The Appropriation of the Figure of Nietzsche in Heidegger, Deleuze… and Badiou?
Ok, here's my first post, its a bit long and is essentially an essay. But it does fit into our current discussion, and I hope it sheds some light on Badiou's position. The symbols have all gone to pot, but I've managed as best I can.
The Appropriation of the Figure of Nietzsche in Heidegger, Deleuze… and Badiou?
[1] In this short piece of work I want to cover a fairly ambitious range of thinkers. Taking as my impetus a single aphorism of Nietzsche’s, number 374 in the Gay Science entitled Our New Infinite, I want to develop the theme of this aphorism by exploring the character that Nietzsche takes in the works of Heidegger and Deleuze; as the ‘last metaphysician’ and the ‘superior empiricist’ respectively. This examination will centre on the nature and importance of the ‘event’ in both of these thinkers’ work, and will be contrasted against the backdrop of an essentially Badiouian framework. This study will restrict itself to the middle period work of Heidegger, which sees him struggle with Nietzsche’s conception of truth, and culminates with the accusation of a falling back into Platonism through the back door. I will also restrict myself to the early works of Deleuze, concentrating almost entirely on Nietzsche and Philosophy. In essence this essay will be a survey of my concerns with philosophy, motivated by the aforementioned aphorism, and using the figure of Nietzsche as a common theme to traverse the theory of the event in Heidegger, Deleuze and Badiou.
[2] To begin with let me quote the passage in aphorism number 374 that I find most important:
[T]he world has become "infinite" for us all over again, inasmuch as we cannot reject the possibility that it may include infinite interpretations. Once more we are seized by a great shudder; but who would feel inclined immediately to deify again after the old manner this monster of an unknown world? And to worship the unknown henceforth as "the Unknown One"? (The Gay Science No. 374)
[3] I think in this small section we see Nietzsche’s commitment to a philosophy of multiplicity, with the strong rejection of deifying this infinity of the world; to subject it, in the old manner of metaphysics, to a totalising concept of the ‘One’. The infinity of the world cannot be taken as a ‘One’, but only as a pure multiplicity; but what are the consequences of this move? The consequences of rejecting the possibility of an overarching concept, or totalising form, is what I believe lies at the heart of the three philosophers I want to link through this aphorism. The affirmation of a multiplicity which is not a one links fundamentally to a second theme; that between relation and non-relation.
[4] This denial of a single fundamental ground for metaphysical philosophical thought sees a division between philosophy’s two main historical concerns of absolute foundations, and the relations between beings. Historically some form of connection had always been sought between immediate appearance, and fundamental grounds, but with this statement the possibility of a ground has disappeared. Here we find the familiar themes of the absence of the ground of metaphysics in such terms as the groundless ground, abgrund, void and abyss. The absolute ground of metaphysics is not a ground, but the absence of ground. Hence the new infinite that Nietzsche talks about is a pure multiplicity that is completely unrelated to the immediate relations between beings. This is the division that I want to concentrate on between pure inconsistent multiplicity, non-relation, and collections of related ones, or beings; consistent multiplicities.
[5] The history of philosophy, according to Heidegger, has always sought to bridge this gap between an absent ground and existent beings. Here the immediate relations between beings have always sought to become more general, and as such more abstract, this is an attempt to reach the concept of Being through a more and more minimal description of beings. But if the groundless ground is non-related then no level of abstraction will bridge the gap, as it will always remain fundamentally a relation, and through greater degrees of sophisticated abstraction this method of trying to understand being actually covers over and moves away from the simplicity of non-relation. Heidegger’s notion of the event ultimately revolves around the unveiling of this fundamental difference, as the recognition of something that cannot be directly expressed. But both Badiou and Deleuze have a more positive conception of the possibilities of ‘metaphysical’ thought in the wake of this revelation. Their respective philosophies seem to gravitate more toward a Nietzschean notion of incorporation. These themes will be developed throughout this essay.
[6] Here then we can see the two types of infinity, that of Nietzsche’s new infinite, in the form of the non-related pure inconsistent multiplicity, and that which follows after the old manner; related consistent multiplicity, which up until now had always been linked to a One. Contemporary continental philosophy is almost exclusively concerned with these two types of multiplicity, and the manner in which they belong together, to form an identity of fundamental difference. This belonging together is almost always articulated by some form of ‘event’, a singular and unpredictable occurrence that articulates this difference. This articulation of fundamental difference is the site, or source of novelty and creation. Badiou’s philosophy does not follow this pattern, but provides a framework in which we can understand both Heidegger and Deleuze in terms of the above description. Such uses of the ‘event’ Badiou see as a return to a deified praise for an ‘Unknown One’.
[7] Before examining the figure of Nietzsche in both Heidegger and Deleuze I will begin with a characterisation of the two forms of infinity given above in the work of Badiou. Badiou’s concern to warn off a return of the ‘Unknown One’ in his philosophy is made clear in meditation one of his major work L’être et l’événtment, here he states: ‘what must be said is that the one, which is not, exists solely as operation. Or; there is no one, there is only the count-for-one.’ The ‘one’ is not is the fundamental wager of Badiou’s position, only pure inconsistent multiplicity ‘is’, every being as one is this inconsistent multiplicity subjected to the operation of a count-for-one. Here we see the first form of infinity; the pseudo-foundational aspect of a pure inconsistent multiplicity, all consistent being is subtracted via a count-for-one from this inconsistency. These subtracted consistent ‘ones’ form no relation to this inconsistent multiplicity which is unaffected by this operation.
[8] The two forms of multiplicity, inconsistent and consistent, can now be distinguished. The first is the inconsistent multiple, an uncounted and unstructured infinity; the second is consistent multiplicity, a multiple constructed from a number of ‘ones’ taken together as a whole, or one. Every counting-for-one, that is any structured multiple, is a presentation of the inconsistent multiple from which it has been subtracted. Therefore every structured multiple is split between the structure it presents, as a multiple of ones, and the inconsistency from which, as a one, it has been subtracted. Badiou equates these consistent multiples with mathematical sets, and it is in this way that they can form a consistent concept of the infinite. This introduction of set theory also introduces another interesting aspect of Badiou’s philosophy, that of his axiomatic approach.
