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”.

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.

Time and World-Lines

Let γ: [s1, s2] → M be a smooth, future-directed timelike curve in M with tangent field ξa. We associate with it an elapsed proper time (relative to gab) given by

∥γ∥= ∫s1s2 (gabξaξb)1/2 ds

This elapsed proper time is invariant under reparametrization of γ and is just what we would otherwise describe as the length of (the image of) γ . The following is another basic principle of relativity theory:

Clocks record the passage of elapsed proper time along their world-lines.

Again, a number of qualifications and comments are called for. We have taken for granted that we know what “clocks” are. We have assumed that they have worldlines (rather than worldtubes). And we have overlooked the fact that ordinary clocks (e.g., the alarm clock on the nightstand) do not do well at all when subjected to extreme acceleration, tidal forces, and so forth. (Try smashing the alarm clock against the wall.) Again, these concerns are important and raise interesting questions about the role of idealization in the formulation of physical theory. (One might construe an “ideal clock” as a point-size test object that perfectly records the passage of proper time along its worldline, and then take the above principle to assert that real clocks are, under appropriate conditions and to varying degrees of accuracy, approximately ideal.) But they do not have much to do with relativity theory as such. Similar concerns arise when one attempts to formulate corresponding principles about clock behavior within the framework of Newtonian theory.

Now suppose that one has determined the conformal structure of spacetime, say, by using light rays. Then one can use clocks, rather than free particles, to determine the conformal factor.

Let g′ab be a second smooth metric on M, with g′ab = Ω2gab. Further suppose that the two metrics assign the same lengths to timelike curves – i.e., ∥γ∥g′ab = ∥γ∥gab ∀ smooth, timelike curves γ: I → M. Then Ω = 1 everywhere. (Here ∥γ∥gab is the length of γ relative to gab.)

Let ξoa be an arbitrary timelike vector at an arbitrary point p in M. We can certainly find a smooth, timelike curve γ: [s1, s2] → M through p whose tangent at p is ξoa. By our hypothesis, ∥γ∥g′ab = ∥γ∥gab. So, if ξa is the tangent field to γ,

s1s2 (g’ab ξaξb)1/2 ds = ∫s1s2 (gabξaξb)1/2 ds

∀ s in [s1, s2]. It follows that g′abξaξb = gabξaξb at every point on the image of γ. In particular, it follows that (g′ab − gab) ξoa ξob = 0 at p. But ξoa was an arbitrary timelike vector at p. So, g′ab = gab at our arbitrary point p. The principle gives the whole story of relativistic clock behavior. In particular, it implies the path dependence of clock readings. If two clocks start at an event p and travel along different trajectories to an event q, then, in general, they will record different elapsed times for the trip. This is true no matter how similar the clocks are. (We may stipulate that they came off the same assembly line.) This is the case because, as the principle asserts, the elapsed time recorded by each of the clocks is just the length of the timelike curve it traverses from p to q and, in general, those lengths will be different.

Suppose we consider all future-directed timelike curves from p to q. It is natural to ask if there are any that minimize or maximize the recorded elapsed time between the events. The answer to the first question is “no.” Indeed, one then has the following proposition:

Let p and q be events in M such that p ≪ q. Then, for all ε > 0, there exists a smooth, future directed timelike curve γ from p to q with ∥γ ∥ < ε. (But there is no such curve with length 0, since all timelike curves have non-zero length.)

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If there is a smooth, timelike curve connecting p and q, there is also a jointed, zig-zag null curve connecting them. It has length 0. But we can approximate the jointed null curve arbitrarily closely with smooth timelike curves that swing back and forth. So (by the continuity of the length function), we should expect that, for all ε > 0, there is an approximating timelike curve that has length less than ε.

The answer to the second question (“Can one maximize recorded elapsed time between p and q?”) is “yes” if one restricts attention to local regions of spacetime. In the case of positive definite metrics, i.e., ones with signature of form (n, 0) – we know geodesics are locally shortest curves. The corresponding result for Lorentzian metrics is that timelike geodesics are locally longest curves.

Let γ: I → M be a smooth, future-directed, timelike curve. Then γ can be reparametrized so as to be a geodesic iff ∀ s ∈ I there exists an open set O containing γ(s) such that , ∀ s1, s2 ∈ I with s1 ≤ s ≤ s2, if the image of γ′ = γ|[s1, s2] is contained in O, then γ′ (and its reparametrizations) are longer than all other timelike curves in O from γ(s1) to γ(s2). (Here γ|[s1, s2] is the restriction of γ to the interval [s1, s2].)

