Embedding Branes in Minkowski Space-Time Dimensions To Decipher Them As Particles Or Otherwise

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The physics treatment of Dirichlet branes in terms of boundary conditions is very analogous to that of the “bulk” quantum field theory, and the next step is again to study the renormalization group. This leads to equations of motion for the fields which arise from the open string, namely the data (M, E, ∇). In the supergravity limit, these equations are solved by taking the submanifold M to be volume minimizing in the metric on X, and the connection ∇ to satisfy the Yang-Mills equations.

Like the Einstein equations, the equations governing a submanifold of minimal volume are highly nonlinear, and their general theory is difficult. This is one motivation to look for special classes of solutions; the physical arguments favoring supersymmetry are another. Just as supersymmetric compactification manifolds correspond to a special class of Ricci-flat manifolds, those admitting a covariantly constant spinor, supersymmetry for a Dirichlet brane will correspond to embedding it into a special class of minimal volume submanifolds. Since the physical analysis is based on a covariantly constant spinor, this special class should be defined using the spinor, or else the covariantly constant forms which are bilinear in the spinor.

The standard physical arguments leading to this class are based on the kappa symmetry of the Green-Schwarz world-volume action, in which one finds that the subset of supersymmetry parameters ε which preserve supersymmetry, both of the metric and of the brane, must satisfy

φ ≡ Re εt Γε|M = Vol|M —– (1)

In words, the real part of one of the covariantly constant forms on M must equal the volume form when restricted to the brane.

Clearly dφ = 0, since it is covariantly constant. Thus,

Z(M) ≡ ∫φ —– (2)

depends only on the homology class of M. Thus, it is what physicists would call a “topological charge”, or a “central charge”.

If in addition the p-form φ is dominated by the volume form Vol upon restriction to any p-dimensional subspace V ⊂ Tx X, i.e.,

φ|V ≤ Vol|V —– (3)

then φ will be a calibration in the sense of implying the global statement

φ ≤ ∫Vol —– (4)

for any submanifold M . Thus, the central charge |Z (M)| is an absolute lower bound for Vol(M).

A calibrated submanifold M is now one satisfying (1), thereby attaining the lower bound and thus of minimal volume. Physically these are usually called “BPS branes,” after a prototypical argument of this type due, for magnetic monopole solutions in nonabelian gauge theory.

For a Calabi-Yau X, all of the forms ωp can be calibrations, and the corresponding calibrated submanifolds are p-dimensional holomorphic submanifolds. Furthermore, the n-form Re eΩ for any choice of real parameter θ is a calibration, and the corresponding calibrated submanifolds are called special Lagrangian.

This generalizes to the presence of a general connection on M, and leads to the following two types of BPS branes for a Calabi-Yau X. Let n = dimR M, and let F be the (End(E)-valued) curvature two-form of ∇.

The first kind of BPS D-brane, based on the ωp calibrations, is (for historical reasons) called a “B-type brane”. Here the BPS constraint is equivalent to the following three requirements:

  1. M is a p-dimensional complex submanifold of X.
  2. The 2-form F is of type (1, 1), i.e., (E, ∇) is a holomorphic vector bundle on M.
  3. In the supergravity limit, F satisfies the Hermitian Yang-Mills equation:ω|p−1M ∧ F = c · ω|pMfor some real constant c.
  4. F satisfies Im e(ω|M + ils2F)p = 0 for some real constant φ, where ls is the correction.

The second kind of BPS D-brane, based on the Re eΩ calibration, is called an “A-type” brane. The simplest examples of A-branes are the so-called special Lagrangian submanifolds (SLAGs), satisfying

(1) M is a Lagrangian submanifold of X with respect to ω.

(2) F = 0, i.e., the vector bundle E is flat.

(3) Im e Ω|M = 0 for some real constant α.

More generally, one also has the “coisotropic branes”. In the case when E is a line bundle, such A-branes satisfy the following four requirements:

(1)  M is a coisotropic submanifold of X with respect to ω, i.e., for any x ∈ M the skew-orthogonal complement of TxM ⊂ TxX is contained in TxM. Equivalently, one requires ker ωM to be an integrable distribution on M.

(2)  The 2-form F annihilates ker ωM.

(3)  Let F M be the vector bundle T M/ ker ωM. It follows from the first two conditions that ωM and F descend to a pair of skew-symmetric forms on FM, denoted by σ and f. Clearly, σ is nondegenerate. One requires the endomorphism σ−1f : FM → FM to be a complex structure on FM.

