Utopia as Emergence Initiating a Truth. Thought of the Day 104.0

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It is true that, in our contemporary world, traditional utopian models have withered, but today a new utopia of canonical majority has taken over the space of any action transformative of current social relations. Instead of radicalness, conformity has become the main expression of solidarity for the subject abandoned to her consecrated individuality. Where past utopias inscribed a collective vision to be fulfilled for future generations, the present utopia confiscates the future of the individual, unless she registers in a collective, popularized expression of the norm that reaps culture, politics, morality, and the like. The ideological outcome of the canonical utopia is the belief that the majority constitutes a safety net for individuality. If the future of the individual is bleak, at least there is some hope in saving his/her present.

This condition reiterates Ernst Bloch’s distinction between anticipatory and compensatory utopia, with the latter gaining ground today (Ruth Levitas). By discarding the myth of a better future for all, the subject succumbs to the immobilizing myth of a safe present for herself (the ultimate transmutation of individuality to individualism). The world can surmount Difference, simply by taking away its painful radicalness, replacing it with a non-violent, pluralistic, and multi-cultural present, as Žižek harshly criticized it for its anti-rational status. In line with Badiou and Jameson, Žižek discerns behind the multitude of identities and lifestyles in our world the dominance of the One and the eradication of Difference (the void of antagonism). It would have been ideal, if pluralism were not translated to populism and the non-violent to a sanctimonious respect of Otherness.

Badiou also points to the nihilism that permeates modern ethicology that puts forward the “recognition of the other”, the respect of “differences”, and “multi-culturalism”. Such ethics is supposed to protect the subject from discriminatory behaviours on the basis of sex, race, culture, religion, and so on, as one must display “tolerance” towards others who maintain different thinking and behaviour patterns. For Badiou, this ethical discourse is far from effective and truthful, as is revealed by the competing axes it forges (e.g., opposition between “tolerance” and “fanaticism”, “recognition of the other” and “identitarian fixity”).

Badiou denounces the decomposed religiosity of current ethical discourse, in the face of the pharisaic advocates of the right to difference who are “clearly horrified by any vigorously sustained difference”. The pharisaism of this respect for difference lies in the fact that it suggests the acceptance of the other, in so far as s/he is a “good other”; in other words, in so far as s/he is the same as everyone else. Such an ethical attitude ironically affirms the hegemonic identity of those who opt for integration of the different other, which is to say, the other is requested to suppress his/her difference, so that he partakes in the “Western identity”.

Rather than equating being with the One, the law of being is the multiple “without one”, that is, every multiple being is a multiple of multiples, stretching alterity into infinity; alterity is simply “what there is” and our experience is “the infinite deployment of infinite differences”. Only the void can discontinue this multiplicity of being, through the event that “breaks” with the existing order and calls for a “new way of being”. Thus, a radical utopian gesture needs to emerge from the perspective of the event, initiating a truth process.

Utopia Banished. Thought of the Day 103.0

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In its essence, utopia has nothing to do with imagining an impossible ideal society; what characterizes utopia is literally the construction of a u-topic space, a space outside the existing parameters, the parameters of what appears to be “possible” in the existing social universe. The “utopian” gesture is the gesture that changes the coordinates of the possible. — (Slavoj Žižek- Iraq The Borrowed Kettle)

Here, Žižek discusses Leninist utopia, juxtaposing it with the current utopia of the end of utopia, the end of history. How propitious is the current anti-utopian aura for future political action? If society lies in impossibility, as Laclau and Mouffe (Hegemony and Socialist Strategy Towards a Radical Democratic Politics) argued, the field of politics is also marked by the impossible. Failing to fabricate an ideological discourse and incapable of historicizing, psychoanalysis appears as “politically impotent” and unable to encumber the way for other ideological narratives to breed the expectation of making the impossible possible, by promising to cover the fissure of the real in socio-political relations. This means that psychoanalysis can interminably unveil the impossible, only for a recycling of ideologies (outside the psychoanalytic discourse) to attempt to veil it.

Juxtaposing the possibility of a “post-fantasmatic” or “less fantasmatic” politics accepts the irreducible ambiguity of democracy and thus fosters the prospect of a radical democratic project. Yet, such a conception is not uncomplicated, given that one cannot totally go beyond fantasy and still maintain one’s subjectivity (even when one traverses it, another fantasy eventually grows), precisely because fantasy is required for the coherence of the subject and the upholding of her desire. Furthermore, fantasy is either there or not; we cannot have “more” or “less” fantasy. Fantasy, in itself, is absolute and totalizing par excellence. It is the real and the symbolic that always make it “less fantasmatic”, as they impose a limit in its operation.

So, where does “perversion” fit within this frame? The encounter with the extra-ordinary is an encounter with the real that reveals the contradiction that lies at the heart of the political. Extra-ordinariness suggests the embodiment of the real within the socio-political milieu; this is where the extra-ordinary subject incarnates the impossible object. Nonetheless, it suggests a fantasmatic strategy of incorporating the real in the symbolic, as an alternative to the encircling of the real through sublimation. In sublimation we still have an (artistic) object standing for the object a, so the lack in the subject is still there, whereas in extra-ordinariness the subject occupies the locus of the object a, in an ephemeral eradication of his/her lack. Extra-ordinariness may not be a condition that subverts or transforms socio-political relations, yet it can have a certain political significance. Rather than a direct confrontation with the impossible, it suggests a fantasmatic embracing of the impossible in its inexpressible totality, which can be perceived as a utopian aspiration.

