Badiou’s Vain Platonizing, or How the World is a Topos? Note Quote.

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As regards the ‘logical completeness of the world’, we need to show that Badiou’s world of T-sets does indeed give rise to a topos.

Badiou’s world consisting of T-Sets – in other words pairs (A, Id) where Id : A × A → T satisfies the particular conditions in respect to the complete Heyting algebra structure of T—is ‘logically closed’, that is, it is an elementary topos. It thus encloses not only pull-backs but also the exponential functor. These make it possible for it to internalize a Badiou’s infinity arguments that operate on the power-functor and which can then be expressed from insde the situation despite its existential status.

We need to demonstrate that Badiou’s world is a topos. Rather than beginning from Badiou’s formalism of T -sets, we refer to the standard mathematical literature based on which T-sets can be regarded as sheaves over the particular Grothendieck-topology on the category T: there is a categorical equivalence between T-sets satisfying the ‘postulate of materialism’ and S hvs(T,J). The complications Badiou was caught up with while seeking to ‘Platonize’ the existence of a topos thus largely go in vain. We only need to show that Shvs(T,J) is a topos.

Consider the adjoint sheaf functor that always exists for the category of presheaves

Idα : SetsCop → Shvs(Cop,J)

, where J is the canonical topology. It then amounts to an equivalence of categories. Thus it suffices to replace this category by the one consisting of presheaves SetsTop. This argument works for any category C rather than the specific category related to an external complete Heyting algebra T. In the category of Sets define YX as the set of functions X → Y. Then in the category of presheaves,

SetsCopYX(U) ≅ Hom(hU,YX) ≅ Hom(hU × X,Y)

, where hU is the representable sheaf hU(V) = Hom(V,U). The adjunction on the right side needs to be shown to exist for all sheaves – not just the representable ones. The proof then follows by an argument based on categorically defined limits, which has an existence. It can also be verified directly that the presheaf YX is actually a sheaf. Finally, for the existence of the subobject-classifier ΩSetsCop, it can be defined as

ΩSetsCop(U) ≅ Hom(hU,Ω) ≅ {sub-presheaves of hU} ≅ {sieves on U}, or alternatively, for the category of proper sheaves Shvs(C,J), as

ΩShvs(C,J)(U) = {closed sieves on U}

Here it is worth reminding ourselves that the topology on T is defined by a basis K(p) = {Θ ⊂ T | ΣΘ = p}. Therefore, in the case of T-sets satisfying the strong ‘postulate of materialism’, Ω(p) consists of all sieves S (downward dense subsets) of T bounded by relation ΣS ≤ p. These sieves are further required to be closed. A sieve S with an envelope ΣS = s is closed if for any other r ≤ s, ie. for all r ≤ s, one has the implication

frs(S) ∈ J(r) ⇒ frs ∈ S

, where frs : r → s is the unique arrow in the poset category. In particular, since ΣS = s for the topology whose basis consists of territories on s, we have the equation 1s(S) = fss(S) = S ∈ J(s). Now the condition that the sieve is closed implies 1s ∈ S. This is only possible when S is the maximal sieve on s—namely it consists of all arrows r → s for r ≤ s. In such a case S itself is closed. Therefore, in this particular case

Ω(p)={↓(s)|s ≤ p} = {hs | s ≤ p}

This is indeed a sheaf whose all amalgamations are ‘real’ in the sense of Badiou’s postulate of materialism. Thus it retains a suitable T-structure. Let us assume now that we are given an object A, which is basically a functor and thus a T-graded family of subsets A(p). For there to exist a sub-functor B ֒→ A comes down to stating that B(p) ⊂ A(p) for each p ∈ T. For each q ≤ p, we also have an injection B(q) ֒→ B(p) compatible (through the subset-representation with respect to A) with the injections A(q) ֒→ B(q). For any given x ∈ A(p), we can now consider the set

φp(x) = {q | q ≤ p and x q ∈ B(q)}

This is a sieve on p because of the compatibility condition for injections, and it is furthermore closed since the map x → Σφp(x) is in fact an atom and thus retains a real representative b ∈ B. Then it turns out that φp(x) =↓ (Eb). We now possess a transformation of functors φ : A → Ω which is natural (diagrammatically compatible). But in such a case we know that B ֒→ A is in turn the pull-back along φ of the arrow true, which is equivalent to the category of T-Sets.

