The biogrammatic interface, generates a political aesthetic in which action is felt through the affective modulations and/or tonalities it incites. A doubling occurs in the moving towards realization, the rearticulation, of the becoming-thing/gesture. This doubling divides in a central differentiation, referencing a voluminous vocabulary of the interstitial – fissure, gap, disjunction, in-between, crack, interface, fold, non-place – descriptors of a bifurcating rift between content and expression, necessary for realization. Deleuze sums up the crux of this Foucauldian argument:

Things can be realized only through doubling or dissociation, creating divergent forms among which they can be distributed. It is here that we see the great dualities: between different classes, or the governing and the governed, or the public and the private. But more than this, it is here that the two forms of realization diverge or become differentiated: a form of expression and a form of content, a discursive and a non- discursive form, the form of the visible and the form of the articulable. It is precisely because the immanent cause, in both its matter and its functions, disregards form, that it is realized on the basis of a central differentiation which, on the one hand will form visible matter, and on the other will formalize articulable functions.’ (

)Gilles Deleuze, Sean Hand-Foucault

It can be argued that this central differentiation or interface distinguishes between the movements of two diagrammatic registers: outside from inside and the forms of realization. Transductive processes between these registers mark portals of entry through which all points of the diagram are in superposition, in passage as intensities of non-localizable relations from one point to another. The diagram distributes affective intensities within the context it maps.

Deleuze elasticizes Foucault’s reach by translating his oeuvre within the folding/unfolding of a knowledge-power-subjectivity continuum, mapping Foucault’s relays between the bifurcating polarities of content/expression, visibilities/statements as they differentiate and integrate through the folding ‘zone of subjectification’. The biogramming interface. The ‘event’ of rearticulation, of knowledge-capture and distribution, takes place through the perceptual filter of differential relations becoming-actual as a perception or thought. This is a topological dynamic mapped by the diagram, affected through the central differentiation (biogram) ‘or the ‘non-place’, as Foucault puts it, where the informal diagram is swallowed up and becomes embodied instead in two different directions that are necessarily divergent and irreducible. The concrete assemblages are therefore opened up by a crack that determines how the abstract machine performs’. It’s the process of swallowing up the relational intensities of a milieu and spitting back out certain selected somethings to be swallowed again that’s of particular interest to political aesthetics of the performative event. Foucault imagined a cartographic container of forces, affects, attractions and repulsions that modulate the diagram, excite the disjunction that separates forms of realization. The abstract machine begins to actualize its virtual potential as it distributes its relational dust.

…….

**Diagrams As Piloting Devices…**

]]>

Risk management is important in the practices of financial institutions and other corporations. Derivatives are popular instruments to hedge exposures due to currency, interest rate and other market risks. An important step of risk management is to use these derivatives in an optimal way. The most popular derivatives are forwards, options and swaps. They are basic blocks for all sorts of other more complicated derivatives, and should be used prudently. Several parameters need to be determined in the processes of risk management, and it is necessary to investigate the influence of these parameters on the aims of the hedging policies and the possibility of achieving these goals.

The problem of determining the optimal strike price and optimal hedging ratio is considered, where a put option is used to hedge market risk under a constraint of budget. The chosen option is supposed to finish in-the-money at maturity in the, such that the predicted loss of the hedged portfolio is different from the realized loss. The aim of hedging is to minimize the potential loss of investment under a specified level of confidence. In other words, the optimal hedging strategy is to minimize the Value-at-Risk (VaR) under a specified level of risk.

A stock is supposed to be bought at time zero with price S_{0}, and to be sold at time T with uncertain price S_{T}. In order to hedge the market risk of the stock, the company decides to choose one of the available put options written on the same stock with maturity at time τ, where τ is prior and close to T, and the n available put options are specified by their strike prices K_{i} (i = 1, 2,··· , n). As the prices of different put options are also different, the company needs to determine an optimal hedge ratio h (0 ≤ h ≤ 1) with respect to the chosen strike price. The cost of hedging should be less than or equal to the predetermined hedging budget C. In other words, the company needs to determine the optimal strike price and hedging ratio under the constraint of hedging budget.

