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

Two Conceptions of Morphogenesis – World as a Dense Evolutionary Plasma of Perpetual Differentiation and Innovation. Thought of the Day 57.0

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Sanford Kwinter‘s two conceptions of morhpogenesis, of which, one is appropriate to a world capable of sustaining transcendental ontological categories, while the other is inherent in a world of perfect immanence. According to the classical, hylomorphic model, a necessarily limited number of possibilities (forms or images) are reproduced (mirrored in reality) over a substratum, in a linear time-line. The insufficiency of such a model, however, is evident in its inability to find a place for novelty. Something either is or is not possible. This model cannot account for new possibilities and it fails to confront the inevitable imperfections and degradations evident in all of its realizations. It is indeed the inevitability of corruption and imperfection inherent in classical creation that points to the second mode of morphogenesis. This mode is dependent on an understanding of the world as a ceaseless pullulation and unfolding, a dense evolutionary plasma of perpetual differentiation and innovation. In this world forms are not carried over from some transcendent realm, but instead singularities and events emerge from within a rich plasma through the continual and dynamic interaction of forces. The morphogenetic process at work in such a world is not one whereby an active subject realizes forms from a set of transcendent possibilities, but rather one in which virtualities are actualized through the constant movement inherent in the very forces that compose the world. Virtuality is understood as the free difference or singularity, not yet combined with other differences into a complex ensemble or salient form. It is of course this immanentist description of the world and its attendant mode of morphogenesis that are viable. There is no threshold beneath which classical objects, states, or relations cease to have meaning yet beyond which they are endowed with a full pedigree and privileged status. Indeed, it is the nature of real time to ensure a constant production of innovation and change in all conditions. This is evidenced precisely by the imperfections introduced in an act of realizing a form. The classical mode of morphogenesis, then, has to be understood as a false model which is imposed on what is actually a rich, perpetually transforming universe. But the sort of novelty which the enactment of the classical model produces, a novelty which from its own perspective must be construed as a defect is not a primary concern if the novelty is registered as having emerged from a complex collision of forces. Above all, it is a novelty uncontaminated by procrustean notions of subjectivity and creation.

Rhizomatic Topology and Global Politics. A Flirtatious Relationship.

 

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Deleuze and Guattari see concepts as rhizomes, biological entities endowed with unique properties. They see concepts as spatially representable, where the representation contains principles of connection and heterogeneity: any point of a rhizome must be connected to any other. Deleuze and Guattari list the possible benefits of spatial representation of concepts, including the ability to represent complex multiplicity, the potential to free a concept from foundationalism, and the ability to show both breadth and depth. In this view, geometric interpretations move away from the insidious understanding of the world in terms of dualisms, dichotomies, and lines, to understand conceptual relations in terms of space and shapes. The ontology of concepts is thus, in their view, appropriately geometric, a multiplicity defined not by its elements, nor by a center of unification and comprehension and instead measured by its dimensionality and its heterogeneity. The conceptual multiplicity, is already composed of heterogeneous terms in symbiosis, and is continually transforming itself such that it is possible to follow, and map, not only the relationships between ideas but how they change over time. In fact, the authors claim that there are further benefits to geometric interpretations of understanding concepts which are unavailable in other frames of reference. They outline the unique contribution of geometric models to the understanding of contingent structure:

Principle of cartography and decalcomania: a rhizome is not amenable to any structural or generative model. It is a stranger to any idea of genetic axis or deep structure. A genetic axis is like an objective pivotal unity upon which successive stages are organized; deep structure is more like a base sequence that can be broken down into immediate constituents, while the unity of the product passes into another, transformational and subjective, dimension. (Deleuze and Guattari)

