Impasse to the Measure of Being. Thought of the Day 47.0

The power set p(x) of x – the state of situation x or its metastructure (Alain Badiou – Being and Event) – is defined as the set of all subsets of x. Now, basic relations between sets can be expressed as the following relations between sets and their power sets. If for some x, every element of x is also a subset of x, then x is a subset of p(x), and x can be reduced to its power set. Conversely, if every subset of x is an element of x, then p(x) is a subset of x, and the power set p(x) can be reduced to x. Sets that satisfy the first condition are called transitive. For obvious reasons the empty set is transitive. However, the second relation never holds. The mathematician Georg Cantor proved that not only p(x) can never be a subset of x, but in some fundamental sense it is strictly larger than x. On the other hand, axioms of set theory do not determine the extent of this difference. Badiou says that it is an “excess of being”, an excess that at the same time is its impasse.

In order to explain the mathematical sense of this statement, recall the notion of cardinality, which clarifies and generalizes the common understanding of quantity. We say that two sets x and y have the same cardinality if there exists a function defining a one-to-one correspondence between elements of x and elements of y. For finite sets, this definition agrees with common intuitions: if a finite set y has more elements than a finite set x, then regardless of how elements of x are assigned to elements of y, something will be left over in y precisely because it is larger. In particular, if y contains x and some other elements, then y does not have the same cardinality as x. This seemingly trivial fact is not always true outside of the domain of finite sets. To give a simple example, the set of all natural numbers contains quadratic numbers, that is, numbers of the form n², as well as some other numbers but the set of all natural numbers, and the set of quadratic numbers have the same cardinality. The correspondence witnessing this fact assigns to every number n a unique quadratic number, namely n².

Counting finite sets has always been done via natural numbers 0, 1, 2, . . . In set theory, the concept of such a canonical measure can be extended to infinite sets, using the notion of cardinal numbers. Without getting into details of their definition, let us say that the series of cardinal numbers begins with natural numbers, which are directly followed by the number ω₀, that is, the size of the set of all natural numbers , then by ω₁, the first uncountable cardinal numbers, etc. The hierarchy of cardinal numbers has the property that every set x, finite or infinite, has cardinality (i.e. size) equal to exactly one cardinal number κ. We say then that κ is the cardinality of x.

The cardinality of the power set p(x) is 2ⁿ for every finite set x of cardinality n. However, something quite paradoxical happens when infinite sets are considered. Even though Cantor’s theorem does state that the cardinality of p(x) is always larger than x – similarly as in the case of finite sets – axioms of set theory never determine the exact cardinality of p(x). Moreover, one can formally prove that there exists no proof determining the cardinality of the power sets of any given infinite set. There is a general method of building models of set theory, discovered by the mathematician Paul Cohen, and called forcing, that yields models, where – depending on construction details – cardinalities of infinite power sets can take different values. Consequently, quantity – “a fetish of objectivity” as Badiou calls it – does not define a measure of being but it leads to its impasse instead. It reveals an undetermined gap, where an event can occur – “that-which-is-not being-qua-being”.

Noneism. Part 2.

Noneism is a very rigourous and original philosophical doctrine, by and large superior to the classical mathematical philosophies. But there are some problems concerning the different ways of characterizing a universe of objects. It is very easy to understand the way a writer characterizes the protagonists of the novels he writes. But what about the characterization of the universe of natural numbers? Since in most kinds of civilizations the natural numbers are characterized the same way, we have the impression that the subject does not intervene in the forging of the characteristics of natural numbers. These numbers appear to be what they are, with total independence of the creative activity of the cognitive subject. There is, of course, the creation of theorems, but the potentially infinite sequence of natural numbers resists any effort to subjectivize its characteristics. It cannot be changed. A noneist might reply that natural numbers are non-existent, that they have no being, and that, in this respect, they are identical with mythological Objects. Moreover, the formal system of natural numbers can be interpreted in many ways: for instance, with respect to a universe of Skolem numbers. This is correct, but it does not explain why the properties of some universes are independent from subjective creation. It is an undeniable fact that there are two kinds of objectual characteristics. On the one hand, we have the characteristics created by subjective imagination or speculative thought; on the other hand, we find some characteristics that are not created by anybody; their corresponding Objects are, in most cases, non-existent but, at the same time, they are not invented. They are just found. The origin of the former characteristics is very easy to understand; the origin of the last ones is, a mystery.

