Coarse Philosophies of Coarse Embeddabilities: Metric Space Conjectures Act Algorithmically On Manifolds – Thought of the Day 145.0

A coarse structure on a set X is defined to be a collection of subsets of X × X, called the controlled sets or entourages for the coarse structure, which satisfy some simple axioms. The most important of these states that if E and F are controlled then so is

E ◦ F := {(x, z) : ∃y, (x, y) ∈ E, (y, z) ∈ F}

Consider the metric spaces Zn and Rn. Their small-scale structure, their topology is entirely different, but on the large scale they resemble each other closely: any geometric configuration in Rn can be approximated by one in Zn, to within a uniformly bounded error. We think of such spaces as “coarsely equivalent”. The other axioms require that the diagonal should be a controlled set, and that subsets, transposes, and (finite) unions of controlled sets should be controlled. It is accurate to say that a coarse structure is the large-scale counterpart of a uniformity than of a topology.

Coarse structures and coarse spaces enjoy a philosophical advantage over coarse metric spaces, in that, all left invariant bounded geometry metrics on a countable group induce the same metric coarse structure which is therefore transparently uniquely determined by the group. On the other hand, the absence of a natural gauge complicates the notion of a coarse family, while it is natural to speak of sets of uniform size in different metric spaces it is not possible to do so in different coarse spaces without imposing additional structure.

Mikhail Leonidovich Gromov introduced the notion of coarse embedding for metric spaces. Let X and Y be metric spaces.

A map f : X → Y is said to be a coarse embedding if ∃ nondecreasing functions ρ1 and ρ2 from R+ = [0, ∞) to R such that

• ρ1(d(x,y)) ≤ d(f(x),f(y)) ≤ ρ2(d(x,y)) ∀ x, y ∈ X.
• limr→∞ ρi(r) = +∞ (i=1, 2).

Intuitively, coarse embeddability of a metric space X into Y means that we can draw a picture of X in Y which reflects the large scale geometry of X. In early 90’s, Gromov suggested that coarse embeddability of a discrete group into Hilbert space or some Banach spaces should be relevant to solving the Novikov conjecture. The connection between large scale geometry and differential topology and differential geometry, such as the Novikov conjecture, is built by index theory. Recall that an elliptic differential operator D on a compact manifold M is Fredholm in the sense that the kernel and cokernel of D are finite dimensional. The Fredholm index of D, which is defined by

index(D) = dim(kerD) − dim(cokerD),

has the following fundamental properties:

(1) it is an obstruction to invertibility of D;

(2) it is invariant under homotopy equivalence.

The celebrated Atiyah-Singer index theorem computes the Fredholm index of elliptic differential operators on compact manifolds and has important applications. However, an elliptic differential operator on a noncompact manifold is in general not Fredholm in the usual sense, but Fredholm in a generalized sense. The generalized Fredholm index for such an operator is called the higher index. In particular, on a general noncompact complete Riemannian manifold M, John Roe (Coarse Cohomology and Index Theory on Complete Riemannian Manifolds) introduced a higher index theory for elliptic differential operators on M.

The coarse Baum-Connes conjecture is an algorithm to compute the higher index of an elliptic differential operator on noncompact complete Riemannian manifolds. By the descent principal, the coarse Baum-Connes conjecture implies the Novikov higher signature conjecture. Guoliang Yu has proved the coarse Baum-Connes conjecture for bounded geometry metric spaces which are coarsely embeddable into Hilbert space. The metric spaces which admit coarse embeddings into Hilbert space are a large class, including e.g. all amenable groups and hyperbolic groups. In general, however, there are counterexamples to the coarse Baum-Connes conjecture. A notorious one is expander graphs. On the other hand, the coarse Novikov conjecture (i.e. the injectivity part of the coarse Baum-Connes conjecture) is an algorithm of determining non-vanishing of the higher index. Kasparov-Yu have proved the coarse Novikov conjecture for spaces which admit coarse embeddings into a uniformly convex Banach space.

Ideological Morphology. Thought of the Day 105.1

When applied to generic fascism, the combined concepts of ideal type and ideological morphology have profound implications for both the traditional liberal and Marxist definitions of fascism. For one thing it means that fascism is no longer defined in terms of style, for e.g. spectacular politics, uniformed paramilitary forces, the pervasive use of symbols like fasces and Swastika, or organizational structure, but in terms of ideology. Moreover, the ideology is not seen as essentially nihilistic or negative (anti-liberalism, anti-Marxism, resistance to transcendence etc.), or as the mystification and aestheticization of capitalist power. Instead, it is constructed in the positive, but not apologetic or revisionist terms of the fascists’ own diagnosis of society’s structural crisis and the remedies they propose to solve it, paying particular attention to the need to separate out the ineliminable, definitional conceptions from time- or place- specific adjacent or peripheral ones. However, for decades the state of fascist studies would have made Michael Freeden’s analysis well-nigh impossible to apply to generic fascism, because precisely what was lacking was any conventional wisdom embedded in common-sense usage of the term about what constituted the ineliminable cluster of concepts at its non-essentialist core. Despite a handful of attempts to establish its definitional constituents that combined deep comparative historiographical knowledge of the subject with a high degree of conceptual sophistication, there was a conspicuous lack of scholarly consensus over what constituted the fascist minimum. Whether there was such an entity as generic fascism even was a question to think through. Or whether Nazism’s eugenic racism and the euthanasia campaign it led to, combined with a policy of physically eliminating racial enemies that led to the systematic persecution and mass murder, was simply unique, and too exceptional to be located within the generic category was another question to think through. Both these positions suggest a naivety about the epistemological and ontological status of generic concepts most regrettable among professional intellectuals, since every generic entity is a utopian heuristic construct, not a real thing and every historically singularity is by definition unique no matter how many generic terms can be applied to it. Other common positions that implied considerable naivety were the ones that dismissed fascism’s ideology as too irrational or nihilistic to be part of the fascist minimum, or generalized about its generic traits by blending fascism and nazism.