Of all clocks passing locally from p to q, the one that will record the greatest elapsed time is the one that “falls freely” from p to q. To get a clock to read a smaller elapsed time than the maximal value, one will have to accelerate the clock. Now, acceleration requires fuel, and fuel is not free. So the above proposition has the consequence that (locally) “saving time costs money.” And proposition before that may be taken to imply that “with enough money one can save as much time as one wants.” The restriction here to local regions of spacetime is essential. The connection described between clock behavior and acceleration does not, in general, hold on a global scale. In some relativistic spacetimes, one can find future-directed timelike geodesics connecting two events that have different lengths, and so clocks following the curves will record different elapsed times between the events even though both are in a state of free fall. Furthermore – this follows from the preceding claim by continuity considerations alone – it can be the case that of two clocks passing between the events, the one that undergoes acceleration during the trip records a greater elapsed time than the one that remains in a state of free fall. (A rolled-up version of two-dimensional Minkowski spacetime provides a simple example)

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Two-dimensional Minkowski spacetime rolledup into a cylindrical spacetime. Three timelike curves are displayed: γ1 and γ3 are geodesics; γ2 is not; γ1 is longer than γ2; and γ2 is longer than γ3.

The connection we have been considering between clock behavior and acceleration was once thought to be paradoxical. Recall the so-called “clock paradox.” Suppose two clocks, A and B, pass from one event to another in a suitably small region of spacetime. Further suppose A does so in a state of free fall but B undergoes acceleration at some point along the way. Then, we know, A will record a greater elapsed time for the trip than B. This was thought paradoxical because it was believed that relativity theory denies the possibility of distinguishing “absolutely” between free-fall motion and accelerated motion. (If we are equally well entitled to think that it is clock B that is in a state of free fall and A that undergoes acceleration, then, by parity of reasoning, it should be B that records the greater elapsed time.) The resolution of the paradox, if one can call it that, is that relativity theory makes no such denial. The situations of A and B here are not symmetric. The distinction between accelerated motion and free fall makes every bit as much sense in relativity theory as it does in Newtonian physics.

A “timelike curve” should be understood to be a smooth, future-directed, timelike curve parametrized by elapsed proper time – i.e., by arc length. In that case, the tangent field ξa of the curve has unit length (ξaξa = 1). And if a particle happens to have the image of the curve as its worldline, then, at any point, ξa is called the particle’s four-velocity there.

Universal Turing Machine: Algorithmic Halting

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A natural number x will be identified with the x’th binary string in lexicographic order (Λ,0,1,00,01,10,11,000…), and a set X of natural numbers will be identified with its characteristic sequence, and with the real number between 0 and 1 having that sequence as its dyadic expansion. The length of a string x will be denoted |x|, the n’th bit of an infinite sequence X will be noted X(n), and the initial n bits of X will be denoted Xn. Concatenation of strings p and q will be denoted pq.

We now define the information content (and later the depth) of finite strings using a universal Turing machine U. A universal Turing machine may be viewed as a partial recursive function of two arguments. It is universal in the sense that by varying one argument (“program”) any partial recursive function of the other argument (“data”) can be obtained. In the usual machine formats, program, data and output are all finite strings, or, equivalently, natural numbers. However, it is not possible to take a uniformly weighted average over a countably infinite set. Chaitin’s universal machine has two tapes: a read-only one-way tape containing the infinite program; and an ordinary two-way read/write tape, which is used for data input, intermediate work, and output, all of which are finite strings. Our machine differs from Chaitin’s in having some additional auxiliary storage (e.g. another read/write tape) which is needed only to improve the time efficiency of simulations.

We consider only terminating computations, during which, of course, only a finite portion of the program tape can be read. Therefore, the machine’s behavior can still be described by a partial recursive function of two string arguments U(p, w), if we use the first argument to represent that portion of the program that is actually read in the course of a particular computation. The expression U (p, w) = x will be used to indicate that the U machine, started with any infinite sequence beginning with p on its program tape and the finite string w on its data tape, performs a halting computation which reads exactly the initial portion p of the program, and leaves output data x on the data tape at the end of the computation. In all other cases (reading less than p, more than p, or failing to halt), the function U(p, w) is undefined. Wherever U(p, w) is defined, we say that p is a self-delimiting program to compute x from w, and we use T(p, w) to represent the time (machine cycles) of the computation. Often we will consider computations without input data; in that case we abbreviate U(p, Λ) and T(p, Λ) as U(p) and T(p) respectively.

