# Intuitive Algebra (Groupoid/Categorical Structure) of Open Strings As Morphisms

A geometric Dirichlet brane is a triple (L, E, ∇E) – a submanifold L ⊂ M, carrying a vector bundle E, with connection ∇E.

The real dimension of L is also often brought into the nomenclature, so that one speaks of a Dirichlet p-brane if p = dimRL.

An open string which stretches from a Dirichlet brane (L, E, ∇E) to a Dirichlet brane (K, F, ∇F), is a map X from an interval I ≅ [0,1] to M, such that X(0) ∈ L and X(1) ∈ K. An “open string history” is a map from R into open strings, or equivalently a map from a two-dimensional surface with boundary, say Σ ≡ I × R, to M , such that the two boundaries embed into L and K.

The quantum theory of these open strings is defined by a functional integral over these histories, with a weight which depends on the connections ∇E and ∇F. It describes the time evolution of an open string state which is a wave function in a Hilbert space HB,B′ labelled by the two choices of brane B = (L, E, ∇E) and B′ = (K, F, ∇F).

Distinct Dirichlet branes can embed into the same submanifold L. One way to represent this would be to specify the configurations of Dirichlet branes as a set of submanifolds with multiplicity. However, we can also represent this choice by using the choice of bundle E. Thus, a set of N identical branes will be represented by tensoring the bundle E with CN. The connection is also obtained by tensor product. An N-fold copy of the Dirichlet brane (L, E, ∇E) is thus a triple (L, E ⊗CN, ∇E ⊗ idN).

In physics, one visualizes this choice by labelling each open string boundary with a basis vector of CN, which specifies a choice among the N identical branes. These labels are called Chan-Paton factors. One then uses them to constrain the interactions between open strings. If we picture such an interaction as the joining of two open strings to one, the end of the first to the beginning of the second, we require not only the positions of the two ends to agree, but also the Chan-Paton factors. This operation is the intuitive algebra of open strings.

Mathematically, an algebra of open strings can always be tensored with a matrix algebra, in general producing a noncommutative algebra. More generally, if there is more than one possible boundary condition, then, rather than an algebra, it is better to think of this as a groupoid or categorical structure on the boundary conditions and the corresponding open strings. In the language of groupoids, particular open strings are elements of the groupoid, and the composition law is defined only for pairs of open strings with a common boundary. In the categorical language, boundary conditions are objects, and open strings are morphisms. The simplest intuitive argument that a non-trivial choice can be made here is to call upon the general principle that any local deformation of the world-sheet action should be a physically valid choice. In particular, particles in physics can be charged under a gauge field, for example the Maxwell field for an electron, the color Yang-Mills field for a quark, and so on. The wave function for a charged particle is then not complex-valued, but takes values in a bundle E.

Now, the effect of a general connection ∇E is to modify the functional integral by modifying the weight associated to a given history of the particle. Suppose the trajectory of a particle is defined by a map φ : R → M; then a natural functional on trajectories associated with a connection ∇ on M is simply its holonomy along the trajectory, a linear map from E|φ(t1) to E|φ(t2). The functional integral is now defined physically as a sum over trajectories with this holonomy included in the weight.

The simplest way to generalize this to a string is to consider the ls → 0 limit. Now the constraint of finiteness of energy is satisfied only by a string of vanishingly small length, effectively a particle. In this limit, both ends of the string map to the same point, which must therefore lie on L ∩ K.

The upshot is that, in this limit, the wave function of an open string between Dirichlet branes (L, E, ∇) and (K, F, ∇F) transforms as a section of E ⊠ F over L ∩ K, with the natural connection on the direct product. In the special case of (L, E, ∇E) ≅ (K, F, ∇F), this reduces to the statement that an open string state is a section of EndE. Open string states are sections of a graded vector bundle End E ⊗ Λ•T∗L, the degree-1 part of which corresponds to infinitesimal deformations of ∇E. In fact, these open string states are the infinitesimal deformations of ∇E, in the standard sense of quantum field theory, i.e., a single open string is a localized excitation of the field obtained by quantizing the connection ∇E. Similarly, other open string states are sections of the normal bundle of L within X, and are related in the same way to infinitesimal deformations of the submanifold. These relations, and their generalizations to open strings stretched between Dirichlet branes, define the physical sense in which the particular set of Dirichlet branes associated to a specified background X can be deduced from string theory.