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Music Territorializes Time: Deleuze. Thought of the Day 19.0

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Deleuze speaks of music expressed through pulsed and non-pulsed time. Pulsed time, or Chronos, is “territorialized time,” the “marking” of time through measure and repetition/return (“ritornello”) that is one expression of musically performative “time because it’s fundamentally the way in which a sonorous form, however simple it may be, marks a territory.” He continues, “Each time that there is a marking of a territoriality, there will be a pulsation of time.” If pulsed time is also musical time, and if both pulsed and musical time is territorial, what can be inferred of proximity?

Territorialization is the assemblage of proximities, in Deleuze and Guattari’s formation, bodies formed through not merely the association but the complicity of parts in their nearness to one another that make up bodies/territories distinct from other bodies/territories. Musical performance territorializes time because the performance of music takes (“appropriates”) embedded times in proximity (the curvature of the rate of change) and creates a body of aesthetic sound and practice distinct from non-territorialized or deterritorialized sound (the noises of traffic and machines or of digestive processes may have musically performative possibilities but are not themselves music or part of a musical territory without coming together through proximities to shared and changing time, to shared and changing space).

Deleuze reminds his listeners that territorialization (and thus pulsed time) may be embedded in measurement but is also contained in “development.” “[A]s soon as you can fix a sonorous form, determinable by its internal coordinates, for example melody-harmony, as soon as you can fix a sonorous form endowed with intrinsic properties, this form is subject to developments, by which it is transformed into other forms or enters into relation or again is connected to other forms, and here, following these transformations and these connections, you can fix pulsations of time.” Pulsations of time, or Chronos, then become even more subject to musical proximities that change in response to one another, an echo of borderswerving’s relationship to borderlinking.

Non-pulsed time, Aion, is defined rather by deterritorialization and the taking apart of “sonorous form.” Deleuze relates non-pulsed time to velocity, recalling the rate of change described by DeLanda and the dromoscopy of Paul Virilio. Aionic time is part of what Deleuze terms the “mixture” of time; Aion and Chronos blend together in a musical territorializing that is also a deterritorializing, and in these opposite yet proximal movements are proximities of becoming and unmaking. The experience of participating in a musical performance as a whole (instruments, performers, audience, composition, context, etc.) and as its component parts (notes, phrases, measures, dynamics, individual characteristics of performers and instruments) is the experience of this mixed time.

And yet, not all mixed time is musical, and not all sounds in proximity are musical. For music to occur, aesthetic proximities must become aware of their possibilities for becoming and unmaking, must perceive the friction and soothing of their near surfaces. As Deleuze questions, “¿Cuando [sic] deviene musical una voz? Yo diría, desde el punto de vista de la expresión, que la voz musical es esencialmente una voz desterrito- rializada. ¿Qué quiere decir eso? Pienso que hay cosas que aún no son música y que, sin embargo, están muy próximas a la música.” (When does a voice become musical? I would say, from the point of view of expression, that the musical voice is essentially a deterritorialized voice. Why does one want to say that? I think that there are things that are not music, and that nevertheless are very close to music.)

Husserl’s Melodies of the Absolute Flux. Note Quote.

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Husserl elaborates the basic problem of time-consciousness by taking the simple example of a melody. Observing that what we perceive endures – i.e., a melody is experienced as a unity of discrete tones, with each tone and the melody as a whole grasped as unified enduring objects – he sets out to examine how this can occur. Clearly, more than one tone must be retained in consciousness, since if each disappeared entirely after it had sounded then their succession, and therefore the melody as a whole, could never be grasped: “in each moment we would have a tone, or perhaps an empty pause in the interval between the sounding of two tones, but never the representation of a melody.” And each tone must also undergo some form of modification in consciousness, enabling it to appear “as more or less past, as pushed back in time, as it were,” since otherwise “instead of a melody we would have a chord of simultaneous tones, or rather a disharmonious tangle of sound, as if we had struck simultaneously all the notes that had previously sounded“. It is in order to account for our ability to experience such temporally extended objects as temporally extended that Husserl takes an immanent tone as his phenomenological datum.