Gothic: Once Again Atheistic Materialism and Hedonistic Flirtations. Drunken Risibility.

The machinery of the Gothic, traditionally relegated to both a formulaic and a sensational aesthetic, gradually evolved into a recyclable set of images, motifs and narrative devices that surpass temporal, spatial and generic categories. From the moment of its appearance the Gothic has been obsessed with presenting itself as an imitation.

Recent literary theory has extensively probed into the power of the Gothic to evade temporal and generic limits and into the aesthetic, narratological and ideological implications this involves. Officially granting the Gothic the elasticity it has always entailed has resulted in a reconfiguration of its spectrum both synchronically – by acknowledging its influence on numerous postmodern fictions – and diachronically – by rescripting, in hindsight, the history of its canon so as to allow space for ambiguous presences.

Both transgressive and hybrid in form and content, the Gothic has been accepted as a malleable genre, flexible enough to create more freely, in Borgesian fashion, its own precursors. The genre flouted what are considered the basic principles of good prose writing: adherence to verisimilitude and avoidance of both narrative diversions and moralising – all of which are, of course, made to be deliberately upset. Many merely cite the epigrammatic power of the essay’s most renowned phrase, that the rise of the Gothic “was the inevitable result of the revolutionary shocks which all of Europe has suffered”.

The eighteenth-century French materialist philosophy purported the displacement of metaphysical investigations into the meaning of life by materialist explorations. Julien Offray de La Mettrie, a French physician and philosopher, the earliest of materialist writers of the Enlightenment, published the materialist manifesto L’ Homme machine (Man a Machine), that did away with the transcendentalism of the soul, banished all supernatural agencies by claiming that mind is as mechanical as matter and equated humans with machines. In his words: “The human body is a machine that winds up its own springs: it is a living image of the perpetual motion”. French materialist thought resulted in the publication of the great 28-volume Encyclopédie, ou Dictionnaire raisonné des sciences, des arts et des méttrie par une société de gens de lettres by Denis Diderot and Jean Le Rond d’ Alembert, and which was grounded on purely materialist principles, against all kinds of metaphysical thinking. Diderot’s atheist materialism set the tone of the Encyclopédie, which, for both editors, was the ideal vehicle […] for reshaping French high culture and attitudes, as well as the perfect instrument with which to insinuate their radical Weltanschauung surreptitiously, using devious procedures, into the main arteries of French Society, embedding their revolutionary philosophic manifesto in a vast compilation ostensibly designed to provide plain information and basic orientation but in fact subtly challenging and transforming attitudes in every respect. While materialist thinkers ultimately disowned La Mettrie because he ran counter to their systematic moral, political and social naturalism, someone like Sade remained deeply influenced and inspired for his indebtedness to La Mettrie’s atheism and hedonism, particularly to the perception of virtue and vice as relative notions − the result of socialisation and at odds with nature.

Suspicion on Consciousness as an Immanent Derivative

The category of the subject (like that of the object) has no place in an immanent world. There can be no transcendent, subjective essence. What, then, is the ontological status of a body and its attendant instance of consciousness? In what would it exist? Sanford Kwinter (conjuncted here) here offers:

It would exist precisely in the ever-shifting pattern of mixtures or composites: both internal ones – the body as a site marked and traversed by forces that converge upon it in continuous variation; and external ones – the capacity of any individuated substance to combine and recombine with other bodies or elements (ensembles), both influencing their actions and undergoing influence by them. The ‘subject’ … is but a synthetic unit falling at the midpoint or interface of two more fundamental systems of articulation: the first composed of the fluctuating microscopic relations and mixtures of which the subject is made up, the second of the macro-blocs of relations or ensembles into which it enters. The image produced at the interface of these two systems – that which replaces, yet is too often mistaken for, subjective essence – may in turn have its own individuality characterized with a certain rigor. For each mixture at this level introduces into the bloc a certain number of defining capacities that determine both what the ‘subject’ is capable of bringing to pass outside of itself and what it is capable of receiving (undergoing) in terms of effects.

