Heroico Desembarazo: H μαθηματική αποτύπωση του "Ενός" του Πρόκλου

Στο Heroico Desembarazo δημοσιεύθηκε μια ανάρτηση η οποία "μαθηματικοποιεί" το "εν" του Πρόκλου μέσω της συνολοθεωρίας.
Ενδιαφέρουσα προσπάθεια η οποία διευρύνει τις αναγνώσεις μέσω της θεωρίας συνόλων
Στην ανάρτηση έγινε και ένα παραγωγικός διάλογος για την κατεύθυνση αυτή¨
Όλη η ανάρτηση και ο διάλογος εδώ

Καστοριάδης -Μάγματα-Θεωρία Συνόλων

Στο έργο του Κ.Καστοριάδη αναπτύσσεται η έννοια του μάγματος.Η έννοια αυτή διατυπώνεται ως αντίρρηση και υπέρβαση των εννοιών της αξιωματικής συνολοθεωρίας ZF.Μπορούμε να  θεωρήσουμε ότι αντιτίθεται προς την έννοια του συνόλου.

Ο καθηγητής Μαθηματικών Α.Τζουβάρας προχωρεί σε μια ενδιαφέρουσα ανάλυση.
Ταξινομεί και αναλύει λογικά την έννοια του μάγματος και κατόπιν την ενσωματώνει σε μια "αναθεωρημένη ευρεία" συνολοθεωρία πέραν των αξιωμάτων ΖF,όπως αυτή τη θεωρία κατέθεσε η μαθηματικός Helen Skala;.


To κείμενο του Α.Τζουβάρα εδώ

Αποσπάσματα από το ίδιο κείμενο στα Ελληνικά εδώ

Badioumathematics: Οδηγός Σπουδών

Εδώ και αρκετούς μήνες στο ιστολόγιο αυτό έχουν σωρευθεί μια σειρά κειμένων με ένα στόχο : πως μπορεί ένας μη μαθηματικός αναγνώστης μπορεί να καταλάβει τα θεμελιακά έργα του A.Badiou :Being & Event και Logic of Worlds.

Αυτά τα  δύο έργα είναι τελικά  εκτεταμένα δοκίμια μαθηματικής λογικής με αναφορές στις πιο ανεπτυγμένες πλευρές της Θεωρίας Συνόλων και της Θεωρίας Κατηγοριών.Οποίος δεν έχει σχέση με τα μαθηματικά αυτά , ουσιαστικά διαβάζει τα έργα αυτά αποσπασματικά.

Μετά από τον  αποθησαυρισμό τόσων κειμένων , εδώ παραθέτω μια   προτάση που θα διευκολύνει ένα μη μαθηματικό να καταλάβει τα θεμελιώδη μαθηματικά του Badiou.

1.-Λοιπόν άρχισε με το " Το μυστήριο του Αλεφ "του Amir Aczel.Μεταφρασμένο θαυμάσια στα ελληνικά σε εισάγει με απλό εύληπτο τρόπο σε εικονοποιημένες έννοιες της θεωρίας συνόλων, στις μαθηματικές θεωρίες για το άπειρο και τους αριθμούς.

2.-Μετά κοίταξε από το ιστολόγιο το video εδώ .Είναι ο καλύτερος τρόπος να καταλάβεις με εικόνες τα επίδικα της συνολοθεωρίας.

3.-Το επόμενο κείμενο ευρίσκεται εδώ.Είναι η πιο απλή εισαγωγή για να δεις πως σχετίζονται τα μαθηματικά και η πολιτική φιλοσοφία

4.-Το επόμενο βήμα είναι να διαβάσεις το " Αφελής Συνολοθεωρία" του Paul Halmos. Είναι για μαθηματικούς , σε ωραία Ελληνικά, αλλά είναι τόσο καλά δομημένο , όπου τα βασικά γίνονται κατανοητά.Είναι τόσο κατανοητή η αρχιτεκτονική της θεωρίας, για το πως "κτίζονται " οι έννοιες σιγά σιγά.

5.-Κατόπιν πάμε  στο "Badiou - A Subject to Truth" του Peter Hallward.Έχει ένα ολόκληρο επεξηγηματικό τμήμα για τα μαθηματικά αυτά. Δεν είναι υποχρεωτικό αλλά βοηθά

Τώρα μπορείς να αρχίσεις το Being and Event 'κάνοντας αναφορές σε όλα τα προηγούμενα.Έτσι θα δεις μερικά εκκεντρικά ζητήματα:

-Είναι δυνατόν να υπάρχει θεωρία του κράτους και των τάξεων με βάση τις θεμελιώδεις μαθηματικές διάφορες του " ανήκειν" και " εγκλείεσθαι";

-Γιατί η μαθηματική τεχνική Forcing , μπορεί να τεκμηριώσει την κοινωνική αλλαγή;

-Γιατί η μαθηματική τεχνική Labelling σηματοδότεί τη σημασία που έχουν τα συνθήματα και ονοματοθεσίες στην πολιτική;

-Πως κατασκευάζονται πολιτικομαθηματικες εξισώσεις;

Και τέλος η έκπληξη
]
Μετά από όλα αυτά, δοκίμασε να διαβάσεις ένα από τα πιο προχωρημένα βιβλία των μαθηματικών: το "The set theory and the continuum hypothesis" του P.Cohen.Η γοητεία οφείλεται στο ότι δομείται επαγωγικά , ως εάν, ο αναγνώστης δεν είναι μαθηματικός.Δεν μπορώ να φανταστώ άλλο ιστορικό τεχνικό κείμενο το οποίο να δομείται με τόσο φιλικό προς τον αναγνώστη τρόπο.Ένα από τα πιο περισπούδαστα βιβλία των μαθηματικών είναι ουσιαστικά για μη μαθηματικούς: αποκάλυψη!

Forcing in Being and Event

Πηγή: Form & Formalism

 

Forcing in Being and Event

Between the theory of forcing presented in ‘Infinitesimal Subversion’ and the one we find in Being and Event, intervenes the a new and decisive condition: a technique developed by Paul J. Cohen in his proof of the independence of the Generalized Continuum Hypothesis and the Axiom of Choice from the axioms of Zermelo-Fraenkel set theory (ZF), which likewise appears under the name of ‘forcing’. Before we address its incorporation into Badiou’s philosophical apparatus, we will take a quick look at forcing in its native, mathematical terrain.