# Tantric Reality

आत्मा त्वं गिरिजा मतिः सहचराः प्राणाः शरीरं गृहं पूजा ते विषयोपभोगरचना निद्रा समाधिस्थितिः।
सञ्चारः पदयोः प्रदक्षिणविधिः स्तोत्राणि सर्वा गिरो यद्यत्कर्म करोमि तत्तदखिलं शम्भो तवाराधनम्॥

Ātmā tvaṃ Girijā matiḥ sahacarāḥ prāṇāḥ śarīraṃ gṛham
Pūjā te viṣayopabhoga-racanā nidrā samādhi-sthitiḥ |

You (tvam) (are) the Self (ātmā) and Girijā –an epithet of Pārvatī, Śiva’s wife, meaning “mountain-born”– (girijā) (is) the intelligence (matiḥ). The vital energies (prāṇāḥ) (are Your)companions (sahacarāḥ). The body (śarīram) (is Your) house (gṛham). Worship (pūjā) of You (te) is prepared (racanā) with the objects (viṣaya) (known as sensual) enjoyments (upabhoga). Sleep (nidrā) (is Your) state (sthitiḥ) of Samādhi –i.e. perfect concentration or absorption– (samādhi). (My) wandering (sañcāraḥ) (is) the ceremony (vidhiḥ) of circumambulation from left to right (pradakṣiṇa) of (Your) feet (padayoḥ) –this act is generally done as a token of respect–. All (sarvāḥ) (my) words (giraḥ) (are) hymns of praise (of You) (stotrāṇi). Whatever (yad yad) action (karma) I do (karomi), all (akhilam) that (tad tad) is adoration (ārādhanam) of You (tava), oh Śambhu — an epithet of Śiva meaning “beneficent, benevolent”.

This Self is an embodiment of the Light of Consciousness; it is Śiva, free and autonomous. As an independent play of intense joy, the Divine conceals its own true nature [by manifesting plurality], and may also choose to reveal its fullness once again at any time. All that exists, throughout all time and beyond, is one infinite divine Consciousness, free and blissful, which projects within the field of its awareness a vast multiplicity of apparently differentiated subjects and objects: each object an actualization of a timeless potentiality inherent in the Light of Consciousness, and each subject the same plus a contracted locus of self-awareness. This creation, a divine play, is the result of the natural impulse within Consciousness to express the totality of its self-knowledge in action, an impulse arising from love. The unbounded Light of Consciousness contracts into finite embodied loci of awareness out of its own free will. When those finite subjects then identify with the limited and circumscribed cognitions and circumstances that make up this phase of their existence, instead of identifying with the transindividual overarching pulsation of pure Awareness that is their true nature, they experience what they call “suffering.” To rectify this, some feel an inner urge to take up the path of spiritual gnosis and yogic practice, the purpose of which is to undermine their misidentification and directly reveal within the immediacy of awareness the fact that the divine powers of Consciousness, Bliss, Willing, Knowing, and Acting comprise the totality of individual experience as well – thereby triggering a recognition that one’s real identity is that of the highest Divinity, the Whole in every part. This experiential gnosis is repeated and reinforced through various means until it becomes the nonconceptual ground of every moment of experience, and one’s contracted sense of self and separation from the Whole is finally annihilated in the incandescent radiance of the complete expansion into perfect wholeness. Then one’s perception fully encompasses the reality of a universe dancing ecstatically in the animation of its completely perfect divinity.”

# Expressivity of Bodies: The Synesthetic Affinity Between Deleuze and Merleau-Ponty. Thought of the Day 54.0

It is in the description of the synesthetic experience that Deleuze finds resources for his own theory of sensation. And it is in this context that Deleuze and Merleau-Ponty are closest. For Deleuze sees each sensation as a dynamic evolution, sensation is that which passes from one ‘order’ to another, from one ‘level’ to another. This means that each sensation is at diverse levels, of different orders, or in several domains….it is characteristic of sensation to encompass a constitutive difference of level and a plurality of constituting domains. What this means for Deleuze is that sensations cannot be isolated in a particular field of sense; these fields interpenetrate, so that sensation jumps from one domain to another, becoming-color in the visual field or becoming-music on the auditory level. For Deleuze (and this goes beyond what Merleau-Ponty explicitly says), sensation can flow from one field to another, because it belongs to a vital rhythm which subtends these fields, or more precisely, which gives rise to the different fields of sense as it contracts and expands, as it moves between different levels of tension and dilation.

If, as Merleau-Ponty says (and Deleuze concurs), synesthetic perception is the rule, then the act of recognition that identifies each sensation with a determinate quality or sense and operates their synthesis within the unity of an object, hides from us the complexity of perception, and the heterogeneity of the perceiving body. Synesthesia shows that the unity of the body is constituted in the transversal communication of the senses. But these senses are not pre given in the body; they correspond to sensations that move between levels of bodily energy – finding different expression in each other. To each of these levels corresponds a particular way of living space and time; hence the simultaneity in depth that is experienced in vision is not the lateral coexistence of touch, and the continuous, sensuous and overlapping extension of touch is lost in the expansion of vision. This heterogenous multiplicity of levels, or senses, is open to communication; each expresses its embodiment in its own way, and each expresses differently the contents of the other senses.