(4)  Let r be the complex dimension of FM. r is even and that r + n = dimR M. Let Ω be the holomorphic trivialization of KX. One requires that Im eΩ|M ∧ Fr/2 = 0 for some real constant α.

Coisotropic A-branes carrying vector bundles of higher rank are still not fully understood. Physically, one must also specify the embedding of the Dirichlet brane in the remaining (Minkowski) dimensions of space-time. The simplest possibility is to take this to be a time-like geodesic, so that the brane appears as a particle in the visible four dimensions. This is possible only for a subset of the branes, which depends on which string theory one is considering. Somewhat confusingly, in the type IIA theory, the B-branes are BPS particles, while in IIB theory, the A-branes are BPS particles.

Complete Manifolds’ Pure Logical Necessity as the Totality of Possible Formations. Thought of the Day 124.0

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In Logical Investigations, Husserl called his theory of complete manifolds the key to the only possible solution to how in the realm of numbers impossible, non-existent, meaningless concepts might be dealt with as real ones. In Ideas, he wrote that his chief purpose in developing his theory of manifolds had been to find a theoretical solution to the problem of imaginary quantities (Ideas Pertaining to a Pure Phenomenology and to a Phenomenological Philosophy).

Husserl saw how questions regarding imaginary numbers come up in mathematical contexts in which formalization yields constructions which arithmetically speaking are nonsense, but can be used in calculations. When formal reasoning is carried out mechanically as if these symbols have meaning, if the ordinary rules are observed, and the results do not contain any imaginary components, these symbols might be legitimately used. And this could be empirically verified (Philosophy of Arithmetic_ Psychological and Logical Investigations with Supplementary Texts).

In a letter to Carl Stumpf in the early 1890s, Husserl explained how, in trying to understand how operating with contradictory concepts could lead to correct theorems, he had found that for imaginary numbers like √2 and √-1, it was not a matter of the possibility or impossibility of concepts. Through the calculation itself and its rules, as defined for those fictive numbers, the impossible fell away, and a genuine equation remained. One could calculate again with the same signs, but referring to valid concepts, and the result was again correct. Even if one mistakenly imagined that what was contradictory existed, or held the most absurd theories about the content of the corresponding concepts of number, the calculation remained correct if it followed the rules. He concluded that this must be a result of the signs and their rules (Early Writings in the Philosophy of Logic and Mathematics). The fact that one can generalize, produce variations of formal arithmetic that lead outside the quantitative domain without essentially altering formal arithmetic’s theoretical nature and calculational methods brought Husserl to realize that there was more to the mathematical or formal sciences, or the mathematical method of calculation than could be captured in purely quantitative analyses.

Understanding the nature of theory forms, shows how reference to impossible objects can be justified. According to his theory of manifolds, one could operate freely within a manifold with imaginary concepts and be sure that what one deduced was correct when the axiomatic system completely and unequivocally determined the body of all the configurations possible in a domain by a purely analytical procedure. It was the completeness of the axiomatic system that gave one the right to operate in that free way. A domain was complete when each grammatically constructed proposition exclusively using the language of the domain was determined from the outset to be true or false in virtue of the axioms, i.e., necessarily followed from the axioms or did not. In that case, calculating with expressions without reference could never lead to contradictions. Complete manifolds have the

distinctive feature that a finite number of concepts and propositions – to be drawn as occasion requires from the essential nature of the domain under consideration –  determines completely and unambiguously on the lines of pure logical necessity the totality of all possible formations in the domain, so that in principle, therefore, nothing further remains open within it.

In such complete manifolds, he stressed, “the concepts true and formal implication of the axioms are equivalent (Ideas).

Husserl pointed out that there may be two valid discipline forms that stand in relation to one another in such a way that the axiom system of one may be a formal limitation of that of the other. It is then clear that everything deducible in the narrower axiom system is included in what is deducible in the expanded system, he explained. In the arithmetic of cardinal numbers, Husserl explained, there are no negative numbers, for the meaning of the axioms is so restrictive as to make subtracting 4 from 3 nonsense. Fractions are meaningless there. So are irrational numbers, √–1, and so on. Yet in practice, all the calculations of the arithmetic of cardinal numbers can be carried out as if the rules governing the operations are unrestrictedly valid and meaningful. One can disregard the limitations imposed in a narrower domain of deduction and act as if the axiom system were a more extended one. We cannot arbitrarily expand the concept of cardinal number, Husserl reasoned. But we can abandon it and define a new, pure formal concept of positive whole number with the formal system of definitions and operations valid for cardinal numbers. And, as set out in our definition, this formal concept of positive numbers can be expanded by new definitions while remaining free of contradiction. Fractions do not acquire any genuine meaning through our holding onto the concept of cardinal number and assuming that units are divisible, he theorized, but rather through our abandonment of the concept of cardinal number and our reliance on a new concept, that of divisible quantities. That leads to a system that partially coincides with that of cardinal numbers, but part of which is larger, meaning that it includes additional basic elements and axioms. And so in this way, with each new quantity, one also changes arithmetics. The different arithmetics do not have parts in common. They have totally different domains, but an analogous structure. They have forms of operation that are in part alike, but different concepts of operation.