Suppose the market price of the stock is S_{0} at time zero, the hedge ratio is h, the price of the put option is P_{0}, and the riskless interest rate is r. At time T, the time value of the hedging portfolio is

S_{0}e^{rT} + hP_{0}e^{rT} —– (1)

and the market price of the portfolio is

S_{T} + h(K − S_{τ})^{+} e^{r(T−τ)} —– (2)

therefore the loss of the portfolio is

L = (S_{0}e^{rT} + hP_{0}e^{rT}) − (S_{T} +h(K−S_{τ})^{+} e^{r(T−τ)}) —– (3)

where x^{+} = max(x, 0), which is the payoff function of put option at maturity.

For a given threshold v, the probability that the amount of loss exceeds v is denoted as

α = Prob{L ≥ v} —– (4)

in other words, v is the Value-at-Risk (VaR) at α percentage level. There are several alternative measures of risk, such as CVaR (Conditional Value-at-Risk), ESF (Expected Shortfall), CTE (Conditional Tail Expectation), and other coherent risk measures. The criterion of optimality is to minimize the VaR of the hedging strategy.

The mathematical model of stock price is chosen to be a geometric Brownian motion, i.e.

dS_{t}/S_{t} = μdt + σdB_{t} —– (5)

where S_{t} is the stock price at time t (0 < t ≤ T), μ and σ are the drift and the volatility of stock price, and B_{t} is a standard Brownian motion. The solution of the stochastic differential equation is

S_{t} = S_{0} e^{σBt + (μ−1/2σ2)t} —– (6)

where B_{0} = 0, and S_{t} is lognormally distributed.

Proposition:

For a given threshold of loss v, the probability that the loss exceeds v is

Prob {L ≥ v} = E [I_{{X ≤ c1}} F_{Y} (g(X) − X)] + E [I_{{X ≥ c1}} F_{Y} (c_{2} − X)] —– (7)

where E[X] is the expectation of random variable X. I_{{X < c}} is the index function of X such that I_{{X < c}} = 1 when {X < c} is true, otherwise I_{{X < c}} = 0. F_{Y} (y) is the cumulative distribution function of random variable Y , and

c_{1} = 1/σ [ln(K/S_{0}) − (μ−1/2σ^{2})τ] ,

g(X) = 1/σ [(ln (S_{0} + hP_{0})e^{rT} − h (K − f(X)) e^{r(T−τ)} −v)/S_{0} − (μ − 1/2σ^{2}) T],

f(X) = S_{0} e^{σX} + (μ−1/2σ^{2})τ,

c_{2} = 1/σ [(ln (S_{0} + hP_{0}) e^{rT} − v)/S_{0} − (μ− 1/2σ^{2}) T

X and Y are both normally distributed, where X ∼ N(0,√τ), Y ∼ N(0,√(T−τ).

For a specified hedging strategy, Q(v) = Prob {L ≥ v} is a decreasing function of v. The VaR under α level can be obtained from equation

Q(v) = α —– (8)

The expectations in Proposition can be calculated with * Monte Carlo simulation methods*, and the optimal hedging strategy which has the smallest VaR can be obtained from equation (8) by numerical searching methods….

]]>

In Foucault’s most famous example, the prison acts as the confluence of content (prisoners) and expression (law, penal code) (* Gilles Deleuze, Sean Hand-Foucault*). Informal Diagrams are proliferate. As abstract machines they contain the transversal vectors that cut across a panoply of features (such as institutions, classes, persons, economic formation, etc), mapping from point to relational point, the generalized features of power economies. The disciplinary diagram explored by Foucault, imposes “a particular conduct upon a particular human multiplicity”. The imposition of force upon force affects and effectuates the felt experience of a life, a living. Deleuze has called the abstract machine “pure matter/function” in which relations between forces are nonetheless very real.

[…] the diagram acts as a non-unifying immanent cause that is co-extensive with the whole social field: the abstract machine is like the cause of the concrete assemblages that execute its relations; and these relations between forces take place ‘not above’ but within the very tissue of the assemblages they produce.

