Disjointed Regularity in Open Classes of Elementary Topology

Let x, y, … denote first-order structures in St_𝜏, x ≈ y will denote isomorphism.

x ∼_n,𝜏 y means that there is a sequence 0 ≠ I₀ ⊆ …. ⊆ I_n of sets of 𝜏-partial isomorphism of finite domain so that, for i < j ≤ n, f ∈ I_i and a ∈ x (respectively, b ∈ y), there is g ∈ I_j such that g ⊇ f and a ∈ Dom(g) (respectively, b ∈ Im(g)). The later is called the extension property.

x ∼_𝜏 y means the above holds for an infinite chain 0 ≠ I₀ ⊆ …. ⊆ I_n ⊆ …

Fraïssé’s characterization of elementary equivalence says that for finite relational vocabularies: x ≡ y iff x ∼_n,𝜏 y. To have it available for vocabularies containing function symbols add the complexity of terms in atomic formulas to the quantifier rank. It is well known that for countable x, y : x ∼_𝜏 y implies x ≈ y.

Given a vocabulary 𝜏 let 𝜏^∗ be a disjoint renaming of 𝜏. If x, y ∈ St_𝜏 have the same power, let y^∗ be an isomorphic copy of y sharing the universe with x and renamed to be of type 𝜏^∗. In this context, (x, y^∗) will denote the 𝜏 ∪ 𝜏^∗-structure that results of expanding x with the relations of y^∗.

Lemma: There is a vocabulary 𝜏⁺ ⊇ 𝜏 ∪ 𝜏^∗ such that for each finite vocabulary 𝜏₀ ⊆ 𝜏 there is a sequence of elementary classes 𝛥₁ ⊇ 𝛥₂ ⊇ 𝛥₃ ⊇ …. in St_𝜏⁺ such that if 𝜋 = 𝜋_{𝜏⁺,𝜏∪𝜏^∗} then (1) 𝜋(𝛥_𝑛) = {(x,y^∗) : |x| = |y| ≥ 𝜔, x ≡_n,𝜏0 y}, (2) 𝜋(⋂_n 𝛥_n) = {(x, y^∗) : |x| = |y| ≥ 𝜔, x ∼_𝜏0 y}. Moreover, ⋂_n𝛥_n is the reduct of an elementary class.

Proof. Let 𝛥 be the class of structures (x, y^∗, <, a, I) where < is a discrete linear order with minimum but no maximum and I codes for each c ≤ a a family I_c = {I(c, i, −, −)}_i∈x of partial 𝜏₀-𝜏₀^∗–isomorphisms from x into y^∗, such that for c < c’ ≤ a : I_c ⊆ I_c and the extension property holds. Describe this by a first-order sentence 𝜃_𝛥 of type 𝜏⁺ ⊇ 𝜏₀ ∪ 𝜏₀^∗ and set 𝛥_𝑛 = Mod_L(𝜃_𝛥 ∧ ∃^≥n x(x ≤ a)}. Then condition (1) in the Lemma is granted by Fraïssé’s characterization and the fact that x being (2) is granted because (x, y^∗, <, a, I) ∈ ⋂_n𝛥_n iff < contains an infinite increasing 𝜔-chain below a, a ∑₁¹ condition.

A topology on St_𝜏 is invariant if its open (closed) classes are closed under isomorphic structures. Of course, it is superfluous if we identify isomorphic structures.

Theorem: Let Γ be a regular compact topology finer than the elementary topology on each class St_𝜏 such that the countable structures are dense in St_𝜏 and reducts and renamings are continuous for these topologies. Then Γ_𝜏 is the elementary topology ∀ 𝜏.