The self-delimiting convention for the program tape forces the domain of U and T, for each data input w, to be a prefix set, that is, a set of strings no member of which is the extension of any other member. Any prefix set S obeys the Kraft inequality

p∈S 2−|p| ≤ 1

Besides being self-delimiting with regard to its program tape, the U machine must be efficiently universal in the sense of being able to simulate any other machine of its kind (Turing machines with self-delimiting program tape) with at most an additive constant constant increase in program size and a linear increase in execution time.

Without loss of generality we assume that there exists for the U machine a constant prefix r which has the effect of stacking an instruction to restart the computation when it would otherwise end. This gives the machine the ability to concatenate programs to run consecutively: if U(p, w) = x and U(q, x) = y, then U(rpq, w) = y. Moreover, this concatenation should be efficient in the sense that T (rpq, w) should exceed T (p, w) + T (q, x) by at most O(1). This efficiency of running concatenated programs can be realized with the help of the auxiliary storage to stack the restart instructions.

Sometimes we will generalize U to have access to an “oracle” A, i.e. an infinite look-up table which the machine can consult in the course of its computation. The oracle may be thought of as an arbitrary 0/1-valued function A(x) which the machine can cause to be evaluated by writing the argument x on a special tape and entering a special state of the finite control unit. In the next machine cycle the oracle responds by sending back the value A(x). The time required to evaluate the function is thus linear in the length of its argument. In particular we consider the case in which the information in the oracle is random, each location of the look-up table having been filled by an independent coin toss. Such a random oracle is a function whose values are reproducible, but otherwise unpredictable and uncorrelated.

Let {φAi (p, w): i = 0,1,2…} be an acceptable Gödel numbering of A-partial recursive functions of two arguments and {φAi (p, w)} an associated Blum complexity measure, henceforth referred to as time. An index j is called self-delimiting iff, for all oracles A and all values w of the second argument, the set { x : φAj (x, w) is defined} is a prefix set. A self-delimiting index has efficient concatenation if there exists a string r such that for all oracles A and all strings w, x, y, p, and q,if φAj (p, w) = x and φAj (q, x) = y, then φAj(rpq, w) = y and φAj (rpq, w) = φAj (p, w) + φAj (q, x) + O(1). A self-delimiting index u with efficient concatenation is called efficiently universal iff, for every self-delimiting index j with efficient concatenation, there exists a simulation program s and a linear polynomial L such that for all oracles A and all strings p and w, and

φAu(sp, w) = φAj (p, w)

and

ΦAu(sp, w) ≤ L(ΦAj (p, w))

The functions UA(p,w) and TA(p,w) are defined respectively as φAu(p, w) and ΦAu(p, w), where u is an efficiently universal index.

For any string x, the minimal program, denoted x∗, is min{p : U(p) = x}, the least self-delimiting program to compute x. For any two strings x and w, the minimal program of x relative to w, denoted (x/w)∗, is defined similarly as min{p : U(p,w) = x}.

By contrast to its minimal program, any string x also has a print program, of length |x| + O(log|x|), which simply transcribes the string x from a verbatim description of x contained within the program. The print program is logarithmically longer than x because, being self-delimiting, it must indicate the length as well as the contents of x. Because it makes no effort to exploit redundancies to achieve efficient coding, the print program can be made to run quickly (e.g. linear time in |x|, in the present formalism). Extra information w may help, but cannot significantly hinder, the computation of x, since a finite subprogram would suffice to tell U to simply erase w before proceeding. Therefore, a relative minimal program (x/w)∗ may be much shorter than the corresponding absolute minimal program x∗, but can only be longer by O(1), independent of x and w.

A string is compressible by s bits if its minimal program is shorter by at least s bits than the string itself, i.e. if |x∗| ≤ |x| − s. Similarly, a string x is said to be compressible by s bits relative to a string w if |(x/w)∗| ≤ |x| − s. Regardless of how compressible a string x may be, its minimal program x∗ is compressible by at most an additive constant depending on the universal computer but independent of x. [If (x∗)∗ were much smaller than x∗, then the role of x∗ as minimal program for x would be undercut by a program of the form “execute the result of executing (x∗)∗.”] Similarly, a relative minimal program (x/w)∗ is compressible relative to w by at most a constant number of bits independent of x or w.

The algorithmic probability of a string x, denoted P(x), is defined as {∑2−|p| : U(p) = x}. This is the probability that the U machine, with a random program chosen by coin tossing and an initially blank data tape, will halt with output x. The time-bounded algorithmic probability, Pt(x), is defined similarly, except that the sum is taken only over programs which halt within time t. We use P(x/w) and Pt(x/w) to denote the analogous algorithmic probabilities of one string x relative to another w, i.e. for computations that begin with w on the data tape and halt with x on the data tape.