This description is sufficient to explain the immanent nature of the subjective bloc as something entirely embedded in and conditioned by its surroundings. What it does not offer – and what is not offered in any detail in the entirety of the work – is an in-depth account of what, exactly, these “defining capacities” are. To be sure, it would be unfair to demand a complete description of these capacities. Kwinter himself has elsewhere referred to the states of the nervous system as “magically complex”. Regardless of the specificity with which these capacities can presently be defined, we must nonetheless agree that it is at this interface, as he calls it, at this location where so many systems are densely overlaid, that consciousness is produced. We may be convinced that this consciousness, this apparent internal space of thought, is derived entirely from immanent conditions and can only be granted the ontological status of an effect, but this effect still manages to produce certain difficulties when attempting to define modes of behavior appropriate to an immanent world.

There is a palpable suspicion of the role of consciousness throughout Kwinter’s work, at least insofar as it is equated with some kind of internal, subjective space. (In one text he optimistically awaits the day when this space will “be left utterly in shreds.”) The basis of this suspicion is multiple and obvious. Among the capacities of consciousness is the ability to attribute to itself the (false) image of a stable and transcendent essence. The workings of consciousness are precisely what allow the subjective bloc to orient itself in a sequence of time, separating itself from an absolute experience of the moment. It is within consciousness that limiting and arbitrary moral categories seem to most stubbornly lodge themselves. (To be sure this is the location of all critical thought.) And, above all, consciousness may serve as the repository for conditioned behaviors which believe themselves to be free of external determination. Consciousness, in short, contains within itself an enormous number of limiting factors which would retard the production of novelty. Insofar as it appears to possess the capacity for self-determination, this capacity would seem most productively applied by turning on itself – that is, precisely by making the choice not to make conscious decisions and instead to permit oneself to be seized by extra-subjective forces.

Capital as a Symbolic Representation of Power. Nitzan’s and Bichler’s Capital as Power: A Study of Order and Creorder.

The secret to understanding accumulation, lies not in the narrow confines of production and consumption, but in the broader processes and institutions of power. Capital, is neither a material object nor a social relationship embedded in material entities. It is not ‘augmented’ by power. It is, in itself, a symbolic representation of power….

Unlike the elusive liberals, Marxists try to deal with power head on – yet they too end up with a fractured picture. Unable to fit power into Marx’s value analysis, they have split their inquiry into three distinct branches: a neo-Marxian economics that substitutes monopoly for labour values; a cultural analysis whose extreme versions reject the existence of ‘economics’ altogether (and eventually also the existence of any ‘objective’ order); and a state theory that oscillates between two opposite positions – one that prioritizes state power by demoting the ‘laws’ of the economy, and another that endorses the ‘laws’ of the economy by annulling the autonomy of the state. Gradually, each of these branches has developed its own orthodoxies, academic bureaucracies and barriers. And as the fractures have deepened, the capitalist totality that Marx was so keen on uncovering has dissipated….

The commodified structure of capitalism, Marx argues, is anchored in the labour process: the accumulation of capital is denominated in prices; prices reflect labour values; and labour values are determined by the productive labour time necessary to make the commodities. This sequence is intuitively appealing and politically motivating, but it runs into logical and empirical impossibilities at every step of the way. First, it is impossible to differentiate productive from unproductive labour. Second, even if we knew what productive labour was, there would still be no way of knowing how much productive labour goes into a given commodity, and therefore no way of knowing the labour value of that commodity and the amount of surplus value it embodies. And finally, even if labour values were known, there would be no consistent way to convert them into prices and surplus value into profit. So, in the end, Marxism cannot explain the prices of commodities – not in detail and not even approximately. And without a theory of prices, there can be no theory of profit and accumulation and therefore no theory of capitalism….

Modern capitalists are removed from production: they are absentee owners. Their ownership, says Veblen, doesn’t contribute to industry; it merely controls it for profitable ends. And since the owners are absent from industry, the only way for them to exact their profit is by ‘sabotaging’ industry. From this viewpoint, the accumulation of capital is the manifestation not of productive contribution but of organized power.

To be sure, the process by which capitalists ‘translate’ qualitatively different power processes into quantitatively unified measures of earnings and capitalization isn’t very ‘objective’. Filtered through the conventional assessments of accountants and the future speculations of investors, the conversion is deeply inter-subjective. But it is also very real, extremely imposing and, as we shall see, surprisingly well-defined.

These insights can be extended into a broader metaphor of a ‘social hologram’: a framework that integrates the resonating productive interactions of industry with the dissonant power limitations of business. These hologramic spectacles allow us to theorize the power underpinnings of accumulation, explore their historical evolution and understand the ways in which various forms of power are imprinted on and instituted in the corporation…..

Business enterprise diverts and limits industry instead of boosting it; that ‘business as usual’ needs and implies strategic limitation; that most firms are not passive price takers but active price makers, and that their autonomy makes ‘pure’ economics indeterminate; that the ‘normal rate of return’, just like the ancient rate of interest, is a manifestation not of productive yield but of organized power; that the corporation emerged not to enhance productivity but to contain it; that equity and debt have little to do with material wealth and everything to do with systemic power; and, finally, that there is little point talking about the deviations and distortions of ‘financial capital’ simply because there is no ‘productive capital’ to deviate from and distort.

Jonathan Nitzan, Shimshon Bichler- Capital as Power:_ A Study of Order and Creorder