The processual conjunction of content and expression; the cutting edge of deterritorialization:

The relations of power and resistance between theory and practice resonate – becoming-form; diagrammatics as praxis, integrates and differentiates the immanent cause and quasi-cause of the actualized occasions of research/creation. What do we mean by immanent cause? It is a cause which is realized, integrated and distinguished in its effect. Or rather, the immanent cause is realized, integrated and distinguished by its effect. In this way there is a correlation or mutual presupposition between cause and effect, between abstract machine and concrete assemblages

Memory is the real name of the relation to oneself, or the affect of self by self […] Time becomes a subject because it is the folding of the outside…forces every present into forgetting but preserves the whole of the past within memory: forgetting is the impossibiltiy of return and memory is the necessity of renewal.

The figure on the left is Henri Bergson’s diagram of an infinitely contracted past that directly intersects with the body at point S – a mobile, sensorimotor present where memory is closest to action. Plane P represents the actual present; plane of contact with objects. The AB segments represent repetitive compressions of memory. As memory contracts it gets closer to action. In it’s more expanded forms it is closer to dreams. The figure on the right extrapolates from Bergson’s memory model to describe the Biogrammatic ontological vector of the Diagram as it moves from abstract (informal) machine in the most expanded form “A” through the cone “tissue” to the phase-shifting (formal), arriving at the Strata of the P plane to become artefact. The ontological vector passes through the stratified, through the interval of difference created in the phase shift (the same phase shift that separates and folds content and expression to move vertically, transversally, back through to the abstract diagram.)

A spatio-temporal-material contracting-expanding of the abstract machine is the processual thinking-feeling-articulating of the diagram becoming-cartographic; synaesthetic conceptual mapping. A play of forces, a series of relays, affecting a tendency toward an inflection of the informal diagram becoming-form. The inflected diagram/biogram folds and unfolds perception, appearances; rides in the gap of becoming between content and expression; intuitively transduces the actualizing (thinking, drawing, marking, erasing) of matter-movement, of expressivity-movement. “* To follow the flow of matter… is intuition in action.*” A processual stage that prehends the process of the virtual actualizing;

the creative construction of a new reality. The biogrammatic stage of the diagrammatic is paradoxically double in that it is both the actualizing of the abstract machine (contraction) and the recursive counter-actualization of the formal diagram (détournement); virtual and actual.

It is the event-dimension of potential – that is the effective dimension of the interrelating of elements, of their belonging to each other. That belonging is a dynamic corporeal “abstraction” – the “drawing off” (transductive conversion) of the corporeal into its dynamism (yielding the event) […] In direct channeling. That is, in a directional channeling: ontological vector. The transductive conversion is an ontological vector that in-gathers a heterogeneity of substantial elements along with the already-constituted abstractions of language (“meaning”) and delivers them together to change. (* Brian Massumi Parables for the Virtual Movement, Affect, Sensation*)

Skin is the space of the body the BwO that is interior and exterior. Interstitial matter of the space of the body.

The material markings and traces of a diagrammatic process, a ‘capturing’ becoming-form. A diagrammatic capturing involves a transductive process between a biogrammatic form of content and a form of expression. The formal diagram is thus an individuating phase-shift as Simondon would have it, always out-of-phase with itself. A becoming-form that inhabits the gap, the difference, between the wave phase of the biogrammatic that synaesthetically draws off the intermix of substance and language in the event-dimension and the drawing of wave phase in which partial capture is formalized. The phase shift difference never acquires a vectorial intention. A pre-decisive, pre-emptive drawing of phase-shifting with a “drawing off” the biogram.

]]>

DeLanda:

…while an ontology based on relations between general types and particular instances is hierarchical, each level representing a different ontological category (organism, species, genera) [or strings, quarks, baryons], an approach in terms of interacting parts and emergent wholes leads to a flat ontology, one made exclusively of unique, singular individuals, differing in spatio-temporal scale but not in ontological status.