Now, the subject-independence of a universe, suggests that it belongs to a Platonic realm. And as far as transafinite set theory is concerned, the subject-independence of its characteristics is much less evident than the characteristic subject-independence of the natural numbers. In the realm of the finite, both characteristics are subject-independent and can be reduced to combinatorics. The only difference between both is that, according to the classical Platonistic interpretation of mathematics, there can only be a single mathematical universe and that, to deductively study its properties, one can only employ classical logic. But this position is not at all unobjectionable. Once the subject-independence of the natural numbers system’s characteristics is posited, it becomes easy to overstep the classical phobia concerning the possibility of characterizing non-classical objective worlds. Euclidean geometry is incompatible with elliptical and hyperbolic geometries and, nevertheless, the validity of the first one does not invalidate the other ones. And vice versa, the fact that hyperbolic and other kinds of geometry are consistently characterized, does not invalidate the good old Euclidean geometry. And the fact that we have now several kinds of non-Cantorian set theories, does not invalidate the classical Cantorian set theory.

Of course, an universally non-Platonic point of view that includes classical set theory can also be assumed. But concerning natural numbers it would be quite artificial. It is very difficult not to surrender to the famous Kronecker’s dictum: God created natural numbers, men created all the rest. Anyhow, it is not at all absurd to adopt a whole platonistic conception of mathematics. And it is quite licit to adopt a noneist position. But if we do this, the origin of the natural numbers’ characteristics becomes misty. However, forgetting this cloudiness, the leap from noneist universes to the platonistic ones, and vice versa, becomes like a flip-flop connecting objectological with ontological (ideal) universes, like a kind of rabbit-duck Gestalt or a Sherrington staircase. So, the fundamental question with respect to the subject-dependent or subject-independent mathematical theories, is: are they created, or are they found? Regarding some theories, subject-dependency is far more understandable; and concerning other ones, subject-independency is very difficult, if not impossible, to negate.

From an epistemological point of view, the fact of non-subject dependent characteristic traits of a universe would mean that there is something like intellectual intuition. The properties of natural numbers, the finite properties of sets (or combinatorics), some geometric axioms, for instance, in Euclidean geometry, the axioms of betweenness, etc., would be apprehended in a manner, that pretty well coincides with the (nowadays rather discredited) concept of synthetical a priori knowledge. This aspect of mathematical knowledge shows that the old problem concerning the analytic and the a priori synthetical knowledge, in spite of the prevailing Quinean pragmatic conception, must be radically reset.

Irrationality. Note Quote.

To mathematics it is unique, that two absolutely contrary opinions do not logically exclude each other but exist simultaneously while there seems to be no chance to pick out a false one and to establish a remaining truth. This case is realised by the philosophy and mathematics of the infinite. While transfinite set theory is impossible without different degrees of infinity, constructivists and intuitionists deny this notion without running into inconsistencies as is admitted by some of the foremost set theorists:

… the attitude of the (neo-)intuitionists that there do not exist altogether non-equivalent infinite sets is consistent, though almost suicidal for mathematics. [p. 62]

It would not be astonishing if in different axiomatic systems different results were obtained with respect to peculiarities of those systems. But set theorists on one side and constructivists and intuitionists on the other are certainly believing to address the same entities when speaking of “rational numbers” or of “irrational numbers”. In spite of that, the former are convinced that there are infinitely many more irrational numbers than rational numbers while the latter deny that:

Hence the continua of Weyl, Lebesgue, Lusin, etc. are denumerable … [p. 255]

This situation yields bewildering results:

Feferman and Levy showed that one cannot prove that there is any non-denumerable set of real numbers which can be well ordered. … Moreover, they also showed that the statement that the set of all real numbers is the union of a denumerable set of denumerable sets cannot be refuted. [p. 62]

Nevertheless, the great majority of mathematicians refuse to accept the thesis that Cantor’s ideas were but a pathological fancy. Though the foundations of set theory are still somewhat shaky. Most surprising and by no means to be expected of a pupil of Fraenkel’s is that Robinson states:

Infinite totalities do not exist in any sense of the word (i.e. either really or ideally). More precisely, any mention, or purported mention, of infinite totalities is, literally, meaningless. Nevertheless, we should act as if infinite totalities really existed. [3]

Does there exist a correct and an incorrect position? And, if so, who is right, who is wrong?