The algorithmic entropy H(x) is defined as the least integer greater than −log2P(x), and the conditional entropy H(x/w) is defined similarly as the least integer greater than − log2P(x/w). Among the most important properties of the algorithmic entropy is its equality, to within O(1), with the size of the minimal program:

∃c∀x∀wH(x/w) ≤ |(x/w)∗| ≤ H(x/w) + c

The first part of the relation, viz. that algorithmic entropy should be no greater than minimal program size, is obvious, because of the minimal program’s own contribution to the algorithmic probability. The second half of the relation is less obvious. The approximate equality of algorithmic entropy and minimal program size means that there are few near-minimal programs for any given input/output pair (x/w), and that every string gets an O(1) fraction of its algorithmic probability from its minimal program.

Finite strings, such as minimal programs, which are incompressible or nearly so are called algorithmically random. The definition of randomness for finite strings is necessarily a little vague because of the ±O(1) machine-dependence of H(x) and, in the case of strings other than self-delimiting programs, because of the question of how to count the information encoded in the string’s length, as opposed to its bit sequence. Roughly speaking, an n-bit self-delimiting program is considered random (and therefore not ad-hoc as a hypothesis) iff its information content is about n bits, i.e. iff it is incompressible; while an externally delimited n-bit string is considered random iff its information content is about n + H(n) bits, enough to specify both its length and its contents.

For infinite binary sequences (which may be viewed also as real numbers in the unit interval, or as characteristic sequences of sets of natural numbers) randomness can be defined sharply: a sequence X is incompressible, or algorithmically random, if there is an O(1) bound to the compressibility of its initial segments Xn. Intuitively, an infinite sequence is random if it is typical in every way of sequences that might be produced by tossing a fair coin; in other words, if it belongs to no informally definable set of measure zero. Algorithmically random sequences constitute a larger class, including sequences such as Ω which can be specified by ineffective definitions.

The busy beaver function B(n) is the greatest number computable by a self-delimiting program of n bits or fewer. The halting set K is {x : φx(x) converges}. This is the standard representation of the halting problem.

The self-delimiting halting set K0 is the (prefix) set of all self-delimiting programs for the U machine that halt: {p : U(p) converges}.

K and K0 are readily computed from one another (e.g. by regarding the self-delimiting programs as a subset of ordinary programs, the first 2n bits of K0 can be recovered from the first 2n+O(1) bits of K; by encoding each n-bit ordinary program as a self-delimiting program of length n + O(log n), the first 2n bits of K can be recovered from the first 2n+O(log n) bits of K0.)

The halting probability Ω is defined as {2−|p| : U(p) converges}, the probability that the U machine would halt on an infinite input supplied by coin tossing. Ω is thus a real number between 0 and 1.

The first 2n bits of K0 can be computed from the first n bits of Ω, by enumerating halting programs until enough have halted to account for all but 2−n of the total halting probability. The time required for this decoding (between B(n − O(1)) and B(n + H(n) + O(1)) grows faster than any computable function of n. Although K0 is only slowly computable from Ω, the first n bits of Ω can be rapidly computed from the first 2n+H(n)+O(1) bits of K0, by asking about the halting of programs of the form “enumerate halting programs until (if ever) their cumulative weight exceeds q, then halt”, where q is an n-bit rational number…

Highest Reality. Thought of the Day 70.0

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यावचिन्त्यावात्मास्य शक्तिश्चैतौ परमार्थो भवतः॥१॥

Yāvacintyāvātmāsya śaktiścaitau paramārtho bhavataḥ

These two (etau), the Self (ātmā) and (ca) His (asya) Power (śaktiḥ) —who (yau) (are) inconceivable (acintyau)—, constitute (bhavataḥ) the Highest Reality (parama-arthaḥ)

The Self is the Core of all, and His Power has become all. I call the Core “the Self” for the sake of bringing more light instead of more darkness. If I had called Him “Śiva”, some people might consider Him as the well-known puranic Śiva who is a great ascetic living in a cave and whose main task consists in destroying the universe, etc. Other people would think that, as Viṣṇu is greater than Śiva, he should be the Core of all and not Śiva. In turn, there is also a tendency to regard Śiva like impersonal while Viṣṇu is personal. There is no end to spiritual foolishness indeed, because there is no difference between Śiva and Viṣṇu really. Anyway, other people could suggest that a better name would be Brahman, etc. In order not to fall into all that ignorant mess of names and viewpoints, I chose to assign the name “Self” to the Core of all. In the end, when spiritual enlightenment arrives, one’s own mind is withdrawn (as I will tell by an aphorism later on), and consequently there is nobody to think about if “This Core of all” is personal, impersonal, Śiva, Viṣṇu, Brahman, etc. Ego just collapses and This that remains is the Self as He essentially is.