For Husserl, formal constraints banning meaningless expressions, meaningless imaginary concepts, reference to non-existent and impossible objects restrict us in our theoretical, deductive work, but that resorting to the infinity of pure forms and transformations of forms frees us from such conditions and explains why having used imaginaries, what is meaningless, must lead, not to meaningless, but to true results.

The Second Trichotomy. Thought of the Day 120.0

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The second trichotomy (here is the first) is probably the most well-known piece of Peirce’s semiotics: it distinguishes three possible relations between the sign and its (dynamical) object. This relation may be motivated by similarity, by actual connection, or by general habit – giving rise to the sign classes icon, index, and symbol, respectively.

According to the second trichotomy, a Sign may be termed an Icon, an Index, or a Symbol.

An Icon is a sign which refers to the Object that it denotes merely by virtue of characters of its own, and which it possesses, just the same, whether any such Object actually exists or not. It is true that unless there really is such an Object, the Icon does not act as a sign; but this has nothing to do with its character as a sign. Anything whatever, be it quality, existent individual, or law, is an Icon of anything, in so far as it is like that thing and used as a sign of it.

An Index is a sign which refers to the Object that it denotes by virtue of being really affected by that Object. It cannot, therefore, be a Qualisign, because qualities are whatever they are independently of anything else. In so far as the Index is affected by the Object, it necessarily has some Quality in common with the Object, and it is in respect to these that it refers to the Object. It does, therefore, involve a sort of Icon, although an Icon of a peculiar kind; and it is not the mere resemblance of its Object, even in these respects which makes it a sign, but it is the actual modification of it by the Object. 

A Symbol is a sign which refers to the Object that it denotes by virtue of a law, usually an association of general ideas, which operates to cause the Symbol to be interpreted as referring to that Object. It is thus itself a general type or law, that is, a Legisign. As such it acts through a Replica. Not only is it general in itself, but the Object to which it refers is of general nature. Now that which is general has its being in the instances it will determine. There must, therefore, be existent instances of what the Symbol denotes, although we must here understand by ‘existent’, existent in the possibly imaginary universe to which the Symbol refers. The Symbol will indirectly, through the association or other law, be affected by those instances; and thus the Symbol will involve a sort of Index, although an Index of a peculiar kind. It will not, however, be by any means true that the slight effect upon the Symbol of those instances accounts for the significant character of the Symbol.

The icon refers to its object solely by means of its own properties. This implies that an icon potentially refers to an indefinite class of objects, namely all those objects which have, in some respect, a relation of similarity to it. In recent semiotics, it has often been remarked by someone like Nelson Goodman that any phenomenon can be said to be like any other phenomenon in some respect, if the criterion of similarity is chosen sufficiently general, just like the establishment of any convention immediately implies a similarity relation. If Nelson Goodman picks out two otherwise very different objects, then they are immediately similar to the extent that they now have the same relation to Nelson Goodman. Goodman and others have for this reason deemed the similarity relation insignificant – and consequently put the whole burden of semiotics on the shoulders of conventional signs only. But the counterargument against this rejection of the relevance of the icon lies close at hand. Given a tertium comparationis, a measuring stick, it is no longer possible to make anything be like anything else. This lies in Peirce’s observation that ‘It is true that unless there really is such an Object, the Icon does not act as a sign ’ The icon only functions as a sign to the extent that it is, in fact, used to refer to some object – and when it does that, some criterion for similarity, a measuring stick (or, at least, a delimited bundle of possible measuring sticks) are given in and with the comparison. In the quote just given, it is of course the immediate object Peirce refers to – it is no claim that there should in fact exist such an object as the icon refers to. Goodman and others are of course right in claiming that as ‘Anything whatever ( ) is an Icon of anything ’, then the universe is pervaded by a continuum of possible similarity relations back and forth, but as soon as some phenomenon is in fact used as an icon for an object, then a specific bundle of similarity relations are picked out: ‘ in so far as it is like that thing.’

