Why Can’t There Be Infinite Descending Chain Of Quotient Representations? – Part 3



For a quiver Q, the category Rep(Q) of finite-dimensional representations of Q is abelian. A morphism f : V → W in the category Rep(Q) defined by a collection of morphisms fi : Vi → Wi is injective (respectively surjective, an isomorphism) precisely if each of the linear maps fi is.

There is a collection of simple objects in Rep(Q). Indeed, each vertex i ∈ Q0 determines a simple object Si of Rep(Q), the unique representation of Q up to isomorphism for which dim(Vj) = δij. If Q has no directed cycles, then these so-called vertex simples are the only simple objects of Rep(Q), but this is not the case in general.

If Q is a quiver, then the category Rep(Q) has finite length.

Given a representation E of a quiver Q, then either E is simple, or there is a nontrivial short exact sequence

0 → A → E → B → 0

Now if B is not simple, then we can break it up into pieces. This process must halt, as every representation of Q consists of finite-dimensional vector spaces. In the end, we will have found a simple object S and a surjection f : E → S. Take E1 ⊂ E to be the kernel of f and repeat the argument with E1. In this way we get a filtration

… ⊂ E3 ⊂ E2 ⊂ E1 ⊂ E

with each quotient object Ei−1/Ei simple. Once again, this filtration cannot continue indefinitely, so after a finite number of steps we get En = 0. Renumbering by setting Ei := En−i for 1 ≤ i ≤ n gives a Jordan-Hölder filtration for E. The basic reason for finiteness is the assumption that all representations of Q are finite-dimensional. This means that there can be no infinite descending chains of subrepresentations or quotient representations, since a proper subrepresentation or quotient representation has strictly smaller dimension.

In many geometric and algebraic contexts, what is of interest in representations of a quiver Q are morphisms associated to the arrows that satisfy certain relations. Formally, a quiver with relations (Q, R) is a quiver Q together with a set R = {ri} of elements of its path algebra, where each ri is contained in the subspace A(Q)aibi of A(Q) spanned by all paths p starting at vertex aiand finishing at vertex bi. Elements of R are called relations. A representation of (Q, R) is a representation of Q, where additionally each relation ri is satisfied in the sense that the corresponding linear combination of homomorphisms from Vai to Vbi is zero. Representations of (Q, R) form an abelian category Rep(Q, R).

A special class of relations on quivers comes from the following construction, inspired by the physics of supersymmetric gauge theories. Given a quiver Q, the path algebra A(Q) is non-commutative in all but the simplest examples, and hence the sub-vector space [A(Q), A(Q)] generated by all commutators is non-trivial. The vector space quotientA(Q)/[A(Q), A(Q)] is seen to have a basis consisting of the cyclic paths anan−1 · · · a1 of Q, formed by composable arrows ai of Q with h(an) = t(a1), up to cyclic permutation of such paths. By definition, a superpotential for the quiver Q is an element W ∈ A(Q)/[A(Q), A(Q)] of this vector space, a linear combination of cyclic paths up to cyclic permutation.

The Second Trichotomy. Thought of the Day 120.0


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.

Surplus of Jouissance Framing the Feminine and the Pervert. Drunken Risibility.


The feminine position towards jouissance also moves beyond the phallic signifier. The woman does not come under the auspice of the paternal constraint of the phallic word, going as far as to sacrifice herself to unlimited jouissance suggesting a thorny ostracism of the paternal despot in his barbarous Will-to-Jouissance. Confronting the risk of turning these two parallel positions into a hazardous equation is locating the difference in the woman’s efforts to deviate from the function of the phallic signifier, where the woman still tries to relate her jouissance to the signifier as she tries to talk about it. This means that she does not disavow the phallic signifier as the pervert does, which explains why she is not placed completely outside the phallic function, on the side of unlimited fatal jouissance, something that would turn her into a callous figure.