[9] In this essay I do not want to enter into a detailed discussion of the technical complexities of Badiou’s system, as these are considerable, and I do not have the space to examine them sufficiently. But what the introduction of set theory allows Badiou to do is to deal with infinite consistent multiples, and use the empty set axiom coupled to the distinction between belonging, being an element, and inclusion, being a subset, to highlight how both the consistent and inconsistent aspects of a consistent multiple operate within a set.
[10] The empty set axiom is the only existential axiom in the system of modern set theory, the ZF system named after its two key developers Zermelo and Franknel, and it simply asserts the existence of a set, a consistent multiple, with no elements. The empty set collects together nothing, it is asserted precisely as Badiou conceives it must; as a pure operation. A set M is usually the result of drawing together an number of elements to form a whole; M={m1, m2, m3,…}. The empty set, usually represented by the symbol 0 (the best I can do), is simply this operation of gathering, or the gathering of nothing; 0={}. This holds to Badiou’s initial assertion that the one is not, here we see that the one is essentially nothing, or void, in its initial existential assertion it appears only as an operation. For Badiou this empty set, 0 , which he calls the void, is the proper name of being, it is the presentation of the inconsistent multiple. It is not the inconsistent multiple itself, but its presentation, its being made consistent through the operation of a count-for-one. But what the empty set presents is precisely nothing; it is a name without reference.
[11] Another axiom, and the last that I’ll discuss in detail, is used to generate from this unpromising beginning the whole variety of possible sets, this is the power set axiom. This axiom allows all other sets to be constructed from the empty set. The power set of a set is the set formed from all the possible parts of the original set; this is best explained with an example. Take the simple set of two elements M={m1, m2}, there are four possible subsets, or parts: {m1}, {m2}, {m1, m2} and finally {0 }, which is equivalent to {{}}. Therefore the set formed by taking all of these subsets together is {{m1}, {m2}, {m1, m2}, {0 }}, this is the power set P(M). In general if a set N has n elements then the power set, P (N), has 2 to the power n elements. What is interesting to note is the appearance of the empty set, which does not appear as an element in the initial set and highlights a peculiar quality of this set. There is a distinct difference between belonging to a set as an element, and being included in a set as a subset. This can be seen from the example above, m1 and m2 are both elements and subsets of M. But the subsets {m1, m2} and {0 } are not elements of M, they are not presented in M. The peculiar quality of the empty set is that it is included in every set, but belongs to no set; it is never presented in a set.
[12] This is how the inconsistent aspect of multiplicity is included in every consistent multiplicity, but it is never presented as an element, as something which belongs to a consistent multiple. Here in this presentation of nothing we see expressed the non-relation of the inconsistent multiple to consistent multiples. What is presented is only ever the proper name of this inconsistent multiple, and never the multiple itself; the void, 0 , is the proper name of being. The empty set, or void, the pure operation of making consistent, is also the foundational element of every set. It is included in every set, and also all of its elements can be considered as constructions from the void. This is achieved through the repetition of the power set operation. If the power set operation is repeated on the empty set then a string of successive sets is generated, each with one more element than the last, this succession can then be considered as the generation of the natural numbers, the starting point for the construction of many important
sets:
o = 0
P(0)={0}= 1
P ({0})={{0}, {{0}}}= 2
P ({{0}, {{0}}})= {{0}, {{0}}, {{0}, {{0}}}}= 3
[13] One of Cantor’s main triumphs in the development of set theory was that the infinite, or as he called them transfinite, sets were mathematically consistent. The infinite set of all natural numbers, for example, was a completed consistent entity. In modern set theory this is codified in an axiom, the axiom of infinity which Badiou sees as a wager on the infinite. Therefore, so far we have seen the first type of infinity that I wanted to characterise, the inconsistent multiple, which is named by the empty set axiom. This axiom, in conjunction with the power set axiom and the axiom of infinity, can be used to create a second form of the infinite; consistent transfinite sets. The smallest of these sets, the natural numbers, are well behaved but if the operation of the power set is reapplied to these transfinite sets a new type of undecidable, and immeasurable, but not inconsistent, transfinite set is created. For the sake of brevity my engagement with this problem, essentially the problem of Cantor’s continuum hypothesis, will have to remain descriptive and un-technical. The cardinal number of the smallest transfinite set is called Aleph 0, but the cardinal number associated with its power set P(Aleph 0) is unknown. What it is important to note is that although this value cannot be measured in an absolute way, this does not mean that it sits in non-relation to consistent sets, such as Aleph o, it is also itself a consistent set, or multiplicity. The power set P(Aleph o) always maintains a minimal relation to Aleph o, a relation of being immeasurably greater: P(Aleph 0)> Aleph 0, this is a determinate relation and distinctly different from the non-relation of inconsistent multiplicity.
[14] What this introduces into mathematics is a field of indetermination, something not usually associated with its severely rigorous and analytical image. What Cohen achieved in 1963, with the development of his method of set theoretical forcing, was to exploit this indeterminacy. Cohen’s theory of forcing provides the central model for Badiou’s theory of the event, and possibly the greatest barrier to the reception of his work. The idea that Badiou is utilising though is simple. Cohen’s theory of forcing is used to prove independence theorems, independence theorems demonstrate that certain assertions made about a model of set theory can neither be proved nor disproved within that model. Such assertions, although consistent with the model, cannot be proved and so cannot be theorems of the model, but can only be adopted as independent axioms, who’s inclusion depends simply on a decision.
[15] The method of forcing was first used to demonstrate the independence of Cantor’s continuum hypothesis by constructing a model of set theory in which this hypothesis fails. Cantor’s continuum hypothesis is an attempt to put a precise measure on the value of P(Aleph 0), this value is usually called Aleph 1. This value is the smallest value that can be assumed for P(Aleph 0), and the purpose of adopting it is to precisely rule out the space of indetermination opened up through the use of the power set axiom, it is a last attempt to sure up the realm of mathematics as something completely determinable. Cohen’s proof is a proof by contradiction, assuming that p(Aleph 0)=Aleph 1 he then forces the conclusion that P(Aleph 0)> Aleph 1. This is achieved by assuming the existence of a certain number of sets from this field of indeterminacy, these sets are known as non-constructible sets, and using these sets to rearrange the elements of À 0 in such a way as to form an excess of new subsets such that these new subsets combined with those generated by normal methods taken together form the power set of Aleph 0 such that P(Aleph 0)> Aleph 1.
