conditions are often less defined by antagonistic relations to an other. Following Badiou's discussions of the event of Galileo in _Infinite Thought_, it doesn't seem to me that a truth-procedure is so much *against* something, as it rather subtracts itself from other discourses characterizing the situation. If I'm a follower of Galileo I don't trouble myself too much, arguing with the Aristotleans. Rather, I'm too busy seeing what new natural phenomena can be mathematized. As a result, something like "speciation through geographical isolation" eventually takes place, producing something new that wasn't in the situation before (I realize Badiou would probably detest this biological metaphor).
In this regard, the theory of subtraction strikes me as absolute vital to Badiou's theory of creation. If a situation overdetermines all elements belong to that situation, if it's impossible for an element to be isolated from a situation, then it's difficult to see how anything can change at all. Rather, we get an ontology where all elements are products of the structured field in which they occur, determined by a rigid necessity much like some readings of Hegel's or Marx's dialectic.
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