Thus sensation is not the causal process, but the communication and synchronization of senses within my body, and of my body with the sensible world; it is, as Merleau-Ponty says, a communion. And despite frequent appeal in the Phenomenology of Perception to the sameness of the body and to the common world to ground the diversity of experience, the appeal here goes in a different direction. It is the differences of rhythm and of becoming, which characterize the sensible world, that open it up to my experience. For the expressive body is itself such a rhythm, capable of synchronizing and coexisting with the others. And Merleau-Ponty refers to this relationship between the body and the world as one of sympathy. He is close here to identifying the lived body with the temporization of existence, with a particular rhythm of duration; and he is close to perceiving the world as the coexistence of such temporalizations, such rhythms. The expressivity of the lived body implies a singular relation to others, and a different kind of intercorporeity than would be the case for two merely physical bodies. This intercorporeity should be understood as inter-temporality. Merleau-Ponty proposes this at the end of the chapter on perception in his Phenomenology of Perception, when he says,

But two temporalities are not mutually exclusive as are two consciousnesses, because each one knows itself only by projecting itself into the present where they can interweave.

Thus our bodies as different rhythms of duration can coexist and communicate, can synchronize to each other – in the same way that my body vibrated to the colors of the sensible world. But, in the case of two lived bodies, the synchronization occurs on both sides – with the result that I can experience an internal resonance with the other when the experiences harmonize, or the shattering disappointment of a  miscommunication when the attempt fails. The experience of coexistence is hence not a guarantee of communication or understanding, for this communication must ultimately be based on our differences as expressive bodies and singular durations. Our coexistence calls forth an attempt, which is the intuition.

# Rhizomatic Topology and Global Politics. A Flirtatious Relationship.

Deleuze and Guattari see concepts as rhizomes, biological entities endowed with unique properties. They see concepts as spatially representable, where the representation contains principles of connection and heterogeneity: any point of a rhizome must be connected to any other. Deleuze and Guattari list the possible benefits of spatial representation of concepts, including the ability to represent complex multiplicity, the potential to free a concept from foundationalism, and the ability to show both breadth and depth. In this view, geometric interpretations move away from the insidious understanding of the world in terms of dualisms, dichotomies, and lines, to understand conceptual relations in terms of space and shapes. The ontology of concepts is thus, in their view, appropriately geometric, a multiplicity defined not by its elements, nor by a center of unification and comprehension and instead measured by its dimensionality and its heterogeneity. The conceptual multiplicity, is already composed of heterogeneous terms in symbiosis, and is continually transforming itself such that it is possible to follow, and map, not only the relationships between ideas but how they change over time. In fact, the authors claim that there are further benefits to geometric interpretations of understanding concepts which are unavailable in other frames of reference. They outline the unique contribution of geometric models to the understanding of contingent structure:

Principle of cartography and decalcomania: a rhizome is not amenable to any structural or generative model. It is a stranger to any idea of genetic axis or deep structure. A genetic axis is like an objective pivotal unity upon which successive stages are organized; deep structure is more like a base sequence that can be broken down into immediate constituents, while the unity of the product passes into another, transformational and subjective, dimension. (Deleuze and Guattari)

The word that Deleuze and Guattari use for ‘multiplicities’ can also be translated to the topological term ‘manifold.’ If we thought about their multiplicities as manifolds, there are a virtually unlimited number of things one could come to know, in geometric terms, about (and with) our object of study, abstractly speaking. Among those unlimited things we could learn are properties of groups (homological, cohomological, and homeomorphic), complex directionality (maps, morphisms, isomorphisms, and orientability), dimensionality (codimensionality, structure, embeddedness), partiality (differentiation, commutativity, simultaneity), and shifting representation (factorization, ideal classes, reciprocity). Each of these functions allows for a different, creative, and potentially critical representation of global political concepts, events, groupings, and relationships. This is how concepts are to be looked at: as manifolds. With such a dimensional understanding of concept-formation, it is possible to deal with complex interactions of like entities, and interactions of unlike entities. Critical theorists have emphasized the importance of such complexity in representation a number of times, speaking about it in terms compatible with mathematical methods if not mathematically. For example, Foucault’s declaration that: practicing criticism is a matter of making facile gestures difficult both reflects and is reflected in many critical theorists projects of revealing the complexity in (apparently simple) concepts deployed both in global politics.  This leads to a shift in the concept of danger as well, where danger is not an objective condition but “an effect of interpretation”. Critical thinking about how-possible questions reveals a complexity to the concept of the state which is often overlooked in traditional analyses, sending a wave of added complexity through other concepts as well. This work seeking complexity serves one of the major underlying functions of critical theorizing: finding invisible injustices in (modernist, linear, structuralist) givens in the operation and analysis of global politics.

In a geometric sense, this complexity could be thought about as multidimensional mapping. In theoretical geometry, the process of mapping conceptual spaces is not primarily empirical, but for the purpose of representing and reading the relationships between information, including identification, similarity, differentiation, and distance. The reason for defining topological spaces in math, the essence of the definition, is that there is no absolute scale for describing the distance or relation between certain points, yet it makes sense to say that an (infinite) sequence of points approaches some other (but again, no way to describe how quickly or from what direction one might be approaching). This seemingly weak relationship, which is defined purely ‘locally’, i.e., in a small locale around each point, is often surprisingly powerful: using only the relationship of approaching parts, one can distinguish between, say, a balloon, a sheet of paper, a circle, and a dot.