[16] Badiou translates this procedure into his theory of the event. The fundamental concerns about systematic, especially mathematical, approaches to philosophy have always centred on their determinism, but with Cohen’s theory of forcing, following on from Gödel’s incompleteness theorems, Badiou thinks that mathematics is not a hermetically sealed deterministic project, but one that can generate novelty in a systematic fashion, in response to a singular occasion. To put it simply, Badiou conceives of situations in terms of set theoretical models, they operate with a few basic axioms and also with a number of independent axioms which have been assumed. The axiomatically infinite nature of such situations always means that there is a field of indeterminacy permeating a situation, certain indiscernible ontological entities which cannot be identified according to the current model governing the situation. As we saw above, Cohen’s theory of forcing can be used to assume the existence of some set of indiscernible beings, and use their assumed existence to rearrange the current situation in such a way that some of the independent axioms assumed to govern that system are proved not to hold. This is a dynamic ever changing view of existence, with their being no ultimate set of axioms, precisely because they are independent. An event for Badiou is precisely the coming to prominence of something that had been previously indiscernible, something invisible at the borders of the situation. And a subject is born in affirming the existence of this indiscernible event, and holding to it in such a way as to constitute a procedure of forcing that will transform the situation.
[17] Events appear, in Badiou’s system, as a void for a situation. Their indeterminacy acts as a void or an inconsistency only relatively to a situation. Through a truth procedure this inconsistency is incorporated into a transformed situation, and its existence, although still not ultimately decidable, is accepted. The situation has been transformed ‘as if’ the undecidable aspect belonged. This is probably the most important aspect of Badiou’s theory, in no way can the event be seen as the actual appearance of the inconsistent multiple, the event only acts like a void in relation to a specific situation. This possible confusion between an event, and an appearance of the inconsistent multiple is what lies at the heart of Badiou’s criticism of Deleuze.
[18] I can now conclude this section on Badiou, as I have developed the two forms of infinity that will be vital for characterising the approaches of Heidegger and Deleuze. The two forms of infinity are associated with the two types of multiplicity; the first is the absolutely inconsistent multiplicity, which sits in non-relation to consistent multiplicity. The second is consistent multiplicity, which always carries a trace of the inconsistent multiple within itself in the form of its proper name; the void or empty set. Consistent multiplicity can then form the second form of infinite, associated with the transfinite sets first developed by Cantor, but most importantly the immeasurable excess of possible parts over the whole embodied in the power set of any transfinite cardinal number. This second form of infinity is consistent, and maintains a definite relation to other consistent multiplicities, although it is a minimal relation which embodies a degree of indeterminability.
[19] The notion that something un-presentable somehow permeates the whole of being, and forms its groundless ground is no longer a revelation, as it is for Heidegger, but a simple consequence of the empty set axiom. Heidegger’s characterisation of Nietzsche as the last metaphysician highlights his theory of the event based solely on the revelation of the separate and irreconcilable difference between consistent and inconsistent multiples. Only when these two aspects, of a first and an other beginning are allowed to be, that is to belong together in an identity of difference, can novelty or creation occur. The idea is that once this distinction has been allowed the realm of beings, in consistent multiplicity can act with a certain freedom, free from being chained to a project of domination that saw it try to represent its other side. For both Badiou and Deleuze this is not enough.
[20] Heidegger characterises Nietzsche as the ‘last metaphysician’ as despite his correct diagnosis of the inherent theme of nihilism at the heart of Western philosophy, he did not turn away from it but simply affirmed it for what it was. This was an affirmative form of nihilism that affirmed the ultimately groundless and purposeless nature of existence, but no longer lamented this position as if some purpose or ground had been lost. It is no longer a melancholy desire for some lost golden age. For Heidegger, Nietzsche’s will to power is nothing more than a will to will, and as such rejects the possibility of a new approach to the question of the truth of being, in denying any ultimate or teleological aim for being, will can only will itself.
[21] As was mentioned above, Heidegger’s preoccupation with the history of Western metaphysics revolves around the idea that the development of philosophy, science and mathematics leads to a domination of the themes of the first beginning; this approach is concerned only with the essence of being conceived solely in terms of what a being is, what is present or presented. Rather than the concern of the other beginning, coextensive with the first but always dominated by it, this position is concerned with the existential aspect of beings, the ‘thatness’ of their being, or their being there. This clearly corresponds to the notion of consistent and inconsistent multiplicities introduced above. If consistent multiplicity is linked with the first beginning’s concern with what a being is, as what is presented, and inconsistent multiplicity as the beings existence, making it be there. A consistent multiplicity is a presentation of its elements, and the trace of the inconsistent multiple is the foundational empty set, from which all other sets are constructed.
[22] The movement of the first beginning as a move to cover over and suppress the other beginning is a movement that not only ignores inconsistent multiplicity but one that tries to represent it in terms of a being. For Heidegger this is the history of Western metaphysics, which introduces the concept of transcendence in order to present something like the inconsistent multiple in terms of presentation. This concept of transcendence is motivated by a move to the infinite, and usually associated with a divine one. But this move to the infinite is an intrinsic property of consistent multiplicity introduced above. This is where the second form of the infinite emerges, with the introduction of the immeasurability of most transfinite sets. The consequences for Heidegger’s position are interesting; initially this transfinitude can be seen as a move toward presenting the inconsistent multiple. The indeterminacy introduced at this level certainly reduces the determinacy of the consistent multiple, this minimal relation can no longer be absolutely measured but it retains the minimal relation of being comparably bigger. This indeterminacy could be seen as presenting the inconsistency of the inconsistent multiple, with the greater generality acting as some sort of notion of progression, or approximation of the ultimately inconsistent and non-relational multiple: an approximation that would render this pure multiplicity as a ‘one’.
[23] In recognising that the inconsistent multiple is coextensive with consistent multiplicity Heidegger rejects this whole movement to infinity as inextricably linked to a project of transcendence, a project which would hope to present the inconsistent multiple in the form of some form of totalising ‘one’. The truth of be-ing lies in a turn away from such transcendent or infinite consideration of being and recognises its immanent determination in an identity of difference. Therefore when Nietzsche diagnoses the development of philosophy and science as a striving to represent a transcendent one through a consideration of infinite being, but fails to turn away from these methods toward the new, semi-mystical revelation of the truth of be-ing he becomes the ‘last metaphysician’.