Sade occupies the perverse frame in terms of jouissance, which is different from feminine jouissance. Although the woman slips away from the phallic function, she still tries to discover channels for relating her jouissance to the symbolic and manage to speak about it. The woman is not fully inscribed in the symbolic, for their structures are marked by a nucleus that persists and goes beyond symbolic boundaries: this is the object a, the remainder of lost jouissance. The pervert situates himself in the position of the object of the drive, whereas the woman tries to pertain not to this object, but its lack, namely the phallus, without fully succeeding in this. There is a surplus enjoyment in both positions pointing towards the new possibilities that the feminine position opens for ethics.

However, even if the woman tries to fasten her jouissance to the phallic function, unlike the pervert, it is precisely this surplus of jouissance that frames both the feminine and the perverse position. Moreover, given that lack and excess are tautological notions for Lacan, in what way did a pervert embody the lack in the drive and how is it different from embodying the excess of the feminine? Despite efforts to separate the two, one thing remains: both the pervert and the woman bear upon a jouissance beyond the limits of the symbolic, where common moral designations become impaired.

Utopia Banished. Thought of the Day 103.0


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”.

Why the Political needs the Pervert? Thought of the Day 102.1


Thus perverts’ desire does not have the opportunity to be organized around finding a fantasmatic solution to the real of sexual difference. The classical scenario of Oedipal dynamics, with its share of lies, make believe, and sexual theories, is not accessible to them. This is why they will search desperately to access symbolic castration that could bring solace to their misery. — Judith Feher-Gurewich (Jean-Michel Rabaté – The Cambridge Companion to Lacan)

Nonetheless, it is contradictory to see the extra-ordinary’s goal as the reinsertion of castration, when in fact there is nothing in his perverse scenarios that incarcerates him in misery. It is more a fantasmatic solution to the deciphering of the enigma of sexual difference, precisely by veiling difference. The extra-ordinary wishes to maintain this veiling, in as much as his jouissance is derived this way. Even if the extra-ordinary efforts to infinitize jouissance are eventually sealed by castration, this is more a side effect of the “perverse” act. At the end, desire always reinscribes itself. Symbolic guilt is inserted in the extra-ordinary’s world through castration, not because the latter relieves him, but because his fantasy has failed. This failure is what creates the misery of the pervert, as in any other subject.

His main target is centred in filling the Other with jouissance. However, it is not something he produces, but more something he unlocks. The pervert unleashes a jouissance, already present in the Other, by eradicating the primacy of the phallic signifier and revealing the Other’s jouissance (the emptiness, the feminine). The neurotic’s anxiety concerns the preservation of desire through the duplication of castration, whereas the pervert’s anxiety emerges from the reverse condition. This is the question of how to extract jouissance from the object without it falling. He does not want to let the object fall, not for fear of castration, but because of the wish to retain jouissance. Inexorably, the nagging question of how to obstruct desire from returning to its initial place grips the pervert because, together with desire, the lack in the Other returns, restoring and maintaining his desiring status, instead of his enjoying status. Without doubt, these are fantasmatic relations that sustain “perverse” desire for jouissance and, at the same time, impose a safe distance from the horror of the Thing’s return.

Anxiety intervenes as the mediating term between desire and jouissance. The desiring subject seeks jouissance, but not in its pure form. Jouissance has to be related to the Other, to occupy a space within the Other of signification, to be put into words. This is what phallic jouissance, the jouissance of the idiot, aims at. The idiocy of it lies in its vain and limited character, since jouissance always fails signification and only a residue is left behind. The remainder is the object a, which perpetuates the desire of the subject. But the object is desired as absent. Coming too close to it, one finds this absence occupied by a real presence. In that case, the object has to fall, like the phallus in its exhausted stage, in order to maintain the desiring status of the subject. The moment desire returns, the object falls, or, better, the moment the object falls, desire returns.

While the subject is engaged in an impossible task (that of inscribing jouissance in the place of the Other) she draws closer to the object. The closer she gets, the more anxiety surfaces, alerting the subject about the presence of a real Other, a primitive pre- symbolic being. In the case of the pervert, things are somehow different. It is not so much the inscription of jouissance in the Other that troubles him, but more the erasure of desire from the field of the Other and its return to a state of unconstrained enjoyment. So, for the pervert, it is essential that the object maintains its potency, not in the service of desire but in the service of jouissance. The anxiety of the extra-ordinary becomes an erotic signal that calls the Other to abandon the locus of desire and indulge in jouissance. But, eventually, desire puts an end to it.