In the characteristic phenomenological move Husserl proposes, at the outset of his lectures, “the complete exclusion of every assumption, stipulation and conviction with respect to objective time”. This suspension of the “natural attitude” towards time leaves – as the phenomenological residue – the indisputable immanent time (succession and duration) of lived experience (erlebnis). And immanent temporal objects within the immanent time of the flow of consciousness will enable reflection of the phenomenon of temporal experience free of all transcendent presuppositions. Husserl can therefore declare his task as being to “exclude all transcendent apprehension and positing and take the tone purely as a hyletic datum”.

Posing the question of “How, in addition to ‘temporal objects,’ immanent and transcendent, does time itself – the duration and succession of objects – become constituted?” Husserl points out that these are “different lines of description….” For example: “When a tone sounds … [we] can make the tone itself, which endures and fades away, into an object and yet not make the duration of the tone or the tone in its duration into an object”. Focusing on the latter, we can observe that the tone appears in “a continuity of ‘modes’ in a ‘continual flow'” – that is, appears in the mode of (as) ‘now’ or as ‘immediately past’ – even though “‘Throughout’ this whole flow of consciousness, one and the same tone is intended as enduring, as now enduring”. Because the tone itself is the same but the manner in which it appears is continually different, then description of the tone itself must be distinguished from description of “the way in which we are ‘conscious’ of … the ‘appearing’ of the immanent tone”. It is this latter that the phenomenology of time-consciousness will analyze.

Husserl accounts for our experience of the duration of the tone by distinguishing the intended temporal determinations of ‘now,’ ‘just-past,’ and ‘about-to-be’ from the consciousness that intends them: the impressional, retentional and protentional consciousness which constitute present, past, and future, respectively. As he describes it, the “source-point” (Quellpunkt) of the enduring object in the flowing stream of consciousness is the “primal impression” – consciousness of the (constantly changing) “tone-now” (Tonjetzt). And as this ‘tone-now’ is modified into ‘something that has been,’ so the primal impression passes over into retention: “the tone-now changes into a tone-having-been; the impressional consciousness, constantly flowing, passes over into an ever new retentional consciousness”. Retention not only “holds in consciousness what has been produced and stamps on it the character of the ‘just-past'” – ensuring that consciousness is always “consciousness of what has just been and not merely consciousness of the now-point of the object that appears as enduring” – but each retention is also retention of the elapsed tone retention, including in itself “the entire series of elapsed intentions in the form of a chain of mediate intentions”. In this way, retention “extends the now-consciousness” such that the “now-apprehension is, as it were, the head attached to the comet’s tail of retentions”.

This description of the extended moment is completed with the addition of protention as the symmetrical futural counterpart of retention. Protention, the intuition of the immediate future, is “just as original and unique as the intuition of the past,” Husserl writes. “Every process that constitutes its object originally is animated by protentions that emptily constitute what is coming as coming”. Retention and protention together combine to form “the living horizon of the now,” for every primal impression “has its retentional and protentional halo” ensuring that “The now point … [always] has for consciousness a temporal fringe”. The punctual now is therefore only an ideal limit, which cannot be phenomenologically given or encountered. And this description of the now as an ideal abstraction therefore applies equally to the primal impression of which it is the correlate: “In the ideal sense … perception (impression) would be the phase of consciousness that constitutes the pure now…. But the now is precisely only an ideal limit, something abstract, which can be nothing by itself”.

The temporal phases of the immanent object are, then, on a different stratum of analysis than the consciousness of those phases; the impressional, retentional, and protentional consciousness which, in intending the object as ‘now,’ ‘just-past,’ or ‘about-to-be’ “constitute the very differences belonging to time”. Husserl reaches the heart of his phenomenological account of time-consciousness with his description of how these “acts that create time” – primal impression, retention, and protention – “can be understood as time-constituting consciousness, as moments of the flow”. The ‘flow’ is made up of these partial intentions which are not fully fledged acts as such because their correlates are not objects but the temporal phases of objects. Retention, for example, “is an intentionality” but it “is not an ‘act’ (that is, an immanent duration-unity constituted in a series of retentional phases)”. The intentionality of these elements of the primal flux differs from that of apprehending or perceptual acts – they in fact constitute as a unity the apprehending act: “In perception a complex of sensation-contents, which are themselves unities constituted in the original temporal flow, undergo unity of apprehension. And this unitary apprehension is again a constituted unity”.