He and His Power are completely inconceivable, i.e. beyond the mental sphere. The Play of names, viewpoints and such is performed by His Power, which is always so frisky. All in all, the constant question is always: “Is oneself completely free like the Self?”. If the answer is “Yes”, one has accomplished the goal of life. And if the answer is “No”, one must get rid of his own bondage somehow then. The Self and His Power constitute the Highest Reality. Once you can attain them, so to speak, you are completely free like Them both. The Self and His Power are “two” only in the sphere of words, because as a matter of fact they form one compact mass of Absolute Freedom and Bliss. Just as the sun can be divided into “core of the sun, surface of the sun, crown”, etc.

तयोरुभयोः स्वरूपं स्वातन्त्र्यानन्दात्मकैकघनत्वेनापि तत्सन्तताध्ययनाय वचोविषय एव द्विधाकृतम्

Tayorubhayoḥ svarūpaṁ svātantryānandātmakaikaghanatvenāpi tatsantatādhyayanāya vacoviṣaya eva dvidhākṛtam

Even though (api) the essential nature (sva-rūpam) of Them (tayoḥ) both (ubhayoḥ) (is) one compact mass (eka-ghanatvena) composed of (ātmaka) Absolute Freedom (svātantrya)(and) Bliss (ānanda), it is divided into two (dvidhā-kṛtam) —only (eva) in the sphere (viṣaye) of words (vacas)— for its close study (tad-santata-adhyayanāya)

The Self is Absolute Freedom and His Power is Bliss. Both form a compact mass (i.e. omnipresent). In other words, the Highest Reality is always “One without a second”, but, in the world of words It is divided into two for studying It in detail. When this division occurs, the act of coming to recognize or realize the Highest Reality is made easier. So, the very Highest Reality generates this division in the sphere of words as a help for the spiritual aspirants to realize It faster.

आत्मा प्रकाशात्मकशुद्धबोधोऽपि सोऽहमिति वचोविषये स्मृतः

Ātmā prakāśātmakaśuddhabodho’pi so’hamiti vacoviṣaye smṛtaḥ

Although (api) the Self (ātmā) (is) pure (śuddha) Consciousness (bodhaḥ) consisting of (ātmaka) Prakāśa or Light (prakāśa), He (saḥ) is called (smṛtaḥ) “I” (aham iti) in the sphere (viṣaye) of words (vacas)

The Self is pure Consciousness, viz. the Supreme Subject who is never an object. Therefore, He cannot be perceived in the form of “this” or “that”. He cannot even be delineated in thought by any means. Anyway, in the world of words, He is called “I” or also “real I” for the sake of showing that He is higher than the false “I” or ego.

Gnostic Semiotics. Thought of the Day 63.0

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The question here is what is being composed? For the deferment and difference that is always already of the Sign, suggests that perhaps the composition is one that lies not within but without, a creation that lies on the outside but which then determines – perhaps through the reader more than anything else, for after all the meaning of a particular sign, be it a word or anything else, requires a form of impregnation by the receiver – a particular meaning.

Is there any choice but to assume a meaning in a sign? Only through the simulation, or ‘belief’ if you prefer (there is really no difference in the two concepts), of an inherent meaning in the sign can any transference continue. For even if we acknowledge that all communication is merely the circulation of empty signifiers, the impregnation of the signified (no matter how unconnected it may be to the other person’s signified) still ensures that the sign carries with it a meaning. Only through this simulation of a meaning is circulation possible – even if one posits that the sign circulates itself, this would not be possible if it were completely empty.

Since it is from without (even if meaning is from the reader, (s)he is external to the signification), this suggests that the meaning is a result, a consequence of forces – its signification is a result of the significance of various forces (convention, context, etc) which then means that inherently, the sign remains empty; a pure signifier leading to yet another signifier.

The interesting element though lies in the fact that the empty signifier then sucks the Other (in the form of the signified, which takes the form of the Absolute Other here) into it, in order to define an existence, but essentially remains an empty signifier, awaiting impregnation with meaning from the reader. A void: always full and empty or perhaps (n)either full (n)or empty. For true potentiality must always already contain the possibility of non-potentiality. Otherwise there would be absolutely no difference between potentiality and actualization – they would merely be different ends of the same spectrum.

Dialectics of God: Lautman’s Mathematical Ascent to the Absolute. Paper.