Following the advice of Fraenkel, namely to judge about the value and necessity of the basic axioms, in particular of the axiom of choice, by considering its consequences, in order to settle this question. These consequences will turn out to entail what, in an euphemistic way, by set theorists usually is called a “paradoxical result”, in order to avoid the term self-contradiction.

Apart from the well-ordering theorem some statements of quite different character – in particular geometrical statements – have been proved by means of the axiom of choice, which because of their paradoxical character induced some mathematicians to reject the axiom. Presumably the earliest statement of this kind is Hausdorff’s discovery that half of the sphere’s surface is congruent to a third of it. … It may surprise scholars working in the field … that even after more than half a century of utilising the axiom of choice and well-ordering theorem, a number of first-rate mathematicians (especially French) have not essentially changed their distrustful attitude.

Transfinite set theory arises from Cantor’s observation that the set of all irrational numbers has infinitely many more members than the set of all rational numbers. While the latter has the same cardinality χ₀ as the set N of all natural numbers n, the cardinality χ of the set of all irrational numbers is larger, χ = 2^χ₀. It is proven to be uncountable, i.e., any bijection with N can be excluded.

Human Rights, (Badiou + Rancière)

“Human Rights are axioms. They can co-exist on the market with many other axioms, notably those concerning security or property, which are unaware of or suspend them even more than they contradict them: “the impure mixture or the impure side by side,” said Nietzsche. Who but the police and armed forces that co-exist with democracies can control and manage poverty and the deterritorialization-reterritorialization of shanty towns? What social democracy has not given the order to fire when the poor come out of their territory or ghetto? Rights save neither men nor a philosophy that is reterritorialized on the democratic State. Human rights will not make us bless capitalism. A great deal of innocence or cunning is needed by a philosophy of communication that claims to restore the society of “consensus” to moralize nations, States, and the market. Humans rights say nothing about the immanent modes of existence of people provided with rights.” 

— Deleuze and Guattari (1996, 107)

The quote from Deleuze and Guattari’s ‘What is Philosophy?’ nicely sums up the abstraction, pure, empty abstraction that Deleuze calls the party line for odious intellectuals. Deleuze does not pay much heed to the notion of human rights, but, instead broods over life rights.

My intention is not to delve into the Deleuzean version of Human/Life Rights, but to look at the conception of human rights from the point of view of two French philosophers who have made an impact in the English speaking world. The thinkers in question here are  Badiou and Rancière who with their delivery of political agencies are not only complexly similar on many grounds, but provide many insights into the differences between one another. Their writings  delve into human rights as a base for their versions of political agencies.

The time was February, 2008, when a group of the so called new philosophers signed a petition in Le Monde calling the practices of the United Nations as diametrically opposed to the ideals of human rights. It was further commented that there was a cause of concern with the institution becoming a caricature. Although we are ensconced in a multicultural world, the level of tolerance could be said to be reaching a nadir of sorts as was substantially proved in the petition detailing religious criticism as a form of racism, thus highlighting the tide against the basic ideas of Human Rights. The petition then called for the return to the Universal Ideals on Human Rights.

It would be far fetched, but still appropriate to call the new philosophers as sandwiched between viewing the ideals of 1948 as problematic with their emphasis on a return to ideals. That the new philosophers are sandwiched between the two poles is attributable to one pole being that of Arendt and Agamben with their insistence on Human Rights as infringing of the political into the private sphere and the other of Badiou and Rancière calling for a political agency critiquing both the view points. Arendt dismisses the idea of Human rights by calling it necessity based as action by the nation state to impose its control over the huge mass of refugees created in the aftermath of the second world war, when the refugees that had been rendered stateless had nothing left but their humanity. This way, for Arendt, the nation state gets to determining who gets the rights and who doesn’t or who is part of humanity and who is not, thus quashing the ideals meant to protecting rights. Despite the good intentions behind the formulation of the Universal Ideals, it is nothing but an apparatus through which, the state exercises its total power over the stateless by making the latter submissive to it and other allied organs of power. This indeed proves the thesis that an interventionist approach is taken up by the nation state into the private sphere.