The word that Deleuze and Guattari use for ‘multiplicities’ can also be translated to the topological term ‘manifold.’ If we thought about their multiplicities as manifolds, there are a virtually unlimited number of things one could come to know, in geometric terms, about (and with) our object of study, abstractly speaking. Among those unlimited things we could learn are properties of groups (homological, cohomological, and homeomorphic), complex directionality (maps, morphisms, isomorphisms, and orientability), dimensionality (codimensionality, structure, embeddedness), partiality (differentiation, commutativity, simultaneity), and shifting representation (factorization, ideal classes, reciprocity). Each of these functions allows for a different, creative, and potentially critical representation of global political concepts, events, groupings, and relationships. This is how concepts are to be looked at: as manifolds. With such a dimensional understanding of concept-formation, it is possible to deal with complex interactions of like entities, and interactions of unlike entities. Critical theorists have emphasized the importance of such complexity in representation a number of times, speaking about it in terms compatible with mathematical methods if not mathematically. For example, Foucault’s declaration that: practicing criticism is a matter of making facile gestures difficult both reflects and is reflected in many critical theorists projects of revealing the complexity in (apparently simple) concepts deployed both in global politics.  This leads to a shift in the concept of danger as well, where danger is not an objective condition but “an effect of interpretation”. Critical thinking about how-possible questions reveals a complexity to the concept of the state which is often overlooked in traditional analyses, sending a wave of added complexity through other concepts as well. This work seeking complexity serves one of the major underlying functions of critical theorizing: finding invisible injustices in (modernist, linear, structuralist) givens in the operation and analysis of global politics.

In a geometric sense, this complexity could be thought about as multidimensional mapping. In theoretical geometry, the process of mapping conceptual spaces is not primarily empirical, but for the purpose of representing and reading the relationships between information, including identification, similarity, differentiation, and distance. The reason for defining topological spaces in math, the essence of the definition, is that there is no absolute scale for describing the distance or relation between certain points, yet it makes sense to say that an (infinite) sequence of points approaches some other (but again, no way to describe how quickly or from what direction one might be approaching). This seemingly weak relationship, which is defined purely ‘locally’, i.e., in a small locale around each point, is often surprisingly powerful: using only the relationship of approaching parts, one can distinguish between, say, a balloon, a sheet of paper, a circle, and a dot.

To each delineated concept, one should distinguish and associate a topological space, in a (necessarily) non-explicit yet definite manner. Whenever one has a relationship between concepts (here we think of the primary relationship as being that of constitution, but not restrictively, we ‘specify’ a function (or inclusion, or relation) between the topological spaces associated to the concepts). In these terms, a conceptual space is in essence a multidimensional space in which the dimensions represent qualities or features of that which is being represented. Such an approach can be leveraged for thinking about conceptual components, dimensionality, and structure. In these terms, dimensions can be thought of as properties or qualities, each with their own (often-multidimensional) properties or qualities. A key goal of the modeling of conceptual space being representation means that a key (mathematical and theoretical) goal of concept space mapping is

associationism, where associations between different kinds of information elements carry the main burden of representation. (Conceptual_Spaces_as_a_Framework_for_Knowledge_Representation)

To this end,

objects in conceptual space are represented by points, in each domain, that characterize their dimensional values. A concept geometry for conceptual spaces

These dimensional values can be arranged in relation to each other, as Gardenfors explains that

distances represent degrees of similarity between objects represented in space and therefore conceptual spaces are “suitable for representing different kinds of similarity relation. Concept

These similarity relationships can be explored across ideas of a concept and across contexts, but also over time, since “with the aid of a topological structure, we can speak about continuity, e.g., a continuous change” a possibility which can be found only in treating concepts as topological structures and not in linguistic descriptions or set theoretic representations.

Cartographies of Disjunction’s Relational Dust. Thought of the Day 27.0

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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.

…….

Kenneth Knoespel notes that diagramma in the original Greek does ‘not simply mean something that is marked out by lines, a figure, a form or a plan, but also carries a second connotation of marking or crossing out,’ suggesting not only ephemerality but also an incompleteness that carries an expectation of potential. ‘What is interesting is that the diagram participates in a geneology of figures that moves from the wax tablet to the computer screen […] the Greek setting of diagram suggests that any figure that is drawn is accompanied by an expectancy that it will be redrawn […] Here a diagram may be thought of as a relay. While a diagram may have been used visually to reinforce an idea one moment, the next it may provide a means of seeing something never seen before. Diagrams As Piloting Devices…

 

Indian Classical Music

किन्तु वयमिदानीं ते न शक्नुमः परिचर्यां कर्तुम् : भूयिष्ठां बहुतरां ते नमउक्तिं नमस्कारवचनं विधेम नमस्कारेण परिचरेम ।

kintu vayamidānīṃ te na śaknumaḥ paricaryāṃ kartum : bhūyiṣṭhāṃ bahutarāṃ te namauktiṃ namaskāravacanaṃ vidhema namaskāreṇa paricarema |

But now I am not in a position to serve you; I offer you many verbal salutations; I serve you through salutations.