The following discussion, however, seeks to go further than DeLanda’s account of hierarchy, extending it to all entities of spatio-temporal entities, thus the interjection of “strings, quarks, baryons” into the quote. That this extension is natural should be clear once van Fraassen’s role in this level-denying consilience. Furthermore, van Fraassen’s account will be employed to illustrate why any level-like organization attributed to the components of an explanation has no bearing on the explanation, and arises due to two things:

1) erroneously clumping together all types of belief statements into a single branch of philosophy that deals with knowledge, and

2) attempting to stratify the causal thicket that is the world, so as conform the produce of scientific enterprise with monism and fundamentalist predilections.

Finally, there is one other way in which the account differs from DeLanda’s flat ontology, when touching upon Kant’s antinomy of teleology: the idea that levels of mechanisms telescopes to a flat ontology, as every part and whole enjoys the same status in a scientific explanation, and only differ in size.

*Figure: The larger arrows point toward the conclusion of consilience. The smaller arrows suggest relationships that create cooperation toward the thesis. The blue bubbles represent the positive web of notions that cohere, and the red bubbles are those notions that are excluded from web. The red arrows indicate where the ideas not included in the web arise, and the totality of this paper works toward a final explication as to why these are to be excluded. There are a few thin blue lines that are not included, because they would make a mess of the image, such as a line connecting abstraction and James’ pragmatism. However, the paper ( Abstraction as Dissection of a Flat Ontology. through) endeavors to make these connections clear, for example, quoting James to show that James presents an idea that seems a precursor to Cartwright’s notion of abstraction. (Note: Yellow lines are really blue lines, but are yellow to avoid confusion that might ensue from blue lines passing through blue bubbles. Green lines are to indicate additivity. The red lines denote notions not connected to the web, yet bear some relation to ideas in the web.) *

]]>

The word fornix means “an archway” or “vault” (in Rome, prostitutes could be solicited there). More directly, fornicatio means “done in the archway”; thus a euphemism for prostitution.

Diagrammatic praxis proposes a contractual (push, pull) approach in which the movement between abstract machine, biogram (embodied, inflected diagram), formal diagram (drawing of, drawing off) and artaffect (realized thing) is topologically immanent. It imagines the practice of writing, of this writing, interleaved with the mapping processes with which it folds and unfolds – forming, deforming and reforming both processes. The relations of non-relations that power the diagram, the thought intensities that resonate between fragments, between content ad expression, the seeable and the sayable, the discursive and the non-discursive, mark entry points; portals of entry through which all points of the diagram pass – push, pull, fold, unfold – without the designation of arrival and departure, without the input/output connotations of a black boxed confection. Ports, as focal points of passage, attract lines of resistance or lines of flight through which the diagram may become both an effectuating concrete assemblage (thing) and remain outside the stratified zone of the audiovisual. It’s as if the port itself is a bifurcating point, a figural inflected archway. The port, as a bifurcation point of resistance (contra black box), modulates and changes the unstable, turbulent interplay between pure Matter and pure Function of the abstract machine. These ports are marked out, localized, situated, by the continuous movement of power-relations:

These power-relations … simultaneously local, unstable and diffuse, do not emanate from a central point or unique locus of sovereignty, but at each moment move from one point to another in a field of forces, marking inflections, resistances, twists and turns when one changes direction or retraces one’s steps… (

)Gilles Deleuze, Sean Hand-Foucault

An inflection point, marked out by the diagram, is not a symmetrical form but the difference between concavity and convexity, a pure temporality, a “true atom of form, the true object of geography.” (* Bernard Cache*)

Cache’s absolute exteriority is equivalent to Deleuze’s description of the Outside “more distant than any exterior […] ‘twisted’, folded and doubled by an Inside that is deeper than any interior, and alone creates the possibility of the derived relation between the interior and the exterior”. This folded and doubled interior is diagrammed by Deleuze in the folds chapter of Foucault.

