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

# Conjuncted: Gross Domestic Product. Part 2.

Conjuncted here.

The topology of the World Trade, which is encapsulated in its adjacency matrix aij defined by

aij(t) ≡ 1 if fij(t) > 0

aij(t) ≡ 0 if fij(t) = 0

, strongly depends on the GDP values wi. Indeed, the problem can be mapped onto the so-called fitness model where it is assumed that the probability pij for a link from i to j is a function p(xi, xj) of the values of a fitness variable x assigned to each vertex and drawn from a given distribution. The importance of this model relies in the possibility to write all the expected topological properties of the network (whose specification requires in principle the knowledge of the N2 entries of its adjacency matrix) in terms of only N fitness values. Several topological properties including the degree distribution, the degree correlations and the clustering hierarchy are determined by the GDP distribution. Moreover, an additional understanding of the World Trade as a directed network comes from the study of its reciprocity, which represents the strong tendency of the network to form pairs of mutual links pointing in opposite directions between two vertices. In this case too, the observed reciprocity structure can be traced back to the GDP values.

The probability that at time t a link exists from i to j (aij = 1) is empirically found to be

pt [xi(t), xj(t)] = [α(t) xi(t) xj(t)]/[1 + β(t) xi(t) xj(t)]

where xi is the rescaled GDP and the parameters α(t) and β(t) can be fixed by imposing that the expected number of links

Lexp(t) = ∑i≠j pt [xi(t), xj(t)]

equals its empirical value

L(t) = ∑i≠j aij(t)

and that the expected number of reciprocated links

Lexp(t) = ∑i≠j pt[xi(t), xj(t)] pt[xj(t), xi(t)]

equals its observed value

L(t) = ∑i≠j aij(t) aji(t)

This particular structure of the World Trade topology can be tested by comparing various expected topological properties with the empirical ones. For instance, we can compare the empirical and the theoretical plots of vertex degrees (at time t) versus their rescaled GDP xi(t). Note that since pt [xi(t), xj(t)] is symmetric under the exchange of i and j, at any given time the expected in-degree and the expected out-degree of a vertex i are equal. We denote both by kexpi, which can be expressed as

kexpi(t) = ∑j≠i pt[xi(t), xj(t)]

Since the number of countries N(t) increases in time, we define the rescaled degrees

k ̃i(t) ≡ ki(t)/[N(t) − 1]

that always represent the fraction of vertices which are connected to i (the term −1 comes from the fact that there are no self-loops in the network, hence the maximum degree is always N − 1). In this way, we can easily compare the data corresponding to different years and network sizes. The results are shown in the figure below for various snapshots of the system. Figure: Plot of the rescaled degrees versus the rescaled GDP at four different years, and comparison with the expected trend.

The empirical trends are in accordance with the expected ones. Then we can also compare the cumulative distribution Pexp>(k ̃exp) of the expected degrees with the empirical degree distributions Pin>(k ̃in) and Pout>(k ̃out). The results are shown in the following figure and are in conformity to a good agreement between the theoretical prediction and the observed behavior. Figure: Cumulative degree distributions of the World Trade topology for four different years and comparison with the expected trend.

Note that the accordance with the predicted behaviour is extremely important since the expected quantities are computed by using only the N GDP values of all countries, with no information regarding the N2 trade values. On the other hand, the empirical properties of the World Trade topology are extracted from trade data, with no knowledge of the GDP values. The agreement between the properties obtained by using these two independent sources of information is therefore surprising. This also shows that the World Trade topology crucially depends on the GDP distribution ρ(x).

# 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”.

# Carnap’s Topological Properties and Choice of Metric. Note Quote.

Husserl’s system is ontologically, a traditional double hierarchy. There are regions or spheres of being, and perfectly traditional ones, except that (due to Kant’s “Copernican revolution”) the traditional order is reversed: after the new Urregion of pure consciousness come the region of nature, the psychological region, and finally a region (or perhaps many regions) of Geist. Each such region is based upon a single highest genus of concrete objects (“individua”), corresponding to the traditional highest genera of substances: in pure consciousness, for example, Erlebnisse; in nature, “things” (Dinge). But each region also contains a hierarchy of abstract genera – genera of singular abstracta and of what Husserl calls “categorial” or “syntactic” objects (classes and relations). This structure of “logical modifications,” found analogously in each region, is the concern of logic. In addition, however, to the “formal essence” which each object has by virtue of its position in the logical hierarchy, there are also truths of “material” (sachliche) essence, which apply to objects as members of some species or genus – ultimately, some region of being. Thus the special sciences, which are individuated (as in Aristotle) by the regions they study, are each broadly divided into two parts: a science of essence and a science of “matters of fact.” Finally, there are what might be called matters of metaphysical essence: necessary truths about objects which apply in virtue of their dependence on objects in prior regions, and ultimately within the Urregion of pure consciousness.