Arendt’s thesis is pressed upon by Agamben, when he calls the peril of our present time as lying alongside an intercourse of the political power into the bare public life as omnipresent. Agamben links the notional intercourse as no different from what the refugees had to face in concentration camps. For him, the human rights act in a totalizing manner as now the most basic human existence is intricately surrendered to power structures thus making existence politicized. To quote Agamben,

“…until a completely new politics – that is, a politics no longer founded on the exception of bare life – is at hand, every theory and every praxis will remain imprisoned and immobile, and the “beautiful day” of life will be given citizenship only either through blood and death or in the perfect senselessness to which the society of the spectacle condemns it.”

Ernst Hemel reads the following quote in a dual way, viz. the seizure of private bare life by the structures of power and the deprivations of the individual in engaging with true emancipatory politics. The institutionalization of human rights is therefore seen as a part of the imprisoned and immobile life that somehow fails in its approach to reach the blunt political situation we are all faced up with. This reaches its aporetic limit in a way to invent a new political situation after criticizing the entire idea underlining human rights. So for both Arendt and Agamben, the codification of the Universal Declaration of Human Rights is fraught with a critique that runs counter to new philosopher’s insistence on attaining the ideal.

Badiou and Rancière both show their aversion to these readings and in their own ways of constructing the political agency exhibit displeasure in treating human rights as an ideal on the one hand and refusing to believe in the all encompassing political dominion on the other. Rancière brilliantly unearths the tautology in Arendt’s version of human rights by noting that the rights of man are the rights of the unpoliticized person, or they are the rights of those who have no rights, thus amounting to nothing and rights of man are the rights of the citizen, that is, they are being attached to the fact of being a citizen, thus connoting rights of man as rights of citizens. This in conflation amounts to a tautology. In effect, there is abandonment of human rights in Arendt according to Rancière since it is based on state power who has the discretion of providing rights to those who are excluded. This argument is taken forward to deal with Agamben, wherein it is noted that any kind of emancipatory political action is in retreat. Rancière quotes from his ‘The Politics of Aesthetics’,

“There was at least one point where ‘bare life’ proved to be ‘political’: there were women sentenced to death as enemies of the revolution. If they could lose their ‘bare life’ out of the public judgment based on political reasons, this meant that even their bare life – their life doomed to death – was political. If, under the guillotine, they were as equal, so to speak, as men, they had the right to the whole of equality, including equal participation to political life.”

It is only in these situations that the totality is fissured in that there is a sense of inclusion but not belonging that is governed by exclusion that is only brought to light through acts of dissensions. This Rancièrean point is closely linked up with what Badiou has been maintaining with his ‘Event’. ‘Event’ is the coming into being of what was never thought of (accidental) in the conceptual structuring of the present scenario. To explicate on the coming into existence of the ‘Event’, one needs to change the conceptuality and his idea behind this is borrowed from Cantorian set theory of placing the element inside the set, but at the same time not belonging to the set. This is philosophically pertinent to the distinction between the political inclusion but non-belonging, in that, inclusion shares the possibilities in the world, whereas belonging-ness presents a systemic snapshot congruent with the given world view. In the moment of the ‘Event’, a person is faced with an ethical choice, by either denying what happened as new and trying to fit it in the existing template or by accepting it and building upon new consequences. To draw on these consequences is brought about by the act of naming. For Badiou, the notion of human rights is incapable of accommodating truth and is an attempt on the part of the dominant structure to be be able to account for all elements of the set. As he writes,

“The refrain of “human rights” is nothing other than the ideology of modern liberal capitalism: We won’t massacre you, we won’t torture you in caves, so keep quiet and worship the golden calf. As for those, who don’t want to worship it, or who don’t believe in our superiority, there’s always the American army and its European minions to make them be quiet.”