[24] This need not be the case, if the aim of a consideration of infinite consistent multiplicity is to present a totalising ‘one’ then it is doomed to failure. But, if as Nietzsche does, one affirms this movement for what it is, then one can recognise the indeterminacy opened up by this consideration of the transfinite realm of consistent multiplicity as a realm of possibility for invention and experimentation. It is in no way implicitly necessary that any such pursuit of the possibilities inherent in this indeterminacy of being be understood as an attempt to found being, or present the un-presentable. This is clear from the examination of Badiou given above. Heidegger equates all such scientific procedures linked with analytical and systematic approaches to be linked to the consideration of infinite being, and therefore metaphysics. The operations of consistent multiplicity freed from scientific practice operate in a completely new and different way, characterised by such terms as poetic thought, poetry and poesis. Therefore Nietzsche’s insistence on transforming these practices into a gay science, one that might explore this realm of indeterminacy freed from any conscious totalising, or teleological, objective is simply impossible, and is a relapse into a metaphysical position which has dominated the West since Plato.
[25] Deleuze’s early preoccupation with Nietzsche is markedly different. Deleuze positively affirms Nietzsche’s position, heralding him as a ‘superior empiricist’, and a practitioner of gay science. Deleuze’s understanding of the will to power, the eternal return, and their connection, allows him to characterise Nietzsche as precisely free from the constraints of following a totalising teleological science, and left to explore and experiment with out conscious aim. Here the realm of indeterminacy opened up by transfinite consistent multiplicity is the indeterminate border of consistent being, forming its unconscious limit. As with Badiou, it is from this border territory, the indiscernible limit of continuity and consistency that creation, and novelty spring. Badiou’s criticism is that the singular moments which appear in this indeterminacy which motivate the multiple events of expression, are taken by Deleuze to be expressions of the founding inconsistency of consistent unities.
[26] The will to power in Deleuze’s book is both the hidden foundational element/aspect in every consistent being, and that which appears at its indiscernible limit expressed as an unconscious desire for expansion, and novelty: ‘The will to power is both the genetic element of force and the principle of synthesis of forces’. Thus every unity is founded on this disunity and incoherence of will to power, and every being is an expression of will to power. This is hidden to the extent that a being considers itself as final, as having defined its own limits. The true expression of will to power is only found in the unique novel transformation of a being. It cannot be expressed either in some final being, or in a state of pure becoming that would expand without limit. A process of becoming must be checked by occasional static moments, or pauses, in which a being can still affirm itself as a being and consolidate itself. This process of continual becoming is not the limit of some unity, or some being, but the limit of unity itself. The degree of expansion, and growth which a being can maintain and still retain a sense of identity, not in some external form, but in the immanent unity holding together these moments of transformation. This is the notion of incorporation, the extent to which a being can act in a dominant way and impose its form on other forces, testing the limits of its unity.
[27] In this we can still see that this idea follows the pattern of analysis started above. If the will to power is a genetic element of force, which we associate with the inconsistent multiple, it is filled with an irrepressible desire to express itself. But expression is only possible in the form of a being; therefore this inconsistent multiple must express itself through a consistent multiple. Although every consistent multiple is an expression of this inconsistency, through the trace of the empty set, this is far from sufficient. It seeks to express itself through an expansion of this consistent multiplicity, to create a new multiplicity that expresses more. But as the inconsistent multiple is completely inexhaustible and unaffected by the subtractions of consistent being the will to expansion, or power is insatiable and unending. The will to power is then seen as the founding aspect, in terms of the empty set, and also as the impetus for an expansion and transformation of this set, as it always desires to express more. Therefore it finds its true expression in this continual movement of becoming, the transformation from each determinate stage to the next as the unity which traverses these transformations. It is this return of the will to power that finally allows for its expression; the expression of the will to power is fundamentally linked to its recurrence, as both founding and transformational. And as such will to power cannot be separated from force, i.e. beings, and cannot be thought of as a principle which is pre-existent or transcendent. It is the immanent condition for being, or force; the will to power ‘is’ only to the extent that beings are. Therefore the eternal return is the eternal recurrence of the will to power. This analysis is criminally brief, but it will have to suffice for the time being.
[28] As there is no transcendent notion of the will to power, but only its multiple expression in multiple beings, or aggregations of force, each such expression is considered in its own right an event. Here again we can reintroduce Badiou’s framework to see what Badiou calls a return of a metaphysical ‘one’ in Deleuze’s work. Here the singularity that appears in the indeterminate, and indiscernible, realm of being is equated with a return of the inconsistency at the heart of being. Badiou sees this as clear misapprehension of what events are; they are not recurrences of the inconsistency of being, i.e. the void, but are only what appear as void-like relative to a situation. They are in fact perfectly legitimate and consistent sets/beings in their own right. This is a confusion between inconsistency and indeterminability; the equation of the two is a simple return of a ‘one’, which is kept from becoming transcendent through the introduction a vital expressive urge inherent in this inconsistency. The multiple events of Deleuze are nothing more than recurrent expressions of inconsistent multiplicity, essential a new ‘one’. Badiou’s events are completely independent multiplicities; each is affirmed in its own right through an ongoing truth procedure which affirms this specificity.
[29] I myself am not sure to what extent we should take the criticisms of Badiou seriously, with respect to Deleuze. We could just as easily read the expression of the inconsistency at the heart of being to be precisely not an actual presentation of this inconsistency, but as simply an occasion for transformation. At bottom the argument between Deleuze and Badiou relates to their reception of a positive Nietzschean inheritance. Both want to affirm the possibility, in the form of multiple events, of a positive role for philosophy after the ‘end of philosophy’ heralded by Heidegger, and his characterisation of Nietzsche as the last metaphysician. But to do this they must show how ‘metaphysical’ practices can operate outside of the worship of some ‘Unknown One’. Deleuze introduces the notion of inconsistency, which lies at the heart of being, in terms of an unconscious vital urge for expression. Whereas Badiou sees events as extra-philosophical, the response to events, in terms of a truth procedure are realised in purely metaphysical operation. Although the actual function of philosophy in Badiou’s system is far from clear What is the best way to accept this inheritance; through vitalism or through a simple fidelity to the specificity of an event? I am in no position to answer this question, and this essay has been less than satisfactory in its analysis, especially with regards to Deleuze. What it has been though is a survey and a statement of intent for my future research, which will seek to follow through this analysis of the function of the event in contemporary philosophy, between the two positions suggested here of a ‘superior empiricism’ and an ‘aleatory rationalism’.