It is, once again, set-theoretical model theory that provides Badiou with the requisite conceptual (scientific) material. Like Robinson’s procedure for the making of ‘non-standard’ models, Cohen’s forcing technique is, at bottom, a systematic way of generating a new model from a model already given. The main thrust of Cohen’s proof is to take a countable, transitive model[4] of ZF and ‘force’ the existence of a new model by supplementing it with a generic element included in, but not belonging to, to initial model—together with all the sets which can be constructed on the supplement’s basis by licensed by the ZF axiomatic. Considered in its logical structure, forcing is a relation of the form ‘a forces P’, where a is a set and P a proposition that will hold in the generic extension of the initial model—provided that a turns out to belong to the generic supplement on which that extension is based. In this respect, forcing resembles a logical inference relation, but one that differs markedly from the inference relation of classical logic—the law of the excluded middle, in particular, does not hold for the forcing relation, and the logic it generates is essentially intuitionistic.[5]

As Cohen has shown, the consequences this supplementation can be quite extraordinary, and go far beyond simply adding a new set’s name to the census. The generic supplement, for instance, may be structured so as to induce a one-to-one correspondence between transfinite ordinals that, in the initial model/situation, counted as distinct orders of infinity, thereby collapsing them onto one another and making them effectively equal. Cohen exploited this possibility to great effect by taking the model that Gödel had built in order to show that the Generalized Continuum Hypothesis (GCH)—the thesis that the size of the set of subsets set of any transfinite cardinal number À_n is equal in size to the next greatest cardinal À_n+1—is consistent with ZF (a model in which the continuum hypothesis holds), and on its basis forcing a generic extension in which the continuum hypothesis fails (the extension being a model in which the set of subsets of À_n is demonstrably equal to almost any cardinal whatsoever, so long as it’s larger than À_n), thereby demonstrating the consistency of GCH’s negation with the theory, and hence the independence, or undecidability, of GCH with respect to ZF.

Being and Event recovers Cohen’s concept and enlists it in a re-articulation of the existing category of forcing: the set underlying the model is now seized upon as the situation that forcing will transform, and faithful Miller’s cartography of change, Badiou adds that the whole procedure—both the articulation of the generic truth and the forcing of its consequences for the situation to come—must in every case proceed from an anomalous occurrence in the ‘utopic point’ of the situation in question, now rechristened ‘evental site’. Though it is now Cohen rather than Robinson whose mathematics condition Badiou’s theory of change, the new category of forcing preserves most of the features familiar to us from “Infinitesimal Subversion.” One crucial difference, however, is that the whole process is now seized as a logic of subjective action: Forcing is now names “the law of the subject” [CITE], the form by which a subject faithful to an event transforms her situation into one to which a still-unknown[6] truth (understood as a generic subset of the initial situation) well and truly belongs, by deriving consequences that the inscription of this new constant will have brought about.

In light of Being and Event’s decision to interpret ZF as the theory of being qua being, and forcing as the form of a subject’s truth-bearing practice, Badiou extracts two lessons from Cohen’s proof of the undecidability of GCH: first, that it demonstrates the existence of a radical ontological gap or ‘impasse’ between infinite multiplicities and the sets of their subsets (to which Badiou associated the notions of ‘representation’ or ‘state of a situation’), the exact measure of which is indeterminate at the level of being-in-itself; second, that this ontological undecidability is nevertheless decidable in practice, but only through the faithful effectuation of a truth, suspended from the anomalous occurrence of an event.

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


Όλο το κείμενο σε PDF εδώ

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)

πηγή :εδώ

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.

Is Badiou's Ontology Consistent With Materialism?

Πηγή:Larval Subjects


21 AUGUST 2006

Is Badiou's Ontology Consistent With Materialism?
Since last week I've been largely out of commission cognitively due to health issues, but I'm slowly regaining the ability to think. During this time, I happened to come across a brief statement by Badiou, explaining why he considers himself a materialist. Here my remarks will be less than elegant, though I hope to localize a problem that seems to emerge with regard to Badiou's ontology, and a place where it might become possible to think questioningly with Badiou (in tandem with Deleuze and what might motivate a Deleuzian ontology).

In an interview accompanying Badiou's Ethics: An Essay on the Understanding of Evil, Hallward asks Badiou to clarify the relation between his mathematical ontology and the nature of material reality. Despite my great love of mathematics and my tendency to advocate some form of Platonic realism where the ontological status of mathematical entities is concerned, I think this question naturally emerges insofar as mathematical entities are often thought as idealities and possibilities, independent of the actuality characterizing the material world. Consequently, there seems to be something of a gulf or chasm between the infinite possibilities of the mathematical world and the actualities characterizing this world. In response, Badiou remarks that,
If we accept that there exists a situation in which what is at stake for thought as being-as-being [viz., ontology]-- and for me, this is simply one situation of thought, among others --then I would say that this situation is the situation defined by mathematics. Mathematics, because if we abstract all presentative predicates little by little, we are left with the multiple, pure and simple. The "that which is presented" can be absolutely anything. Pure presentation as such, abstracting all reference to "that which" --which is to say, then, being-as-being, being as pure multiplicity --can be thought only through pure mathematics.

To the extent that we abstract the "that which is presented" in the diversity of situations, to consider the presentation of presentation itself-- that is to say, in the end, pure multiplicity-- then the real and the possible are rendered necessarily indistinct. What I call ontology is the generic form of presentation as such, considered independently of the question as to whether what is presented is real or possible... Are they real, do they exist somewhere, are they merely possible, are they linguistic products...? I think we have to abandon these questions simply because it is of the essence of ontology, as I conceive it, to be beneath the distinction of the real and the possible. What we will necessarily be left with is a science of the multiple in general, such that the question of knowing what is effectively presented in a particular situation remains suspended. A contrario, every time we examine something that is presented, from the strict point of view of its objective presentation, we will have a horizon of mathematicity, which is, in my opinion, the only thing that can be clear. In the final analysis, physics-- that is to say, the theory of matter --is mathematical. (Ethics: An Essay on the Understanding of Evil, pgs. 127-128, italics mine)
A moment later Badiou reminds us of what ontology is, remarking that, "all of this simply confirms a very old and somewhat inevitable ontological programme: that ontology always gathers up what remains to thought once we abandon the predicative, particular determination of 'that which is presented'" (129). Ontology, then, is what remains once we subtract all other predicates characterizing an existent or an entity. According to Badiou, all that remains after such an operation of subtraction occurs is pure multiplicity or multiplicity qua multiplicity sans any other predicates or qualities:
Now, the existent qua existent is absolutely unbound [I read this as "un-related"]. This is a fundamental characteristic of the pure manifold as it is thought in Set theory. There are only multiplicities and nothing else. None of them on their own is connected to another. In Set Theory even functions have to be thought as pure multiplicities or manifolds. This is why we identify them by their graph [I'm not sure what he's getting at here with his reference to the graph of a function]. The "beingness" of the existent does not presuppose anything else than its immanent composition, that is, that it might be a manifold of manifolds. Strictly speaking, this excludes the possibility that there might be a being of the relation. When thought as such, and therefore purely generically, Being is subtracted from any connection. (Briefings on Existence: A Short Treatise on Transitory Ontology, "Being and Appearing", 162. This essay can also be found in Theoretical Writings)
The key sentence in this passage is "manifold of manifolds", which should be translated as "multiplicities of multiplicities". In thinking Being as manifold of manifolds Badiou is effectively claiming that there are no ultimate or irreducible terms of which sets would be composed, but only endless multiplicities without ultimate identities, and that these manifolds are unrelated or unconnected to one another. As such, Badiou's ontology is an ontology of infinite dissemination without One or an overarching unity at the level of either the whole or the part. Now I take it that the great merit of Badiou's mathematical thesis is two-fold:
Badiou's ontology effectively allows us to escape any epistemological orientation in ontology, by sidestepping any questions of how it is possible for a subject to relate to being. That is, we need raise no questions of how a mind is able to know or represent being. This point might be obscure to those who have no background in philosophy of mathematics; however, ever since Frege and Husserl, questions of the psychology through which mathematics is known have been staunchly excluded from the thinking of the mathematical qua mathematical. Those unfamiliar with these arguments and the manner in which they trenchantly depose any psychologism would do well to refer to Frege's Foundations of Arithmetic. This is why it's so important to Badiou that ontology evade the distinction between actuality and possibility (as it must not be a matter of mind representing reality, but of a "common being" to possibility and actuality). It is also one reason that Badiou's ontology diverges radically from Zizek's Hegelian orientation. The consequences of this move are far reaching. In one fell swoop, Badiou is able to escape all questions revolving around the subject and anthropology. If mathematics truly is ontology, and if mathematics is independent of questions of knowledge, then all questions about differing subjective perspective on reality, different cultural perspectives on reality, etc., fall to the wayside as interesting psychological, anthropological, and sociological speculations, but speculations that are quite irrelevant to ontological researches. In short, ontology and philosophy no longer need concern themselves with cultural studies, linguistics, or the social sciences. This is what it means, for Badiou, to say that math inscribes the real.