Just like the qualisign, the icon is a limit category. ‘A possibility alone is an Icon purely by virtue of its quality; and its object can only be a Firstness.’ (Charles S. PeirceThe Essential Peirce_ Selected Philosophical Writings). Strictly speaking, a pure icon may only refer one possible Firstness to another. The pure icon would be an identity relation between possibilities. Consequently, the icon must, as soon as it functions as a sign, be more than iconic. The icon is typically an aspect of a more complicated sign, even if very often a most important aspect, because providing the predicative aspect of that sign. This Peirce records by his notion of ‘hypoicon’: ‘But a sign may be iconic, that is, may represent its object mainly by its similarity, no matter what its mode of being. If a substantive is wanted, an iconic representamen may be termed a hypoicon’. Hypoicons are signs which to a large extent makes use of iconical means as meaning-givers: images, paintings, photos, diagrams, etc. But the iconic meaning realized in hypoicons have an immensely fundamental role in Peirce’s semiotics. As icons are the only signs that look-like, then they are at the same time the only signs realizing meaning. Thus any higher sign, index and symbol alike, must contain, or, by association or inference terminate in, an icon. If a symbol can not give an iconic interpretant as a result, it is empty. In that respect, Peirce’s doctrine parallels that of Husserl where merely signitive acts require fulfillment by intuitive (‘anschauliche’) acts. This is actually Peirce’s continuation of Kant’s famous claim that intuitions without concepts are blind, while concepts without intuitions are empty. When Peirce observes that ‘With the exception of knowledge, in the present instant, of the contents of consciousness in that instant (the existence of which knowledge is open to doubt) all our thought and knowledge is by signs’ (Letters to Lady Welby), then these signs necessarily involve iconic components. Peirce has often been attacked for his tendency towards a pan-semiotism which lets all mental and physical processes take place via signs – in the quote just given, he, analogous to Husserl, claims there must be a basic evidence anterior to the sign – just like Husserl this evidence before the sign must be based on a ‘metaphysics of presence’ – the ‘present instant’ provides what is not yet mediated by signs. But icons provide the connection of signs, logic and science to this foundation for Peirce’s phenomenology: the icon is the only sign providing evidence (Charles S. Peirce The New Elements of Mathematics Vol. 4). The icon is, through its timeless similarity, apt to communicate aspects of an experience ‘in the present instant’. Thus, the typical index contains an icon (more or less elaborated, it is true): any symbol intends an iconic interpretant. Continuity is at stake in relation to the icon to the extent that the icon, while not in itself general, is the bearer of a potential generality. The infinitesimal generality is decisive for the higher sign types’ possibility to give rise to thought: the symbol thus contains a bundle of general icons defining its meaning. A special icon providing the condition of possibility for general and rigorous thought is, of course, the diagram.

The index connects the sign directly with its object via connection in space and time; as an actual sign connected to its object, the index is turned towards the past: the action which has left the index as a mark must be located in time earlier than the sign, so that the index presupposes, at least, the continuity of time and space without which an index might occur spontaneously and without any connection to a preceding action. Maybe surprisingly, in the Peircean doctrine, the index falls in two subtypes: designators vs. reagents. Reagents are the simplest – here the sign is caused by its object in one way or another. Designators, on the other hand, are more complex: the index finger as pointing to an object or the demonstrative pronoun as the subject of a proposition are prototypical examples. Here, the index presupposes an intention – the will to point out the object for some receiver. Designators, it must be argued, presuppose reagents: it is only possible to designate an object if you have already been in reagent contact (simulated or not) with it (this forming the rational kernel of causal reference theories of meaning). The closer determination of the object of an index, however, invariably involves selection on the background of continuities.

On the level of the symbol, continuity and generality play a main role – as always when approaching issues defined by Thirdness. The symbol is, in itself a legisign, that is, it is a general object which exists only due to its actual instantiations. The symbol itself is a real and general recipe for the production of similar instantiations in the future. But apart from thus being a legisign, it is connected to its object thanks to a habit, or regularity. Sometimes, this is taken to mean ‘due to a convention’ – in an attempt to distinguish conventional as opposed to motivated sign types. This, however, rests on a misunderstanding of Peirce’s doctrine in which the trichotomies record aspects of sign, not mutually exclusive, independent classes of signs: symbols and icons do not form opposed, autonomous sign classes; rather, the content of the symbol is constructed from indices and general icons. The habit realized by a symbol connects it, as a legisign, to an object which is also general – an object which just like the symbol itself exists in instantiations, be they real or imagined. The symbol is thus a connection between two general objects, each of them being actualized through replicas, tokens – a connection between two continua, that is:

Definition 1. Any Blank is a symbol which could not be vaguer than it is (although it may be so connected with a definite symbol as to form with it, a part of another partially definite symbol), yet which has a purpose.