Hindustani Classical Music (2)

Music has been a cultivated art in India for at least three thousand years. It flows from the essential element of chant in ancient Vedic religious expression. More than any other musical form, the Indian raga tradition structurally and acoustically corresponds to and embodies the spiritual/religious experience. It offers a direct experience of the consciousness of the ancient world, with a range of expression rarely accessible today. All Indian instruments are played as extensions of the ultimate, because most natural, instrument — the human voice — that chants the sacred poems, mantras, and invocations of the gods.

In India music is practiced by members of hereditary guilds, often families, whose traditions remain unbroken for hundreds of years. It is the chamber music of an aristocratic society where the livelihood of the artist does not depend upon his ability and will to amuse the crowd. The musician’s education begins in infancy and he must absorb, thoroughly understand, and reproduce all that preceded him before adding his unique perspective to the living tradition. The listener is expected to respond with an art of his own: he must be technically critical, schooled in appreciation of the spirit of musical experience, contribute an attitude of reverence for the tradition, have a desire to “commune with the gods,” a preference for conviction over prettiness, authenticity over legitimacy, and an appreciation of the song apart from the singer/player.

The European musical scale has been reduced to twelve fixed notes by merging close intervals such as D sharp and E flat — a compromise of necessity in the development of the mathematical harmony that made possible the triumphs of Western orchestration, causing the Western keyboard, unlike instruments from other musical cultures, to be inherently “out of tune.” The Indian scale, on the other hand, covers the same tonal range using a twenty-two note scale to develop a purely melodic art which retains the advantages of pure intonation and modal coloring. What is fixed in Indian music is a group of intervals. The precise vibration value of a note depends on its position in a progression, not on its relation to a tonic. Following the Eastern idea that the emptiness enclosed by the form of a vessel is the actual purpose, essence, or soul of that vessel, the interval more than the note is heard as producing the continuity of sound that is the essence of music. In the Indian tradition the interval is what is sung or played as distinct from the vertical harmonic division of European song and the nature of the sound of keyed and fixed-key instruments. The quarter-tone or sruti is the microtonal interval between two successive scale notes, but as raga themes rarely employ two and never three of the seven primary scale notes in succession, microtones are heard only in ornamentation of the theme. They reveal that which lies unmanifest in the emptiness that is the heart of the vessel of melody composed of the primary scale notes. Sruti also designates the word of the guru, impossible to write but revealed by teacher to student in hushed tones or, more often, as an expression of the essence of understanding from one heart or consciousness to another.

The Indian song form, or raga (literally, coloring or passion), may be best defined as a melody-mold or ground plan of a song. Origins of the ragas are varied but all flow directly from human experience of the spiritual or religious and the responsive feeling (rasa) of love, joy, longing, or devotion. The ragas evoke feelings both human and spiritual. A myth tells of the bird Musikar or Dipaka-Lotus whose beak has seven apertures. Through each of these openings it blows a different note, and at different seasons of the year it combines them to produce ragas specific to the hour of the day and season. An egg was created from the ashes of a fire ignited by the magic sound of a raga; from this egg another Musikar was born, followed by many others. Like all myths, this conveys a truth, that of the ideal of raga — a form growing naturally, like ripples on water, a flower toward the sun, or ice crystals on a leaf of grass, whose beauty and meaning are enhanced by a sympathetic human response to the movement of spirit in the world of matter. The myth suggests the numinous, sacred qualities embodied in the raga form.

We can hear in Indian music the richest correlation of sound with the origins and manifestations of spiritual consciousness. The idea of nonmanifest sound — the essence in the interval between notes — is akin to the New Testament conception of the Word, and underlies and pervades the music. It lies beneath all that is manifest in nature, cosmic and microcosmic, and realizes itself as the multiplicities and differentiations of existence.