The Appropriation of the Figure of Nietzsche in Heidegger, Deleuze… and Badiou?
[1] In this short piece of work I want to cover a fairly ambitious range of thinkers. Taking as my impetus a single aphorism of Nietzsche’s, number 374 in the Gay Science entitled Our New Infinite, I want to develop the theme of this aphorism by exploring the character that Nietzsche takes in the works of Heidegger and Deleuze; as the ‘last metaphysician’ and the ‘superior empiricist’ respectively. This examination will centre on the nature and importance of the ‘event’ in both of these thinkers’ work, and will be contrasted against the backdrop of an essentially Badiouian framework. This study will restrict itself to the middle period work of Heidegger, which sees him struggle with Nietzsche’s conception of truth, and culminates with the accusation of a falling back into Platonism through the back door. I will also restrict myself to the early works of Deleuze, concentrating almost entirely on Nietzsche and Philosophy. In essence this essay will be a survey of my concerns with philosophy, motivated by the aforementioned aphorism, and using the figure of Nietzsche as a common theme to traverse the theory of the event in Heidegger, Deleuze and Badiou.
[2] To begin with let me quote the passage in aphorism number 374 that I find most important:
[T]he world has become "infinite" for us all over again, inasmuch as we cannot reject the possibility that it may include infinite interpretations. Once more we are seized by a great shudder; but who would feel inclined immediately to deify again after the old manner this monster of an unknown world? And to worship the unknown henceforth as "the Unknown One"? (The Gay Science No. 374)
[3] I think in this small section we see Nietzsche’s commitment to a philosophy of multiplicity, with the strong rejection of deifying this infinity of the world; to subject it, in the old manner of metaphysics, to a totalising concept of the ‘One’. The infinity of the world cannot be taken as a ‘One’, but only as a pure multiplicity; but what are the consequences of this move? The consequences of rejecting the possibility of an overarching concept, or totalising form, is what I believe lies at the heart of the three philosophers I want to link through this aphorism. The affirmation of a multiplicity which is not a one links fundamentally to a second theme; that between relation and non-relation.
[4] This denial of a single fundamental ground for metaphysical philosophical thought sees a division between philosophy’s two main historical concerns of absolute foundations, and the relations between beings. Historically some form of connection had always been sought between immediate appearance, and fundamental grounds, but with this statement the possibility of a ground has disappeared. Here we find the familiar themes of the absence of the ground of metaphysics in such terms as the groundless ground, abgrund, void and abyss. The absolute ground of metaphysics is not a ground, but the absence of ground. Hence the new infinite that Nietzsche talks about is a pure multiplicity that is completely unrelated to the immediate relations between beings. This is the division that I want to concentrate on between pure inconsistent multiplicity, non-relation, and collections of related ones, or beings; consistent multiplicities.
[5] The history of philosophy, according to Heidegger, has always sought to bridge this gap between an absent ground and existent beings. Here the immediate relations between beings have always sought to become more general, and as such more abstract, this is an attempt to reach the concept of Being through a more and more minimal description of beings. But if the groundless ground is non-related then no level of abstraction will bridge the gap, as it will always remain fundamentally a relation, and through greater degrees of sophisticated abstraction this method of trying to understand being actually covers over and moves away from the simplicity of non-relation. Heidegger’s notion of the event ultimately revolves around the unveiling of this fundamental difference, as the recognition of something that cannot be directly expressed. But both Badiou and Deleuze have a more positive conception of the possibilities of ‘metaphysical’ thought in the wake of this revelation. Their respective philosophies seem to gravitate more toward a Nietzschean notion of incorporation. These themes will be developed throughout this essay.
[6] Here then we can see the two types of infinity, that of Nietzsche’s new infinite, in the form of the non-related pure inconsistent multiplicity, and that which follows after the old manner; related consistent multiplicity, which up until now had always been linked to a One. Contemporary continental philosophy is almost exclusively concerned with these two types of multiplicity, and the manner in which they belong together, to form an identity of fundamental difference. This belonging together is almost always articulated by some form of ‘event’, a singular and unpredictable occurrence that articulates this difference. This articulation of fundamental difference is the site, or source of novelty and creation. Badiou’s philosophy does not follow this pattern, but provides a framework in which we can understand both Heidegger and Deleuze in terms of the above description. Such uses of the ‘event’ Badiou see as a return to a deified praise for an ‘Unknown One’.
[7] Before examining the figure of Nietzsche in both Heidegger and Deleuze I will begin with a characterisation of the two forms of infinity given above in the work of Badiou. Badiou’s concern to warn off a return of the ‘Unknown One’ in his philosophy is made clear in meditation one of his major work L’être et l’événtment, here he states: ‘what must be said is that the one, which is not, exists solely as operation. Or; there is no one, there is only the count-for-one.’ The ‘one’ is not is the fundamental wager of Badiou’s position, only pure inconsistent multiplicity ‘is’, every being as one is this inconsistent multiplicity subjected to the operation of a count-for-one. Here we see the first form of infinity; the pseudo-foundational aspect of a pure inconsistent multiplicity, all consistent being is subtracted via a count-for-one from this inconsistency. These subtracted consistent ‘ones’ form no relation to this inconsistent multiplicity which is unaffected by this operation.
[8] The two forms of multiplicity, inconsistent and consistent, can now be distinguished. The first is the inconsistent multiple, an uncounted and unstructured infinity; the second is consistent multiplicity, a multiple constructed from a number of ‘ones’ taken together as a whole, or one. Every counting-for-one, that is any structured multiple, is a presentation of the inconsistent multiple from which it has been subtracted. Therefore every structured multiple is split between the structure it presents, as a multiple of ones, and the inconsistency from which, as a one, it has been subtracted. Badiou equates these consistent multiples with mathematical sets, and it is in this way that they can form a consistent concept of the infinite. This introduction of set theory also introduces another interesting aspect of Badiou’s philosophy, that of his axiomatic approach.
[9] In this essay I do not want to enter into a detailed discussion of the technical complexities of Badiou’s system, as these are considerable, and I do not have the space to examine them sufficiently. But what the introduction of set theory allows Badiou to do is to deal with infinite consistent multiples, and use the empty set axiom coupled to the distinction between belonging, being an element, and inclusion, being a subset, to highlight how both the consistent and inconsistent aspects of a consistent multiple operate within a set.