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Figure and Translation, visit Fractal Ontology

The first of Lautman’s two theses (On the unity of the mathematical sciences) takes as its starting point a distinction that Hermann Weyl made on group theory and quantum mechanics. Weyl distinguished between ‘classical’ mathematics, which found its highest flowering in the theory of functions of complex variables, and the ‘new’ mathematics represented by (for example) the theory of groups and abstract algebras, set theory and topology. For Lautman, the ‘classical’ mathematics of Weyl’s distinction is essentially analysis, that is, the mathematics that depends on some variable tending towards zero: convergent series, limits, continuity, differentiation and integration. It is the mathematics of arbitrarily small neighbourhoods, and it reached maturity in the nineteenth century. On the other hand, the ‘new’ mathematics of Weyl’s distinction is ‘global’; it studies the structures of ‘wholes’. Algebraic topology, for example, considers the properties of an entire surface rather than aggregations of neighbourhoods. Lautman re-draws the distinction:

In contrast to the analysis of the continuous and the infinite, algebraic structures clearly have a finite and discontinuous aspect. Though the elements of a group, field or algebra (in the restricted sense of the word) may be infinite, the methods of modern algebra usually consist in dividing these elements into equivalence classes, the number of which is, in most applications, finite.

In his other major thesis, (Essay on the notions of structure and existence in mathematics), Lautman gives his dialectical thought a more philosophical and polemical expression. His thesis is composed of ‘structural schemas’ and ‘origination schemas’ The three structural schemas are: local/global, intrinsic properties/induced properties and the ‘ascent to the absolute’. The first two of these three schemas close to Lautman’s ‘unity’ thesis. The ‘ascent to the absolute’ is a different sort of pattern; it involves a progress from mathematical objects that are in some sense ‘imperfect’, towards an object that is ‘perfect’ or ‘absolute’. His two mathematical examples of this ‘ascent’ are: class field theory, which ‘ascends’ towards the absolute class field, and the covering surfaces of a given surface, which ‘ascend’ towards a simply-connected universal covering surface. In each case, there is a corresponding sequence of nested subgroups, which induces a ‘stepladder’ structure on the ‘ascent’. This dialectical pattern is rather different to the others. The earlier examples were of pairs of notions (finite/infinite, local/global, etc.) and neither member of any pair was inferior to the other. Lautman argues that on some occasions, finite mathematics offers insight into infinite mathematics. In mathematics, the finite is not a somehow imperfect version of the infinite. Similarly, the ‘local’ mathematics of analysis may depend for its foundations on ‘global’ topology, but the former is not a botched or somehow inadequate version of the latter. Lautman introduces the section on the ‘ascent to the absolute’ by rehearsing Descartes’s argument that his own imperfections lead him to recognise the existence of a perfect being (God). Man (for Descartes) is not the dialectical opposite of or alternative to God; rather, man is an imperfect image of his creator. In a similar movement of thought, according to Lautman, reflection on ‘imperfect’ class fields and covering surfaces leads mathematicians up to ‘perfect’, ‘absolute’ class fields and covering surfaces respectively.

Albert Lautman Dialectics in mathematics

Iain Hamilton Grant’s Schelling in Opposition to Fichte. Note Quote.

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The stated villain of Philosophies of Nature is not Hegelianism but rather ‘neo-Fichteanism’. It is Grant’s ‘Philosophies of Nature After Schelling‘, which takes up the issue of graduating Schelling to escape the accoutrements of Kantian and Fichtean narrow transcendentalism. Grant frees Schelling from the grips of narrow minded inertness and mechanicality in nature that Kant and Fichte had presented nature with. This idea is the Deleuzean influence on Grant. Manuel De Landa makes a vociferous case in this regard. According to De Landa, the inertness of matter was rubbished by Deleuze in the way that Deleuze sought for a morphogenesis of form thereby launching a new kind of materialism. This is the anti-essentialist position of Deleuze. Essentialism says that matter and energy are inert, they do not have any morphogenetic capabilities. They cannot give rise to new forms on their own. Disciplines like complexity theory, non-linear dynamics do give matter its autonomy over inertness, its capabilities in terms of charge. But its account of the relationship between Fichte and Schelling actually obscures the rich meaning of speculation in Hegel and after. Grant quite accurately recalls that Schelling confronted Fichte’s identification of the ‘not I’ with passive nature – the consequence of identifying all free activity with the ‘I’ alone. For Grant, that which Jacobi termed ‘speculative egotism’ becomes the nightmare of modern philosophy and of technological modernity at large. The ecological concern is never quite made explicit in Philosophies of Nature. Yet Grant’s introduction to Schelling’s On the World Soul helps to contextualise the meaning of his ‘geology of morals’.