Husserl can therefore distinguish and outline the three levels of his analysis of time and consciousness as follows: Firstly, “the things of empirical experience in objective time”; secondly, “the constituting multiplicities of appearance … the immanent unities in pre-empirical time”; and lastly, “the absolute time-constituting flow of consciousness” which, as that which “lies before all constitution,” is the ultimate stratum of the constitutive process. This absolute consciousness “is not itself content or object in phenomenological time”. It is a ‘flow’ of “continuous ‘change'” which cannot be described as having constancy or duration, nor even as a ‘process,’ since the concept of process presupposes persistence and a ‘something’ that persists and endures through change. However, the flow does possess, in a sense, something abiding: “What abides, above all, is the formal structure of the flow, the form of the flow”. This unchanging form of the absolute flux is the retentional/impressional/protentional structure by which “a now becomes constituted by means of an impression and … a trail of retentions and a horizon of protentions are attached to the impression”. The question remains, of course, of how we can know this flow which is neither content nor object:

Every temporal appearance, after phenomenological reduction, dissolves into … a flow. But I cannot perceive in turn this consciousness itself into which all of this is dissolved. For this new percept would again be something temporal that points back to a constituting consciousness of a similar sort, and so in infinitum. Hence the question arises: How do I come to know the constituting flow?

To deal with this question Husserl recalls the ‘double intentionality’ of retention. One of these is the “‘primary memory’ of the (just sensed) tone” which “serves for the constitution of the immanent object”. But there is also the other, the second retentional intentionality which “is constitutive of the unity of this primary memory in the flow”. This “retention of retention” ensures that “each past now retentionally shelters within itself all earlier stages” and also therefore that “there extends throughout the flow a horizontal intentionality [Längsintentionalität] that, in the course of the flow, continuously coincides with itself”. By means of this, the unity of the flow becomes itself “constituted in the flow of consciousness as a one-dimensional quasi-temporal order”. The absolute flux is, therefore, self-constituting. It constitutes the unity of immanent objects in a unitary immanent time and thereby, “as shocking (when not initially even absurd) as it may seem,” also its own unity:

two inseparably united intentionalities, requiring one another like two sides of one and the same thing, are interwoven with one another in the one, unique flow of consciousness. By virtue of on limits of language of the intentionalities, immanent time becomes constituted…. In the other intentionality, it is the quasi-temporal arrangement of the phases of the flow that becomes constituted…. This prephenomenal, preimmanent temporality becomes constituted intentionally as the form of the time-constituting consciousness and in itself.

And it is this second retentional intentionality that gives us our oblique self-awareness of the flux, removing the problem of infinite regress whilst simultaneously resolving the difficulty of knowing the flow. “The self-appearance of the flow does not require a second flow: on the contrary, it constitutes itself as a phenomenon in itself”. It requires no second flow because this is a non-objectivating awareness – experienced in the same way as we experience acts, in a perceptual objectivation, without thematizing them. Unlike such acts, however, it cannot itself be made an object of reflection. Because there is no object or substance that endures, and no ‘time’ here as such, our ability to speak of the absolute flux runs up against the limits of language and conceptual thought. “We can say nothing other than the following: This flow is something we speak of in conformity with what is constituted, but it is not ‘something in objective time.’ It is absolute subjectivity and has the absolute properties of something to be designated metaphorically as ‘flow’.” Husserl is blunt about the inescapable inadequacy of his vocabulary here: “For all of this, we lack names”.

In a sense Husserl’s project in the lectures on time-consciousness can be understood as an inquiry into the constitution of constitution; into the way in which intentional acts of consciousness are constituted as temporal unities able to have as their correlate the transcendent temporally extended object. As he observes: “It is certainly evident that the perception of a temporal object itself has temporality, that the perception of duration itself presupposes duration of perception, that the perception of any temporal form itself has temporal form”. Yet Ricoeur writes that “The fact that the perception of duration never ceases to presuppose the duration of perception did not seem to trouble Husserl“, implying that Husserl was blind to the significance of his own observation. This rather offhand remark plays little role in Ricoeur’s argument for the conflict between Kant and Husserl’s respective treatments of time, but given that Husserl was clearly sorely troubled by this ‘fact’ – that it is arguably the very observation that led him beyond Kant’s standpoint to explore the temporality of the constitutive act itself.