In a closely related vein, Badiou's thesis allows us to depart, once and for all, from Heidegger's endless preparations for properly posing the question of being. There is no need to engage in an elaborate hermeneutic of Dasein as that being that is "ontic-ontological" and who has a pre-ontological understanding of being, as the being of being is exhausted in its mathematicity.
To my thinking, these consequences cannot but come as a breath of fresh air to philosophy insofar as it has increasingly come to be dominated by cultural studies, rhetorical analysis, and pseudo-pious phenomenological discourses.

However, I wonder nonetheless whether Badiou hasn't moved a bit too quickly. Badiou's thesis regarding materialism seems to be that insofar as science always approaches matter mathematically, a mathematical ontology is necessarily a materialist ontology:
...it [the thesis that mathematics = ontology] is a fully materialist thesis, because everyone can see that the investigation of matter, the very concept of matter, is a concept whose history shows it to be at the edge of mathematicity... 'Matter' would simply be, immediately after being, the most general possible name of the presented (of 'what is presented'). Being-as-being would be that point of indistinction between the possible and the real that only mathematics apprehends in the exploration of the general configuration of the purely multiple. Matter, in the sense in which it is at stake in physics, is matter as enveloping any particular presentation--and I am a materialist in the sense that I think that any presentation is material. (Ethics: An Essay on the Understanding of Evil, 130, italics mine)
What seems to be missing in Badiou's account of materialism is precisely any discussion of this "concept at the edge of mathematicity". I have highlighted these passages on how being-qua-being is the point of indistinction between the possible and the real to indicate that the moment we enter the realm of the material, we are no longer dealing with something merely possible, but rather with something actual. That is, something has been added to what we're talking about, what we're investigating, that isn't strictly mathematical. Although mathematics is certainly an essential dimension of physics or any science (I would accept Kant's thesis that mathematization is a necessary condition of scientificity), the object of any particular science is an object that cannot itself be mathematically deduced.

Doesn't this edge of mathematicity, this element that is enveloped by mathematicity, itself have some claim to being? Hallward hits the nail on the head when he responds to Badiou's elaboration of materialism, by remarking that, "It seems, however, that your most basic concept, the concept of a situation, oscillates somewhat between an essentially mathematical order and what appears to be a no less essentially eclectic order, combining heterogeneous elements of actuality" (129). The problem as I see it is that unlike being-qua-being, a situation (what Badiou now calls a "world"), does not straddle the distinction between the possible and the real. A situation is real, it is actual, it is this situation and no other.

It thus seems to me that for Badiou there is a tremendous gap between ontology as the "presentation of presentation" or pure multiplicity "without connection", and the ordered situations of the world. One might respond by arguing that ontology is not in the business of explaining situations as it only studies pure multiplicities, not "consistent multiplicities". However, Badiou himself says otherwise:
What links a being to the constraint of a local or situated exposure of its manifold-being is something we call this existent's "appearing." It is the existent's being to appear insofar as Being as a whole does not exist. Every being is being-there. This is the essence of appearing. Appearing is the site, the "there" of the multiple-existent insofar as it is thought in its being. Appearing in no way depends on space or time, or more generally on a transcendental field. It does not depend on a Subject whose constitution would be presupposed. The manifold-being does not appear for a subject. Instead, it is more in line with the essence of the existent to appear. As soon as it falls short of being localized according to the whole, it has to assert its manifold-being from the point of view of a non-whole, that is, of another particular existent determining the being of the there of being-there [incidentally, Hegel already conceives appearing as appearing to another existent or Relation in the "Doctrine of Essence", Science of Logic].

Appearing is an intrinsic determination of Being. The localization of the existent, which is its appearing, involves another particular being: its site or situation. This is why it can be seen immediately that appearing is as such what connects or reconnects an existent or its site. The essence of appearing is the relation. (Briefings on Existence, pg. 162)
I confess that I find these remarks exhilerating, though I am unable to understand Badiou's thesis or the logical entailment necessitating that "because the whole is not, the existent must appear. " It seems to me that there is a fundamental ontological axiom here that is currently the is a central theme of a good deal of contemporary theory (Deleuze's thesis that the Whole is not giveable in Cinema 1 and that this is a condition for the given, Zizek's thesis that the One is not, Lacan's thesis that the world does not exist, etc. One of my central questions is that of how to understand the relation between the in-existence of the Whole (not simply that we cannot know the whole, but that the whole does not exist --and the givenness of the given. I am not sure why this question strikes me as so important, but there's something there. Now, when Badiou glosses category theory in an earlier essay "Group, Category, Subject", he largely describes my own ontological project:
In category theory, the initial data are particularly meager. We merely dispose of undifferentiated objects (in fact, simple letters deprived of any interiority) and of 'arrows' (or morphisms) 'going' from one object to another. Basically, the only material we have is oriented relations. A linkage (the arrow) has its source in one object and target in another. Granted, the aim is ultimately for the 'objects' to become mathematical structures and the 'arrows' the connection between these structures. But the purely logical initial grasping renders the determination of an object's sense entirely extrinsic or positional. It all depends on what we can learn from the arrows going toward that object (whose object is the target), or of those coming from it (whose objects is the source). An object is but the marking of a network of actions, a cluster of connections. Relation precedes Being. This is why at this point of our inquiry we have established ourselves in logic, and not ontology. It is not a determined universe of thought we are formalizing, but the formal possibility of a universe" (Briefings on Existence, "Group, Category, Subject", 144).
From my perspective, this is the place to begin ontologically, as I see no way that can account for the emergence of Relation on the basis of pure, unconnected multiplicity as described by Badiou. I am not certain why Badiou refers to this as a logic (generally I'm fuzzy on his conception of logic overall and why he is so hostile to placing math under logic), nor am I sure why he refers to this as the articulation of a possible universe, rather than a determined universe. However, when Badiou suggests that Relation precedes being, and proposes that we conceive entities as "networks of action", "clusters of connections", "bundles of relations", I think this is the right direction to move in ontologically. It is this move towards conceiving beings as activities, as networks, as doings, that allows us to do away with substance ontology. This, then, is the central difficulty. One the one hand, Badiou wishes to claim that ontology is indifferent to the distinction between the possible and the real. Yet on the other hand, Badiou wishes to argue that it belongs to the essence of Being, that it is intrinsic to being, to appear. How can this be? How can we simultaneously affirm both of these theses without undermining the thesis that ontology is indifferent to the distinction between the possible and the real?