It is not the extra-ordinary that aims at castration, so that he lets loose some of his anxiety. As an integral part of sexual jouissance, the extra-ordinary does not want to give up anxiety, which is what the neurotic does with his symptom, in the reverse way. The Other’s anxiety, the exposition of its truth, requests the confinement of the jouissance operating in perversion. Castration has to be imposed because of the contaminating nature of the object’s jouissance. The more it maintains its omnipotent character, the more it threatens the Other’s consistency, as provided by desire. The extra-ordinary dramatizes the staging of castration. It is not an actual event, as the phallus does not belong to the order of the cosmic world. None the less, politics and power locate the phallus in the imaginary realm. Emblems of patriarchal power are handed from one authority figure to the next, propelling the replication of the same power mechanism and concealing the absence of the phallus.

The social and the political world needs the “pervert” in order to redefine and reinscribe the imaginary boundaries of its morality and, hence, since the patriarchal orientation of the majority is taken as a gnomon, enhance the existing moral code. This reflects the underlying imaginary dynamics of what social constructionism has long now described: the exception of the pervert makes the rule for the “normality” of the present moral, social, political, and cultural organization of the world. As long as the pervert remains outside of this world, the safety from the perilous obscenity and odiousness of real jouissance is ensured. Concomitantly, this is translated to further distance from desire and its permanent endurance, something that nourishes guilt, as was previously argued. As if guilt suggested a privileged moral state, power uses it as an essential demagogic tool, in order to secure its good and further vilify the “pervert”, who also experiences guilt for “betraying” desire, not in the sense of staying away from jouissance, but failing to fully consummate it.

Desire of the Pervert. Thought of the Day 102.0


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.

Negation. Thought of the Day 99.0


Negation reveals more a neurotic attitude towards jouissance, denounced as a perverse desire, that dominates both political and social life. Negation presupposes the acquisition of the meaning of “No” and it suggests a vigorous and compromising attitude between an idea remaining unconscious (repressed) and conscious at the same time. Thus, to negate means to go against the law and succumb to jouissance in a concealed way. Negating castration releases a destructive force against the paternal function, a force fuelled with jouissance. It is not the symbolic reality, but the non-symbolic real as a threatening source that is being negated. This means that the real is actually expressed through symbolic means, but in a negative form. Disavowal, involves a sexualization of the object precluding the threat of castration as punishment. But the threat is still there in the unconscious, whereas negation means that castration is negated even in the unconscious. Negation does not suggest a compromise (in the form of a splitting of the ego) between the denial of something and its acceptance, as disavowal does. Rather, it maintains the repressed status of castration by allowing the latter to be unconsciously expressed in its negated status. So, negation has a more hostile and aggressive attitude (originating in the death drive) towards castration, whereas disavowal originates in Eros. Disavowal does not go against castration, but keeps it at bay by not acknowledging it, which is different from negating it. In this way, the sexualization of the object (the mother’s phallus) remains intact. Therefore, the responsibility for extracting jouissance is also negated.

Constructivism. Note Quote.


Constructivism, as portrayed by its adherents, “is the idea that we construct our own world rather than it being determined by an outside reality”. Indeed, a common ground among constructivists of different persuasion lies in a commitment to the idea that knowledge is actively built up by the cognizing subject. But, whereas individualistic constructivism (which is most clearly enunciated by radical constructivism) focuses on the biological/psychological mechanisms that lead to knowledge construction, sociological constructivism focuses on the social factors that influence learning.