Philosophically, this cosmic nonmanifest sound continually creates, destroys, and recreates manifold universes. Its capacities are infinite, therefore measureless. For those who can “hear” it, it brings news of vast starry firmaments and interstellar spaces, of all universes past and all possible worlds of the future, whether those firmaments are galactic, atomic, physical, or spiritual. It is a potency, presence, possibility, and performance all at the same time. In India, music is heard not as a thing that humans make but as an aspect of the divine revealing itself (revelation/sruti) to which the musician and listeners contribute by their skill, understanding, acceptance, and appreciation.

The Dipaka-Lotus bird with its seven natural tones which make up the octave is an analogy of the seven principles or souls of sound, the seven veils of Isis or Prakriti, the seven spheres of resonance which constitute a grama (village or brotherhood), and the seven aspects not only of human but of universal nature.

The ancient Indians and their modern musical descendants believe that to one who understands fully the complex nature of a tone, the innermost secrets of our universe are revealed. Each tone in the raga is considered to have a specific spiritual and emotional charge in relation to the whole. The word svara (tone) is often defined as that which shines by itself. Tones are said by Indian musicians, as well as their ancient Chinese and Sufi brethren, to originate in the heart that responds with a spontaneous sensitivity to the movement of purusha (spirit) through prakriti (matter). The purpose of Indian song is not to dwell on and confirm the confusions of life, but to express and arouse ideal feelings and passions of body and soul in man and nature in response to the impulse of divine spirit. There is a magical aspect to sound, growing from the Vedic chants invoking the divine, though music is heard as essentially impersonal:

it reflects emotions and experiences which are deeper, wider and older than the emotion or wisdom of any single individual. Its sorrow is without tears, its joy without exultation and it is passionate without any loss of serenity. It is in the deepest sense of the words all-human. (Ananda K Coomaraswamy, The Dance of Shiva

In appreciating Indian music we experience and appreciate the consciousness of the ancient world embodied in it. It focuses and expresses the individual’s organic oneness with cosmic and natural forces that are the world we know. The materialistic focus of modern consciousness sees itself as separate from and threatened by nature. Indian music reflects a social order based in the awareness of unity and cooperation rather than on division and competition which leads to economic, social, and cultural insecurity and alienation. Goods produced and services rendered were not based on a perceived economic need for constant expansion leading to exploitation but were generated to serve needs of the organic whole. Ancient Indian consciousness focused, as does its music, on serving the needs of spirit rather than the demands of matter. Though Indian music is ancient it is not primitive: sophistication, subtlety, and assumption of the experience of spirit as the root and goal of all existence can best be described as primal. To appreciate it modern listeners must expand their ability to perceive and express human and cosmic spiritual nature, in much the same way that a child matures the primitive sing-song approach and simplistic rhythmic insistence of nursery school to include the subtlety of expression of which adults are capable.

The objective of the raga is the rasa — the aesthetic emotion — the motif embodied in the melody. As souls inhabit bodies, so every rasa is embodied in the rupa (form) of a particular raga or ragini (feminine form of raga). To invoke rasa, one meditates on the rupa that is appropriate to that raga’s essence, the distillation of mood, mode, time, and season. This meditation is shared by musician and listeners. The experience invoked by a master musician’s meditation on a fine instrument with a knowledgeable, appreciative audience is the disappearance of player, instrument, and listener — pure song, spirit singing itself into being.

Rather than confining melody to the necessities of an intellectualized harmonic concept, Indian musicians and listeners do not attempt to “chain with the mind the feet of the mysterious bird that goes to and away from the cage” (Indian folk song). The bird is pure melody, song of spirit supported by and interacting with the essential, complex rhythms of life. It is the spontaneous response of the heart, that which shines by itself, the spiritual fire of a soul lit by the radiance of nonmanifest sound, the Word, Brahman, Atman, God — divinity containing all worlds within it and evolving all worlds forth into being. The song of Brahman is AUM. Indian musical art is an imitation of the perfect spontaneity with which gods and enlightened beings understand and acknowledge that which is beyond inner and outer, rises above good and evil, is beyond conflict, is the perfection of compassion love and wisdom — the very heart of All.