Could it be that this is the real source of Badiou's hostility to Deleuze? It will be recalled that Deleuze defines his transcendental empiricism as that ontology that articulates the conditions of real being, and not all possible being. In developing this ontology Deleuze, following Bergson, advances a substantial critique of the category of possibility, arguing that the dialectic between the possible and the real is unable to give any account of how the real is realized insofar as the real in no way differs from the possible (Kant's famous critique of the ontological proof for the existence of God, wherein he argues that "existence is not a real predicate"). That is, accounts of realization are unable to explain what the real contributes to being. It seems to me that Badiou finds himself in a very similar position and that for this reason it is difficult to identify his ontology as being genuinely materialist.

H θεωρία συνόλων και η Υπόθεση του Συνεχούς σε επτά λεπτά για μη μαθηματικούς

Σ,Καπελλίδη:Θεωρία Συνόλων

Οι Σημειώσεις του Σ.Καπελλίδη εδώ

Ενα πολύ κατατοπιστικό βιβλίο, υπο την μορφή σημειώσεων για την Θεωρία Συνόλων

Χάρις στην επιμέλεια του μαθηματικού Μ.Χατζόπουλου διατίθεται σε εύχρηστη τυπόσιμη μορφή

Η επανάσταση του Zermelo και ο Badiou

Σε κείμενο του Lyn Sebastian Purcell για την σχέση Badiou Derrida γίνεται εκτενής ανάλυση της συνολοθεωρίας.Παραθέτω ένα απόσπασμα και σχετικό σύνδεσμο .


ZERMELO'S REVOLUTION We should like to begin our engagement with Badiou by noting a ghostly presence within Badiou’s own thought—a specter (revenant) who haunts the whole of his ontology. Consider the following statement from Being and Event: ‘That it is necessary to tolerate the almost complete arbitrariness of a choice, that quantity, the very paradigm of objectivity, leads to pure subjectivity; such is what I would willingly call the Cantor-Gödel-Cohen-Easton symptom’ (BE 280). We are not here interested in this full itinerary, which is punctuated by the names of four great mathematicians, but only its first point, and the unmentioned name that stands between Cantor and Gödel, namely Ernst Zermelo. This mathematician, who is present only as a dash in Badiou’s thought, we argue forms the symptomal point of his enterprise. If attended to correctly, we argue it is here that one can uncover an alternative appropriation of Cantor. 2.1 Against the Whole The ‘Cantorian Revolution’ in Badiou’s thought is tantamount to the rejection of the whole. After Cantor established that it was possible to think the infinite, reversing more than two millennia’s wisdom on the matter, there was a short period in which set theory operated by use of something like Gottlob Frege’s unlimited abstraction principle, which had the advantage of allowing mathematicians to obtain almost all the sets necessary for mathematics from it alone.3 It was as follows: given a well defined property P, there exists a unique set A that consists of only those things that have the property P. Usually, such a set is expressed with braces as follows: {x | P(x)}, which means ‘the set of all x having the property x’. The difficulties with this principle are well-known: such a principle allows for selfmembership. If some sets can be members of themselves, then others are sets that are not members of themselves. That this distinction results in a logical paradox was an observation Bertrand Russell made (and Zermelo independently), and has come to be known as Russell’s paradox.4 The response of the mathematical community was to try to avoid this inconsistency by addressing or reformulating the abstraction principle. This aim was the point of Russell’s theory of types. Yet, in the end the solution that was provided by Ernst Zermelo (in 1908) proved most acceptable. 3We note that Badiou rightly counts Frege as the second attempt to think a set, while Cantor’s intuition of objects constitutes the first (BE 40). 4 For those interested, Badiou reproduces this paradox in Being and Event pages 40-1, and more thoroughly in Logics of Worlds, trans. Oliver Feltham, New York, Continuum Press, pp. 153-5 (Henceforth: LW

Όλο το κείμενα εδώ

James Juniper:Badiou,Whitehead,Leibniz,Deleuze

Το κείμενο έχει το εξης ενδιαφέρον εύρημα.Διατυπώνεται πως η Μαθηματική "Μεταφυσική" του Badiou, βασίζεται σε μια διαστρεβλωτική ανάγνωση του Leibniz.Αναδεικνύεται πως η  μοναδολογία του Leibniz έχει ενσωματωμένα στοιχεία δυναμικής και ρευστότητας τα οποία όμως ο ΑΒ δεν αναγνωρίζει, έτσι ώστε η δικη του Μαθηματική σύλληψη να εμφανίζει καινοτομίες οι οποίες όμως τελικά δεν υπάρχουν.Το συμπέρασμα είναι αρκετά επικριτικό:O Badiou κατηγορείται για μια "παθολογική άρνηση και σκόπιμη αποσιώπιση" του έργου του Leibniz.
LLS

Mis-readings of Leibniz: Deleuze and Whitehead against Badiou

James Juniper, Lecturer in the School of Economics, Politics and Tourism and Associate of the Centre of Full Employment and Equity, the University of Newcastle

Abstract:

The paper is motivated by the desire to identify exactly what Leibniz has contributed to Deleuze and Whitehead’s particular version of (non-organic) vitalism. This reading of Leibniz is compared with those of Badiou (with a little help from Heidegger, who specifically demonstrates the dependence of logic on ontology rather than of ontology on logic). The paper compares each of these philosopher’s interpretations of the fundamental principles that ground Leibnizian monadology, with the intention of highlighting the implications of these readings for political theory. In particular, Badiou’s notion of a schema of torsion is examined and distinguished from Deleuze’s notions of actualization and realization.

Introduction_________________________________________________________1

Leibniz on Substance and Knowledge___________________________________3

The Debate over Leibniz’s Principles____________________________________8

Badiou’s Reading of Leibniz__________________________________________11

Conclusion_________________________________________________________14

Introduction

The political motivation for this paper is straightforward. In his review of Logic and Existence, Jean Hyppolite’s reconstruction of Hegelian philosophy, Deleuze (1953) foreshadows the necessity to construct a philosophy of difference that does not “take things up to contradiction”. Along with his interpretations of Duns Scotus, Hume, Kant, Spinoza, Whitehead and Bergson, Deleuze’s analysis of Leibniz’s philosophy of expression makes a major contribution to this task of constructing an alternative and non-organic form of Vitalism, designed to ultimately displace the more familiar organicist tradition that can be traced from Rousseau’s conception of the social contract through to the German Idealism of Hegel and his contemporaries.