Following Žižek or Badiou’s contemporary views, the extra-ordinary gesture is not qualified as an authentic utopian act, because it does not traverse fantasy, it does not rewrite social conditions. It is well known that Žižek prioritizes the negativeness of the real in his rhetoric, something that outstrips any positive imaginary or symbolic reflection in his work. But this entails the risk of neglecting the equal importance of all three registers for subjectivity. The imaginary constitutes an essential motive force for any drastic action to take place, as long as the symbolic limit is not thwarted. It is also what keeps us humane and sustains our relation to the other.

It is possible to touch the real, through imaginary means, without becoming a post-human figure (such as Antigone, who remains the figurative conception of Žižek’s traversing of the fantasy). Fantasy (and, therefore, ideology) can be a source of optimism and motivation and it should not be bound exclusively to the static character of compensatory utopia, according to Bloch’s distinction. In as much as fantasy infuses the subject’s effort to grasp the impossible, recognizing it as such and not breeding the futile expectation of turning the impossible into possible (regaining the object, meeting happiness), the imaginary can form the pedestal for an anticipatory utopia.

The imaginary does not operate only as a force that disavows difference for the sake of an impossible unity and completeness. It also suggests an apparatus that soothes the realization of the symbolic fissure, breeding hope and fascination, that is to say, it stirs up emotional states that encircle the lack of the subject. Moreover, it must be noted that the object a, apart from real properties, also has an imaginary hypostasis, as it is screened in fantasies that cover lack. If our image’s coherence is an illusion, it is this illusion that motivates us as individual and social subjects and help us relate to each other.

The anti-imaginary undercurrent in psychoanalysis is also what accounts for renunciation of idealism in the democratic discourse. The point de capiton is not just a common point of reference; it is a master signifier, which means it constitutes an ideal par excellence. The master signifier relies on fantasy and imaginary certainty about its supreme status. The ideal embodied by the master is what motivates action, not only in politics, but also in sciences, and arts. Is there a democratic prospect for the prevalence of an ideal that does not promise impossible jouissance, but possible jouissance, without confining it to the phallus? Since it is possible to touch jouissance, but not to represent it, the encounter with jouissance could endorse an ideal of incompleteness, an ideal of confronting the limits of human experience vis-à-vis unutterable enjoyment.

We need an extra-ordinary utopianism to the extent that it provokes pre-fixed phallic and normative access to enjoyment. The extra-ordinary himself does not go so far as to demand another master signifier, but his act is sufficiently provocative in divulging the futility of the master’s imaginary superiority. However, the limits of the extra-ordinary utopian logic is that its fantasy of embodying the impossible never stops in its embodiment (precisely because it is still a fantasy), and instead it continues to make attempts to grasp it, without accepting that the impossible remains impossible.

An alternative utopia could probably maintain the fantasy of embodying the impossible, acknowledging it as such. So, any time fantasy collapses, violence does not emerge as a response, but we continue the effort to symbolically speculate and represent the impossible, precisely because in this effort resides hope that sustains our reason to live and desire. As some historians say, myths distort “truth”, yet we cannot live without them; myths can form the only tolerable approximation of “truth”. One should see them as “colourful” disguises of the achromous core of his/her existence, and the truth is we need more “colour”.

Conjuncted: Indiscernibles – Philosophical Constructibility. Thought of the Day 48.1

Simulated Reality

Conjuncted here.

“Thought is nothing other than the desire to finish with the exorbitant excess of the state” (Being and Event). Since Cantor’s theorem implies that this excess cannot be removed or reduced to the situation itself, the only way left is to take control of it. A basic, paradigmatic strategy for achieving this goal is to subject the excess to the power of language. Its essence has been expressed by Leibniz in the form of the principle of indiscernibles: there cannot exist two things whose difference cannot be marked by a describable property. In this manner, language assumes the role of a “law of being”, postulating identity, where it cannot find a difference. Meanwhile – according to Badiou – the generic truth is indiscernible: there is no property expressible in the language of set theory that characterizes elements of the generic set. Truth is beyond the power of knowledge, only the subject can support a procedure of fidelity by deciding what belongs to a truth. This key thesis is established using purely formal means, so it should be regarded as one of the peak moments of the mathematical method employed by Badiou.

Badiou composes the indiscernible out of as many as three different mathematical notions. First of all, he decides that it corresponds to the concept of the inconstructible. Later, however, he writes that “a set δ is discernible (…) if there exists (…) an explicit formula λ(x) (…) such that ‘belong to δ’ and ‘have the property expressed by λ(x)’ coincide”. Finally, at the outset of the argument designed to demonstrate the indiscernibility of truth he brings in yet another definition: “let us suppose the contrary: the discernibility of G. A formula thus exists λ(x, a1,…, an) with parameters a1…, an belonging to M[G] such that for an inhabitant of M[G] it defines the multiple G”. In short, discernibility is understood as:

  1. constructibility
  2. definability by a formula F(y) with one free variable and no parameters. In this approach, a set a is definable if there exists a formula F(y) such that b is an element of a if F(b) holds.
  3. definability by a formula F (y, z1 . . . , zn) with parameters. This time, a set a is definable if there exists a formula F(y, z1,…, zn) and sets a1,…, an such that after substituting z1 = a1,…, zn = an, an element b belongs to a iff F(b, a1,…, an) holds.

