Evental Sites. Thought of the Day 48.0

badiou_being_and_appearance1

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.

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.