To each delineated concept, one should distinguish and associate a topological space, in a (necessarily) non-explicit yet definite manner. Whenever one has a relationship between concepts (here we think of the primary relationship as being that of constitution, but not restrictively, we ‘specify’ a function (or inclusion, or relation) between the topological spaces associated to the concepts). In these terms, a conceptual space is in essence a multidimensional space in which the dimensions represent qualities or features of that which is being represented. Such an approach can be leveraged for thinking about conceptual components, dimensionality, and structure. In these terms, dimensions can be thought of as properties or qualities, each with their own (often-multidimensional) properties or qualities. A key goal of the modeling of conceptual space being representation means that a key (mathematical and theoretical) goal of concept space mapping is

associationism, where associations between different kinds of information elements carry the main burden of representation. (Conceptual_Spaces_as_a_Framework_for_Knowledge_Representation)

To this end,

objects in conceptual space are represented by points, in each domain, that characterize their dimensional values. A concept geometry for conceptual spaces

These dimensional values can be arranged in relation to each other, as Gardenfors explains that

distances represent degrees of similarity between objects represented in space and therefore conceptual spaces are “suitable for representing different kinds of similarity relation. Concept

These similarity relationships can be explored across ideas of a concept and across contexts, but also over time, since “with the aid of a topological structure, we can speak about continuity, e.g., a continuous change” a possibility which can be found only in treating concepts as topological structures and not in linguistic descriptions or set theoretic representations.

# Sobolev Spaces

For any integer n ≥ 0, the Sobolev space Hn(R) is defined to be the set of functions f which are square-integrable together with all their derivatives of order up to n:

f ∈ Hn(R) ⇐⇒ ∫-∞ [f2 + ∑k=1n (dkf/dxk)2 dx ≤ ∞

This is a linear space, and in fact a Hilbert space with norm given by:

∥f∥Hn = ∫-∞ [f2 + ∑k=1n (dkf/dxk)2) dx]1/2

It is a standard fact that this norm of f can be expressed in terms of the Fourier transform fˆ (appropriately normalized) of f by:

∥f∥2Hn = ∫-∞ [(1 + y2)n |fˆ(y)|2 dy

The advantage of that new definition is that it can be extended to non-integral and non-positive values. For any real number s, not necessarily an integer nor positive, we define the Sobolev space Hs(R) to be the Hilbert space of functions associated with the following norm:

∥f∥2Hs = ∫-∞ [(1 + y2)s |fˆ(y)|2 dy —– (1)

Clearly, H0(R) = L2(R) and Hs(R) ⊂ Hs′(R) for s ≥ s′ and in particular Hs(R) ⊂ L2(R) ⊂ H−s(R), for s ≥ 0. Hs(R) is, for general s ∈ R, a space of (tempered) distributions. For example δ(k), the k-th derivative of a delta Dirac distribution, is in H−k−1/2</sup−ε(R) for ε > 0.

In the case when s > 1/2, there are two classical results.

Continuity of Multiplicity:

If s > 1/2, if f and g belong to Hs(R), then fg belongs to Hs(R), and the map (f,g) → fg from Hs × Hs to Hs is continuous.

Denote by Cbn(R) the space of n times continuously differentiable real-valued functions which are bounded together with all their n first derivatives. Let Cnb0(R) be the closed subspace of Cbn(R) of functions which converges to 0 at ±∞ together with all their n first derivatives. These are Banach spaces for the norm:

∥f∥Cbn = max0≤k≤n supx |f(k)(x)| = max0≤k≤n ∥f(k)∥ C0b

Sobolev embedding:

If s > n + 1/2 and if f ∈ Hs(R), then there is a function g in Cnb0(R) which is equal to f almost everywhere. In addition, there is a constant cs, depending only on s, such that:

∥g∥Cbn ≤ c∥f∥Hs

From now on we shall always take the continuous representative of any function in Hs(R). As a consequence of the Sobolev embedding theorem, if s > 1/2, then any function f in Hs(R) is continuous and bounded on the real line and converges to zero at ±∞, so that its value is defined everywhere.

We define, for s ∈ R, a continuous bilinear form on H−s(R) × Hs(R) by:

〈f, g〉= ∫-∞ (fˆ(y))’ gˆ(y)dy —– (2)

where z’ is the complex conjugate of z. Schwarz inequality and (1) give that

|< f , g >| ≤ ∥f∥H−s∥g∥Hs —– (3)

which indeed shows that the bilinear form in (2) is continuous. We note that formally the bilinear form (2) can be written as

〈f, g〉= ∫-∞ f(x) g(x) dx

where, if s ≥ 0, f is in a space of distributions H−s(R) and g is in a space of “test functions” Hs(R).

Any continuous linear form g → u(g) on Hs(R) is, due to (1), of the form u(g) = 〈f, g〉 for some f ∈ H−s(R), with ∥f∥H−s = ∥u∥(Hs)′, so that henceforth we can identify the dual (Hs(R))′ of Hs(R) with H−s(R). In particular, if s > 1/2 then Hs(R) ⊂ C0b0 (R), so H−s(R) contains all bounded Radon measures.