Even though in “Being and Event” Badiou does not explain the reasons for this variation, it clearly follows from his other writings (Alain Badiou Conditions) that he is convinced that these notions are equivalent. It should be emphasized then that this is not true: a set may be discernible in one sense, but indiscernible in another. First of all, the last definition has been included probably by mistake because it is trivial. Every set in M[G] is discernible in this sense because for every set a the formula F(y, x) defined as y belongs to x defines a after substituting x = a. Accepting this version of indiscernibility would lead to the conclusion that truth is always discernible, while Badiou claims that it is not so.

Is it not possible to choose the second option and identify discernibility with definability by a formula with no parameters? After all, this notion is most similar to the original idea of Leibniz intuitively, the formula F(y) expresses a property characterizing elements of the set defined by it. Unfortunately, this solution does not warrant indiscernibility of the generic set either. As a matter of fact, assuming that in ontology, that is, in set theory, discernibility corresponds to constructibility, Badiou is right that the generic set is necessarily indiscernible. However, constructibility is a highly technical notion, and its philosophical interpretation seems very problematic. Let us take a closer look at it.

The class of constructible sets – usually denoted by the letter L – forms a hierarchy indexed or numbered by ordinal numbers. The lowest level L0 is simply the empty set. Assuming that some level – let us denote it by Lα – has already been

constructed, the next level Lα+1 is constructed by choosing all subsets of L that can be defined by a formula (possibly with parameters) bounded to the lower level Lα.

Bounding a formula to Lα means that its parameters must belong to Lα and that its quantifiers are restricted to elements of Lα. For instance, the formula ‘there exists z such that z is in y’ simply says that y is not empty. After bounding it to Lα this formula takes the form ‘there exists z in Lα such that z is in y’, so it says that y is not empty, and some element from Lα witnesses it. Accordingly, the set defined by it consists of precisely those sets in Lα that contain an element from Lα.

After constructing an infinite sequence of levels, the level directly above them all is simply the set of all elements constructed so far. For example, the first infinite level Lω consists of all elements constructed on levels L0, L1, L2,….

As a result of applying this inductive definition, on each level of the hierarchy all the formulas are used, so that two distinct sets may be defined by the same formula. On the other hand, only bounded formulas take part in the construction. The definition of constructibility offers too little and too much at the same time. This technical notion resembles the Leibnizian discernibility only in so far as it refers to formulas. In set theory there are more notions of this type though.

To realize difficulties involved in attempts to philosophically interpret constructibility, one may consider a slight, purely technical, extension of it. Let us also accept sets that can be defined by a formula F (y, z1, . . . , zn) with constructible parameters, that is, parameters coming from L. Such a step does not lead further away from the common understanding of Leibniz’s principle than constructibility itself: if parameters coming from lower levels of the hierarchy are admissible when constructing a new set, why not admit others as well, especially since this condition has no philosophical justification?

Actually, one can accept parameters coming from an even more restricted class, e.g., the class of ordinal numbers. Then we will obtain the notion of definability from ordinal numbers. This minor modification of the concept of constructibility – a relaxation of the requirement that the procedure of construction has to be restricted to lower levels of the hierarchy – results in drastic consequences.

Evental Sites. Thought of the Day 48.0

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According to Badiou, the undecidable truth is located beyond the boundaries of authoritative claims of knowledge. At the same time, undecidability indicates that truth has a post-evental character: “the heart of the truth is that the event in which it originates is undecidable” (Being and Event). Badiou explains that, in terms of forcing, undecidability means that the conditions belonging to the generic set force sentences that are not consequences of axioms of set theory. If in the domains of specific languages (of politics, science, art or love) the effects of event are not visible, the content of “Being and Event” is an empty exercise in abstraction.

Badiou distances himself from\ a narrow interpretation of the function played by axioms. He rather regards them as collections of basic convictions that organize situations, the conceptual or ideological framework of a historical situation. An event, named by an intervention, is at the theoretical site indexed by a proposition A, a new apparatus, demonstrative or axiomatic, such that A is henceforth clearly admissible as a proposition of the situation. Accordingly, the undecidability of a truth would consist in transcending the theoretical framework of a historical situation or even breaking with it in the sense that the faithful subject accepts beliefs that are impossible to reconcile with the old mode of thinking.

However, if one consequently identifies the effect of event with the structure of the generic extension, they need to conclude that these historical situations are by no means the effects of event. This is because a crucial property of every generic extension is that axioms of set theory remain valid within it. It is the very core of the method of forcing. Without this assumption, Cohen’s original construction would have no raison d’être because it would not establish the undecidability of the cardinality of infinite power sets. Every generic extension satisfies axioms of set theory. In reference to historical situations, it must be conceded that a procedure of fidelity may modify a situation by forcing undecidable sentences, nonetheless it never overrules its organizing principles.