Axiom 1. It is the nature of every symbol to blank in part. [ ]

Definition 2. Any Sheet would be that element of an entire symbol which is the subject of whatever definiteness it may have, and any such element of an entire symbol would be a Sheet. (‘Sketch of Dichotomic Mathematics’ (The New Elements of Mathematics Vol. 4 Mathematical Philosophy)

The symbol’s generality can be described as it having always blanks having the character of being indefinite parts of its continuous sheet. Thus, the continuity of its blank parts is what grants its generality. The symbol determines its object according to some rule, granting the object satisfies that rule – but leaving the object indeterminate in all other respects. It is tempting to take the typical symbol to be a word, but it should rather be taken as the argument – the predicate and the proposition being degenerate versions of arguments with further continuous blanks inserted by erasure, so to speak, forming the third trichotomy of term, proposition, argument.

Utopia Banished. Thought of the Day 103.0

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

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

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

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

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

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

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

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

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

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

Desire of the Pervert. Thought of the Day 102.0

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The subject’s lack is the cynosure of the analytic process. The psychoanalytic discourse places the object a, the marker of lack, in the dominant position. The analyst embroiders the transferential relationship with the analysand by centralizing the constitutive lack of the object as a precondition for desire, which brings the subject to the locus of the Other. As well as lack, the specular image that takes over it and marks its boundaries, that is, the ego, is the other focal point of analysis. The image has its borders; this is the frame of the mirror. Around the limits of the image is where anxiety will make its appearance as what signals the momentary disruption of all points of identification. The limits of the mirror are symbolized by Lacan’s “little diamond” (<>), the sign which indicates the relation between the subject and the object in the matheme of fantasy ($<>a). This relation is mediated by desire. The role of the specular image, functioning as a sort of filter, is to protect the subject from anxiety by covering lack, but also marking it. The reflection in the mirror functions like a window frame that demarcates the illusory world of recognition (imaginary) from what Lacan calls “stage” (symbolic reality). In this stage, we find the desire of the masochist and the sadist. The extra-ordinary and the ordinary subject stage their desire in the same arena, playing the same part, with diametrically different techniques.

The scenarios of “perverse” desire do not just linger in a fantasmatic frame (as happens with neurosis); the extra-ordinary cross the window, taking fantasy on stage, that is, acting it out in the symbolic. The vacillation between desire and jouissance is absent from the extra-ordinary, because he is certain about what he wants. Contrary to the neurotic, whose desire always remains in doubt (this is the desire of the Other), the pervert does not have the doubt, but the knowledge of what he desires. The enduring question of “what the Other wants from me” is absent; the “pervert” takes the game in his hands, he knows and applies the rules. The desire of the “pervert” is to be passively enjoyed by the Other, as it is best manifested in masochism. Lacan notes that the masochist is supposed to know how to enjoy the Other. The masochist is the one who gives the orders, the commands, the knowledge, to the Other, who has to tackle its limits. The masochist is aiming at the jouissance of the Other . . . the final term he is aiming at is anxiety of the Other.

Perverse Ideologies. Thought of the Day 100.0

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Žižek (Fantasy as a Political Category A Lacanian Approach) says,

What we are thus arguing is not simply that ideology permeates also the alleged extra-ideological strata of everyday life, but that this materialization of ideology in the external materiality renders visible inherent antagonisms that the explicit formulation of ideology cannot afford to acknowledge. It is as if an ideological edifice, in order to function “normally,” must obey a kind of “imp of perversity” and articulate its inherent antagonism in the externality of its material existence.

In this fashion, Žižek recognizes an element of perversity in all ideologies, as a prerequisite for their “normal” functioning. This is because all ideologies disguise lack and thus desire through disavowal. They know that lack is there, but at the same time they believe it is eliminated. There is an object that takes over lack, that is to say the Good each ideology endorses, through imaginary means. If we generalize Žižek’s suggestion, we can either see all ideological relations mediated by a perverse liaison or perversion as a condition that simply helps the subjects relate to each other, when signification fails and they are confronted with the everlasting question of sexual difference, the non-representable dimension. Ideology, then, is just one solution that makes use of the perverse strategy when dealing with Difference. In any case, it is not pathological and cannot be determined mainly by relying on the role of disavowal. Instead of père-vers (this is a Lacanian neologism that denotes the meanings of “perversion” and “vers le père”, referring to the search for jouissance that does not abolish the division of the subject, her desire. In this respect, the père-vers is typical of both neurosis and perversion, where the Name-of-the-Father is not foreclosed and thereby complete jouissance remains unobtainable sexuality, that searches not for absolute jouissance, but jouissance related to desire, the political question is more pertinent to the père-versus, so to say, anything that goes against the recognition of the desire of the Other. Any attempt to disguise lack for instrumental purposes is a père-versus tactic.