[10] The empty set axiom is the only existential axiom in the system of modern set theory, the ZF system named after its two key developers Zermelo and Franknel, and it simply asserts the existence of a set, a consistent multiple, with no elements. The empty set collects together nothing, it is asserted precisely as Badiou conceives it must; as a pure operation. A set M is usually the result of drawing together an number of elements to form a whole; M={m1, m2, m3,…}. The empty set, usually represented by the symbol 0 (the best I can do), is simply this operation of gathering, or the gathering of nothing; 0={}. This holds to Badiou’s initial assertion that the one is not, here we see that the one is essentially nothing, or void, in its initial existential assertion it appears only as an operation. For Badiou this empty set, 0 , which he calls the void, is the proper name of being, it is the presentation of the inconsistent multiple. It is not the inconsistent multiple itself, but its presentation, its being made consistent through the operation of a count-for-one. But what the empty set presents is precisely nothing; it is a name without reference.
[11] Another axiom, and the last that I’ll discuss in detail, is used to generate from this unpromising beginning the whole variety of possible sets, this is the power set axiom. This axiom allows all other sets to be constructed from the empty set. The power set of a set is the set formed from all the possible parts of the original set; this is best explained with an example. Take the simple set of two elements M={m1, m2}, there are four possible subsets, or parts: {m1}, {m2}, {m1, m2} and finally {0 }, which is equivalent to {{}}. Therefore the set formed by taking all of these subsets together is {{m1}, {m2}, {m1, m2}, {0 }}, this is the power set P(M). In general if a set N has n elements then the power set, P (N), has 2 to the power n elements. What is interesting to note is the appearance of the empty set, which does not appear as an element in the initial set and highlights a peculiar quality of this set. There is a distinct difference between belonging to a set as an element, and being included in a set as a subset. This can be seen from the example above, m1 and m2 are both elements and subsets of M. But the subsets {m1, m2} and {0 } are not elements of M, they are not presented in M. The peculiar quality of the empty set is that it is included in every set, but belongs to no set; it is never presented in a set.
[12] This is how the inconsistent aspect of multiplicity is included in every consistent multiplicity, but it is never presented as an element, as something which belongs to a consistent multiple. Here in this presentation of nothing we see expressed the non-relation of the inconsistent multiple to consistent multiples. What is presented is only ever the proper name of this inconsistent multiple, and never the multiple itself; the void, 0 , is the proper name of being. The empty set, or void, the pure operation of making consistent, is also the foundational element of every set. It is included in every set, and also all of its elements can be considered as constructions from the void. This is achieved through the repetition of the power set operation. If the power set operation is repeated on the empty set then a string of successive sets is generated, each with one more element than the last, this succession can then be considered as the generation of the natural numbers, the starting point for the construction of many important
sets:
o = 0
P(0)={0}= 1
P ({0})={{0}, {{0}}}= 2
P ({{0}, {{0}}})= {{0}, {{0}}, {{0}, {{0}}}}= 3
[13] One of Cantor’s main triumphs in the development of set theory was that the infinite, or as he called them transfinite, sets were mathematically consistent. The infinite set of all natural numbers, for example, was a completed consistent entity. In modern set theory this is codified in an axiom, the axiom of infinity which Badiou sees as a wager on the infinite. Therefore, so far we have seen the first type of infinity that I wanted to characterise, the inconsistent multiple, which is named by the empty set axiom. This axiom, in conjunction with the power set axiom and the axiom of infinity, can be used to create a second form of the infinite; consistent transfinite sets. The smallest of these sets, the natural numbers, are well behaved but if the operation of the power set is reapplied to these transfinite sets a new type of undecidable, and immeasurable, but not inconsistent, transfinite set is created. For the sake of brevity my engagement with this problem, essentially the problem of Cantor’s continuum hypothesis, will have to remain descriptive and un-technical. The cardinal number of the smallest transfinite set is called Aleph 0, but the cardinal number associated with its power set P(Aleph 0) is unknown. What it is important to note is that although this value cannot be measured in an absolute way, this does not mean that it sits in non-relation to consistent sets, such as Aleph o, it is also itself a consistent set, or multiplicity. The power set P(Aleph o) always maintains a minimal relation to Aleph o, a relation of being immeasurably greater: P(Aleph 0)> Aleph 0, this is a determinate relation and distinctly different from the non-relation of inconsistent multiplicity.
[14] What this introduces into mathematics is a field of indetermination, something not usually associated with its severely rigorous and analytical image. What Cohen achieved in 1963, with the development of his method of set theoretical forcing, was to exploit this indeterminacy. Cohen’s theory of forcing provides the central model for Badiou’s theory of the event, and possibly the greatest barrier to the reception of his work. The idea that Badiou is utilising though is simple. Cohen’s theory of forcing is used to prove independence theorems, independence theorems demonstrate that certain assertions made about a model of set theory can neither be proved nor disproved within that model. Such assertions, although consistent with the model, cannot be proved and so cannot be theorems of the model, but can only be adopted as independent axioms, who’s inclusion depends simply on a decision.
[15] The method of forcing was first used to demonstrate the independence of Cantor’s continuum hypothesis by constructing a model of set theory in which this hypothesis fails. Cantor’s continuum hypothesis is an attempt to put a precise measure on the value of P(Aleph 0), this value is usually called Aleph 1. This value is the smallest value that can be assumed for P(Aleph 0), and the purpose of adopting it is to precisely rule out the space of indetermination opened up through the use of the power set axiom, it is a last attempt to sure up the realm of mathematics as something completely determinable. Cohen’s proof is a proof by contradiction, assuming that p(Aleph 0)=Aleph 1 he then forces the conclusion that P(Aleph 0)> Aleph 1. This is achieved by assuming the existence of a certain number of sets from this field of indeterminacy, these sets are known as non-constructible sets, and using these sets to rearrange the elements of À 0 in such a way as to form an excess of new subsets such that these new subsets combined with those generated by normal methods taken together form the power set of Aleph 0 such that P(Aleph 0)> Aleph 1.