Another notion which cannot be located within the generic theory of truth without extreme consequences is evental site. An evental site – an element “on the edge of the void” – opens up a situation to the possibility of an event. Ontologically, it is defined as “a multiple such that none of its elements are presented in the situation”. In other words, it is a set such that neither itself nor any of its subsets are elements of the state of the situation. As the double meaning of this word indicates, the state in the context of historical situations takes the shape of the State. A paradigmatic example of a historical evental site is the proletariat – entirely dispossessed, and absent from the political stage.

The existence of an evental site in a situation is a necessary requirement for an event to occur. Badiou is very strict about this point: “we shall posit once and for all that there are no natural events, nor are there neutral events” – and it should be clarified that situations are divided into natural, neutral, and those that contain an evental site. The very matheme of event – its formal definition is of no importance here is based on the evental site. The event raises the evental site to the surface, making it represented on the level of the state of the situation. Moreover, a novelty that has the structure of the generic set but it does not emerge from the void of an evental site, leads to a simulacrum of truth, which is one of the figures of Evil.

However, if one takes the mathematical framework of Badiou’s concept of event seriously, it turns out that there is no place for the evental site there – it is forbidden by the assumption of transitivity of the ground model M. This ingredient plays a fundamental role in forcing, and its removal would ruin the whole construction of the generic extension. As is known, transitivity means that if a set belongs to M, all its elements also belong to M. However, an evental site is a set none of whose elements belongs to M. Therefore, contrary to Badious intentions, there cannot exist evental sites in the ground model. Using Badiou’s terminology, one can say that forcing may only be the theory of the simulacrum of truth.

Conjuncted: Operations of Truth. Thought of the Day 47.1

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Conjuncted here.

Let us consider only the power set of the set of all natural numbers, which is the smallest infinite set – the countable infinity. By a model of set theory we understand a set in which  – if we restrict ourselves to its elements only – all axioms of set theory are satisfied. It follows from Gödel’s completeness theorem that as long as set theory is consistent, no statement which is true in some model of set theory can contradict logical consequences of its axioms. If the cardinality of p(N) was such a consequence, there would exist a cardinal number κ such that the sentence the cardinality of p(N) is κ would be true in all the models. However, for every cardinal κ the technique of forcing allows for finding a model M where the cardinality of p(N) is not equal to κ. Thus, for no κ, the sentence the cardinality of p(N) is κ is a consequence of the axioms of set theory, i.e. they do not decide the cardinality of p(N).

The starting point of forcing is a model M of set theory – called the ground model – which is countably infinite and transitive. As a matter of fact, the existence of such a model cannot be proved but it is known that there exists a countable and transitive model for every finite subset of axioms.

A characteristic subtlety can be observed here. From the perspective of an inhabitant of the universe, that is, if all the sets are considered, the model M is only a small part of this universe. It is deficient in almost every respect; for example all of its elements are countable, even though the existence of uncountable sets is a consequence of the axioms of set theory. However, from the point of view of an inhabitant of M, that is, if elements outside of M are disregarded, everything is in order. Some of M because in this model there are no functions establishing a one-to-one correspondence between them and ω0. One could say that M simulates the properties of the whole universe.

The main objective of forcing is to build a new model M[G] based on M, which contains M, and satisfies certain additional properties. The model M[G] is called the generic extension of M. In order to accomplish this goal, a particular set is distinguished in M and its elements are referred to as conditions which will be used to determine basic properties of the generic extension. In case of the forcing that proves the undecidability of the cardinality of p(N), the set of conditions codes finite fragments of a function witnessing the correspondence between p(N) and a fixed cardinal κ.

In the next step, an appropriately chosen set G is added to M as well as other sets that are indispensable in order for M[G] to satisfy the axioms of set theory. This set – called generic – is a subset of the set of conditions that always lays outside of M. The construction of M[G] is exceptional in the sense that its key properties can be described and proved using M only, or just the conditions, thus, without referring to the generic set. This is possible for three reasons. First of all, every element x of M[G] has a name existing already in M (that is, an element in M that codes x in some particular way). Secondly, based on these names, one can design a language called the forcing language or – as Badiou terms it – the subject language that is powerful enough to express every sentence of set theory referring to the generic extension. Finally, it turns out that the validity of sentences of the forcing language in the extension M[G] depends on the set of conditions: the conditions force validity of sentences of the forcing language in a precisely specified sense. As it has already been said, the generic set G consists of some of the conditions, so even though G is outside of M, its elements are in M. Recognizing which of them will end up in G is not possible for an inhabitant of M, however in some cases the following can be proved: provided that the condition p is an element of G, the sentence S is true in the generic extension constructed using this generic set G. We say then that p forces S.

In this way, with an aid of the forcing language, one can prove that every generic set of the Cohen forcing codes an entire function defining a one-to-one correspondence between elements of p(N) and a fixed (uncountable) cardinal number – it turns out that all the conditions force the sentence stating this property of G, so regardless of which conditions end up in the generic set, it is always true in the generic extension. On the other hand, the existence of a generic set in the model M cannot follow from axioms of set theory, otherwise they would decide the cardinality of p(N).

The method of forcing is of fundamental importance for Badious philosophy. The event escapes ontology; it is “that-which-is-not-being-qua-being”, so it has no place in set theory or the forcing construction. However, the post-evental truth that enters, and modifies the situation, is presented by forcing in the form of a generic set leading to an extension of the ground model. In other words, the situation, understood as the ground model M, is transformed by a post-evental truth identified with a generic set G, and becomes the generic model M[G]. Moreover, the knowledge of the situation is interpreted as the language of set theory, serving to discern elements of the situation; and as axioms of set theory, deciding validity of statements about the situation. Knowledge, understood in this way, does not decide the existence of a generic set in the situation nor can it point to its elements. A generic set is always undecidable and indiscernible.