The omnipresent keynote (Aum) of the universe coming into being swells from the tambura (drone) making a pedal point rich in overtones. Like all that is profound it rewards those who with patient humility seek the divine hidden in the heart of the musical experience. The drone corresponds to Brahman, the Unmanifest Logos, source and ultimate goal of Being. From and against this infinite potentiality the musician draws forth the raga whose rhythm is initially free, with the direction of what is to come subtly implied until the essential elements and graceful implications of this universe/song have been as fully explored as the musician’s inspiration and training allow. At a nod the power of the drums begins slowly to unfold, as Daivi-prakriti (Divine nature; divine will; the vital force of the universe; the “electricity” of cosmic consciousness; the Greek Eros; the Tibetan Fohat. Fohat carries the divine thought to become that which it truly is: a song of wonder at the manifold surprises hidden within and evolving from its Self, a reverential awe at the unmovable serenity from the heart of which dance and flow in waves the myriad, ever-changing aspects of THAT which is one and unchangeable. It is spirit discovering itself. The drone is Brahman, the raga is the world, as artistic microcosmic realization of the macrocosmic experience of spirit.

European rhythms are based on repeated stress, as in marching. Rather than using the bar as the fixed unit and marking its beginning with a stress or accent the Indian musician’s fixed unit is a section, or group of bars which are not necessarily alike. The rhythmic cycle of Ata Tala, for example, is counted as 5 plus 5 plus 2 plus 2. Indian rhythmic complexes count into the fifties, and cycles involving half beats (i.e., 5½, 9½) are now developing in this living musical tradition. But even during the most ecstatic moments of the second stage (gat) of the raga, during which the explicate rhythmic pattern unfolds, the drone remains as the omniscient, omnipotent cause from which proceed the origin, subsistence, and dissolution (Brahma, Vishnu, and Siva) of the raga — of the world. The activity and ecstasy of the musical universe build to a glorious climax then fade away into the drone from which they sprang like myriad bubbles of sunflecked foam that danced briefly on the swells of eternity.

As “one can never step into the same river twice” one can never play a raga exactly the same way twice. The musician seeks to express the uniqueness of the moment: time, season, audience, instrument, planets, musician, and stars will never again occur in the same relationship. Though the river is never the same it is always a river, an aspect of the ocean of divinity made manifest. With the assumption that each dewdrop and river flows from and seeks return to its divine source the musician improvises a spontaneous expression of that journey. The raga form conveys all the joy and grief of being human, yet the final absorption of that experience in Brahman transports all to a state in which the universe is perceived as neither good nor bad but simply as TAT (THAT). The raga manifests this understanding and acceptance in a personal, spontaneous, improvisatory, and fully realized expression of artistic beauty and power. It is the inner reality of things rather than any transient or partial experience that the singer/musician voices.

Computer Algebra Systems (CAS): Mathematica. Note Quote.

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If we are generous, there is one clear analogue to Leibniz’s vision in the contemporary world, and that is a computer algebra system (CAS), such as Mathematica, Maple, or Maxsyma. A computer algebra system is a piece of software which allows one to perform specific mathematical computations, such as differentiation or integration, as well as basic programming tasks, such as list manipulation, and so on. As the development of CAS’s have progressed hand in hand with the growth of the software industry and scientific computing, they have come to incorporate a large amount of functionality, spanning many different scientific and technical domains.

In this sense, CAS’s have some resemblance to Leibniz vision. Mathematica, for example, has all of the special functions of mathematical physics as well as data from many different sources which can be systematically and uniformly manipulated by the symbolic representation of the Wolfram Language. One could reasonably claim that it incorporates many of the basic desiderata of Leibniz’s universal calculus – it has both the structured data of Leibniz’s hypothetical encyclopedia, as well as symbolic means for manipulating this data. 

However, Leibniz’s vision was significantly more ambitious than any contemporary CAS can make claim to have realized. For instance, while a CAS incorporates mathematical knowledge from different domains, these domains are effectively different modules within the software that can be used in a standalone fashion. Consider, for example, that one can use a CAS to perform calculations relevant to both quantum mechanics and general relativity. The existence of both of these capabilities in a single piece of software says nothing about the long-standing theoretical obstacles to creating a unified theory of quantum gravity. Indeed, as has long been bemoaned in the formal verification and theorem proving communities, CAS’s are effectively a large number of small pieces of software neatly packaged into a single bundle that the user interacts with in a monolithic way. This fact has consequences for those interested in the robustness of the underlying computations, but in the present context, it simply serves to highlight a fundamental problem in Leibniz’s agenda.