Thinking does not depend on a beautiful interiority that reunites the visible ad articulable elements, but is carried under the intrusion of an outside that eats into the interval and forces or dismembers the internal […] when there are only environments and whatever lies betwen them, when words and things are opened up by the environment without ever coinciding, there is a liberation of forces which come from the outside and exist only in a mixed up state of agitation, modification and mutation. In truth they are dice throws, for thinking involves throwing the dice. If the outside, farther away than any external world, is also closer than any internal world, is this not a sign that thought affects itself, by revealing the outside to be its own unthought element?

“It cannot discover the unthought […] without immediately bringing the unthought nearer to itself – or even, perhaps, without pushing it farther away, and in any case without causing man’s own being to undergo a change by the very fact, since it is deployed in the distance between them” (* Gilles Deleuze, Sean Hand-Foucault*)

*Figure*: Left: a simulation of Deleuze’s central marking in his diagram of the Foucaultian diagram. This is the line of the Outside as Fold. Right: To best express the relations of diagrammatic praxis between content and expression (theory and practice) the Fold figure needs to be drawn as a double Fold (“twice twice” as Massumi might say) – a folded möbius strip. Here the superinflections between inside/outside and content/expression provide transversal vectors.

A topology or topological becoming-shapeshift retains its connectivity, its interconnectedness to preserve its autonomy as a singularity. All the points of all its matter reshape as difference in itself. A topology does not resemble itself. The möbius strip and the infamous torus-to-coffe cup are examples of 2d and 3d topologies. technically a topological surface is totalized, it can not comprise fragments cut or glued to produce a whole. Its change is continuous. It is not cut-copy-pasted. But the cut and its interval are requisite to an emergent new.

For Deleuze, the essence of meaning, the essence of essence, is best expressed in two infinitives; ‘to cut ” and “to die” […] Definite tenses keeping company in time. In the slash between their future and their past: “to cut” as always timeless and alone (

).Massumi

Add the individuating “to shift” to the infinitives that reside in the timeless zone of indetermination of future-past. Given the paradigm of the topological-becoming, how might we address writing in the age of copy-paste and hypertext? The seamless and the stitched? As potential is it diagram? A linguistic multiplicity whose virtual immanence is the metalanguage potentiality between the phonemes that gives rise to all language?

An overview diagram of diagrammatic praxis based on Deleuze’s diagram of the Foucaultian model shown below. The main modification is to the representation of the Fold. In the top figure, the Fold or zone of subjectification becomes a double-folded möbius strip.

Four folds of subjectification:

1. material part of ourselves which is to be surrounded and folded

2. the fold of the relation between forces always according to a particular rule that the relation between forces is bent back in order to become a relation to oneself (rule ; natural, divine, rational, aesthetic, etc)

3. fold of knowledge constitutes the relation of truth to our being and our being to truth which will serve as the formal condition for any kind of knowledge

4. the fold of the outside itself is the ultimate fold: an ‘interiority of expectation’ from which the subject, in different ways, hopes for immortality, eternity, salvation, freedom or death or detachment.

]]>

*The androcentric scientific and meta-scientific evidence that the penis is the male reproductive organ is considered overwhelming and largely uncontroversial.”*

That’s how we began. We used this preposterous sentence to open a “paper” consisting of 3,000 words of utter nonsense posing as academic scholarship. Then a peer-reviewed academic journal in the social sciences accepted and published it.

This paper should *never* have been published. Titled, “The Conceptual Penis as a Social Construct,” our paper “argues” that “The penis *vis-à-vis* maleness is an incoherent construct. We argue that the *conceptual penis* is better understood not as an anatomical organ but as a gender-performative, highly fluid social construct.” As if to prove philosopher David Hume’s claim that there is a deep gap between what is and what ought to be, our *should-never-have-been*-published paper *was *published in the open-access (meaning that articles are freely accessible and not behind a paywall), peer-reviewed journal *Cogent Social Sciences*.

Assuming the pen names “Jamie Lindsay” and “Peter Boyle,” and writing for the fictitious “Southeast Independent Social Research Group,” we wrote an absurd paper loosely composed in the style of post-structuralist discursive gender theory. The paper was ridiculous by intention, essentially arguing that penises shouldn’t be thought of as male genital organs but as damaging social constructions. We made no attempt to find out what “post-structuralist discursive gender theory” actually means. We assumed that if we were merely clear in our moral implications that maleness is intrinsically bad and that the penis is somehow at the root of it, we could get the paper published in a respectable journal.