Proof: We show that any pair of disjoint closed classes C₁, C₂ of Γ_𝜏 may be separated by an elementary class. Assume this is not the case since C_i are compact in the topology Γ_𝜏 then they are compact for the elementary topology and, by regularity of the latter, ∃ x_i ∈ C_i such that x₁ ≡ x₂ in L_𝜔𝜔(𝜏). The x_i must be infinite, otherwise they would be isomorphic contradicting the disjointedness of the C_i. By normality of Γ_𝜏, there are towers U_i ⊆ C^‘_i ⊆ U^‘_i ⊆ C^”_i, i = 1,2, separating the C_i with U_i, U^‘_i open and C^‘_i, C^”_i closed in Γ_𝜏 and disjoint. Let I be a first-order sentence of type 𝜏^‘ ⊇ 𝜏 such that (z, ..) |= I ⇔ z is infinite, and let π be the corresponding reduct operation. For fixed n ∈ ω and the finite 𝜏₀  ⊆ 𝜏, let t be a first-order sentence describing the common ≡_n,𝜏0 – equivalence class of x₁, x₂. As,

(x_i,..) ∈ Mod𝜏^‘(I) ∩ π^-1 Mod(t) ∩ π^-1U_i, i = 1, 2,..

and this class is open in Γ_𝜏‘ by continuity of π, then by the density hypothesis there are countable x_i ∈ U_i , i = 1, 2, such that x₁ ≡_n,𝜏 x₂. Thus for some expansion of (x₁, x₂^∗),

(x, x^∗,..) ∈ 𝛥_n,𝜏0 ∩ 𝜋₁⁻¹(𝐶^‘₁) ∩ (𝜌𝜋₂)⁻¹(C^‘₂) —– (1)

where 𝛥_𝑛,𝜏0 is the class of Lemma, 𝜋₁, 𝜋₂ are reducts, and 𝜌 is a renaming:

𝜋₁(x₁, x₂^∗, …) = x₁ 𝜋₁ : St_𝜏⁺ → St_{𝜏∪𝜏^∗} → St_𝜏

𝜋₂(x₁, x₂^∗, …) = x₂^∗ 𝜋₂ : St_𝜏⁺ → St_{𝜏∪𝜏^∗} → St_𝜏^∗

𝜌(x₂^∗) = x₂ 𝜌 : St_𝜏^∗ → St_𝜏

Since the classes (1) are closed by continuity of the above functors then ⋂_n𝛥_n,𝜏0 ∩ 𝜋₁⁻¹(C^‘₁) ∩ (𝜌𝜋₂)⁻¹(C^‘₂) is non-emtpy by compactness of Γ_𝜏⁺. But ⋂_n𝛥_n,𝜏0 = 𝜋(V) with V elementary of type 𝜏⁺⁺ ⊇ 𝜏⁺. Then

V ∩ π^-1π₁^-1(U^‘₁) ∩ π^-1(ρπ₂)^-1 (U^‘₂) ≠ 0

is open for Γ_L⁺⁺ and the density condition it must contain a countable structure (x₁, x^*₂, ..). Thus (x₁, x^*₂, ..) ∈ ∩_n 𝛥_𝑛,𝜏0, with x_i ∈ U^‘_i ⊆ C^”_i. It follows that x₁ ~_𝜏0 x₂ and thus x₁ |𝜏₀ ≈ x₂ |𝜏₀. Let δ_𝜏0 be a first-order sentence of type 𝜏 ∪ 𝜏^* ∪{h} such that (x, y^*, h) |= δ_𝜏0 ⇔ h : x |𝜏₀ ≈ y|𝜏₀. By compactness,

(∩_𝜏0⊆_fin𝜏 Mod_{𝜏∪𝜏^*∪{f}} (δ_𝜏0)) ∩ π₁^-1(C^”₁) ∩ (ρπ₂)^-1(C^”₂) ≠ 0

and we have h : x₁ ≈ x₂, x_i ∈ C^”_i, contradicting the disjointedness of C^”_i. Finally, if C is a closed class of Γ_𝜏 and x ∉ C, cl_Γ𝜏{x} is disjoint from C by regularity of Γ_𝜏. Then cl_Γ𝜏{x} and C may be separated by open classes of elementary topology, which implies C is closed in this topology.

Advertisement

Intuition

During his attempt to axiomatize the category of all categories, Lawvere says

Our intuition tells us that whenever two categories exist in our world, then so does the corresponding category of all natural transformations between the functors from the first category to the second (The Category of Categories as a Foundation).