What we miss from Grant’s critique of Fichte is the manner by which the corrective, positive characterisation of nature proceeds from Schelling’s confirmation of Fichte’s rendering of the fact of consciousness (Tatsache) into the act of consciousness (Tathandlung). Schelling, as a consequence, becomes singularly critical of contemplative speculation, since activity now implies working on nature and thereby changing it – along with it, we might say – rather than either simply observing it or even experimenting upon it.

In fact, Grant reads Schelling only in opposition to Fichte, with drastic consequences for his speculative realism: the post-Fichtean element of Schelling’s naturephilosophy allows for the new sense of speculation he will share with Hegel – even though they will indeed turn this against Kant and Fichte. Without this account, we are left with the older, contemplative understanding of metaphysical speculation, which leads to a certain methodologism in Grant’s study. Hence, ‘the principle method of naturephilosophy consists in “unconditioning” the phenomena’. Relatedly, Meillassoux defines the ‘speculative’ as ‘every type of thinking’ – not acting, – ‘that claims to be able to access some form of absolute’.

In direct contrast to this approach, the collective ‘system programme’ of Hegel, Schelling and Hölderlin was not a programme for thinking alone. Their revolutionised sense of speculation, from contemplation of the stars to reform of the worldly, is overlooked by today’s speculative realism – a philosophy that, ‘refuses to interrogate reality through human (linguistic, cultural or political) mediations of it’. We recall that Kant similarly could not extend his Critique to speculative reason precisely on account of his contemplative determination of pure reason (in terms of the hierarchical gap between reason and the understanding). Grant’s ‘geology of morals’ does not oppose ‘Kanto-Fichtean philosophy’, as he has it, but rather remains structurally within the sphere of Kant’s pre-political metaphysics.

Meillassoux, Deleuze, and the Ordinal Relation Un-Grounding Hyper-Chaos. Thought of the Day 41.0

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As Heidegger demonstrates in Kant and the Problem of Metaphysics, Kant limits the metaphysical hypostatization of the logical possibility of the absolute by subordinating the latter to a domain of real possibility circumscribed by reason’s relation to sensibility. In this way he turns the necessary temporal becoming of sensible intuition into the sufficient reason of the possible. Instead, the anti-Heideggerian thrust of Meillassoux’s intellectual intuition is that it absolutizes the a priori realm of pure logical possibility and disconnects the domain of mathematical intelligibility from sensibility. (Ray Brassier’s The Enigma of Realism: Robin Mackay – Collapse_ Philosophical Research and Development. Speculative Realism.) Hence the chaotic structure of his absolute time: Anything is possible. Whereas real possibility is bound to correlation and temporal becoming, logical possibility is bound only by non-contradiction. It is a pure or absolute possibility that points to a radical diachronicity of thinking and being: we can think of being without thought, but not of thought without being.

Deleuze clearly situates himself in the camp when he argues with Kant and Heidegger that time as pure auto-affection (folding) is the transcendental structure of thought. Whatever exists, in all its contingency, is grounded by the first two syntheses of time and ungrounded by the third, disjunctive synthesis in the implacable difference between past and future. For Deleuze, it is precisely the eternal return of the ordinal relation between what exists and what may exist that destroys necessity and guarantees contingency. As a transcendental empiricist, he thus agrees with the limitation of logical possibility to real possibility. On the one hand, he thus also agrees with Hume and Meillassoux that [r]eality is not the result of the laws which govern it. The law of entropy or degradation in thermodynamics, for example, is unveiled as nihilistic by Nietzsche s eternal return, since it is based on a transcendental illusion in which difference [of temperature] is the sufficient reason of change only to the extent that the change tends to negate difference. On the other hand, Meillassoux’s absolute capacity-to-be-other relative to the given (Quentin Meillassoux, Ray Brassier, Alain Badiou – After finitude: an essay on the necessity of contingency) falls away in the face of what is actual here and now. This is because although Meillassoux s hyper-chaos may be like time, it also contains a tendency to undermine or even reject the significance of time. Thus one may wonder with Jon Roffe (Time_and_Ground_A_Critique_of_Meillassou) how time, as the sheer possibility of any future or different state of affairs, can provide the (non-)ground for the realization of this state of affairs in actuality. The problem is less that Meillassoux’s contingency is highly improbable than that his ontology includes no account of actual processes of transformation or development. As Peter Hallward (Levi Bryant, Nick Srnicek and Graham Harman (editors) – The Speculative Turn: Continental Materialism and Realism) has noted, the abstract logical possibility of change is an empty and indeterminate postulate, completely abstracted from all experience and worldly or material affairs. For this reason, the difference between Deleuze and Meillassoux seems to come down to what is more important (rather than what is more originary): the ordinal sequences of sensible intuition or the logical lack of reason.