Let us briefly consider certain fundamental assumptions of individualistic constructivism. The first issue a constructivist theory of cognition ought to elucidate concerns of course the raw materials on which knowledge is constructed. On this issue, von Glaserfeld, an eminent representative of radical constructivism, gives a categorical answer: “from the constructivist point of view, the subject cannot transcend the limits of individual experience” (Michael R. Matthews Constructivism in Science Education_ A Philosophical Examination). This statement presents the keystone of constructivist epistemology, which conclusively asserts that “the only tools available to a ‘knower’ are the senses … [through which] the individual builds a picture of the world”. What is more, the so formed mental pictures do not shape an ‘external’ to the subject world, but the distinct personal reality of each individual. And this of course entails, in its turn, that the responsibility for the gained knowledge lies with the constructor; it cannot be shifted to a pre-existing world. As Ranulph Glanville confesses, “reality is what I sense, as I sense it, when I’m being honest about it” .

In this way, individualistic constructivism estranges the cognizing subject from the external world. Cognition is not considered as aiming at the discovery and investigation of an ‘independent’ world; it is viewed as a ‘tool’ that exclusively serves the adaptation of the subject to the world as it is experienced. From this perspective, ‘knowledge’ acquires an entirely new meaning. In the expression of von Glaserfeld,

the word ‘knowledge’ refers to conceptual structures that epistemic agents, given the range of present experience, within their tradition of thought and language, consider viable….[Furthermore] concepts have to be individually built up by reflective abstraction; and reflective abstraction is not a matter of looking closer but at operating mentally in a way that happens to be compatible with the perceptual material at hand.

To say it briefly, ‘knowledge’ signifies nothing more than an adequate organization of the experiential world, which makes the cognizing subject capable to effectively manipulate its perceptual experience.

It is evident that such insights, precluding any external point of reference, have impacts on knowledge evaluation. Indeed, the ascertainment that “for constructivists there are no structures other than those which the knower forms by its own activity” (Michael R. MatthewsConstructivism in Science Education A Philosophical Examination) yields unavoidably the conclusion drawn by Gerard De Zeeuw that “there is no mind-independent yardstick against which to measure the quality of any solution”. Hence, knowledge claims should not be evaluated by reference to a supposed ‘external’ world, but only by reference to their internal consistency and personal utility. This is precisely the reason that leads von Glaserfeld to suggest the substitution of the notion of “truth” by the notion of “viability” or “functional fit”: knowledge claims are appraised as “true”, if they “functionally fit” into the subject’s experiential world; and to find a “fit” simply means not to notice any discrepancies. This functional adaptation of ‘knowledge’ to experience is what finally secures the intended “viability”.

In accordance with the constructivist view, the notion of ‘object’, far from indicating any kind of ‘existence’, it explicitly refers to a strictly personal construction of the cognizing subject. Specifically, “any item of the furniture of someone’s experiential world can be called an ‘object’” (von Glaserfeld). From this point of view, the supposition that “the objects one has isolated in his experience are identical with those others have formed … is an illusion”. This of course deprives language of any rigorous criterion of objectivity; its physical-object statements, being dependent upon elements that are derived from personal experience, cannot be considered to reveal attributes of the objects as they factually are. Incorporating concepts whose meaning is highly associated with the individual experience of the cognizing subject, these statements form at the end a personal-specific description of the world. Conclusively, for constructivists the term ‘objectivity’ “shows no more than a relative compatibility of concepts” in situations where individuals have had occasion to compare their “individual uses of the particular words”.

From the viewpoint of radical constructivism, science, being a human enterprise, is amenable, by its very nature, to human limitations. It is then naturally inferred on constructivist grounds that “science cannot transcend [just as individuals cannot] the domain of experience” (von Glaserfeld). This statement, indicating that there is no essential differentiation between personal and scientific knowledge, permits, for instance, John Staver to assert that “for constructivists, observations, objects, events, data, laws and theory do not exist independent of observers. The lawful and certain nature of natural phenomena is a property of us, those who describe, not of nature, what is described”. Accordingly, by virtue of the preceding premise, one may argue that “scientific theories are derived from human experience and formulated in terms of human concepts” (von Glaserfeld).