However, if one tries to reduce categorial constructions to set theory, one faces some serious problems in the case of a category of functors. Lawvere (who, according to his aim of axiomatization, is not concerned by such a reduction) relies here on “intuition” to stress that those working with categorial concepts despite these problems have the feeling that the envisaged construction is clear, meaningful and legitimate. Not the reducibility to set theory, but an “intuition” to be specified answers for clarity, meaningfulness and legitimacy of a construction emerging in a mathematical working situation. In particular, Lawvere relies on a collective intuition, a common sense – for he explicitly says “our intuition”. Further, one obviously has to deal here with common sense on a technical level, for the “we” can only extend to a community used to the work with the concepts concerned.

In the tradition of philosophy, “intuition” means immediate, i.e., not conceptually mediated cognition. The use of the term in the context of validity (immediate insight in the truth of a proposition) is to be thoroughly distinguished from its use in the sensual context (the German Anschauung). Now, language is a manner of representation, too, but contrary to language, in the context of images the concept of validity is meaningless.

Obviously, the aspect of cognition guiding is touched on here. Especially the sensual intuition can take the guiding (or heuristic) function. There have been many working situations in history of mathematics in which making the objects of investigation accessible to a sensual intuition (by providing a Veranschaulichung) yielded considerable progress in the development of the knowledge concerning these objects. As an example, take the following account by Emil Artin of Emmy Noether’s contribution to the theory of algebras:

Emmy Noether introduced the concept of representation space – a vector space upon which the elements of the algebra operate as linear transformations, the composition of the linear transformation reflecting the multiplication in the algebra. By doing so she enables us to use our geometric intuition.

Similarly, Fréchet thinks to have really “powered” research in the theory of functions and functionals by the introduction of a “geometrical” terminology:

One can [ …] consider the numbers of the sequence [of coefficients of a Taylor series] as coordinates of a point in a space [ …] of infinitely many dimensions. There are several advantages to proceeding thus, for instance the advantage which is always present when geometrical language is employed, since this language is so appropriate to intuition due to the analogies it gives birth to.

Mathematical terminology often stems from a current language usage whose (intuitive, sensual) connotation is welcomed and serves to give the user an “intuition” of what is intended. While Category Theory is often classified as a highly abstract matter quite remote from intuition, in reality it yields, together with its applications, a multitude of examples for the role of current language in mathematical conceptualization.

This notwithstanding, there is naturally also a tendency in contemporary mathematics to eliminate as much as possible commitments to (sensual) intuition in the erection of a theory. It seems that algebraic geometry fulfills only in the language of schemes that essential requirement of all contemporary mathematics: to state its definitions and theorems in their natural abstract and formal setting in which they can be considered independent of geometric intuition (Mumford D., Fogarty J. Geometric Invariant Theory).

In the pragmatist approach, intuition is seen as a relation. This means: one uses a piece of language in an intuitive manner (or not); intuitive use depends on the situation of utterance, and it can be learned and transformed. The reason for this relational point of view, consists in the pragmatist conviction that each cognition of an object depends on the means of cognition employed – this means that for pragmatism there is no intuitive (in the sense of “immediate”) cognition; the term “intuitive” has to be given a new meaning.

What does it mean to use something intuitively? Heinzmann makes the following proposal: one uses language intuitively if one does not even have the idea to question validity. Hence, the term intuition in the Heinzmannian reading of pragmatism takes a different meaning, no longer signifies an immediate grasp. However, it is yet to be explained what it means for objects in general (and not only for propositions) to “question the validity of a use”. One uses an object intuitively, if one is not concerned with how the rules of constitution of the object have been arrived at, if one does not focus the materialization of these rules but only the benefits of an application of the object in the present context. “In principle”, the cognition of an object is determined by another cognition, and this determination finds its expression in the “rules of constitution”; one uses it intuitively (one does not bother about the being determined of its cognition), if one does not question the rules of constitution (does not focus the cognition which determines it). This is precisely what one does when using an object as a tool – because in doing so, one does not (yet) ask which cognition determines the object. When something is used as a tool, this constitutes an intuitive use, whereas the use of something as an object does not (this defines tool and object). Here, each concept in principle can play both roles; among two concepts, one may happen to be used intuitively before and the other after the progress of insight. Note that with respect to a given cognition, Peirce when saying “the cognition which determines it” always thinks of a previous cognition because he thinks of a determination of a cognition in our thought by previous thoughts. In conceptual history of mathematics, however, one most often introduced an object first as a tool and only after having done so did it come to one’s mind to ask for “the cognition which determines the cognition of this object” (that means, to ask how the use of this object can be legitimized).