# Universal Inclusion of the Void. Thought of the Day 38.0

The universal inclusion of the void means that the intersection between two sets whatsoever is comparable with the void set. That is to say, there is no multiple that does not include within it some part of the “inconsistency” that it structures. The diversity of multiplicity can exhibit multiple modes of articulation, but as multiples, they have nothing to do with one another, they are two absolutely heterogeneous presentations, and this is why this relation – of non-relation – can only be thought under the signifier of being (of the void), which indicates that the multiples in question have nothing in common apart from being multiples. The universal inclusion of the void thus guarantees the consistency of the infinite multiplicities immanent to its presentation. That is to say, it underlines the universal distribution of the ontological structure seized at the point of the axiom of the void set. The void does not merely constitute a consistency at a local point but also organises, from this point of difference, a universal structure that legislates on the structure of all sets, the universe of consistent multiplicity.

This final step, the carrying over of the void seized as a local point of the presentation of the unpresentable, to a global field of sets provides us with the universal point of difference, applicable equally to any number of sets, that guarantees the universal consistency of ontological presentation. In one sense, the universal inclusion of the void demonstrates that, as a unit of presentation, the void anchors the set theoretical universe by its universal inclusion. As such, every presentation in ontological thought is situated in this elementary seizure of ontological difference. The void is that which “fills” ontological or set theoretical presentation. It is what makes common the universe of sets. It is in this sense that the “substance” or constitution of ontology is the void. At the same stroke, however, the universal inclusion of the void also concerns the consistency of set theory in a logical sense.

The universal inclusion of the void provides an important synthesis of the consistency of presentation. What is presented is necessarily consistent but its consistency gives way to two distinct senses. Consistency can refer to its own “substance,” its immanent presentation. Distinct presentations constitute different presentations principally because “what” they present are different. Ontology’s particularity is its presentation of the void. On the other hand, a political site might present certain elements just as a scientific procedure might present yet others. The other sense of consistency is tied to presentation as such, the consistency of presentation in its generality. When one speaks loosely about the “world” being consistent, where natural laws are verifiable against a background of regularity, it is this consistency that is invoked and not the elements that constitute the particularity of their presentation. This sense of consistency, occurring across presentations would certainly take us beyond the particularity of ontology. That is to say, ontological presentation presents a species of this consistency. However, the possibility of multiple approaches does not exclude an ontological treatment of this consistency.

# Biogrammatic Vir(Ac)tuality. Note Quote.

In Foucault’s most famous example, the prison acts as the confluence of content (prisoners) and expression (law, penal code) (Gilles Deleuze, Sean Hand-Foucault). Informal Diagrams are proliferate. As abstract machines they contain the transversal vectors that cut across a panoply of features (such as institutions, classes, persons, economic formation, etc), mapping from point to relational point, the generalized features of power economies. The disciplinary diagram explored by Foucault, imposes “a particular conduct upon a particular human multiplicity”. The imposition of force upon force affects and effectuates the felt experience of a life, a living. Deleuze has called the abstract machine “pure matter/function” in which relations between forces are nonetheless very real.

[…] the diagram acts as a non-unifying immanent cause that is co-extensive with the whole social field: the abstract machine is like the cause of the concrete assemblages that execute its relations; and these relations between forces take place ‘not above’ but within the very tissue of the assemblages they produce.

The processual conjunction of content and expression; the cutting edge of deterritorialization:

The relations of power and resistance between theory and practice resonate – becoming-form; diagrammatics as praxis, integrates and differentiates the immanent cause and quasi-cause of the actualized occasions of research/creation. What do we mean by immanent cause? It is a cause which is realized, integrated and distinguished in its effect. Or rather, the immanent cause is realized, integrated and distinguished by its effect. In this way there is a correlation or mutual presupposition between cause and effect, between abstract machine and concrete assemblages

Memory is the real name of the relation to oneself, or the affect of self by self […] Time becomes a subject because it is the folding of the outside…forces every present into forgetting but preserves the whole of the past within memory: forgetting is the impossibiltiy of return and memory is the necessity of renewal.

The figure on the left is Henri Bergson’s diagram of an infinitely contracted past that directly intersects with the body at point S – a mobile, sensorimotor present where memory is closest to action. Plane P represents the actual present; plane of contact with objects. The AB segments represent repetitive compressions of memory. As memory contracts it gets closer to action. In it’s more expanded forms it is closer to dreams. The figure on the right extrapolates from Bergson’s memory model to describe the Biogrammatic ontological vector of the Diagram as it moves from abstract (informal) machine in the most expanded form “A” through the cone “tissue” to the phase-shifting (formal), arriving at the Strata of the P plane to become artefact. The ontological vector passes through the stratified, through the interval of difference created in the phase shift (the same phase shift that separates and folds content and expression to move vertically, transversally, back through to the abstract diagram.)