Therefore, from the perspective of knowledge, it is not possible to establish, whether a situation is still the ground-model or it has undergone a generic extension resulting from the occurrence of an event; only the subject can interventionally decide this. And it is only the subject who decides about the belonging of particular elements to the generic set (i.e. the truth). A procedure of truth or procedure of fidelity (Alain Badiou – Being and Event) supported in this way gives rise to the subject language. It consists of sentences of set theory, so in this respect it is a part of knowledge, although the veridicity of the subject language originates from decisions of the faithful subject. Consequently, a procedure of fidelity forces statements about the situation as it is after being extended, and modified by the operation of truth.

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

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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”.

Badiou’s Vain Platonizing, or How the World is a Topos? Note Quote.

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As regards the ‘logical completeness of the world’, we need to show that Badiou’s world of T-sets does indeed give rise to a topos.

Badiou’s world consisting of T-Sets – in other words pairs (A, Id) where Id : A × A → T satisfies the particular conditions in respect to the complete Heyting algebra structure of T—is ‘logically closed’, that is, it is an elementary topos. It thus encloses not only pull-backs but also the exponential functor. These make it possible for it to internalize a Badiou’s infinity arguments that operate on the power-functor and which can then be expressed from insde the situation despite its existential status.

We need to demonstrate that Badiou’s world is a topos. Rather than beginning from Badiou’s formalism of T -sets, we refer to the standard mathematical literature based on which T-sets can be regarded as sheaves over the particular Grothendieck-topology on the category T: there is a categorical equivalence between T-sets satisfying the ‘postulate of materialism’ and S hvs(T,J). The complications Badiou was caught up with while seeking to ‘Platonize’ the existence of a topos thus largely go in vain. We only need to show that Shvs(T,J) is a topos.

Consider the adjoint sheaf functor that always exists for the category of presheaves

Idα : SetsCop → Shvs(Cop,J)

, where J is the canonical topology. It then amounts to an equivalence of categories. Thus it suffices to replace this category by the one consisting of presheaves SetsTop. This argument works for any category C rather than the specific category related to an external complete Heyting algebra T. In the category of Sets define YX as the set of functions X → Y. Then in the category of presheaves,

SetsCopYX(U) ≅ Hom(hU,YX) ≅ Hom(hU × X,Y)

, where hU is the representable sheaf hU(V) = Hom(V,U). The adjunction on the right side needs to be shown to exist for all sheaves – not just the representable ones. The proof then follows by an argument based on categorically defined limits, which has an existence. It can also be verified directly that the presheaf YX is actually a sheaf. Finally, for the existence of the subobject-classifier ΩSetsCop, it can be defined as

ΩSetsCop(U) ≅ Hom(hU,Ω) ≅ {sub-presheaves of hU} ≅ {sieves on U}, or alternatively, for the category of proper sheaves Shvs(C,J), as

ΩShvs(C,J)(U) = {closed sieves on U}

Here it is worth reminding ourselves that the topology on T is defined by a basis K(p) = {Θ ⊂ T | ΣΘ = p}. Therefore, in the case of T-sets satisfying the strong ‘postulate of materialism’, Ω(p) consists of all sieves S (downward dense subsets) of T bounded by relation ΣS ≤ p. These sieves are further required to be closed. A sieve S with an envelope ΣS = s is closed if for any other r ≤ s, ie. for all r ≤ s, one has the implication

frs(S) ∈ J(r) ⇒ frs ∈ S

, where frs : r → s is the unique arrow in the poset category. In particular, since ΣS = s for the topology whose basis consists of territories on s, we have the equation 1s(S) = fss(S) = S ∈ J(s). Now the condition that the sieve is closed implies 1s ∈ S. This is only possible when S is the maximal sieve on s—namely it consists of all arrows r → s for r ≤ s. In such a case S itself is closed. Therefore, in this particular case

Ω(p)={↓(s)|s ≤ p} = {hs | s ≤ p}

This is indeed a sheaf whose all amalgamations are ‘real’ in the sense of Badiou’s postulate of materialism. Thus it retains a suitable T-structure. Let us assume now that we are given an object A, which is basically a functor and thus a T-graded family of subsets A(p). For there to exist a sub-functor B ֒→ A comes down to stating that B(p) ⊂ A(p) for each p ∈ T. For each q ≤ p, we also have an injection B(q) ֒→ B(p) compatible (through the subset-representation with respect to A) with the injections A(q) ֒→ B(q). For any given x ∈ A(p), we can now consider the set

φp(x) = {q | q ≤ p and x q ∈ B(q)}

This is a sieve on p because of the compatibility condition for injections, and it is furthermore closed since the map x → Σφp(x) is in fact an atom and thus retains a real representative b ∈ B. Then it turns out that φp(x) =↓ (Eb). We now possess a transformation of functors φ : A → Ω which is natural (diagrammatically compatible). But in such a case we know that B ֒→ A is in turn the pull-back along φ of the arrow true, which is equivalent to the category of T-Sets.