The idea that it could depend on the situation whether validity is questioned or not has formerly been overlooked, perhaps because one always looked for a reductionist epistemology where the capacity called intuition is used exclusively at the last level of regression; in a pragmatist epistemology, to the contrary, intuition is used at every level in form of the not thematized tools. In classical systems, intuition was not simply conceived as a capacity; it was actually conceived as a capacity common to all human beings. “But the power of intuitively distinguishing intuitions from other cognitions has not prevented men from disputing very warmly as to which cognitions are intuitive”. Moreover, Peirce criticises strongly cartesian individualism (which has it that the individual has the capacity to find the truth). We could sum up this philosophy thus: we cannot reach definite truth, only provisional; significant progress is not made individually but only collectively; one cannot pretend that the history of thought did not take place and start from scratch, but every cognition is determined by a previous cognition (maybe by other individuals); one cannot uncover the ultimate foundation of our cognitions; rather, the fact that we sometimes reach a new level of insight, “deeper” than those thought of as fundamental before, merely indicates that there is no “deepest” level. The feeling that something is “intuitive” indicates a prejudice which can be philosophically criticised (even if this does not occur to us at the beginning).

In our approach, intuitive use is collectively determined: it depends on the particular usage of the community of users whether validity criteria are or are not questioned in a given situation of language use. However, it is acknowledged that for example scientific communities develop usages making them communities of language users on their own. Hence, situations of language use are not only partitioned into those where it comes to the users’ mind to question validity criteria and those where it does not, but moreover this partition is specific to a particular community (actually, the community of language users is established partly through a peculiar partition; this is a definition of the term “community of language users”). The existence of different communities with different common senses can lead to the following situation: something is used intuitively by one group, not intuitively by another. In this case, discussions inside the discipline occur; one has to cope with competing common senses (which are therefore not really “common”). This constitutes a task for the historian.

Agamben and the Biopolitical – Nihilistic and Thanatopolitical Expressions. Thought of the Day 56.0

Agamben’s logic of biopolitics as the logic of the symmetry between sovereign power and the sacredness of bare life should be understood in terms of its historico-ontological destiny. Although this theme is only hinted at in Homo Sacer and the volumes that follow it, Agamben resolutely maintains that biopolitics is inherently metaphysical. If on the one hand ‘the inclusion of bare life in the political realm constitutes the original […] nucleus of sovereign power’ and ‘biopolitics is at least as old as the sovereign exception’, on the other hand, this political nexus cannot be dissociated from the epochal situation of metaphysics. Here Agamben openly displays his Heideggerian legacy; bare life, that which in history is increasingly isolated by biopolitics as Western politics, must be strictly related to ‘pure being’, that which in history is increasingly isolated by Western metaphysics:

Politics [as biopolitics] appears as the truly fundamental structure of Western metaphysics insofar as it occupies the threshold on which the relation between the living being and the logos is realized. In the ‘politicization’ of bare life – the metaphysical task par excellence – the humanity of living man is decided.

Commentators have not as yet sufficiently emphasized how biopolitics is consequently nothing else than Agamben’s name for metaphysics as nihilism. More specifically, while bare life remains for him the ‘empty and indeterminate’ concept of Western politics – which is thus as such originally nihilistic – its forgetting goes together with the progressive coming to light of what it conceals. From this perspective, nihilism will therefore correspond to the modern and especially post-modern generalisation of the state of exception: ‘the nihilism in which we are living is […] nothing other than the coming to light of […] the sovereign relation as such’. In other words, nihilism reveals the paradox of the inclusive exclusion of bare life, homo sacer, qua foundation of sovereign power, as well as the fact that sovereign power cannot recognize itself for what it is. Beyond Foucault’s biopolitical thesis according to which modernity is increasingly characterized by the way in which power directly captures life as such as its object, what interests Agamben the most is:

the decisive fact that, together with the process by which exception everywhere becomes the rule, the realm of bare life – which is originally situated at the margins of the political order – gradually begins to coincide with the political realm.