So in effect, one way to describe Leibniz’s universal calculus, was an attempt to create something like a modern computer algebra system, but which extended across all areas of human knowledge. This goal itself would be quite an ambitious one, but in addition Leibniz wanted the additional property that the symbolic representation should have a transparent relationship to the corresponding encyclopedia, as well as possess the capacity of mnemonics to be memorized with ease. To quote Leibniz (caution: German) himself,

My invention contains all the functions of reason: it is a judge for controversies; an interpreter of notions; a scale for weighing probabilities; a compass which guides us through the ocean of experience; an inventory of things; a table of thoughts; a microscope for scrutinizing things close at hand; an innocent magic; a non-chimerical cabala; a writing which everyone can read in his own language; and finally a language which can be learnt in a few weeks, traveling swiftly across the world, carrying the true religion with it, wherever it goes.

It difficult to not be swept away by the beauty of Leibniz’s imagery. And yet, from our modern vantage point, there is hardly a doubt that this agenda could not possibly have worked.

Psychological Approaches to Cognition and Rationality in Political Science

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The theoretical basis of information processing in politics comes largely from psychologists studying other issues and from fields outside the realm of psychology. We assume that the task of translating available information into sets of focused, legally consistent beliefs, judgement and commitments is a subtle process, not at all a straightforward or an obvious issue, and furthermore, although political reasoning may take place largely outside a person’s awareness, political cognition is a very active mental process. Cognitive theories in politics are largely bent on understanding as to how people selectively attend to, interpret and organise information in ways that allow everyone to reach coherent understandings. For various reasons known or unknown to all of us now, such understandings may deviate substantially from the true set of affairs and from whatever mix of information or disinformation is available to be considered.

The two terms ‘belief’ and ‘system’ have been a familiar part of the language of attitude psychology for decades. Let us define ‘belief system’ in a three point structure:

  • a set of related beliefs, attitudes together
  • rules of how these contents of mind are linked to one another
  • linkages between the related beliefs and ideologies.

Now to model a belief system is an attempt to create an abstract representation of how someone’s set of related beliefs are formed, organised, maintained and modified.

In the cognition of psychology, or vice versa, a commonality of two terms is being used nowadays in the form of schemas and scripts with the former being the building blocks about peoples’ beliefs about the globe, and the latter being the working assemblages of those building blocks, with an adequate emphasis on sequences of actions involving various schematised objects. Scripts are our images of events routinely take place; events involve objects and activities for which schematic representations are available. For e.g., we might have a script of leadership in mixed age groups. Men routinely expect to exercise leadership over women and bristly at having to be subordinate, while women expect the same status differentiation and fear the wrath of others if women acquire leadership responsibilities. This hypothesised script incorporates schemas about the nature of leadership, about gender differences and about emotions such as resentment and fear. Just as schemas organise our understanding of concepts and objects, scripts organise our understanding about about activities and events  that involve and link those objects and concepts.

Much of the modern social psychology is concerned with the attribution processes. These refer to subjective connections people make between cause and effect. Attribution processes, by their nature involve going beyond the ‘information given’ in the direct observation of events. They are inferential process that allow us to understand what we think are the meaningful causes, observations and motivations underlying the observable behaviour directly. they are the central elements of the broader constructive processes through which people find meaning in ongoing events. Regardless of how well our attributive reasoning corresponds with objective reality, attribution process provide us with an enhanced sense of confidence that we understand what is going on around us. two kinds of attributive processes are heuristics and biases, when the former can be considered as mental short-cuts, by which one is able to circumvent the tediousness of logically exhaustive reasoning, or to fill in lacunae in our knowledge base and reach conclusions that make sense to our already made up assumptions.