This already damning characterization of our hoax understates our paper’s lack of fitness for academic publication by orders of magnitude. We didn’t try to make the paper coherent; instead, we stuffed it full of jargon (like “discursive” and “isomorphism”), nonsense (like arguing that hypermasculine men are both inside and outside of certain discourses at the same time), red-flag phrases (like “pre-post-patriarchal society”), lewd references to slang terms for the penis, insulting phrasing regarding men (including referring to some men who choose not to have children as being “unable to coerce a mate”), and allusions to rape (we stated that “manspreading,” a complaint levied against men for sitting with their legs spread wide, is “akin to raping the empty space around him”). After completing the paper, we read it carefully to ensure it didn’t say anything meaningful, and as neither one of us could determine what it is actually about, we deemed it a success.

Why did Boghossian and Lindsay *do* this?

Sokal exposed an infatuation with academic puffery that characterizes the entire project of academic postmodernism. Our aim was smaller yet more pointed. We intended to test the hypothesis that flattery of the academic Left’s moral architecture in general, and of the moral orthodoxy in gender studies in particular, is the overwhelming determiner of publication in an academic journal in the field. That is, we sought to demonstrate that a desire for a certain moral view of the world to be validated could overcome the critical assessment required for legitimate scholarship. Particularly, we suspected that gender studies is crippled academically by an overriding almost-religious belief that maleness is the root of all evil. On the evidence, our suspicion was justified.

In the words of Graham Harman,

Endless turf war….

The authors worry that gender studies folk will believe that, “…men do often suffer from machismo braggadocio, and that there is an isomorphism between these concepts via some personal toxic hypermasculine conception of their penises.” But I don’t really see why a gender studies academic wouldn’t believe this… This is NOT a case of cognitive dissonance.

As much as the authors like to pretend like they have “no idea” what they are talking about, they clearly do. They are taking existing gender study ideas and just turning up the volume and adding more jargon. As if this proves a point against the field.

The author’s biases are on their sleeve. Their arguments are about as effective as a Men’s Rights Activist on Reddit. By using a backhanded approach in an attempt to give a *coup de grace* to gender studies academaniacs, all they’ve done is blow $625 and “exposed” the already well known issue of pay-to-play. If they wanted to make an actual case against the “feminazis” writ large, I suggest they “man” up and actually make a real argument rather than show a bunch of fancy words can fool some people. Ah!, but far from being a meta-analytical multiplier of defense, quantum homeomorphism slithers through the conceptual penis!

]]>

In his book on Foucault first published in 1986, Deleuze drew a diagram in the last chapter, Foldings, that depicts in overview the Outside as abstract machine, defined by the line of the outside (1), which separates the unformed interplay of forces and resistance from the strategies and strata that filter the affects of power relations to become “the world of knowledge”.

The central Fold of subjectification, of ‘Life’ is “hollowed out” and ignored by the forces of the outside as they are realized in the strata fulfilling the obligation of the diagram to “come to fruition in the archive.” This is dual process of integration and differentiation. The residual dust of the affective relations produced by force upon force, integrate into the strata even as they differentiate to forms of realization – visible or articulable. The ‘empty’ fissure/fold attracts and repels these moving curvilinear strategies as they differentiate and ”hop over” it. Ostensibly, the Fold of subjectification effectuates change as both continuously topological, and as discontinuously catastrophic (as in leaping over). So, the process of crystallization from informal to formal paradoxically integrates as it differentiates. Deleuze’s somewhat paradoxical description follows:

The informal relations between forces differentiate from one another by creating heterogeneous curves which pass through the neighborhood of particular features (statements) and that of the scenes which distribute them into figures of light (visibilities). And at the same time the relations between forces became integrated, precisely in the formal relations between the two, from one side to the other of differentiation. This is because the relations between forces ignored the fissure within the strata, which begins only below them. They are apt to hollow out the fissure by being actualized in the strata, but also to hop over it in both senses of the term by becoming differentiated even as they become integrated.