Under the latter form of vitalism, the distinction between the (externally caused and atomistic) machine and the (self-caused and internally related) organism carries over to that holding between the state and civil society respectively (Quadrio, 2009). Under the former, the genesis of products (Modes) from their respective constitutive essences (Attributes) is (integrated) through Substance as the infinite power to act, which in theoretical terms is conceived genealogically as both an immanent cause and as self-cause (Substance acting through its essence). A closely related paper (Juniper, forthcoming), examines how Deleuze constructs this genealogy from a close reading of Kant’s three Critiques (specifically, the Third Critique’s “Analytic of the Beautiful”, and both the Transcendental argument and the “Transcendental Dialectic” in the second half of the First Critique) distinguishing it from Badiou’s conception of Kantian philosophy as

1

a “subtractive ontology”

1. This paper will compare Deleuze’s reading of Leibniz with that of Badiou, with the objective of identifying the specifically Leibnizean sources of this (non-organic) vitalism contrasting this reading with that of Badiou. A more detailed examination of the political implications of such a vitalism, both in theoretical and practical terms, must remain the task of subsequent research2.

In his work on Leibniz, Heidegger (1984) comments explicitly on the diversity of interpretations of Leibniz’s monadological metaphysics, suggesting that diversity of this kind is true of all authentic philosophy. The result of any effort to distil the essence of Leibniz in such a way that it would be agreed upon by all, he reasons, would surely be something dead.

Alain Badiou (2005: 315), for one, finds in Leibniz the paradoxical combination of a “prodigiously modern thought” with a “conscious conservative will”. In actuality, however, he suggests that this will served as a prelude to Leibniz’s more radical anticipations because inventive freedom could only be exercised once an adequate foundation for thinking could be guaranteed.

In

Being and Event Badiou provides us with a series of philosophical, poetic and political examples of what he calls the “generic event”. Examples of such “truth-events” range from the October Revolution for Lenin, Crick and Watson’s discovery of the double-helix, and St. Paul’s revelatory “road to Damascus” experience of the truth of the resurrection, through to Picasso and Braque’s discovery of synthetic cubism. As Bosteel (2001, 2002) reveals, Badiou’s manner of grasping the event is closely related to his earlier thinking about ‘schemas of torsion’. For him, Rousseau’s notion of the social pact provides one of the clearest political examples of torsion at play. Badiou homes in on the apparent contradiction in Rousseau’s conception of the social pact as something that is both presupposed and constituted by the ‘general will’. He discerns in this paradoxical torsion the signature of the evental form: namely, once constituted, it’s being becomes what is always and already presupposed (Badiou, 2005: 345-6). Thus, for Rousseau the body politic operates as a supernumerary multiple for which the ultra-one of the event is the social pact! The prospect of a self-belonging of the body politic to the very multiple that it is, could never violate the pact as it would destroy itselfi! In this manner the pact supplements the state of nature insofar as it is interposed between nature (void) and itself.

Badiou discerns the presence of a similar “schema in torsion” at play in the work of other philosophers and poets, including Leibniz. In the work of the latter philosopher, Badiou suggests that torsion is instituted by the Leibnitzian question,

par excellence, “Why is there something rather than nothing?” On the basis of there being something rather than

1

Similar interpretations of Deleuze along these Kantian lines can be found in Kerslake (2002), Shaviro (2009) and Smith (1997). Badiou’s work, however, is not examined in any of these papers.

2

In Juniper (forthcoming) it is argued that Deleuze’s notion of double causality provides vitalism with a materialist framework for transforming the traditional organicist dichotomy between the freedom and autonomy of noumenal spirit and the mechanism of phenomenal nature. Moreover, it is shown that this transformative notion can be infused with a mathematical robustness through the (non-metaphorical) deployment of the category theoretic concept of an adjunction, insofar as the latter is assigned the role of an Idea in both the Kantian and Lautmanian sense of this term.

2

nothing, he observes that Leibniz is able to infer that essence, in and of itself, strives for existence or that logic desires the being of what conforms to it. While the axioms or principles that Leibniz constructs impose the question, Badiou observes, the complete response afforded by the Leibnizian system of monadology, which supposes the axioms, also confirms them. By way of justifying this observation the following section of the paper will largely draw on Heidegger’s reading of Leibniz, examining the latter’s conceptions of substance and knowledge. It will lay bare Heidegger’s core objective, which is to demonstrate the dependence of logic on ontology rather than of ontology on logic. This section sets the context for an interrogation of different interpretations of Leibniz’s underlying principles or axioms on the part of Heidegger, Russell, Broad, and Deleuze. Badiou’s reading of Leibniz is examined in the fourth section of the paper and conclusions follow.

Leibniz on Substance and Knowledge

This section of the paper draws on Heidegger’s masterful interpretation of Leibniz for two purposes. First, Heidegger clearly demonstrates how Leibniz takes over from Thomas Aquinas and the Scholastics the crucial distinction they make between active and passive power. Specifically, Heidegger shows how this distinction informs Leibniz’s crucial notion of

conatus or striving, the conceptualisation of pre-hension as a “gripping-in-advance”, and his subsequent analysis of processes of actualisation and realization. Second, in clarifying the step-by-step unfolding of knowledge and judgement in Leibniz’s logic, Heidegger argues, against both Hegel and the seemingly opposed tradition of logical positivism, that logic must be grounded in metaphysics rather than metaphysics in logic. This claim justifies the efforts of Deleuze and Whitehead to fashion a metaphysics of difference that does not take things up to contradiction.

Heidegger contends that, at the centre of Leibniz’s monadology is an attempt to construct a positive rather than a negative definition of substance. While Descartes had defined substance as the thing, which so exists that it needs no other thing in order to exist, Spinoza had, in like manner, defined substance as that which is in itself and is conceived by itself, that of which the concept does not require the concept of another thing from which it needs to be formed. Leibniz discovered the positive conception that he required in Giordano Bruno’s notion of the monad (μοναδ ): a Greek term conveying the characteristics of simplicity, unity, oneness, individuality, and solitude.

Monads are conceived as formal “atoms”, ‘little gods”, or “animating points”: far from requiring any unification in themselves, they are that which gives unity insofar as they are primordially simple points of active force (

vis primitiva) or principles of formation (forma or ειδοσ) (Heidegger, 1984, 77; citing Leibniz, 1969, 482). Monads grasp themselves along with perception, presenting the world from a unique viewpoint so that each monad is the universe in concentrated form. Crucially, although monads are oriented in advanced towards a pre-disposed harmony, they share with all finite substance a passivity or resistance, which is correlated with what the monad is not but could well be. This negative and finite aspect of the drive characterises what Leibniz understands by prime matter (materia prima) and, in addition, conditions the monad’s relationship to the resistance or weight associated with secondary matter (materia secunda, massa).

3

For both Leibniz and the Scholastics Divine knowledge becomes the cognitive ideal for humans. This knowledge, however, includes both what is possible and what is or will become actual. The distinction Aquinas made between the active disposition to act and the passive disposition towards being formed carries over to a twofold distinction between active and passive forms of power or potentia. In Leibniz, this active force (vis primitiva) is embodied in his conception of the drive (conatus), conceived as a self-propulsive striving

ΣΥΝΕΧΙΖΕΤΑΙ ΕΔΩ

The Limits of The Subject in Badiou’s Being and Event

The Limits of The Subject in Badiou’s Being and Event

Brian Anthony Smith

Abstract: This essay is an examination of the limits of the model of the subject that Badiou establishes in Being and Event. This will concentrate on both Being and Event, and the later ethical developments introduced in Ethics: An Essay on the Understanding of Evil. My aim will be to show that there is a possible subjective figure, based on the independence of the Axiom of Choice, which remains unexamined in both these works. The introduction of this new subjective figure not only complicates Badiou’s ethical categories of Good and Evil, but it also raises questions about the nature of the subject in general in his philosophy.