A spatio-temporal-material contracting-expanding of the abstract machine is the processual thinking-feeling-articulating of the diagram becoming-cartographic; synaesthetic conceptual mapping. A play of forces, a series of relays, affecting a tendency toward an inflection of the informal diagram becoming-form. The inflected diagram/biogram folds and unfolds perception, appearances; rides in the gap of becoming between content and expression; intuitively transduces the actualizing (thinking, drawing, marking, erasing) of matter-movement, of expressivity-movement. “To follow the flow of matter… is intuition in action.” A processual stage that prehends the process of the virtual actualizing;

the creative construction of a new reality. The biogrammatic stage of the diagrammatic is paradoxically double in that it is both the actualizing of the abstract machine (contraction) and the recursive counter-actualization of the formal diagram (détournement); virtual and actual.

It is the event-dimension of potential – that is the effective dimension of the interrelating of elements, of their belonging to each other. That belonging is a dynamic corporeal “abstraction” – the “drawing off” (transductive conversion) of the corporeal into its dynamism (yielding the event) […] In direct channeling. That is, in a directional channeling: ontological vector. The transductive conversion is an ontological vector that in-gathers a heterogeneity of substantial elements along with the already-constituted abstractions of language (“meaning”) and delivers them together to change. (Brian Massumi Parables for the Virtual Movement, Affect, Sensation)

Skin is the space of the body the BwO that is interior and exterior. Interstitial matter of the space of the body.

The material markings and traces of a diagrammatic process, a ‘capturing’ becoming-form. A diagrammatic capturing involves a transductive process between a biogrammatic form of content and a form of expression. The formal diagram is thus an individuating phase-shift as Simondon would have it, always out-of-phase with itself. A becoming-form that inhabits the gap, the difference, between the wave phase of the biogrammatic that synaesthetically draws off the intermix of substance and language in the event-dimension and the drawing of wave phase in which partial capture is formalized. The phase shift difference never acquires a vectorial intention. A pre-decisive, pre-emptive drawing of phase-shifting with a “drawing off” the biogram.

If effects realize something this is because the relations between forces or power relations, are merely virtual, potential, unstable vanishing and molecular, and define only possibilities of interaction so long as they do not enter a macroscopic whole capable of giving form to their fluid manner and diffuse function. But realization is equally an integration, a collection of progressive integrations that are initially local and then become or tend to become global, aligning, homogenizing and summarizing relations between forces: here law is the integration of illegalisms.

# Topological Drifts in Deleuze. Note Quote.

Brion Gysin: How do you get in… get into these paintings?

William Burroughs: Usually I get in by a port of entry, as I call it. It is often a face through whose eyes the picture opens into a landscape and I go literally right through that eye into that landscape. Sometimes it is rather like an archway… a number of little details or a special spot of colours makes the port of entry and then the entire picture will suddenly become a three-dimensional frieze in plaster or jade or some other precious material.

The word fornix means “an archway” or “vault” (in Rome, prostitutes could be solicited there). More directly, fornicatio means “done in the archway”; thus a euphemism for prostitution.

Diagrammatic praxis proposes a contractual (push, pull) approach in which the movement between abstract machine, biogram (embodied, inflected diagram), formal diagram (drawing of, drawing off) and artaffect (realized thing) is topologically immanent. It imagines the practice of writing, of this writing, interleaved with the mapping processes with which it folds and unfolds – forming, deforming and reforming both processes. The relations of non-relations that power the diagram, the thought intensities that resonate between fragments, between content ad expression, the seeable and the sayable, the discursive and the non-discursive, mark entry points; portals of entry through which all points of the diagram pass – push, pull, fold, unfold – without the designation of arrival and departure, without the input/output connotations of a black boxed confection. Ports, as focal points of passage, attract lines of resistance or lines of flight through which the diagram may become both an effectuating concrete assemblage (thing) and remain outside the stratified zone of the audiovisual. It’s as if the port itself is a bifurcating point, a figural inflected archway. The port, as a bifurcation point of resistance (contra black box), modulates and changes the unstable, turbulent interplay between pure Matter and pure Function of the abstract machine. These ports are marked out, localized, situated, by the continuous movement of power-relations:

These power-relations … simultaneously local, unstable and diffuse, do not emanate from a central point or unique locus of sovereignty, but at each moment move from one point to another in a field of forces, marking inflections, resistances, twists and turns when one changes direction or retraces one’s steps… (Gilles Deleuze, Sean Hand-Foucault)

An inflection point, marked out by the diagram, is not a symmetrical form but the difference between concavity and convexity, a pure temporality, a “true atom of form, the true object of geography.” (Bernard Cache)

Cache’s absolute exteriority is equivalent to Deleuze’s description of the Outside “more distant than any exterior […] ‘twisted’, folded and doubled by an Inside that is deeper than any interior, and alone creates the possibility of the derived relation between the interior and the exterior”. This folded and doubled interior is diagrammed by Deleuze in the folds chapter of Foucault.