Biases can be thought of as tendencies to come to some kind of conclusions more often than others. We have to often take the short cut of relying on representations of some bit of information, while ignoring other factors that also should be taken into account. We have to attach probabilities. Suppose a foreign service analyst anted to know whether a move by a foreign government to increase security at border was part of a larger plan to prepare for a surprise military attack across the border. the cue for the analyst is border clampdown; one possible meaning is that a military invasion is about to begin. The analyst must decide how likely it is. If the analyst uses the representativeness heuristic, she would decide how typical a border crackdown is as a early sign of a coming invasion. The more typically she feels the border clampdown is a sign of coming invasion, the more credibility she would attach to that interpretation of the change. In fact, representativeness, the degree to which some cue resembles or fits as part of the typical form of an interpretation, is an important and a legitimate aspect of assessing probabilities. The representativeness of heuristic however is the tendency to ignore other relevant information and thereby overemphasise the role of representativeness. Representativeness is one of the most prominently and actively investigated cognitive heuristics. Of course in most real life settings it cannot be proven that we credit or blame the actor too much for her behaviour or consequences. However, in carefully designed experiments in which hapless actors obviously have very little control over what happens to them, observers nonetheless hold the actors responsible for their actions and circumstances.

Now moving on to integrative complexity, it is a combination of two distinct mental activities, differentiation and integration. Differentiation refers to person’s recognition of multiple issues or facets in thinking about a political problem. Undifferentiated thinking occurs when an individual sees a problem as involving very few distinct issues, or that all of the issues nearly lead to the same conclusion. Differentiating one’s understanding of political situation gives one a better grasp on that situation, but it can cause difficulties too. different aspects of a political problem may contradict each other or may lead to contradictory actions. differentiating a problem can also lead a decision maker to the discovery that she really does not have a full grasp on the relevant information, which can be an unpleasant awareness, especially when decisions are to made immediately.

Integration on the other hand refers to the careful consideration of the relationships among parts of the problem. as a political actor formulates opinions and possible choices, integrated thinking allows the person to see how various courses of action may lead to contradictory outcomes, how goals might be well set by actions that violate one’s presuppositions or outcomes. Integration moves the thinker away from all or nothing oversimplification of issues. thus it improves the chances for political compromise, the heart of successful diplomacy. furthermore, by opening up the eyes of the decision maker to the complex interconnections of many political problems, it enables her to anticipate the complicated consequences that may follow from her choices. Obviously, high levels of integration can occur when an individual or a group has successfully differentiated the various issues involved in a problem. without the identification of the issue, there is nothing to integrate. however, simple awareness of all of the potentially conflicting aspects of a problem does not guarantee that a decision maker will pull these elements meaningfully. On can recognise any number of ambiguous qualifications, contradictions and non-sequitors, yet ignore most of them in deciding what to believe and what to do. Thus integration requires differentiation, but generally vice versa does not follow.

Integrating complexity may affect the careers of political leaders. It may also help shape the outcome of entire political and military conflicts, not just the future carer of leaders. For e.g., intense diplomatic activity between the US and the USSR averted a potential WW3, which arose in 1962 when the US objected to the Soviet missile deployment in Cuba. Taking the above case, it was hypothesised that in very complex political situations, highly integrated thinking is necessary in order for leaders to discover the availability and superiority of non-military solutions.

Everyone knows that attitudes about a political problem influence our political actions. Exceptions are there, but people usually act in ways that further their beliefs avoid acting in ways that contradict their beliefs. We no longer claim that the causal link from beliefs to behaviour is simple; instead, attention is now directed towards understanding the complex and subtle ways in which beliefs influence decision-making. General beliefs are considered to be less general in predicting actions such as voting behaviour. Some also maintain that general beliefs are important influences on specific actions, though the influence is not a direct cause-effect link. Instead, general beliefs produce subtle tendencies to favour some interpretation of events over other plausible interpretations, and to favour some general styles of political action over others when choosing a specific political action. Talking of political actor’s operational code, there are diagnostic propensities which are tendencies to interpret ambiguous events in some ways rather that in others, to search for certain kinds of information rather than others, and to exaggerate or ignore the probable role of chance and uncontrollable events. For eg. one national leader may immediately look for the hostile intentions behind any important diplomatic move on the part of arrival nation. Such a person would search for other evidence confirming his or her initial presumption, by contrast, another leader might be aware that the rival nation has severe internal problems, and presume that any important foreign policy initiatives from that nation are attempts to distract its citizens from those problems. Choice propensities are tendencies to prefer certain modes of political action to others. Diagnostic propensities are the expressions in political reasoning of leader’s general views about how to act effectively in political arena.