Keywords: Badiou; Axiom of Choice; Subject; Individual; Non-constructible Sets; Temporality

The figure of the subject in Badiou’s Being and Event[1] is key to understanding the link between his revival of a systematic ontology, in the form of set theoretical mathematics, and his wider philosophical and ethical concerns. Through a critical examination of the subject, as it appears in Being and Event, and an evaluation of the categories of subjective Good and Evil, developed in his book Ethics: an Essay on the Understanding of Evil[2], I hope to probe the limits of this subjective model and to propose a new subjective figure that appears possible, but unexamined, in either of these works.

My analysis will focus on two main points: first, Badiou’s use of the Axiom of Choice, as a key factor in his philosophy that allows for the possibility of a subject, and, second, his selective use of set theoretical forcing, which concentrates mainly on the independence of the Continuum Hypothesis.

Badiou’s ethics is based on the capacity of individuals to distinguish themselves from their finite animal nature and to become immortal; to become immortal is to become a subject (E 12, 132). What constitutes this singular ability, our rationality, is the use of mathematics (E 132). Specifically it is the Axiom of Choice that elevates the human animal to the level of a potential subject. This axiom expresses an individual’s freedom, a freedom equivalent to the affirmation of pure chance. [3] It is this capacity that allows an individual to affirm its chance encounter with an event; the moment of this affirmation is called intervention and marks the birth of a subject (BE Meditations 20 and 22).

The importance of the Axiom of Choice is clear; it provides the connection between the individual, the event and the subject. It defines the individual and provides the condition under which subjectivity is possible.

Badiou’s appeal to Paul Cohen’s theory of forcing is predominately directed toward his proof of the independence of Georg Cantor’s Continuum Hypothesis. But in Cohen’s book, Set Theory and the Continuum Hypothesis, the method of forcing is used equally to prove the independence of the Axiom of Choice.[4] For Badiou, the Continuum Hypothesis is a restrictive theorem of ontology; it confines ontology to the merely constructible and neuters the individual by reducing the power of the Axiom of Choice (BE Meditations 28 and 9). Under such a restriction the Axiom of Choice loses its independence as an axiom and becomes a theorem, a mere consequence of the system (BE 305-7). Cohen’s theory of forcing is important as it shows that it is possible to construct a model of set theory in which the Continuum Hypothesis fails, thus liberating us from its restrictive bonds. In the process it not only reinstates the full power of the Axiom of Choice, the freedom of the individual, but also, through the use of this axiom, a subject emerges.

The mathematical theory of forcing, as it is applied to the Continuum Hypothesis, provides Badiou with the paradigmatic model for the subjective response to an event. The subjective process emancipates the individual, through a correct use of their freedom in the face of an event, from some restrictive condition of their situation. This production of a truth introduces true novelty that expands, or extends, the subject’s situation. This forms the basis of Badiou’s theory of ethics. Subjective endeavour, forcing the truth of an event, forms the positive concept of the Good and ‘It is from our positive capability for Good… that we are to identify Evil’ (E 16). The range of types of Evil can be identified with false, abortive or totalizing activities that try to subvert a truth procedure, the Good being the practice of the virtues of discernment, courage and moderation (E 91). The subject remains faithful to the event and its consequences.

The clarity and decisive character of Badiou’s ethics is refreshing, but is it the case that the subject is always intrinsically good? What would happen if we examined the consequences of a valid subjective process, based on the mathematical model of forcing, which instead of liberating an individual, in the process of their subjective action, condemned them? I think the independence of the Axiom of Choice provides such an occasion. What would be the consequences of forcing a situation in which the Axiom of Choice fails, in which the freedom of the individual is denied and the competence of the subject questioned?

H συνέχεια και όλο το άρθρο έδώ

Βασικά Badioumathematics-Από το "Στο ντιβάνι με το Λακάν"

 

Αρχική Δημοσίευση:Στο ντιβάνι με το Λακάν.Σύνδεση Εδώ

 

Απόσπασμα από το άρθρο Generic sovereignty: the philosophy of Alain Badiou του Peter Hallward.

Mετάφραση : Kraftwerk

 

 

Η οντολογία του Μπαντιού

Η θεωρία των συνόλων έτσι όπως διατυπώθηκε από τον Καντόρ και εμπλουτίστηκε από τους Ζερμέλο, Φράνκελ και Γκέντελ αναπτύσσει την αφηρημένη μορφή του πολλαπλού ως πολλαπλού. Σύμφωνα με τον Μπαντιού τα μαθηματικά είναι ο λόγος της οντολογίας, η περιγραφή του είναι-ως –είναι ή ως καθαρής πολλαπλότητας (πράγμα το οποίο δεν σημαίνει ότι τα μαθηματικά προσφέρουν τον λόγο που ερμηνεύει ή βγάζει νόημα από αυτή την οντολογία). Η μαθηματική απόδειξη εγγυάται απευθείας από το είναι. Αν τα μαθηματικά είναι ένας διαυγής τομέας γνώσης, τελείως άμεσος στο είναι, αυτό οφείλεται στο ότι δεν υπάρχει αυθύπαρκτο μαθηματικό αντικείμενο. Τα μαθηματικά κάνουν ξεκάθαρα τη λογική που παρουσιάζει κάθε διανοητό ( ή αδιανόητο) αντικείμενο, την λογική της παρουσίασης όπως τυποποιείται στα αξιώματα και τα θεωρήματα της θεωρίας των συνόλων.

Ένα πολλαπλό είναι συνώνυμο με ένα σύνολο. Αυτό που η θεωρία των συνόλων περιγράφει είναι η διάκριση μιας απόλυτης πολλαπλότητας στοιχείων χωρίς αναφορά σε οποιεσδήποτε θετικά προσδιορισμένες δομικές μονάδες ή ορισμούς. Η πολλαπλότητα όπως προσδιορίζεται μέσα από την θεωρία των συνόλων είναι “sans-un”. Πρόκειται για μια πολλαπλότητα που συναντάται- ως το μόνο συμπέρασμα συνεκτικό με τον θάνατο του θεού- με σαφήνεια και επάρκεια στο κενό σύνολο ή το κενό. «Εν αρχή ην το κενό». Σύμφωνα με τον Μπαντιού η ατομικότητα δεν μπορεί παρά να είναι το αποτέλεσμα μιας δράσης. Ένα Είναι-μετρημένο σαν ένα. Αυτή η ενότητα ενός οποιουδήποτε δοθέντος στοιχείου ενός συνόλου είναι αποτέλεσμα του γεγονότος ότι ανήκει σε αυτό το σύνολο, όπως η ενότητα ενός μέρους ή υποσυνόλου ενός συνόλου σχετίζεται με τον εγκλεισμό σε αυτό το σύνολο. Παράγεται από ένα ιδιαίτερο μέτρημα ως ένα.