To the extent that this external materialization of ideology is subjected to fantasmatic processes, it divulges nothing more than the perversity that organizes all social and political relations far from the sexual pathology associated with the pervert. The Other of power, this fictional Other that any ideology fabricates, is the One who disavows the discontinuities of the normative chain of society. Expressed through the signifiers used by leadership, this Other knows very well the cul-de-sac of the fictional view of society as a unified body, but still believes that unity is possible, substantiating this ideal.

The ideological Other disregards the impossibility of bridging Difference; therefore, it meets the perversion that it wants to associate with the extra-ordinary. Disengaging it from pathology, disavowal can be stated differently, as a prompt that says: “let’s pretend!” Pretend as if a universal harmony, good, and unity are feasible. Symbolic Difference is replaced with imaginary difference, which nourishes antagonism and hostility by fictionalizing an external threat that jeopardizes the unity of the social body. Thus, fantasy of the obscene extra-ordinary, who offends the conformist norm, is in itself a perverse fantasy. The Other knows very well that the pervert constitutes no threat, but still requires his punishment, moral reformation, or treatment.

Žižek’s Dialectical Coincidentia Oppositorium. Thought of the 98.0

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Without doubt, the cogent interlacing of Lacanian theorization with Hegelianism manifests Žižek’s prowess in articulating a highly pertinent critique of ideology for our epoch, but whether this comes from a position of Marxist orthodoxy or a position of a Lacanian doctrinaire who monitors Marxist politics is an open question.

Through this Lacanian prism, Žižek sees subjectivity as fragmented and decentred, considering its subordinate status to the unsurpassable realm of the signifiers. The acquisition of a consummate identity dwells in impossibility, in as much as it is bound to desire, provoked by a lacuna which is impossible to fill up. Thus, for Žižek, socio-political relations evolve from states of lack, linguistic fluidity, and contingency. What temporarily arrests this fluid state of the subject’s slithering in the realm of the signifiers, giving rise to her self-identity, is what Lacan calls point de capiton. The term refers to certain fundamental “anchoring” points in the signifying chain where the signifier is tied to the signified, providing an illusionary stability in signification. Laclau and Mouffe (Hegemony and Socialist Strategy Towards a Radical Democratic Politics) were the first to make use of the idea of the point de capiton in relation to hegemony and the formation of identities. In this context, ideology is conceptualized as a terrain of firm meanings, determined and comprised by numerous points de capiton (Zizek The Sublime Object of Ideology).

The real is the central Lacanian concept that Žižek implements in his rhetoric. He associates the real with antagonism (e.g., class conflict) as the unsymbolizable and irreducible gap that lies in the heart of the socio-symbolic order and around which society is formed. As Žižek argues, “class struggle designates the very antagonism that prevents the objective (social) reality from constituting itself as a self-enclosed whole” (Renata Salecl, Slavoj Zizek-Gaze and Voice As Love Objects). This logic is indebted to Laclau and Mouffe, who were the first to postulate that social antagonism is what impedes the closure of society, marking thus its impossibility. Žižek expanded this view and associated antagonism with the notion of the real.

Functioning as a hegemonic fantasmatic veil, ideology covers the lacuna of the symbolic, in the form of a fantasy, so that it protracts desire and hence subjectivity. On the imaginary level, ideology functions as the “mirror” that reflects antagonisms, that is to say, the real unrepresentable kernel that undermines the political. Around this emptiness of representation, the fictional narrative of ideology, its meaning, is to unfurl. The role of socio-ideological fantasy is to provide consistency to the symbolic order by veiling its void, and to foster the illusion of a coherent social unity.

Nevertheless, fantasy has both unifying and disjunctive features, as its role is to fill the void of the symbolic, but also to circumscribe this void. According to Žižek, “the notion of fantasy offers an exemplary case of the dialectical coincidentia oppositorium”. On the one side, it provides a “hallucinatory realisation of desire” and on the other side, it evokes disturbing images about the Other’s jouissance to which the subject has no (symbolic or imaginary) access. In so reasoning, ideology promises unity and, at the same time, creates another fantasy, where the failure of acquiring the anticipated ideological unity is ascribed.