In the framework now of social constructivism, if one accepts that the term ‘knowledge’ means no more than “what is collectively endorsed” (David Bloor Knowledge and Social Imagery), he will probably come to the conclusion that “the natural world has a small or non-existent role in the construction of scientific knowledge” (Collins). Or, in a weaker form, one can postulate that “scientific knowledge is symbolic in nature and socially negotiated. The objects of science are not the phenomena of nature but constructs advanced by the scientific community to interpret nature” (Rosalind Driver et al.). It is worth remarking that both views of constructivism eliminate, or at least downplay, the role of the natural world in the construction of scientific knowledge.

It is evident that the foregoing considerations lead most versions of constructivism to ultimately conclude that the very word ‘existence’ has no meaning in itself. It does acquire meaning only by referring to individuals or human communities. The acknowledgement of this fact renders subsequently the notion of ‘external’ physical reality useless and therefore redundant. As Riegler puts it, within the constructivist framework, “an external reality is neither rejected nor confirmed, it must be irrelevant”.

Hegelian Marxism of Lukács: Philosophy as Systematization of Ideology and Politics as Manipulation of Ideology. Thought of the Day 80.0


In the Hegelian Marxism of Lukács, for instance, the historicist problematic begins from the relativisation of theory, whereby that it is claimed that historical materialism is the “perspective” and “worldview” of the revolutionary class and that, in general, theory (philosophy) is only the coherent systematisation of the ideological worldview of a social group. No distinction of kind exists between theory and ideology, opening the path for the foundational character of ideology, expressed through the Lukácsian claim that the ideological consciousness of a historical subject is the expression of objective relations, and that, correlatively, this historical subject (the proletariat) alienates-expresses a free society by means of a transparent grasp of social processes. The society, as an expression of a single structure of social relations (where the commodity form and reified consciousness are theoretical equivalents) is an expressive totality, so that politics and ideology can be directly deduced from philosophical relations. According to Lukács’ directly Hegelian conception, the historical subject is the unified proletariat, which, as the “creator of the totality of [social] contents”, makes history according to its conception of the world, and thus functions as an identical subject-object of history. The identical subject-object and the transparency of praxis therefore form the telos of the historical process. Lukács reduces the multiplicity of social practices operative within the social formation to the model of an individual “making history,” through the externalisation of an intellectual conception of the world. Lukács therefore arrives at the final element of the historicist problematic, namely, a theorisation of social practice on the model of individual praxis, presented as the historical action of a “collective individual”. This structure of claims is vulnerable to philosophical deconstruction (Gasché) and leads to individualist political conclusions (Althusser).

In the light of the Gramscian provenance of postmarxism, it is important to note that while the explicit target of Althusser’s critique was the Hegelian totality, Althusser is equally critical of the aleatory posture of Gramsci’s “absolute historicism,” regarding it as exemplary of the impasse of radicalised historicism (Reading Capital). Althusser argues that Gramsci preserves the philosophical structure of historicism exemplified by Lukács and so the criticism of “expressive totality,” or spiritual holism, also applies to Gramsci. According to Gramsci, “the philosophy of praxis is absolute ‘historicism,’ the absolute secularisation and earthiness of thought, an absolute humanism of history”. Gramsci’s is an “absolute” historicism because it subjects the “absolute knowledge” supposed to be possible at the Hegelian “end of history” to historicisation-relativisation: instead of absolute knowledge, every truly universal worldview becomes merely the epochal totalisation of the present. Consequently, Gramsci rejects the conception that a social agent might aspire to “absolute knowledge” by adopting the “perspective of totality”. If anything, this exacerbates the problems of historicism by bringing the inherent relativism of the position to the surface. Ideology, conceptualised as the worldview of a historical subject (revolutionary proletariat, hegemonic alliance), forms the foundation of the social field, because in the historicist lens a social system is cemented by the ideology of the dominant group. Philosophy (and by extension, theory) represents only the systematisation of ideology into a coherent doctrine, while politics is based on ideological manipulation as its necessary precondition. Thus, for historicism, every “theoretical” intervention is immediately a political act, and correlatively, theory becomes the direct servant of ideology.



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

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

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

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

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

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

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

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

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

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

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

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

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

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