Αυτό που ο Μπαντιού αποκαλεί κατάσταση (situation) είναι συνώνυμο ξανά με ένα σύνολο. Είναι κάθε πολλαπλότητα δομημένη από μια συγκεκριμένη καταμέτρηση, από συγκεκριμένα κριτήρια εγκλεισμού ή αποκλεισμού. Η δομή μιας κατάστασης είναι αυτή που την θεμελιώνει ως συγκεκριμένη κατάσταση. Είναι αυτή που καταμετρά ή διακρίνει τα στοιχεία της ως δικά της στοιχεία. Κάθε κατάσταση είναι το αποτέλεσμα μιας δόμησης. Η δομή μιας κατάστασης πχ το σύνολο της ζωής για παράδειγμα διαχωρίζει τα στοιχεία του συνόλου των ζώντων πραγμάτων από τα στοιχεία από το σύνολο των μη ζώντων, των βιοχημικών στοιχείων. Η δομή της κατάστασης «Γαλλία» διαχωρίζει το εθνικό σύνολο από τους Γάλλους πολίτες κτλ. Είναι πιθανό κάποιος να υποστηρίξει ότι η συνέπεια μιας τέτοιας προσέγγισης είναι ένα είδος αφελούς εμπιστοσύνης σε μια ημι-στατιστική απαρίθμηση. Αυτό θα ίσχυε αν δεν έλεγε κανείς ότι για τον Μπαντιού κάθε ανθρώπινη κατάσταση ή σύνολο (ατομικό ή συλλογικό) χωρίς να σταματά να γίνεται αντιληπτό ως σύνολο συνθέτει ένα μη μετρήσιμο άπειρο. Πάντα αυτό που έχει αυθεντική αξία είναι αυτό που βρίσκεται πέρα από τον αριθμό και διαφεύγει κάθε πιθανής καταμέτρησης.

Τρία στοιχεία μπορούν να θεωρηθούν σημαντικά για την θεμελίωση της οντολογίας του Μπαντιού. Το πρώτο είναι το αξίωμα του άπειρου, η ενεργή θεώρηση ενός ριζικού απείρου πέρα από κάθε πιθανή απόδειξη ή «κατασκευή». Για τον Μπαντιού όπως και για τον Καντόρ το άπειρο είναι το όριο του διανοητού, του ανθρώπινου-του πεπερασμένου. Θεμελιώνοντας την μη συνεκτικότητα μιας «Μεγάλης Ολότητας {Grand Tut} (παρουσιαζόμενη από τον Καντ) ως ένα που τα εγκλείει όλα και ανήκει στον αυτό του (όπως τέθηκε από τον Καντόρ) η θέση της απειρότητας του είναι, είναι αναγκαίως μια οντολογική απόφαση, ή με άλλα λόγια ένα αξίωμα.

Το δεύτερο είναι το αξίωμα της θεμελίωσης ή του κενού. Η θεωρία των συνόλων ξεκινά με την παραδοχή ότι υπάρχει το κενό. Το κενό σύνολο δεν έχει στοιχεία. Η θεωρία των συνόλων υποστηρίζει ότι το κενό ή κενό σύνολο- το σύνολο στο οποίο τίποτα δεν ανήκει- εμπεριέχεται σε κάθε πιθανό σύνολο (περιλαμβανομένου και του εαυτού του). Το κενό είναι το καθολικά εγκλεισμένο υποσύνολο. Αυτό το άδειο υποσύνολο είναι θεμελιώδες με την έννοια ότι για να θεμελιωθεί ένα σύνολο πρέπει να έχει τουλάχιστο ένα στοιχείο που να μην παρουσιάζει τίποτα από αυτά που παρουσιάζει η πολλαπλότητα (το ίδιο το σύνολο). Ο Μπαντιού δίνει ένα παράδειγμα από το σύνολο των ζώντων οργανισμών: αν τα κυτταρικά οργανίδια μπορούν να περιληφθούν σε αυτό το σύνολο , σε ένα συγκεκριμένο σημείο της κυτταρικής οργάνωσης υπάρχουν στοιχεία (πχ μιτοχόνδρια) των οποίων τα στοιχεία (πρωτεϊνικές μεμβράνες, βιοχημικές δομές) δεν είναι οι ίδιες στοιχεία του συνόλου των ζώντων οργανισμών. Τέτοιες δομές είναι «θεμελιώδεις» για το σύνολο των ζώντων οργανισμών με την έννοια ότι πάνω σε αυτές δομούνται τα ζωντανά στοιχεία, αλλά τα ίδια δεν είναι ζωντανά. Στην περίπτωση των μαθηματικών οντοτήτων το κενό είναι αυτός ο θεμελιώδης όρος. Το κενό εγκλείεται σε κάθε μαθηματικό σύνολο αλλά δεν ανήκει σε κανένα. Μια άμεση συνέπεια είναι ότι κανένα θεμελιωμένο σύνολο δεν μπορεί να ανήκει στον εαυτό του. Για παράδειγμα το σύνολο όλων των αριθμών δεν μπορεί να είναι ένας αριθμός. Ένα σύνολο δεν μπορεί να αυτοθεμελιωθεί. Για να το θέσουμε με άλλους όρους το κενό σύνολο δεν μπορεί να δομηθεί αλλά είναι το θεμέλιο όλες τις δυνατές δομήσεις. Το απόλυτα πρωταρχικό σημείο κάθε οντολογίας είναι το όνομα του κενού επειδή σε μια κατάσταση δεν υπάρχει διανοητός τρόπος να συναντήσεις το κενό. Κάθε τι που παρουσιάζεται σε μια κατάσταση απαριθμείται ως ένα. Το κενό είναι αυτό πάνω στο οποίο βασίζεται κάθε καταμέτρηση, εξαιτίας του υπάρχει η παρουσίαση.

Το τρίτο αξίωμα σχετίζεται με την υπέρβαση σε ένα οποιοδήποτε σύνολο (συμπεριλαμβανομένου και το κενό σύνολο) των μερών (υποσυνόλων) έναντι των στοιχείων. Υπάρχουν εμφανώς πολλοί περισσότεροι τρόποι συνδυασμού των στοιχείων ενός συνόλου σε μέρη (ή υποσύνολα) από τον αριθμό των στοιχείων αυτού του συνόλου. Ένα πεπερασμένο σύνολο με ν στοιχεία θα έχει 2 εις την ν υποσύνολα ή μέρη. Ένα σύνολο με άπειρο αριθμό στοιχείων θα έχει ένα απείρως μεγαλύτερο αριθμό από υποσύνολα. Για παράδειγμα το σύνολο με τρία στοιχεία χ,ψ,ζ, έχει 8 υποσύνολα (2 εις την Τρίτη): {χ}. {ψ}, {ζ}, {χ,ψ}, {χ,ζ}. {ζ,ψ}, {χ,ψ,ζ}, {ο}- αυτό το τελευταίο υποσύνολο εγκλείεται σε όλα τα σύνολα. Υπάρχει πάντα – ανεξάρτητα από το τι είναι το α – τουλάχιστο ένα στοιχείο του ρ(α) που δεν είναι στοιχείο του α ακόμη και στην οριακή περίπτωση του κενού συνόλου το οποίο δεν έχει στοιχεία. Αυτό σημαίνει ότι κανένα πολλαπλό δεν είναι σε θέση να κάνει ένα από όλα όσα εγκλείει.