2

Badiou Contra Grothendieck Functorally. Note Quote.

What makes categories historically remarkable and, in particular, what demonstrates that the categorical change is genuine? On the one hand, Badiou fails to show that category theory is not genuine. But, on the other, it is another thing to say that mathematics itself does change, and that the ‘Platonic’ a priori in Badiou’s endeavour is insufficient, which could be demonstrated empirically.

Yet the empirical does not need to stand only in a way opposed to mathematics. Rather, it relates to results that stemmed from and would have been impossible to comprehend without the use of categories. It is only through experience that we are taught the meaning and use of categories. An experience obviously absent from Badiou’s habituation in mathematics.

To contrast, Grothendieck opened up a new regime of algebraic geometry by generalising the notion of a space first scheme-theoretically (with sheaves) and then in terms of groupoids and higher categories. Topos theory became synonymous to the study of categories that would satisfy the so called Giraud’s axioms based on Grothendieck’s geometric machinery. By utilising such tools, Pierre Deligne was able to prove the so called Weil conjectures, mod-p analogues of the famous Riemann hypothesis.

These conjectures – anticipated already by Gauss – concern the so called local ζ-functions that derive from counting the number of points of an algebraic variety over a finite field, an algebraic structure similar to that of for example rational Q or real numbers R but with only a finite number of elements. By representing algebraic varieties in polynomial terms, it is possible to analyse geometric structures analogous to Riemann hypothesis but over finite fields Z/pZ (the whole numbers modulo p). Such ‘discrete’ varieties had previously been excluded from topological and geometric inquiry, while it now occurred that geometry was no longer overshadowed by a need to decide between ‘discrete’ and ‘continuous’ modalities of the subject (that Badiou still separates).

Along with the continuous ones, also discrete variates could then be studied based on Betti numbers, and similarly as what Cohen’s argument made manifest in set-theory, there seemed to occur ‘deeper’, topological precursors that had remained invisible under the classical formalism. In particular, the so called étale-cohomology allowed topological concepts (e.g., neighbourhood) to be studied in the context of algebraic geometry whose classical, Zariski-description was too rigid to allow a meaningful interpretation. Introducing such concepts on the basis of Jean-Pierre Serre’s suggestion, Alexander Grothendieck did revolutionarize the field of geometry, and Pierre Deligne’s proof of the Weil-conjenctures, not to mention Wiles’ work on Fermat’s last theorem that subsequentely followed.

Grothendieck’s crucial insight drew on his observation that if morphisms of varieties were considered by their ‘adjoint’ field of functions, it was possible to consider geometric morphisms as equivalent to algebraic ones. The algebraic category was restrictive, however, because field-morphisms are always monomorphisms which makes geometric morphisms: to generalize the notion of a neighbourhood to algebraic category he needed to embed algebraic fields into a larger category of rings. While a traditional Kuratowski covering space is locally ‘split’ – as mathematicians call it – the same was not true for the dual category of fields. In other words, the category of fields did not have an operator analogous to pull-backs (fibre products) unless considered as being embedded within rings from which pull-backs have a co-dual expressed by the tensor operator ⊗. Grothendieck thus realized he could replace ‘incorporeal’ or contained neighborhoods U ֒→ X by a more relational description: as maps U → X that are not necessarily monic, but which correspond to ring-morphisms instead.

Topos theory applies similar insight but not in the context of only specific varieties but for the entire theory of sets instead. Ultimately, Lawvere and Tierney realized the importance of these ideas to the concept of classification and truth in general. Classification of elements between two sets comes down to a question: does this element belong to a given set or not? In category of S ets this question calls for a binary answer: true or false. But not in a general topos in which the composition of the subobject-classifier is more geometric.

Indeed, Lawvere and Tierney then considered this characteristc map ‘either/or’ as a categorical relationship instead without referring to its ‘contents’. It was the structural form of this morphism (which they called ‘true’) and as contrasted with other relationships that marked the beginning of geometric logic. They thus rephrased the binary complete Heyting algebra of classical truth with the categorical version Ω defined as an object, which satisfies a specific pull-back condition. The crux of topos theory was then the so called Freyd–Mitchell embedding theorem which effectively guaranteed the explicit set of elementary axioms so as to formalize topos theory. The Freyd–Mitchell embedding theorem says that every abelian category is a full subcategory of a category of modules over some ring R and that the embedding is an exact functor. It is easy to see that not every abelian category is equivalent to RMod for some ring R. The reason is that RMod has all small limits and colimits. But for instance the category of finitely generated R-modules is an abelian category but lacks these properties.

But to understand its significance as a link between geometry and language, it is useful to see how the characteristic map (either/or) behaves in set theory. In particular, by expressing truth in this way, it became possible to reduce Axiom of Comprehension, which states that any suitable formal condition λ gives rise to a peculiar set {x ∈ λ}, to a rather elementary statement regarding adjoint functors.

At the same time, many mathematical structures became expressible not only as general topoi but in terms of a more specific class of Grothendieck-topoi. There, too, the ‘way of doing mathematics’ is different in the sense that the object-classifier is categorically defined and there is no empty set (initial object) but mathematics starts from the terminal object 1 instead. However, there is a material way to express the ‘difference’ such topoi make in terms of set theory: for every such a topos there is a sheaf-form enabling it to be expressed as a category of sheaves S etsC for a category C with a specific Grothendieck-topology.