[16] Badiou translates this procedure into his theory of the event. The fundamental concerns about systematic, especially mathematical, approaches to philosophy have always centred on their determinism, but with Cohen’s theory of forcing, following on from Gödel’s incompleteness theorems, Badiou thinks that mathematics is not a hermetically sealed deterministic project, but one that can generate novelty in a systematic fashion, in response to a singular occasion. To put it simply, Badiou conceives of situations in terms of set theoretical models, they operate with a few basic axioms and also with a number of independent axioms which have been assumed. The axiomatically infinite nature of such situations always means that there is a field of indeterminacy permeating a situation, certain indiscernible ontological entities which cannot be identified according to the current model governing the situation. As we saw above, Cohen’s theory of forcing can be used to assume the existence of some set of indiscernible beings, and use their assumed existence to rearrange the current situation in such a way that some of the independent axioms assumed to govern that system are proved not to hold. This is a dynamic ever changing view of existence, with their being no ultimate set of axioms, precisely because they are independent. An event for Badiou is precisely the coming to prominence of something that had been previously indiscernible, something invisible at the borders of the situation. And a subject is born in affirming the existence of this indiscernible event, and holding to it in such a way as to constitute a procedure of forcing that will transform the situation.
[17] Events appear, in Badiou’s system, as a void for a situation. Their indeterminacy acts as a void or an inconsistency only relatively to a situation. Through a truth procedure this inconsistency is incorporated into a transformed situation, and its existence, although still not ultimately decidable, is accepted. The situation has been transformed ‘as if’ the undecidable aspect belonged. This is probably the most important aspect of Badiou’s theory, in no way can the event be seen as the actual appearance of the inconsistent multiple, the event only acts like a void in relation to a specific situation. This possible confusion between an event, and an appearance of the inconsistent multiple is what lies at the heart of Badiou’s criticism of Deleuze.
[18] I can now conclude this section on Badiou, as I have developed the two forms of infinity that will be vital for characterising the approaches of Heidegger and Deleuze. The two forms of infinity are associated with the two types of multiplicity; the first is the absolutely inconsistent multiplicity, which sits in non-relation to consistent multiplicity. The second is consistent multiplicity, which always carries a trace of the inconsistent multiple within itself in the form of its proper name; the void or empty set. Consistent multiplicity can then form the second form of infinite, associated with the transfinite sets first developed by Cantor, but most importantly the immeasurable excess of possible parts over the whole embodied in the power set of any transfinite cardinal number. This second form of infinity is consistent, and maintains a definite relation to other consistent multiplicities, although it is a minimal relation which embodies a degree of indeterminability.
[19] The notion that something un-presentable somehow permeates the whole of being, and forms its groundless ground is no longer a revelation, as it is for Heidegger, but a simple consequence of the empty set axiom. Heidegger’s characterisation of Nietzsche as the last metaphysician highlights his theory of the event based solely on the revelation of the separate and irreconcilable difference between consistent and inconsistent multiples. Only when these two aspects, of a first and an other beginning are allowed to be, that is to belong together in an identity of difference, can novelty or creation occur. The idea is that once this distinction has been allowed the realm of beings, in consistent multiplicity can act with a certain freedom, free from being chained to a project of domination that saw it try to represent its other side. For both Badiou and Deleuze this is not enough.
[20] Heidegger characterises Nietzsche as the ‘last metaphysician’ as despite his correct diagnosis of the inherent theme of nihilism at the heart of Western philosophy, he did not turn away from it but simply affirmed it for what it was. This was an affirmative form of nihilism that affirmed the ultimately groundless and purposeless nature of existence, but no longer lamented this position as if some purpose or ground had been lost. It is no longer a melancholy desire for some lost golden age. For Heidegger, Nietzsche’s will to power is nothing more than a will to will, and as such rejects the possibility of a new approach to the question of the truth of being, in denying any ultimate or teleological aim for being, will can only will itself.
[21] As was mentioned above, Heidegger’s preoccupation with the history of Western metaphysics revolves around the idea that the development of philosophy, science and mathematics leads to a domination of the themes of the first beginning; this approach is concerned only with the essence of being conceived solely in terms of what a being is, what is present or presented. Rather than the concern of the other beginning, coextensive with the first but always dominated by it, this position is concerned with the existential aspect of beings, the ‘thatness’ of their being, or their being there. This clearly corresponds to the notion of consistent and inconsistent multiplicities introduced above. If consistent multiplicity is linked with the first beginning’s concern with what a being is, as what is presented, and inconsistent multiplicity as the beings existence, making it be there. A consistent multiplicity is a presentation of its elements, and the trace of the inconsistent multiple is the foundational empty set, from which all other sets are constructed.
[22] The movement of the first beginning as a move to cover over and suppress the other beginning is a movement that not only ignores inconsistent multiplicity but one that tries to represent it in terms of a being. For Heidegger this is the history of Western metaphysics, which introduces the concept of transcendence in order to present something like the inconsistent multiple in terms of presentation. This concept of transcendence is motivated by a move to the infinite, and usually associated with a divine one. But this move to the infinite is an intrinsic property of consistent multiplicity introduced above. This is where the second form of the infinite emerges, with the introduction of the immeasurability of most transfinite sets. The consequences for Heidegger’s position are interesting; initially this transfinitude can be seen as a move toward presenting the inconsistent multiple. The indeterminacy introduced at this level certainly reduces the determinacy of the consistent multiple, this minimal relation can no longer be absolutely measured but it retains the minimal relation of being comparably bigger. This indeterminacy could be seen as presenting the inconsistency of the inconsistent multiple, with the greater generality acting as some sort of notion of progression, or approximation of the ultimately inconsistent and non-relational multiple: an approximation that would render this pure multiplicity as a ‘one’.
[23] In recognising that the inconsistent multiple is coextensive with consistent multiplicity Heidegger rejects this whole movement to infinity as inextricably linked to a project of transcendence, a project which would hope to present the inconsistent multiple in the form of some form of totalising ‘one’. The truth of be-ing lies in a turn away from such transcendent or infinite consideration of being and recognises its immanent determination in an identity of difference. Therefore when Nietzsche diagnoses the development of philosophy and science as a striving to represent a transcendent one through a consideration of infinite being, but fails to turn away from these methods toward the new, semi-mystical revelation of the truth of be-ing he becomes the ‘last metaphysician’.