This ontological structure translates directly into an epistemological one, because all being in the posterior regions rests on positing Erlebnisse in the realm of pure consciousness, and in particular on originary (immediate) rational theoretical positings, i.e. “intuitions.” The various sciences are therefore based on various types of intuition. Sciences of matters of fact, on the one hand, correspond to the kinds of ordinary intuition, analogous to perception. Sciences of essence, on the other hand, and formal logic, correspond to (formal or material) “essential insight” (Wesensschau). Husserl equates formal- and material-essential insight, respectively, as sources of knowledge, to Kant’s analytic and synthetic a priori, whereas ordinary perceptual intuition, the source of knowledge about matters of fact, corresponds to the Kantian synthetic a posteriori. Phenomenology, finally, as the science of essence in the region of pure consciousness, has knowledge of the way beings in one region are dependent on those in another.

In Carnap’s doctoral thesis, Der Raum, he applies the above Husserlian apparatus to the problem of determining our sources of knowledge about space. Is our knowledge of space analytic, synthetic a priori, or empirical? Carnap answers, in effect: it depends on what you mean by “space.” His answer foreshadows much of his future thought, but is also based directly on Husserl’s remark about this question in Ideen I: that, whereas Euclidean manifold is a formal category (logical modification), our knowledge of geometry as it applies to physical objects is a knowledge of material essence within the region of nature. Der Raum is largely an expansion and explication of that one remark. Our knowledge of “formal space,” Carnap says, is analytic, i.e. derives from “formal ontology in Husserl’s sense,” but our knowledge of the “intuitive space” in which sensible objects are necessarily found is synthetic a priori, i.e. material-essential. There is one important innovation: Carnap claims that essential (a priori) knowledge of intuitive space extends only to its topological properties, whereas the full structure of physical space requires also a choice of metric. This latter choice is informed by the actual behavior of objects (e.g. measuring rods), and knowledge of physical space is thus in part a posteriori – as Carnap also says, a knowledge of “matters of fact.” But such considerations never force the choice of one metric or another: our knowledge of physical space also depends on “free positing”. This last point, which has no equivalent in Husserl, is important. Still more telling is that Carnap compares the choice involved here to a choice of language, although at this stage he sees this as a mere analogy. On the whole, however, the treatment of Der Raum is more or less orthodoxly Husserlian.

# Is the Hierarchical Society Cardinal?

Even the question posed has a stink of the inhuman, or un-human, though it is evident that in theory we might try and flatten such hierarchies, the same never holds true in practice. Although such hierarchies might be held on to surreptitiously, the tendency to be resilient is never really ruled out in matters as sensitive as these, which make us prone to getting branded as fundamentalists or fanatics, or anything which has semblance to the right-wing ideology. So, in a nutshell, hierarchies in the Social are indeed emanating from the right-wing, or are at least given to sway in their descriptions and prescriptions.

So, are these hierarchies important? Well, the answer at first go is a strict ‘no’. But, let us deliberate upon. One way is to look upon hierarchy as dominant, and the other is identity when there is an absence of hierarchies. Now, those that belong to the first camp would imply reciprocity as enabling social order. Reciprocity is a relationship that exists one-to-one, one-to-many and many-to-one as regards the first camp, while one and the many merge into one another as regards the second camp. In the first camp, reciprocity is built up on adherence, while in the second, it is more and more symbiotic. The existing of social strata could advocate the existence of micro-cultural forms characterised by the features that lead to the formation of such strata in the first place and could include notions like religiosity, power (both muscle and economic), cultural and intellectual/ideological. On the flip side, such notions are tolerant towards multiculturalism or pluralism. Dominance becomes a nested approach, while becoming-identity is a web-like structure with nodes of individuals or clusters of societies that interact on a horizontal level, or to put it more politically, act on a democratic level, in theory at least, to say the least. Disturbance in this net is knotted into a nest, where dominance takes over the democratic structure and subsequently forcing the second camp to be evacuated onto the first one. Now here is where the catch lies. From netted to a nested structure would mean classification, and it is classification that gets conceptual authority, thus in a hugely ironical manner ameliorating the potential of conflicts due to centralised authoritarian structure. This is its use value. Hierarchies try to make sense out of the apparent relationships between things with the caveat that orientations that determine those relations are just looming round the corner.

In hierarchical societies there are domains of individuals, clusters, micro-cultures or societies that are instances of isolated-ness from each other, whereas in non-hierarchical societies these domains tendentially overlap into one another, or even across one another making the very study of latter kind of studies difficult in intent. Other use value would lie in mapping domains become easier in dominance or stratified societies as compared with in non-hierarchical societies. # Representation as a Meaningful Philosophical Quandary 