The political is thus reduced to the biopolitical: the original repression of the sovereign relation on which Western politics has always relied is now inextricably bound up with its return in the guise of a radical biopoliticisation of the political. Like nihilism, such a generalisation of the state of exception – the fact that, today, we are all virtually homines sacri – is itself a profoundly ambiguous biopolitical phenomenon. Today’s state of exception both radicalizes – qualitatively and quantitatively – the thanatopolitical expressions of sovereignty (epitomized by the nazis’ extermination of the Jews for a mere ‘capacity to be killed’ inherent in their condition as such) and finally unmasks its hidden logic.

Agamben explicitly relates to the possibility of a ‘new politics’. Conversely, a new politics is unthinkable without an in-depth engagement with the historico-ontological dimension of sacratio and the structural political ambiguity of the state of exception. Although such new politics ‘remains largely to be invented’, very early on in Homo Sacer, Agamben unhesitatingly defines it as ‘a politics no longer founded on the exceptio of bare life’. beyond the exceptionalist logic – by now self-imploded – that unites sovereignty to bare life, Agamben seems to envisage a relaional politics that would succeed in ‘constructing the link between zoe and bios’. This link between the bare life of man and his political existence would ‘heal’ the original ‘fracture’ which is at the same time precisely what causes their progressive indistinction in the generalized state of exception. Having said this, Agamben also conceives of such new politics as a non-relational relation that ‘will […] have to put the very form of relation into question, and to ask if the political fact is not perhaps thinkable beyond relation and, thus, no longer in the form of a connection’.

Diagrammatic Political Via The Exaptive Processes

The principle of individuation is the operation that in the matter of taking form, by means of topological conditions […] carries out an energy exchange between the matter and the form until the unity leads to a state – the energy conditions express the whole system. Internal resonance is a state of the equilibrium. One could say that the principle of individuation is the common allagmatic system which requires this realization of the energy conditions the topological conditions […] it can produce the effects in all the points of the system in an enclosure […]

This operation rests on the singularity or starting from a singularity of average magnitude, topologically definite.

If we throw in a pinch of Gilbert Simondon’s concept of transduction there’s a basis recipe, or toolkit, for exploring the relational intensities between the three informal (theoretical) dimensions of knowledge, power and subjectification pursued by Foucault with respect to formal practice. Supplanting Foucault’s process of subjectification with Simondon’s more eloquent process of individuation marks an entry for imagining the continuous, always partial, phase-shifting resolutions of the individual. This is not identity as fixed and positionable, it’s a preindividual dynamic that affects an always becoming- individual. It’s the pre-formative as performative. Transduction is a process of individuation. It leads to individuated beings, such as things, gadgets, organisms, machines, self and society, which could be the object of knowledge. It is an ontogenetic operation which provisionally resolves incompatibilities between different orders or different zones of a domain.

What is at stake in the bigger picture, in a diagrammatic politics, is double-sided. Just as there is matter in expression and expression in matter, there is event-value in an  exchange-value paradigm, which in fact amplifies the force of its power relations. The economic engine of our time feeds on event potential becoming-commodity. It grows and flourishes on the mass production of affective intensities. Reciprocally, there are degrees of exchange-value in eventness. It’s the recursive loopiness of our current Creative Industries diagram in which the social networking praxis of Web 2.0 is emblematic and has much to learn.

Causality

Causation is a form of event generation. To present an explicit definition of causation requires introducing some ontological concepts to formally characterize what is understood by ‘event’.

The concept of individual is the basic primitive concept of any ontological theory. Individuals associate themselves with other individuals to yield new individuals. It follows that they satisfy a calculus, and that they are rigorously characterized only through the laws of such a calculus. These laws are set with the aim of reproducing the way real things associate. Specifically, it is postulated that every individual is an element of a set s in such a way that the structure S = ⟨s, ◦, ◻⟩ is a commutative monoid of idempotents. This is a simple additive semi-group with neutral element.