[24] This need not be the case, if the aim of a consideration of infinite consistent multiplicity is to present a totalising ‘one’ then it is doomed to failure. But, if as Nietzsche does, one affirms this movement for what it is, then one can recognise the indeterminacy opened up by this consideration of the transfinite realm of consistent multiplicity as a realm of possibility for invention and experimentation. It is in no way implicitly necessary that any such pursuit of the possibilities inherent in this indeterminacy of being be understood as an attempt to found being, or present the un-presentable. This is clear from the examination of Badiou given above. Heidegger equates all such scientific procedures linked with analytical and systematic approaches to be linked to the consideration of infinite being, and therefore metaphysics. The operations of consistent multiplicity freed from scientific practice operate in a completely new and different way, characterised by such terms as poetic thought, poetry and poesis. Therefore Nietzsche’s insistence on transforming these practices into a gay science, one that might explore this realm of indeterminacy freed from any conscious totalising, or teleological, objective is simply impossible, and is a relapse into a metaphysical position which has dominated the West since Plato.
[25] Deleuze’s early preoccupation with Nietzsche is markedly different. Deleuze positively affirms Nietzsche’s position, heralding him as a ‘superior empiricist’, and a practitioner of gay science. Deleuze’s understanding of the will to power, the eternal return, and their connection, allows him to characterise Nietzsche as precisely free from the constraints of following a totalising teleological science, and left to explore and experiment with out conscious aim. Here the realm of indeterminacy opened up by transfinite consistent multiplicity is the indeterminate border of consistent being, forming its unconscious limit. As with Badiou, it is from this border territory, the indiscernible limit of continuity and consistency that creation, and novelty spring. Badiou’s criticism is that the singular moments which appear in this indeterminacy which motivate the multiple events of expression, are taken by Deleuze to be expressions of the founding inconsistency of consistent unities.
[26] The will to power in Deleuze’s book is both the hidden foundational element/aspect in every consistent being, and that which appears at its indiscernible limit expressed as an unconscious desire for expansion, and novelty: ‘The will to power is both the genetic element of force and the principle of synthesis of forces’. Thus every unity is founded on this disunity and incoherence of will to power, and every being is an expression of will to power. This is hidden to the extent that a being considers itself as final, as having defined its own limits. The true expression of will to power is only found in the unique novel transformation of a being. It cannot be expressed either in some final being, or in a state of pure becoming that would expand without limit. A process of becoming must be checked by occasional static moments, or pauses, in which a being can still affirm itself as a being and consolidate itself. This process of continual becoming is not the limit of some unity, or some being, but the limit of unity itself. The degree of expansion, and growth which a being can maintain and still retain a sense of identity, not in some external form, but in the immanent unity holding together these moments of transformation. This is the notion of incorporation, the extent to which a being can act in a dominant way and impose its form on other forces, testing the limits of its unity.
[27] In this we can still see that this idea follows the pattern of analysis started above. If the will to power is a genetic element of force, which we associate with the inconsistent multiple, it is filled with an irrepressible desire to express itself. But expression is only possible in the form of a being; therefore this inconsistent multiple must express itself through a consistent multiple. Although every consistent multiple is an expression of this inconsistency, through the trace of the empty set, this is far from sufficient. It seeks to express itself through an expansion of this consistent multiplicity, to create a new multiplicity that expresses more. But as the inconsistent multiple is completely inexhaustible and unaffected by the subtractions of consistent being the will to expansion, or power is insatiable and unending. The will to power is then seen as the founding aspect, in terms of the empty set, and also as the impetus for an expansion and transformation of this set, as it always desires to express more. Therefore it finds its true expression in this continual movement of becoming, the transformation from each determinate stage to the next as the unity which traverses these transformations. It is this return of the will to power that finally allows for its expression; the expression of the will to power is fundamentally linked to its recurrence, as both founding and transformational. And as such will to power cannot be separated from force, i.e. beings, and cannot be thought of as a principle which is pre-existent or transcendent. It is the immanent condition for being, or force; the will to power ‘is’ only to the extent that beings are. Therefore the eternal return is the eternal recurrence of the will to power. This analysis is criminally brief, but it will have to suffice for the time being.
[28] As there is no transcendent notion of the will to power, but only its multiple expression in multiple beings, or aggregations of force, each such expression is considered in its own right an event. Here again we can reintroduce Badiou’s framework to see what Badiou calls a return of a metaphysical ‘one’ in Deleuze’s work. Here the singularity that appears in the indeterminate, and indiscernible, realm of being is equated with a return of the inconsistency at the heart of being. Badiou sees this as clear misapprehension of what events are; they are not recurrences of the inconsistency of being, i.e. the void, but are only what appear as void-like relative to a situation. They are in fact perfectly legitimate and consistent sets/beings in their own right. This is a confusion between inconsistency and indeterminability; the equation of the two is a simple return of a ‘one’, which is kept from becoming transcendent through the introduction a vital expressive urge inherent in this inconsistency. The multiple events of Deleuze are nothing more than recurrent expressions of inconsistent multiplicity, essential a new ‘one’. Badiou’s events are completely independent multiplicities; each is affirmed in its own right through an ongoing truth procedure which affirms this specificity.
[29] I myself am not sure to what extent we should take the criticisms of Badiou seriously, with respect to Deleuze. We could just as easily read the expression of the inconsistency at the heart of being to be precisely not an actual presentation of this inconsistency, but as simply an occasion for transformation. At bottom the argument between Deleuze and Badiou relates to their reception of a positive Nietzschean inheritance. Both want to affirm the possibility, in the form of multiple events, of a positive role for philosophy after the ‘end of philosophy’ heralded by Heidegger, and his characterisation of Nietzsche as the last metaphysician. But to do this they must show how ‘metaphysical’ practices can operate outside of the worship of some ‘Unknown One’. Deleuze introduces the notion of inconsistency, which lies at the heart of being, in terms of an unconscious vital urge for expression. Whereas Badiou sees events as extra-philosophical, the response to events, in terms of a truth procedure are realised in purely metaphysical operation. Although the actual function of philosophy in Badiou’s system is far from clear What is the best way to accept this inheritance; through vitalism or through a simple fidelity to the specificity of an event? I am in no position to answer this question, and this essay has been less than satisfactory in its analysis, especially with regards to Deleuze. What it has been though is a survey and a statement of intent for my future research, which will seek to follow through this analysis of the function of the event in contemporary philosophy, between the two positions suggested here of a ‘superior empiricism’ and an ‘aleatory rationalism’.
posted by Brian at 11:39 am
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