Αυτό που ο Μπαντιού αποκαλεί κράτος (state) μιας κατάστασης ή μιας «μεταδομής μιας κατάστασης» είναι αυτό που παρεμβαίνει για να θέσει υπό έλεγχο αυτή την υπέρβαση, να θεμελιώσει το σύνολο των μερών (υποσυνόλων). Το κράτος οργανώνει τους ποικίλους τρόπους διευθετώντας τα μέρη ενός συνόλου σε ένα ενσωματωμένο όλο. Για την δομή μιας κατάστασης, η αρχή της απαρίθμησής της (απαρίθμηση των στοιχείων της) δεν επαρκεί από μόνη της για να τη διατηρήσει ως δομημένη. Η δομή δεν παρέχει τάξη στην υπέρβαση των μερών πάνω στα στοιχεία. Έτσι αν μια κατάσταση αντιστέκεται- και σύμφωνα με τον Μπαντιού είναι στη φύση των καταστάσεων να αντιστέκονται- πρέπει να υπάρχει συγχρόνως με την δομή της, μια αρχική «μεταδομή» η οποία διευθετεί την σχέση μεταξύ της δομής και των στοιχείων, μεταξύ του συνόλου και αυτού που απαριθμείται. Αυτό που ο Μπαντιού ονομάζει «γνώση» (savoir) (ή η γλώσσα μιας κατάστασης) παρέχει άπειρους τρόπους διευθέτησης των στοιχείων μιας κατάστασης, αλλά δεν μπορεί να παρέχει ενότητα σε αυτές τις διευθετήσεις. Το ότι το σύνολο στην παρουσίαση διαφεύγει της απαρίθμησης απαιτεί κάθε δομή να διπλασιάζεται από μια μεταδομή διαμέσου της οποίας η δομή μιας κατάστασης-κάθε δομημένης παρουσίασης- μετριέται ως ένα. Μια κατάσταση απαριθμεί στοιχεία. Το κράτος μιας κατάστασης απαριθμεί τα μέρη του, τους τρόπους συνδυασμού των στοιχείων. Σε ένα εθνικό σύνολο, για παράδειγμα, τα στοιχεία του οποίου εγκλείουν τον πληθυσμό που απαριθμείται ως εθνικός, το κράτος είναι αυτό που οργανώνει τα μέρη του, όπως οι φορολογούμενοι, οι εκλογείς, εγκληματίες και άλλα.

Με άλλα λόγια, το κράτος διασφαλίζει ότι η εν δυνάμει αναρχική οργάνωση των κοινωνικών συνδυασμών παραμένει δομημένη με τέτοιο τρόπο έτσι ώστε να διατηρείται η κατάσταση. Το κράτος κρατά τα πράγματα στην θέση τους. Είναι ένα είδος πρωτογενούς αντίδρασης στην οντολογική αναρχία. Το κράτος κυριολεκτικά είναι η αρχή μιας εύτακτης αντικειμενικότητας, η βίαιη επιβολή της τάξης είναι εγγενές χαρακτηριστικό της αντικειμενικότητας. Ή ξανά το κράτος απαγορεύει την αποκάλυψη του κενού που είναι θεμελιώδες σε κάθε κατάσταση. Σε μια καπιταλιστική κατάσταση , για παράδειγμα, αυτό το θεμελιωτικό κενό , που εγκλείεται αλλά δεν παρουσιάζεται, δεν ανήκει, είναι το προλεταριάτο, η αόρατη αλλά παραγωγική βάση πάνω στην οποία η κατάσταση υπάρχει ως τέτοια. Το κράτος σε αυτή την κατάσταση απαγορεύει την αποκάλυψη ή την ενδυνάμωση του προλεταριάτου. Στην σύγχρονη γαλλική κατάσταση , το κράτος επιβεβαιώνει έναν παρόμοιο αποκλεισμό μιας αόρατης τάξης μεταναστών.

Πάντα για τον Μπαντιού, το κράτος είναι το σοβαρό ερώτημα, το κεντρικό ερώτημα. Κάθε μεγάλη εξέγερση της εργαζόμενης μάζας, την θέτει ενάντια στην κατάσταση και μια σκέψη (une pensee) δεν είναι τίποτα άλλο από μια επιθυμία για να υπερβούμε το κράτος. Την δεκαετία του 90 όπως και του 60 η από- στατιστικοποίηση της Αλήθειας παραμένει για μας ένα πρόγραμμα σκέψης : η καρδιά του ερωτήματος είναι η εγγύηση του κράτους για την διατήρηση της τάξης, ο διαχωρισμός μεταξύ παρουσίασης και αναπαράστασης, μεταξύ δομής και μεταδομής. Με άλλα λόγια, ο στόχος παραμένει το τέλος της αναπαράστασης (το τέλος της μεταδομής) και την καθολικότητα της απλής παρουσίασης, η ελευθερία των στοιχείων χωρίς την εξαναγκασμένη ενσωμάτωση των συνδυασμών τους, ένα ελεύθερο και ισότιμο ανήκειν χωρίς αναγκαστικούς εγκλεισμούς.

Η παραγωγή γενολογικών αληθειών θα σχετιστεί πρωτίστως και με το πλήρες ρίσκο του κενού, την από- στατιστικοποίηση της σκέψης. Αν θεωρούμε ως statelike (etatique) αυτό που απαριθμεί, ονομάζει και ελέγχει τα μέρη μιας κατάστασης, τότε όταν μια αλήθεια υφαιρείται από το κράτος αυτό σημαίνει ότι είναι εκτός αρίθμησης, πέραν της πρόβλεψης και μη ελεγχόμενη.

Συμβαντικός χώρος είναι ο τόπος από όπου ίσως κάτι καινούργιο συμβεί , κάτι πέρα από την αναγνώριση και τον έλεγχο του κράτους. Σύμφωνα με τον Μπαντιού είναι το κενό που έχει ως αποτέλεσμα την διαραφή (suture) των πολλαπλοτήτων στο είναι τους και αν η φύση του έχει λησμονηθεί αυτό οφείλεται στο ότι το κενό αυτό γεμίζεται (Σπινόζα ή Χάιντεγγερ) ή καλύπτεται (Λάιμπνιτς ή Βιτγκενστάιν) – τόσες πολλές φιλοσοφικές προσπάθειες, θα πει ο Μπαντιού για να αποφύγει κανείς το ερώτημα του κράτους.

Η οντολογία του Μπαντιού περιγράφει με άλλα λόγια τις προϋποθέσεις, που πρέπει να τηρηθούν ώστε να μετακινηθούμε πέρα από το κράτος και τις αναπαραστάσεις του προς μια κατάσταση αγνής παράστασης. Σε αντίθεση με τον Χάιντεγγερ ή με τον Ντελέζ, ο Μπαντιού θεμελιώνει μια οντολογία για να κινηθεί πέρα από αυτή. Η οντολογία του παρέχει μια ακριβή περιγραφή του τι δεν μπορεί να περιληφθεί σε αυτή, αυτού που υπερβαίνει την φύση της αντικειμενικότητας- το σύμπλεγμα συμβάν-αλήθεια-υποκείμενο. Μπορούμε να κατανοήσουμε την ύπαρξη της σκέψης αν κατανοήσουμε πως μπορεί να ξεκινήσει μια ρήξη με τον νόμο του είναι.