Gilles Deleuze, Sean Hand-Foucault

So this “pineal gland” figure of the Fold is the “center of the cyclone”, where life is lived “par excellence” as a “slow Being”.

*qua* gamespace are, among others, the revitalized forces of the military-academic-entertainment complex that fuel economic models such as the Creative Industries that pervade the conditions of play in artistic research. McKenzie Wark concludes his book * GAMER THEORY*, with prescient comments on the black hole quality of a topology of the outside

]]>

Machine learning researchers tend to develop general purpose solutions as self-contained packages. A wide variety of these are available as open-source packages at places like mloss.org, or from in-house code, proprietary packages, and cloud-based platforms. Using self-contained solutions often results in a glue code system design pattern, in which a massive amount of supporting code is written to get data into and out of general-purpose packages.

This glue code design pattern can be costly in the long term, as it tends to freeze a system to the peculiarities of a specific package. General purpose solutions often have different design goals: they seek to provide one learning system to solve many problems, but many practical software systems are highly engineered to apply to one large-scale problem, for which many experimental solutions are sought. While generic systems might make it possible to interchange optimization algorithms, it is quite often refactoring of the construction of the problem space which yields the most benefit to mature systems. The glue code pattern implicitly embeds this construction in supporting code instead of in principally designed components. As a result, the glue code pattern often makes experimentation with other machine learning approaches prohibitively expensive, resulting in an ongoing tax on innovation.

Glue code can be reduced by choosing to re-implement specific algorithms within the broader system architecture. At first, this may seem like a high cost to pay – reimplementing a machine learning package in C++ or Java that is already available in R or matlab, for example, may appear to be a waste of effort. But the resulting system may require dramatically less glue code to integrate in the overall system, be easier to test, be easier to maintain, and be better designed to allow alternate approaches to be plugged in and empirically tested. Problem-specific machine learning code can also be tweaked with problem-specific knowledge that is hard to support in general packages.

]]>

* Bayesianism in game theory can be characterised as the view that it is always possible to define probabilities for anything that is relevant for the players’ decision-making.* In addition, it is usually taken to imply that the players use Bayes’ rule for updating their beliefs. If the probabilities are to be always definable, one also has to specify what players’ beliefs are before the play is supposed to begin.

The CPA however is a methodological injunction to include everything that may affect the players’ behaviour in the game: not just everything that motivates the players, but also everything that affects the players’ beliefs should be explicitly modelled by the game: if players had different priors, this would mean that the game structure would not be completely specified because there would be differences in players’ behaviour that are not explained by the model. In a dispute over the status of the CPA, * Faruk Gul essentially argues that the CPA does not follow from the Harsanyi doctrine*. He does this by distinguishing between two different interpretations of the common prior, the ‘prior view’ and the ‘infinite hierarchy view’. The former is a genuinely dynamic story in which it is assumed that there really is a prior stage in time. The latter framework refers to

* Robert Aumann, however, defends the CPA by arguing that whenever there are differences in beliefs, there must have been a prior stage in which the priors were the same, and from which the current beliefs can be derived by conditioning on the differentiating events. If players differ in their present beliefs, they must have received different information at some previous point in time, and they must have processed this information correctly*. Based on this assumption, he further argues that players cannot ‘agree to disagree’: if a player knows that his opponents’ beliefs are different from his own, he should revise his beliefs to take the opponents’ information into account. The only case where the CPA would be violated, then, is when players have different beliefs, and have common knowledge about each others’ different beliefs and about each others’ epistemic rationality. Aumann’s argument seems perfectly legitimate if it is taken as a metaphysical one, but we do not see how it could be used as a justification for using the CPA as a modelling assumption in this or that application of game theory

]]>

Solving Diophantine equations, that is giving integer solutions to polynomials, is often unapproachably difficult. Weil describes one strategy in a letter to his sister, the philosopher Simone Weil: Look for solutions in richer fields than the rationals, perhaps fields of rational functions over the complex numbers. But these are quite different from the integers:

A solution modulo 5 to a polynomial P(X,Y,..Z) is a list of integers X,Y,..Z making the value P(X,Y,..Z) divisible by 5, or in other words equal to 0 modulo 5. For example, X^{2} + Y^{2} − 3 has no integer solutions. That is clear since X and Y would both have to be 0 or ±1, to keep their squares below 3, and no combination of those works. But it has solutions modulo 5 since, among others, 3^{2} + 3^{2} − 3 = 15 is divisible by 5. Solutions modulo a given prime p are easier to find than integer solutions and they amount to the same thing as solutions in the finite field of integers modulo p.

To see if a list of polynomial equations P_{i}(X, Y, ..Z) = 0 have a solution modulo p we need only check p different values for each variable. Even if p is impractically large, equations are more manageable modulo p. Going farther, we might look at equations modulo p, but allow some irrationals, and ask how the number of solutions grows as we allow irrationals of higher and higher degree—roots of quadratic polynomials, roots of cubic polynomials, and so on. This is looking for solutions in all finite fields, as in Weil’s letter.

Take any good n-dimensional algebraic space (any smooth projective variety of dimension n) defined by integer polynomials on a finite field F_{q}. For each s ∈ N, let N_{s} be the number of points defined on the extension field F_{(qs)}. Define the zeta function Z(t) as an exponential using a formal variable t:

Z(t) = exp(∑_{s=1}^{∞}N_{s}t_{s}/s)

The first Weil conjecture says Z(t) is a rational function:

Z(t) = P(t)/Q(t)

for integer polynomials P(t) and Q(t). This is a strong constraint on the numbers of solutions N_{s}. It means there are complex algebraic numbers a_{1} . . . a_{i} and b_{1} . . . b_{j} such that

N_{s} =(a^{s}_{1} +…+ a^{s}_{i}) − (b^{s}_{1} +…+ b^{s}_{j})

And each algebraic conjugate of an a (resp. b) also an a (resp. b).

The second conjecture is a functional equation:

Z(1/q^{n}t) = ± q^{nE/2}t^{E}Z(t)

This says the operation x → q^{n}/x permutes the a’s (resp. the b’s).The third is a Riemann Hypothesis

Z(t) = (P_{1}(t)P_{3}(t) · · · P_{2n−1}(t))/(P_{0}(t)P_{2}(t) · · · P_{2n}(t))

where each P_{k} is an integer polynomial with all roots of absolute value q^{−k/2}. That means each a has absolute value q^{k} for some 0 ≤ k ≤ n. Each b has absolute value q^{(2k−1)/2} for some 0 ≤ k ≤ n.

Over it all is the conjectured link to topology. Let B_{0}, B_{1}, . . . B_{2n} be the Betti numbers of the complex manifold defined by the same polynomials. That is, each B_{k} gives the number of k-dimensional holes or handles on the continuous space of complex number solutions to the equations. And recall an algebraically n-dimensional complex manifold is topologically 2n-dimensional. Then each P_{k} has degree B_{k}. And E is the Euler number of the manifold, the alternating sum

∑_{k=0}^{2n} (−1)^{k}B_{k}

On its face the topology of a continuous manifold is worlds apart from arithmetic over finite fields. But the topology of this manifold tells how many a’s and b’s there are with each absolute value. This implies useful numerical approximations to the numbers of roots N_{s}. Special cases of these conjectures, with aspects of the topology, were proved before Weil, and he proved more. All dealt with curves (1-dimensional) or hypersurfaces (defined by a single polynomial).

* Weil presented the topology as motivating the conjectures for higher dimensional varieties*. He especially pointed out how the whole series of conjectures would follow quickly if we could treat the spaces of finite field solutions as topological manifolds. The topological strategy was powerfully seductive but seriously remote from existing tools. Weil’s arithmetic spaces were not even precisely defined. To all appearances they would be finite or (over the algebraic closures of the finite fields) countable and so everywhere discontinuous. Topological manifold methods could hardly apply.

]]>