Badiou’s Materiality as Incorporeal Ontology. Note Quote.

incorporeal_by_metal_bender-d5ze12s

Badiou criticises the proper form of intuition associated with multiplicities such as space and time. However, his own ’intuitions’ are constrained by set theory. His intuition is therefore as ‘transitory’ as is the ontology in terms of which it is expressed. Following this constrained line of reasoning, however, let me now discuss how Badiou encounters the question of ‘atoms’ and materiality: in terms of the so called ‘atomic’ T-sets.

If topos theory designates the subobject-classifier Ω relationally, the external, set-theoretic T-form reduces the classificatory question again into the incorporeal framework. There is a set-theoretical, explicit order-structure (T,<) contra the more abstract relation 1 → Ω pertinent to categorical topos theory. Atoms then appear in terms of this operator <: the ‘transcendental grading’ that provides the ‘unity through which all the manifold given in an intuition is united in a concept of the object’.

Formally, in terms of an external Heyting algebra this comes down to an entity (A,Id) where A is a set and Id : A → T is a function satisfying specific conditions.

Equaliser: First, there is an ‘equaliser’ to which Badiou refers as the ‘identity’ Id : A × A → T satisfies two conditions:

1) symmetry: Id(x, y) = Id(y, x) and
2) transitivity: Id(x, y) ∧ Id(y, z) ≤ Id(x, z).

They guarantee that the resulting ‘quasi-object’ is objective in the sense of being distinguished from the gaze of the ‘subject’: ‘the differences in degree of appearance are not prescribed by the exteriority of the gaze’.

This analogous ‘identity’-function actually relates to the structural equalization-procedure as appears in category theory. Identities can be structurally understood as equivalence-relations. Given two arrows X ⇒ Y , an equaliser (which always exists in a topos, given the existence of the subobject classifier Ω) is an object Z → X such that both induced maps Z → Y are the same. Given a topos-theoretic object X and U, pairs of elements of X over U can be compared or ‘equivalized’ by a morphism XU × XUeq ΩU structurally ‘internalising’ the synthetic notion of ‘equality’ between two U-elements. Now it is possible to formulate the cumbersome notion of the ‘atom of appearing’.

An atom is a function a : A → T defined on a T -set (A, Id) so that
(A1) a(x) ∧ Id(x, y) ≤ a(y) and
(A2) a(x) ∧ a(y) ≤ Id(x, y).
As expressed in Badiou’s own vocabulary, an atom can be defined as an ‘object-component which, intuitively, has at most one element in the following sense: if there is an element of A about which it can be said that it belongs absolutely to the component, then there is only one. This means that every other element that belongs to the component absolutely is identical, within appearing, to the first’. These two properties in the definition of an atom is highly motivated by the theory of T-sets (or Ω-sets in the standard terminology of topological logic). A map A → T satisfying the first inequality is usually thought as a ‘subobject’ of A, or formally a T-subset of A. The idea is that, given a T-subset B ⊂ A, we can consider the function
IdB(x) := a(x) = Σ{Id(x,y) | y ∈ B}
and it is easy to verify that the first condition is satisfied. In the opposite direction, for a map a satisfying the first condition, the subset
B = {x | a(x) = Ex := Id(x, x)}
is clearly a T-subset of A.
The second condition states that the subobject a : A → T is a singleton. This concept stems from the topos-theoretic internalization of the singleton-function {·} : a → {a} which determines a particular class of T-subsets of A that correspond to the atomic T-subsets. For example, in the case of an ordinary set S and an element s ∈ S the singleton {s} ⊂ S is a particular, atomic type of subset of S.
The question of ‘elements’ incorporated by an object can thus be expressed externally in Badiou’s local theory but ‘internally’ in any elementary topos. For the same reason, there are two ways for an element to be ‘atomic’: in the first sense an ‘element depends solely on the pure (mathematical) thinking of the multiple’, whereas the second sense relates it ‘to its transcendental indexing’. In topos theory, the distinction is slightly more cumbersome. Badiou still requires a further definition in order to state the ‘postulate of materialism’.
An atom a : A → T is real if if there exists an element x ∈ T so that a(y) = Id(x,y) ∀ y ∈ A.
This definition gives rise to the postulate inherent to Badiou’s understanding of ‘democratic materialism’.
Postulate of Materialism: In a T-set (A,Id), every atom of appearance is real.
What the postulate designates is that there really needs to exist s ∈ A for every suitable subset that structurally (read categorically) appears to serve same relations as the singleton {s}. In other words, what ‘appears’ materially, according to the postulate, has to ‘be’ in the set-theoretic, incorporeal sense of ‘ontology’. Topos theoretically this formulation relates to the so called axiom of support generators (SG), which states that the terminal object 1 of the underlying topos is a generator. This means that the so called global elements, elements of the form 1 → X, are enough to determine any particular object X.
Thus, it is this specific condition (support generators) that is assumed by Badiou’s notion of the ‘unity’ or ‘constitution’ of ‘objects’. In particular this makes him cross the line – the one that Kant drew when he asked Quid juris? or ’Haven’t you crossed the limit?’ as Badiou translates.
But even without assuming the postulate itself, that is, when considering a weaker category of T-sets not required to fulfill the postulate of atomism, the category of quasi-T -sets has a functor taking any quasi-T-set A into the corresponding quasi-T-set of singletons SA by x → {x}, where SA ⊂ PA and PA is the quasi-T-set of all quasi-T-subsets, that is, all maps T → A satisfying the first one of the two conditions of an atom designated by Badiou. It can then be shown that, in fact, SA itself is a sheaf whose all atoms are ‘real’ and which then is a proper T-set satisfying the ‘postulate of materialism’. In fact, the category of T-Sets is equivalent to the category of T-sheaves Shvs(T, J). In the language of T-sets, the ‘postulate of materialism’ thus comes down to designating an equality between A and its completed set of singletons SA.