Solitude: Thought of the Day 18.0

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A reason Nietzsche ponders solitude is that his is largely a philosophy of the future. There is heavy emphasis in Beyond Good and Evil on the temporal nature of the human condition. He posits that “the taste of the time and the virtue of the time weakens and thins down the will.” In order to surpass current modes and fashions in thinking, one must become removed from the present. The new philosopher is necessarily a man of tomorrow and the day after tomorrow and so he is solitary and in contradiction to the ideals of today. Fundamentally, Nietzsche sees current Europe (and especially Germany) as not yet prepared for an overturning of present morality. Although he does predict the time is approaching, there is the overarching sense throughout Beyond Good and Evil that Nietzsche expects (and even embraces) the fact that his philosophy needs a significant passage of time to be understood. His work is lonely. He labors to lay groundwork for the philosophers of the future who will continue on this path someday.

The life of the free spirit is solitary because it requires the recognition of the untruth of life in order to be beyond good and evil. Religion and democratic enlightenment in Europe have forged a herd mentality of mediocrity which has rejected such a possibility. In this society, everyone’s thoughts and morality are given equal merit. Nietzsche despises this because it forces us to reject our nature; both the ugliness and the beauty of it. He tells us that religion is able to teach even the lowliest of people how to place themselves in an illusory higher order of things so they may have the impression that they are content. This herd mentality protects the pack and also makes life palatable. It is also the first enemy of anyone looking to discover their own truths. Nietzsche concludes his book by reflecting on the wonders of solitude. For the free spirit, solitude is life-affirming because the absence of the stifling dogmas of the herd allows for the greatest expansion of one’s sense of self. To be truly beyond good and evil one must be removed from grappling with the order and morality imposed by democratic enlightenment and religion. Only when one stands alone vis-à-vis the herd is greatness and nobleness possible. Upon being removed from the seething torrent of austere and rigid thinking now strangling Europe, the free spirit foments his own morality and thrives.

Speech. Thought of the Day 17.0

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Speech, is a gesture, an indication, or a pointing toward, a certain intended signification. Speech, if it is understood, brings a certain something before us, but what is the status of that something? Firstly, given that language is equivocal, the signified necessarily goes beyond any attempt to signify it. As such, language never affords total expression, but rather, is merely the linguistic embodiment of an attempt to signify. It is therefore the case that these significations have the status of “Ideas,” which target, or aim at total expression but are constantly outstripped by the “things themselves” which they signify. The signified is never present before the act of expression; rather, it is this act of expression which realizes it as an intention. It is, furthermore, appropriate to say that we have, or possess, a language as the sum total of available significations. Language is intrinsically historical, in the sense that any synchronic moment possesses all previous synchronic moments within it. Any particular present carries with it all presents occurring prior to it. The distinction between the synchronic and the diachronic, therefore, cannot be maintained in a language as it is lived. It is the case, therefore, that any particular signification becomes available as a kind of ‘sedimentation’ within the ‘tradition’ of a language. The significative intention, therefore, must draw from available meanings but is also limited by the ‘world’ as the limit of possible meanings. The speaking subject, therefore, through the power of expression, is able to draw from available meaning and in turn, through them, constitute a new meaning. Understanding the meaning, therefore, is a process of taking up the signification of others, or having them “dwell within me,” such that a new ‘style’ of thought has been awakened. What has, thereby, been ‘acquired’ will remain available, without the need to reactivate the original process of constitution. A new ‘sedimentation’ has been constituted, which does not erase, or eliminate, the ‘sedimentations’ previously available. Rather the new ‘acquisition’ is incorporated into the cultural tradition that is language and is added as a new possibility for an expressive intention. The speech of others comes to “dwell” within me in a movement of transcendence, beyond the merely available meanings of the language, and is understood the moment I am able to take it within myself and express it anew. It seems to be the case, therefore, that what is available to me is not solely my ‘own,’ but ‘ours’ in the sense that what is available to me is available to everyone and only becomes mine specifically when, through my mute intention, I take it up into myself and express it anew. The ‘tradition,’ or language, is that which gives us the means of realizing our significative, or mute, intentions, however, at the same time it is constituted as the result of our expressivity.