Pertaining to Jacques Derrida’s work Specters of Marx (Specters of Marx The State of the Debt, The Work of Mourning; the New International), where the typical ontological conception of the living is seen to be incomplete and inseparable from the spectre, namely, a ghostly embodiment that haunts the living present (Derrida introduces the notion of hauntology to refer to this pseudo-material incarnation of the spirit that haunts and challenges ontological present), Žižek elaborates the spectral apparitions of the real in the politico–ideological domain. He makes a distinction between this “spectre” and “symbolic fiction”, that is, reality per se. Both have a common fantasmatic hypostasis, yet they perform antithetical functions. Symbolic fiction forecloses the real antagonism at the crux of reality, only to return as a spectre, as another fantasy.

Prisoner’s Dilemma. Thought of the Day 64.0

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A system suffering from Prisoner’s Dilemma cannot find the optimal solution because the individual driving forces go against the overall driving force. This is called Prisoner’s Dilemma based on the imaginary situation of two prisoners:

Imagine two criminals, named alphabetically A and B, being caught and put in separate prison cells. The police is trying to get confessions out of them. They know that if none will talk, they will both walk out of there for lack of evidence. So the police makes a proposal to each one: “We’ll make it worth your while. If you confess, and your colleague not, we give you 10 thousand euro and your colleague will get 50 years in prison. If you both confess you will each get 20 years in prison”. The decision table for these prisoners is like this:

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As you can see for yourself, the individual option for A, independent of what B decides to do, is confessing; moving from right column to left column, it is either reducing his sentence from 50 to 20 years, or instead of walking out of there even getting a fat bonus on top. The same applies to B, moving from bottom row to top row of the table. So, they wind up both confessing and getting 20 years in prison. That while it is obvious that the optimal situation is both not talking and walking out of prison scot-free (with the loot!). Because A and B cannot come to an agreement, but both optimize their own personal yield instead, they both get severely punished!

The Prisoner’s Dilemma applies to economy. If people in society cannot come to an agreement, but instead let everybody take decisions to optimize the situation for themselves (as in liberalism), they wind up with a non-optimal situation in which all the wealth is condensed on a single entity. This does not even have to be a person, but the capital itself. Nobody will get anything, beyond the alms granted by the system. In fact, the system will tend to reduce these alms – the minimum wages, or unemployment benefit – and will have all kinds of dogmatic justifications for them, but basically is a strategy of divide-and-conquer, inhibiting people to come to agreements, for instance by breaking the trade unions.

An example of a dogmatic reason is “lowering wages will make that more people get hired for work”. Lowering wages will make the distortion more severe. Nothing more. Moreover, as we have seen, work can be done without human labor. So if it is about competition, men will be cut out of the deal sooner or later. It is not about production. It is about who gets the rights to the consumption of the goods produced. That is also why it is important that people should unite, to come to an agreement where everybody benefits. Up to and including the richest of them all! It is better to have 1% of 1 million than 100% of 1 thousand. Imagine this final situation: All property in the world belongs to the final pan-global bank, with their headquarters in an offshore or fiscal paradise. They do not pay tax. The salaries (even of the bank managers) are minimal. So small that it is indeed not even worth it to call them salary.

Noneism. Part 1.

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Noneism was created by Richard Routley. Its point of departure is the rejection of what Routley calls “The Ontological Assumption”. This assumption consists in the explicit or, more frequently, implicit belief that denoting always refers to existing objects. If the object, or objects, on which a proposition is about, do not exist, then these objects can only be one: the null entity. It is incredible that Frege believed that denoting descriptions without a real (empirical, theoretical, or ideal) referent denoted only the null set. And it is also difficult to believe that Russell sustained the thesis that non-existing objects cannot have properties and that propositions about these objects are false.

This means that we can have a very clear apprehension of imaginary objects, and quite clear intellection of abstract objects that are not real. This is possible because to determine an object we only need to describe it through its distinctive traits. This description is possible because an object is always chacterized through some definite notes. The amount of traits necessary to identify an object greatly varies. In some cases we need only a few, for instance, the golden mountain, or the blue bird; in other cases we need more, for instance, the goddess Venus or the centaur Chiron. In other instances the traits can be very numerous, even infinite. For instance the chiliedron, and the decimal number 0,0000…009, in which 9 comes after the first million zeros, have many traits. And the ordinal omega or any Hilbert space have infinite traits (although these traits can be reckoned through finite definitions). These examples show, in a convincing manner, that the Ontological Assumption is untenable. We must reject it and replace it with what Routley dubbs the Characterization Postulate. The Characterization Postulate says that, to be an object means to be characterized by determined traits. The set of the characterizing traits of an object can be called its “characteristic”. When the characteristic of an object is set up, the object is perfectly recognizable.