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Badiou, Heyting Algebras cross the Grothendieck Topoi. Note Quote.

Let us commence by introducing the local formalism that constitutes the basis of Badiou’s own, ‘calculated phenomenology’. Badiou is unwilling to give up his thesis that the history of thinking of being (ontology) is the history of mathematics and, as he reads it, that of set theory. It is then no accident that set theory is the regulatory framework under which topos theory is being expressed. He does not refer to topoi explicitly but rather to the so called complete Heyting algebras which are their procedural equivalents. However, he fails to mention that there are both ‘internal’ and ‘external’ Heyting algebras, the latter group of which refers to local topos theory, while it appears that he only discusses the latter—a reduction that guarantees that indeed that the categorical insight may give nothing new.

Indeed, the external complete Heyting algebras T then form a category of the so called T-sets, which are the basic objects in the ‘world’ of the Logics of Worlds. They local topoi or the so called ‘locales’ that are also ‘sets’ in the traditional sense of set theory. This ‘constitution’ of his worlds thus relies only upon Badiou’s own decision to work on this particular regime of objects, even if that regime then becomes pivotal to his argument which seeks to denounce the relevance of category theory.

This problematic is particularly visible in the designation of the world m (mathematically a topos) as a ‘complete’ (presentative) situation of being of ‘universe [which is] the (empty) concept of a being of the Whole’ He recognises the ’impostrous’ nature of such a ‘whole’ in terms of Russell’s paradox, but in actual mathematical practice the ’whole’ m becomes to signify the category of Sets – or any similar topos that localizable in terms of set theory. The vocabulary is somewhat confusing, however, because sometimes T is called the ‘transcendental of the world’, as if m were defined only as a particular locale, while elsewhere m refers to the category of all locales (Loc).

An external Heyting algebra is a set T with a partial order relation <, a minimal element μ ∈ T , a maximal element M ∈ T . It further has a ‘conjunction’ operator ∧ : T × T → T so that p ∧ q ≤ p and p ∧ q = q ∧ p. Furthermore, there is a proposition entailing the equivalence p ≤ q iff p ∧ q = p. Furthermore p ∧ M = p and μ ∧ p = μ for any p ∈ T .

In the ‘diagrammatic’ language that pertains to categorical topoi, by contrast, the minimal and maximal elements of the lattice Ω can only be presented as diagrams, not as sets. The internal order relation ≤ Ω can then be defined as the so called equaliser of the conjunction ∧ and projection-map

≤Ω →e Ω x Ω →π1 L

The symmetry can be expressed diagrammatically by saying that

IMG_20170417_215019_HDR

is a pull-back and commutes. The minimal and maximal elements, in categorical language, refer to the elements evoked by the so-called initial and terminal objects 0 and 1.

In the case of local Grothendieck-topoi – Grothendieck-topoi that support generators – the external Heyting algebra T emerges as a push-forward of the internal algebra Ω, the logic of the external algebra T := γ ∗ (Ω) is an analogous push-forward of the internal logic of Ω but this is not the case in general.

What Badiou further requires of this ‘transcendental algebra’ T is that it is complete as a Heyting algebra.

A complete external Heyting algebra T is an external Heyting algebra together with a function Σ : PT → T (the least upper boundary) which is distributive with respect to ∧. Formally this means that ΣA ∧ b = Σ{a ∧ b | a ∈ A}.

In terms of the subobject classifier Ω, the envelope can be defined as the map Ωt : ΩΩ → Ω1 ≅ Ω, which is internally left adjoint to the map ↓ seg : Ω → ΩΩ that takes p ∈ Ω to the characteristic map of ↓ (p) = {q ∈ Ω | q ≤ p}27.

The importance the external complete Heyting algebra plays in the intuitionist logic relates to the fact that one may now define precisely such an intuitionist logic on the basis of the operations defined above.

The dependence relation ⇒ is an operator satisfying

p ⇒ q = Σ{t | p ∩ t ≤ q}.

(Negation). A negation ¬ : T → T is a function so that

¬p =∑ {q | p ∩ q = μ},

and it then satisfies p ∧ ¬p = μ.

Unlike in what Badiou calls a ‘classical world’ (usually called a Boolean topos, where ¬¬ = 1Ω), the negation ¬ does not have to be reversible in general. In the domain of local topoi, this is only the case when the so called internal axiom of choice is valid, that is, when epimorphisms split – for example in the case of set theory. However, one always has p ≤ ¬¬p. On the other hand, all Grothendieck-topoi – topoi still materially presentable over Sets – are possible to represent as parts of a Boolean topos.