Thinking does not depend on a beautiful interiority that reunites the visible ad articulable elements, but is carried under the intrusion of an outside that eats into the interval and forces or dismembers the internal […] when there are only environments and whatever lies betwen them, when words and things are opened up by the environment without ever coinciding, there is a liberation of forces which come from the outside and exist only in a mixed up state of agitation, modification and mutation. In truth they are dice throws, for thinking involves throwing the dice. If the outside, farther away than any external world, is also closer than any internal world, is this not a sign that thought affects itself, by revealing the outside to be its own unthought element?

“It cannot discover the unthought […] without immediately bringing the unthought nearer to itself – or even, perhaps, without pushing it farther away, and in any case without causing man’s own being to undergo a change by the very fact, since it is deployed in the distance between them” (Gilles Deleuze, Sean Hand-Foucault)

Figure: Left: a simulation of Deleuze’s central marking in his diagram of the Foucaultian diagram. This is the line of the Outside as Fold. Right: To best express the relations of diagrammatic praxis between content and expression (theory and practice) the Fold figure needs to be drawn as a double Fold (“twice twice” as Massumi might say) – a folded möbius strip. Here the superinflections between inside/outside and content/expression provide transversal vectors.

A topology or topological becoming-shapeshift retains its connectivity, its interconnectedness to preserve its autonomy as a singularity. All the points of all its matter reshape as difference in itself. A topology does not resemble itself. The möbius strip and the infamous torus-to-coffe cup are examples of 2d and 3d topologies. technically a topological surface is totalized, it can not comprise fragments cut or glued to produce a whole. Its change is continuous. It is not cut-copy-pasted. But the cut and its interval are requisite to an emergent new.

For Deleuze, the essence of meaning, the essence of essence, is best expressed in two infinitives; ‘to cut ” and “to die” […] Definite tenses keeping company in time. In the slash between their future and their past: “to cut” as always timeless and alone (Massumi).

Add the individuating “to shift” to the infinitives that reside in the timeless zone of indetermination of future-past. Given the paradigm of the topological-becoming, how might we address writing in the age of copy-paste and hypertext? The seamless and the stitched? As potential is it diagram? A linguistic multiplicity whose virtual immanence is the metalanguage potentiality between the phonemes that gives rise to all language?

An overview diagram of diagrammatic praxis based on Deleuze’s diagram of the Foucaultian model shown below. The main modification is to the representation of the Fold. In the top figure, the Fold or zone of subjectification becomes a double-folded möbius strip.

Four folds of subjectification:

1. material part of ourselves which is to be surrounded and folded

2. the fold of the relation between forces always according to a particular rule that the relation between forces is bent back in order to become a relation to oneself (rule ; natural, divine, rational, aesthetic, etc)

3. fold of knowledge constitutes the relation of truth to our being and our being to truth which will serve as the formal condition for any kind of knowledge

4. the fold of the outside itself is the ultimate fold: an ‘interiority of expectation’ from which the subject, in different ways, hopes for immortality, eternity, salvation, freedom or death or detachment.

# To Err or Not? Neo-Kantianism’s Logical Flaw. Note Quote.

According to Bertrand Russell, the sense in which every object is ‘one’ is a very shadowy sense because it is applicable to everything alike. However, Russell argues, the sense in which a class may be said to have one member is quite precise. “A class u has one member when u is not null, and ‘x and y are us’ implies ‘x is identical with y’.” In this case the one-ness is a property of a class and Russell calls this class a unit-class. Thus, Russell claims further, the number ‘one’ is not to be asserted of terms but of classes having one member in the above-defined sense. The same distinction between the different uses of ‘one’ was also made by Frege and Couturat. Frege says that the sense in which every object is ‘one’ is very imprecise, that is, every single object possesses this property. However, Frege argues that when one speaks of ‘the number one’, one indicates by means of the definite article a definite and unique object of scientific study. In his reply to Poincaré’s critique of the logicist programme, Couturat says that the confusion which exists in Poincaré’s mind arises from the double meaning of the word for ‘one’, that is, it is used both as a name of a number and as an indefinite article:

To sum up, it is not enough to conceive any one object to conceive the number one, nor to think of two objects together to have by that alone the idea of the number two.

According to Couturat, from the fact that the proposition “x and y are the elements of the class u” contains the symbols x and y one should not conclude that the number two is implied in this proposition. As a result, from the viewpoint of Russell, Couturat and Frege, the neo-Kantians are making here an elementary logical mistake. This awakens an interesting question. Why the neo-Kantians did not notice the mistake they had made? The answer is not that they would not have been aware of the opinion of the logicists. Both Cohn and Cassirer discuss the above-mentioned passage in Russell’s Principles. However, although Cohn and Cassirer were familiar with the distinction presented by Russell, it did not convince them. In Cohn’s view, Russell’s unit-class does not define ‘one’ but ‘only one’. As Cohn sees it, ‘only one’ means the limitation of a class to one object. Thus Russell’s ‘unit-class’ already presupposes that an object is seen as a unit. As a result, Russell’s definition of ‘one’ is unsuccessful since it already presupposes the number ‘one’. Cassirer, too, refers to Russell’s explanation, according to which it is naturally incontestable that every member of a class is in some sense one, but, Cassirer says, it does not follow from this that the concept of ‘one’ is presupposed. Cassirer mentions also Russell’s explanation according to which the meaning of the assertion that a class u possesses ‘one’ member is determined by the fact that this class is not null and that if x and y are u, then x is identical with y. According to Cassirer, the logical function of number is here not so much deduced as rather described by a technical circumlocution. Cassirer argues that in order to comprehend Russell’s explanation it is necessary that the term x is understood as identical with itself, and at the same time it is related to another term y and the former is judged as agreeing with or differing from the latter. In Cassirer’s view, if this process of positing and differentiation is accepted, then all that has been done will be to presuppose the number in the sense of the theory of ordinal number.