Once this postulate is adopted, its consequences are far reaching. Since we can characterize objects through any traits whatsoever, an object can not only be inexistent, it can even be absurd or inconsistent. For instance, the “squond” (the circle that is square and round). And we can make perfectly valid logical inferences from the premiss: x is the sqound:

(1) if x is the squond, then x is square
(2) if x is the squond, then x is round

So, the theory of objects has the widest realm of application. It is clear that the Ontological Assumption imposes unacceptable limits to logic. As a matter of fact, the existential quantifier of classical logic could not have been conceived without the Ontological Assumption. The expression “(∃x)Fx” means that there exists at least an object that has the property F (or, in extensional language, that there exists an x that is a member of the extension of F). For this reason, “∃x” is unappliable to non existing objects. Of course, in classical logic we can deny the existence of an Object, but we cannot say anything about Objects that have never existed and shall never exist (we are strictly speaking about classical logic). We cannot quantify individual variables of a first order predicate that do not refer to a real, actual, past or future entity. For instance, we cannot say “(∃x) (x is the eye of Polyphemus)”. This would be false, of course, because Polyphemus does not exist. But if the Ontological Assumption is set aside, it is true, within a mythological frame, that Polyphemus has a single eye and many other properties. And now we can understand why noneism leads to logical material-dependence.

As we have anticipated, there must be some limitations concerning the selection of the contradictory properties; otherwise the whole theory becomes inconsistent and is trivialized. To avoid trivialization neutral (noneist) logic distinguishes between two sorts of negation: the classical propositional negation: “8 is not P”, and the narrower negation: “8 is non-P”. In this way, and by applying some other technicalities (for instance, in case an universe is inconsistent, some kind of paraconsistent logic must be used) trivialization is avoided. With the former provisions, the Characterization Postulate can be applied to create inconsistent universes in which classical logic is not valid. For instance, a world in which there is a mysterious personage, that within determined but very subtle circumstances, is and is not at the same time in two different places. In this case the logic to be applied is, obviously, some kind of paraconsistent logic (the type to be selected depends on the characteristic of the personage). And in another universe there could be a jewel which has two false properties: it is false that it is transparent and it is false that it is opaque. In this kind of world we must use, clearly, some kind of paracomplete logic. To develop naive set theory (in Halmos sense), we must use some type of paraconsistent logic to cope with the paradoxes, that are produced through a natural way of mathematical reasoning; this logic can be of several orders, just like the classical. In other cases, we can use some kind of relevant and, a fortiori, paraconsistent logic; and so on, ad infinitum.

But if logic is content-dependent, and this dependence is a consequence of the Ontological Assumption’s rejection, what about ontology? Because the universes determined through the application of the Characterization Postulate may have no being (in fact, most of them do not), we cannot say that the objects that populate such universes are entities, because entities exist in the empirical world, or in the real world that underpins the phenomena, or (in a somewhat different way), in an ideal Platonic world. Instead of speaking about ontology, we should speak about objectology. In essence objectology is the discipline founded by Meinong (Theory of Objects), but enriched and made more precise by Routley and other noneist logicians. Its main division would be Ontology (the study of real physical and Platonic objects) and Medenology (the study of objects that have no existence).

Conjuncted: Gauge Theory

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Weyl introduced as a phase factor an exponential in which the phase α is preceded by the imaginary unit i, e.g., e+iqα(x), in the wave function for the wave equations (for instance, the Dirac equation is (iγμμ − m)ψ = 0). It is here that Weyl correctly formulated gauge theory as a symmetry principle from which electromagnetism could be derived. It had been shown that for a quantum theory of charged particles interacting with the electromagnetic field, invariance under a gauge transformation of the potentials required multiplication of the wave function by the now well-know phase factor. Yang cited Weyl’s gauge theory results as reported by Pauli as a source for Yang-Mills gauge theory; although Yang didn’t find out until much later that these were Weyl’s results. Moreover, Pauli did not explicitly mention Weyl’s geometric interpretation. It was only much after Yang and Mills published their article that Yang realized the connection between their work and geometry. Yang says