Politics in its very essence is an impersonal activity. The vast bulk of political planning, commitment and actions take place among groups of people, whether these people come together to pool resources, squabble, or negotiate compromises among their conflicting group interests. What is then the psychology of rationality in political groups? But groups are different. Groups do not negate the picture about the nature of political cognition; they complicate it instead. Groups themselves do not think. It is still the individual people who share or hide their personal beliefs and goals.

What is a camel?
It’s a horse driven by a committee.

This old joke is a cynical comment on the creativity of committees. It is easy to point to mediocre decisions made by groups, but there is a more serious problem than middling decisions. Groups are capable of profoundly bad decisions. Some of the worst decisions in world history were made by groups that would seem to have been assembled in producing rational, creative policies and judgments.

What characteristics make groups particularly susceptible to poor decisions? First and foremost, the group is highly cohesive. Group members know, trust and like each other; they often share common or similar histories; they enjoy being part of the group and value working in it. Second, the group isolates itself from possible influencing of the others. A strong sense of identification with the group leads to lost ties with others who might have some valuable information to share. Third, the group lacks any systematic way of doing things. without formal guidelines for procedure, agenda decisions are made casually and are subject to influences that cut full deliberations. Fourth, the leader of such groups tend to be directive. Fifth, the group is experiencing stress, with a sense of urgency about some crises in which acting quickly seems critical. The choice may be among some unpleasant activities, the available information may be very confusing and incomplete and the group members may be fatigued. Thus solidarity, isolation, sloppy procedures and directive leadership in a stressful situation make some groups vulnerable to groupthink. Two features describe groupthink. First set contains working assumptions and styles of interacting that group members carry with them into the work setting. The second set features describe faulty deliberations as the group sets about its task. The group members lack adequate contingency plans to be prepared for quick response if the preferred course of action does not work as the group hopes and believes it will.

To avoid groupthink, first the leader of the group should actively encourage dissent; she should make it known that dissenting opinions are valued and they are valued not just for variety’s sake but because that they may be right. Second, the leader should avoid letting her own initials be known. Third, parallel subgroups can be set up early on to work separately on the same tasks. These subgroups will probably develop different assessments and plans, which can be brought to the whole group for consideration. This neatly disrupts the tendency of groups to focus on just option for the upcoming decision. A choice is rational if it follows certain careful procedures that lead to the selection of the course of action that offers the greatest expected value or benefit or utility for the chooser. The group members making a rational decision first identify the opportunity and need for a choice. They then identify every conceivable course of action available to them. They determine all possible consequences of each course of action available to them. They evaluate each possible consequence in terms of,

1) its likelihood of occurrence,
2) its value if it does occur.

Now the decision making group has a problem, and a set of possible solutions. This information is then distilled into a single choice by working backwards. The probability of each consequence is then multiplied by its value; the products of all consequences for each course of action are then added up. The resulting sums are the expected values of each possible consequence. The group then simply selects the option with the largest possible expected value (or smallest negative value if a choice is a no-win situation).

There is something called the posterior rationality, where the choice process is discovered after the choice is made. The world may be too unpredictable and complicated for most well intended plans to have much chance of success. If so, traditional rationality may be irrelevant as a model for complex organisations. However, goals and intentions can still be inferred in reverse, by reinterpreting earlier choices and redefining one’s original goals.

In conclusion, political actors, groups and institutions such as governments do not simply observe and understand political circumstances in some automatic fashion that accurately captures true political realities. Political realities are for most part social constructions and the construing process is built on the philosophy and psychology of human cognition. Political cognition like any other cognition is extremely complex. It is easy enough to find examples of poor political judgments: the wonder may be that politics often seems to be rational, given all the challenges and limitations. To the extent that we can find a sense of coherence in politics and government, we should acknowledge the importance of the social construction process in shaping that coherence. Although political activists devote much more time to the political agenda than does the average citizen, still they rely on the same cognitive resources and procedures and hence are subject to the same biases and distortions as any thinking person.