In the structure S, s is the set of all individuals, the element ◻ ∈ s is a fiction called the null individual, and the binary operation ◦ is the association of individuals. Although S is a mathematical entity, the elements of s are not, with the only exception of ◻, which is a fiction introduced to form a calculus. The association of any element of s with ◻ yields the same element. The following definitions characterize the composition of individuals.

1. x ∈ s is composed ⇔ (∃ y, z) s (x = y ◦ z)
2. x ∈ s is simple ⇔ ∼ (∃ y, z) s (x = y ◦ z)
3. x ⊂ y ⇔ x ◦ y = y (x is part of y ⇔ x ◦ y = y)
4. Comp(x) ≡ {y ∈ s|y ⊂ x} is the composition of x.

Real things are distinguished from abstract individuals because they have a number of properties in addition to their capability of association. These properties can be intrinsic (P_i) or relational (P_r). The intrinsic properties are inherent and they are represented by predicates or unary applications, whereas relational properties depend upon more than a single thing and are represented by n-ary predicates, with n ≥ 1. Examples of intrinsic properties are electric charge and rest mass, whereas velocity of macroscopic bodies and volume are relational properties.

An individual with its properties make up a thing X : X =< x, P(x) >

Here P(x) is the collection of properties of the individual x. A material thing is an individual with concrete properties, i.e. properties that can change in some respect.

The state of a thing X is a set of functions S(X) from a domain of reference M (a set that can be enumerable or nondenumerable) to the set of properties P_X. Every function in S(X) represents a property in P_X. The set of the physically accessible states of a thing X is the lawful state space of X : S_L(X). The state of a thing is represented by a point in S_L(X). A change of a thing is an ordered pair of states. Only changing things can be material. Abstract things cannot change since they have only one state (their properties are fixed by definition).

A legal statement is a restriction upon the state functions of a given class of things. A natural law is a property of a class of material things represented by an empirically corroborated legal statement.

The ontological history h(X) of a thing X is a subset of S_L(X) defined by h(X) = {⟨t, F(t)⟩|t ∈ M}

where t is an element of some auxiliary set M, and F are the functions that represent the properties of X.

If a thing is affected by other things we can introduce the following definition:

h(Y/X ) : “history of the thing Y in presence of the thing X”.

Let h(X) and h(Y) be the histories of the things X and Y, respectively. Then

h(Y/X) = {⟨t,H(t)⟩|t ∈ M},

where H≠ F is the total state function of Y as affected by the existence of X, and F is the total state function of X in the absence of Y. The history of Y in presence of X is different from the history of Y without X .

We can now introduce the notion of action:

X ▷ Y : “X acts on Y”

X ▷ Y =^def h(Y/X) ≠ h(Y)

An event is a change of a thing X, i.e. an ordered pair of states:

(s₁, s₂) ∈ E_L(X) = S_L(X) × S_L(X)

The space E_L(X) is called the event space of X.

Causality is a relation between events, i.e. a relation between changes of states of concrete things. It is not a relation between things. Only the related concept of ‘action’ is a relation between things. Specifically,

C'(x): “an event in a thing x is caused by some unspecified event e^x_xi“.

C'(x) =^def (∃ e^x_xi) [e^x_xi ∈ E_L(X) ⇔ x_i ▷ x.

C(x, y): “an event in a thing x is caused by an event in a thing y”.

C(x, y) =^def (∃ e^x_y) [e^x_y ∈ E_L(x) ⇔ y ▷ x

In the above definitions, the notation e^x_y indicates in the superscript the thing x to whose event space belongs the event e, whereas the subscript denotes the thing that acted triggering the event. The implicit arguments of both C’ and C are events, not things. Causation is a form of event generation. The crucial point is that a given event in the lawful event space E_L(x) is caused by an action of a thing y iff the event happens only conditionally to the action, i.e., it would not be the case of e^x_y without an action of y upon x. Time does not appear in this definition, allowing causal relations in space-time without a global time orientability or even instantaneous and non-local causation. If causation is non-local under some circumstances, e.g. when a quantum system is prepared in a specific state of polarization or spin, quantum entanglement poses no problem to realism and determinism. The quantum theory describes an aspect of a reality that is ontologically determined and with non-local relations. Under any circumstances the postulates of Special Relativity are violated, since no physical system ever crosses the barrier of the speed of light.