For Badiou, the only universality is that which resists structuring and becomes tangible in the notion of an ‘Event’. If Universality be equated with Truth, then according to his thesis in Manifesto of Philosophy, ‘Truth makes a hole in knowledge‘ and therefore it could now be inferred why for Badiou human rights as a kind of universality in equality, in freedom is anything but a form of dominant western ideology. To quote him again,

“The latest violence, the presumptuous arrogance inherent in the currently prevalent conception of human rights derives from the fact that these are actually the rights of the finitude […]. By way of contrast, the eventual conception of universal singularities requires that human rights be thought of as the rights of the infinite.”

So for Badiou, codification of the situations along the prefixed lines of universality results in redundancy alone and little wonder why he admonishes the case for human rights to be thought of as that which is included but not belonging. He takes a similar viewpoint towards justice by claiming the irrelevance of justice in the creation of anything new and thereby is more concerned with the conditions of possibilities of new politics rather than improving the sphere of juridicalness. In a way, what Badiou is looking for is very similar to what Rancière aims at and that being looking at human rights as an affirmative action. For both the thinkers, it is the exclusive situation where the insight into the human rights is to be taken up, to be formulated in a reconstructive manner. The exclusive situation is normed as disruption by the thinkers and this disruption is then the affirmation for the coming into being of affirmative changes in the socio-political aspect. Since, this aspect of coming into existence is missing in the universal declaration, it becomes non-political in its conception as far as gauging the totality of the situation is concerned. Rancière sees this as the inability of the logic that dictates who is part of the situation, who has the right to voice claims and who forms the basis of political agency. For Badiou, it is the false totality altogether as it is impossible to envisage anything new or radical getting to the surface concretely. Since, there is absence of anything radically new, it is doomed to repeat the dominant power based ideology.

Although there are similarities in the ways the thinkers look at human rights, there are some differences that are stark in nature. For Rancière, it is the un-belongingness that counts cardinally despite the fact of the subject being inclusive in the system under consideration, whereas for Badiou, it is the subject getting called onto witnessing the ‘Event’ and thereby faced up with the radical choice that is ethical in nature. Badiou’s invoking of mathematics to first name the ‘Event’ and thereafter follow it up to rewrite the radicality of the situation differs from reinterpreting human rights as suggested by Rancière. Most importantly, Rancière uses disruption as a singular revelation in that he is constrained in the expansionary vision/force of the dissensus. Badiou on the other hand emphasizes on the extensibility of the ‘Event’. Rancière works within the existing system and is not concerned much with restructuring and vacillates between dissensus and consensus thereby giving it a more democratic feel of basing itself on negotiation, where Badiou aims at a revolutionary agency that he calls militant in nature.

The only universal human right that Badiou and Rancière envision is the right to intervene in the name of infinite universality, and they remain far from any institutionalization of universal human rights. Instead their theories  are geared towards a critical evaluation of the underlying presuppositions  of doing politics, and providing rights. This critical evaluation is done in  preparation of ‘truthful’ politics,   which entails for both Rancière and  Badiou a radical break with notions of politics that are defined in terms  such as citizenship, freedom of speech or a return to ideal enlightenment values. Politics aim at a constant possibility.

Agamben, G. (1998) Homo Sacer: Sovereign Power and Bare Life. Stanford: Stanford University Press.

Arendt, H. (1973) The Origins of Totalitarianism. New York: Harcourt Brace Jovanovich Inc.

Badiou, A. (2004) ‘Huit Thèses sur l’Universel’. http://www.lacan.com/baduniversel.htm

    – (2001/2002) ‘On Evil : An Interview with Alain Badiou’. Cabinet Magizine Online, 5, http://www.egs.edu/faculty/badiou/badiou-on-evil.html

Deleuze, G and Guattari, F (1996) ‘What is Philosophy?’. New York: Columbia University Press.

Hemel, Ernst van den. (2008) Krisis:  Journal for Contemporary Philosophy.

Rancière, J. (2004) The Politics of Aesthetics. New York: Continuum.