The neo-Kantian critique cannot be explained away as a mere logical error. The real reason why they did not accept the distinction is that to accept it would be to accept at least part of the logicist programme. As Warren Goldfarb has pointed out, Poincaré’s argument will be logically in error only if one simultaneously accepts the analysis of notions ‘in no case’ and ‘a class with one object’ that was first made available through modern mathematical logic. In other words, the logicists claim that the appearance of circularity is eliminated when one distinguishes uses of numerical expressions that can be replaced by purely quantificational devices from the full-blooded uses of such expressions that the formal definition is meant to underwrite. Hence the notions ‘in no case’ and ‘a class with one object’ do not presuppose any number theory if one simultaneously accepts the analysis which first-order quantificational logic provides for them. Poincaré does not accept this analysis, and, as result, he can bring the charge of petitio principii.

Like Poincaré, the neo-Kantians were not ready to accept Russell’s analysis of the expression ‘a class with one object’. As they see it, although the notion ‘a class with one object’ does not presuppose the number ‘one’ if one accepts the logicist definition of number, it will presuppose it if one advocates a neo-Kantian theory of number. According to Cassirer, the concept of number is the first and truest expression of rational method in general. Later Cassirer added that number is not merely a product of pure thought but its very prototype and source. It not only originates from the pure regularities of thought but designates the primary and original act to which these regularities ultimately go back. In Natorp’s view, number is the purest and simplest product of thought. Natorp claims that the first precondition for the logical understanding of number is the insight that number has nothing to do with the existing things but that number is only concerned with the pure regularities of thought. Natorp connects number to the fundamental logical function of quantity. In his view, the quantitative function of thought is produced when multiplicity is singled out from the fundamental relation between unity and multiplicity. Moreover, multiplicity is a plurality of distinguishable elements. Plurality, in turn, is necessarily a plurality of unities. Thus unity in the sense of numerical oneness is the unavoidable starting-point, the indispensable foundation of every quantitative positing of pure thought. According to Natorp, the quantitative positing of thought proceeds in three steps. First, pure thought posits something as one. What is posited as one is not important (it can be the world, an atom, and so on). It is only something to which the thought attaches the character of oneness. Second, the positing of the one can be repeated in the sense that while the one remains posited, we can posit always another in comparison with it. This is the way in which we attain plurality. Third and last, we collect the individual positings into a whole, that is, to a new unity in the sense of a unity of several. In this way we attain a definite plurality, that is, “so much” as distinguished from an indefinite set. In other words, one and one and one, and so forth, are here joined to new mental unities (duality, triplicity, and so forth).

According to Cohn, the natural numbers are the most abstract objects possible. Everything thinkable can be an object, and every object has two elements: the thinking-form and the objectivity. The thinking-form belongs to every object, and Cohn calls it “positing”. It can be described by saying that every object is identical with itself. This formal definiteness of an object has nothing to do with the determination of an object with regard to content. Since the thinking-form belongs to every object in the same way, it alone is not enough to form any specific object. The particularity of any individual object, or as Cohn puts it, the objectivity of any individual object, is something new and foreign when compared to the thinking-form of the object. In other words, Cohn argues that the necessary elements of every object are the thinking-form, and the objectivity. As a result, natural numbers are objects which have the thinking-form of identity and the minimum of objectivity, that is, the form of identity must be thought to be filled with something in some way or other. Moreover, Cohn says that his theory of natural numbers presupposes the possibility of arbitrary object-formation, that is, the possibility to construct arbitrarily many objects. On the basis of these two logical presuppositions, Cohn says that we are able to form arbitrarily many objects which are all equal with each other. According to Cohn, all of these objects can be described by the same symbol 1, and after this operation the fundamental equation 1 = 1 can be presented. Cohn says that the two separate symbols 1 in the equation signify different unities and the sign of equality means only that in any arithmetical relation any arbitrary unity can be replaced with any other unity. Moreover, Cohn says that we can collect an arbitrary number of objects into an aggregate, that is, into a new object. This is expressed by the repeated use of the word ‘and’. In arithmetic the combination of unities into a new unity has the form: 1 + 1 + 1 and so on (when ‘and’ is replaced by ‘+’). The most simple combination (1 + 1) can be described as 2, the following one (1 + 1 + 1) as 3, and so on. Thus a new number can always be attained by adding a new unity.