But for Deleuze time as the creatio ex nihilo of pure possibility is not just irrelevant in relation to real processes of chaosmosis, which are both chaotic and probabilistic, molecular and molar. Rather, because it puts the Principle of Sufficient Reason as principle of difference out of real action it is either meaningless with respecting to the real or it can only have a negative or limitative function. This is why Deleuze replaces the possible/real opposition with that of virtual/actual. Whereas conditions of possibility always relate asymmetrically and hierarchically to any real situation, the virtual as sufficient reason is no less real than the actual since it is first of all its unconditioned or unformed potential of becoming-other.

Dissipations – Bifurcations – Synchronicities. Thought of the Day 29.0

Deleuze’s thinking expounds on Bergson’s adaptation of multiplicities in step with the catastrophe theory, chaos theory, dissipative systems theory, and quantum theory of his era. For Bergson, hybrid scientific/philosophical methodologies were not viable. He advocated tandem explorations, the two “halves” of the Absolute “to which science and metaphysics correspond” as a way to conceive the relations of parallel domains. The distinctive creative processes of these disciplines remain irreconcilable differences-in-kind, commonly manifesting in lived experience. Bergson: Science is abstract, philosophy is concrete. Deleuze and Guattari: Science thinks the function, philosophy the concept. Bergson’s Intuition is a method of division. It differentiates tendencies, forces. Division bifurcates. Bifurcations are integral to contingency and difference in systems logic.

The branching of a solution into multiple solutions as a system is varied. This bifurcating principle is also known as contingency. Bifurcations mark a point or an event at which a system divides into two alternative behaviours. Each trajectory is possible. The line of flight actually followed is often indeterminate. This is the site of a contingency, were it a positionable “thing.” It is at once a unity, a dualism and a multiplicity:

Bifurcations are the manifestation of an intrinsic differentiation between parts of the system itself and the system and its environment. […] The temporal description of such systems involves both deterministic processes (between bifurcations) and probabilistic processes (in the choice of branches). There is also a historical dimension involved […] Once we have dissipative structures we can speak of self-organisation.

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Figure: In a dynamical system, a bifurcation is a period doubling, quadrupling, etc., that accompanies the onset of chaos. It represents the sudden appearance of a qualitatively different solution for a nonlinear system as some parameter is varied. The illustration above shows bifurcations (occurring at the location of the blue lines) of the logistic map as the parameter r is varied. Bifurcations come in four basic varieties: flip bifurcation, fold bifurcation, pitchfork bifurcation, and transcritical bifurcation. 

A bifurcation, according to Prigogine and Stengers, exhibits determinacy and choice. It pertains to critical points, to singular intensities and their division into multiplicities. The scientific term, bifurcation, can be substituted for differentiation when exploring processes of thought or as Massumi explains affect:

Affect and intensity […] is akin to what is called a critical point, or bifurcation point, or singular point, in chaos theory and the theory of dissipative structures. This is the turning point at which a physical system paradoxically embodies multiple and normally mutually exclusive potentials… 

The endless bifurcating division of progressive iterations, the making of multiplicities by continually differentiating binaries, by multiplying divisions of dualities – this is the ontological method of Bergson and Deleuze after him. Bifurcations diagram multiplicities, from monisms to dualisms, from differentiation to differenciation, creatively progressing. Manuel Delanda offers this account, which describes the additional technicality of control parameters, analogous to higher-level computer technologies that enable dynamic interaction. These protocols and variable control parameters are later discussed in detail in terms of media objects in the metaphorical state space of an in situ technology:

[…] for the purpose of defining an entity to replace essences, the aspect of state space that mattered was its singularities. One singularity (or set of singularities) may undergo a symmetry-breaking transition and be converted into another one. These transitions are called bifurcations and may be studied by adding to a particular state space one or more ‘control knobs’ (technically control parameters) which determine the strength of external shocks or perturbations to which the system being modeled may be subject.

Another useful example of bifurcation with respect to research in the neurological and cognitive sciences is Francesco Varela’s theory of the emergence of microidentities and microworlds. The ready-for-action neuronal clusters that produce microindentities, from moment to moment, are what he calls bifurcating “break- downs”. These critical events in which a path or microidentity is chosen are, by implication, creative: