Read an interesting experiment in programing using Kurt Gödel’s number theories: (Norman H. Cohen. Gödel numbers: A new approach to structured programming. SIGPLAN Notices 15, No. 4 (April 1980), pp. 70-74; download pdf at bottom part of page):
My reason for researching this had to do with another investigation into Nick Land’s use of Gödel. As Mackay and Brassier note,
One of the tasks of schizoanalysis has now become the decrypting of the ‘tics’ bequeathed to the human frame by the geotraumatic catastrophe, and ‘KataςoniX’ treats vestigial semantic content as a mere vehicle for code ‘from the outside’: the ‘tic’ symptoms of geotraumatism manifested in the shape of sub-linguistic clickings and hissings. Already disintegrated into the number-names of a hyperpagan pantheon, syncretically drawing on the occult, nursery rhyme, anthropology, SF and Lovecraft, among other sources, the ‘subterranean current of impressions, correspondences, and analogies’(Artaud) beneath language is now allowed uninhibited (but rigorously-prepared) development, in an effort to corporeally de-engineer the organicity of logos.
The element of these explorations remains the transformed conception of space vividly exhibited in Gibsonian cyberpunk and which is a crucial component in Land’s writings, a powerful bulwark against Kant’s architectonic ambition to subsume all space under unity. Coding and sequencing mechanisms alone now construct intensive space, and this lies at the core of Land’s typology of number, since dimensionality is a consequence of stratification. Naming and numbering converge in counting, understood as immanent fusion of nomination and sequencing. No longer an index of measure, number becomes diagrammatic rather than metric. From the perspective of Land’s ‘transcendental arithmetic’, the Occidental mathematisation of number is denounced as a repressive mega-machine of knowledge – an excrescent outgrowth of the numbering practices native to exploratory intelligence – and the great discoveries of mathematics are interpreted as misconstrued discoveries about the planomenon (or plane of consistency), as exemplified by Gödel’s ‘arithmetical counterattack against axiomatisation’.Land eschews the orthodox philosophical reception of Gödel as the mathematician who put an end to Hilbert’s dream of absolute formal consistency, thus opening up a space for meta-mathematical speculation. More important, for Land, are the implications of Gödel’s ‘decoded’ approach to number, which builds on the Richard Paradox, generated by the insight that numbers are, at once, indices and data. [my italics]
The Gödel episode also gives Land occasion to expand upon the theme of the ‘stratification’ of number: according to the model of stratification, as the ‘lower strata’ of numbers become ever more consolidated and metrically rigidified, their problematic component reappears at a ‘higher’ strata in the form of ‘angelic’ mathematical entities as-yet resistant to rigorous coding. A sort of apotheosis is reached in this tendency with Gödel’s flattening of arithmetic through the cryptographic employment of prime numbers as numerical ‘particles’, and Cantor’s discovery of ‘absolute cardinality’ in the sequence of transfinites.
Thus for Land the interest of Gödel’s achievement is not primarily ‘mathematical’ but rather belongs to a lineage of the operationalisation of number in coding systems that will pass through Turing and into the technological mega-complex of contemporary techno-capital.
By using arithmetic to code meta-mathematical statements and hypothesising an arithmetical relation between the statements – an essentially qabbalistic procedure – Gödel also indicates the ‘reciprocity between the logicisation of number and the numerical decoding of language’, highlighting a possible revolutionary role for other non-mathematical numerical practices. As well as reappraising numerology in the light of such ‘lexicographic’ insights, the mapping of stratographic space opens up new avenues of investigation – limned in texts such as ‘Introduction to Qwernomics’ – into the effective, empirical effects of culture – chapters of a ‘universal history of contingency’ radicalising Nietzsche’s insight that ‘our writing equipment contributes its part to our thinking’. The varieties of ‘abstract culture’ present in games, rhythms, calendrical systems, etc., become the subject of an attempt at deliberate, micro-cultural insurrection through number, exemplified in the CCRU’s ‘hyperstitional’ spirals and the ‘qwertypological’ diagrams that in the end merge with the qabbalistic tracking of pure coding ‘coincidences’. Ultimately, it is not just a question of conceiving, but of practicing new ways of thinking the naming and numbering of things. Importantly, this allows Land to diagnose the ills of ‘postmodernism’ – the inflation of hermeneutics into a generalised historicist relativism – in a manner that differs from his contemporaries’ predominantly semantic interpretations of the phenomenon, and to propose a rigorous intellectual alternative that does not involve reverting to dogmatic modernism.1
Against Badiou and his followers of Platonic materialist measure, Land’s insight is to follow Deleuze and Guattari: “No longer an index of measure, number becomes diagrammatic rather than metric. From the perspective of Land’s ‘transcendental arithmetic’, the Occidental mathematisation of number is denounced as a repressive mega-machine of knowledge – an excrescent outgrowth of the numbering practices native to exploratory intelligence – and the great discoveries of mathematics are interpreted as misconstrued discoveries about the planomenon (or plane of consistency), as exemplified by Gödel’s ‘arithmetical counterattack against axiomatisation’.”
This leads to a notion of a-signifying systems as opposed to signifying, which brings us back to Land’s “No longer an index of measure, number becomes diagrammatic rather than metric.” We learn from Deleuze’s and Guattari’s Kafka: Toward a Minor Literature that the minor writer engages ‘a machine of expression capable of disorganizing its own forms, and of disorganizing the forms of content, in order to liberate pure contents which mingle with expression in a single intense matter’ (K 51).
Exactly how this revolutionary practice works is not clearly delineated in Kafka, for Deleuze and Guattari offer no satisfactory examples of the process of transformation which leads from deterritorialized sound to a dissolution and reconstruction of content. Some clarification of this process may be gained, however, from a consideration of Deleuze’s analysis of Francis Bacon’s approach to painting in Francis Bacon: The Logic of Sensation (1981). Deleuze notes that for modern artists, the blank canvas is not a tabula rasa, but the space of unconscious visual preconceptions and received conventions of representation, which the artist brings to the canvas and which he struggles against and tries to vanquish, escape, or subvert. For Francis Bacon, the moment of subversion comes during the process of painting when a chance stroke of the brush introduces a small locus of chaos, a limited catastrophe that Bacon calls a ‘diagram’.
‘The diagram’, says Deleuze, ‘is indeed a chaos, a catastrophe, but also a seed of order or of rhythm’ (FB 67). Bacon follows the suggested form, colour or line of this diagram and uses it as a generative device for constructing an intensive set of relations within the painting itself, which simultaneously deform the figure he started to paint and form a new figure of that deformed figure. Deleuze contrasts Bacon’s practice with that of abstract formalists, such as Mondrian and Kandinsky, and abstract expressionists, such as Pollock. The danger of abstract formalism is that the constraints of representation may simply be replaced with those of an abstract code, in which case the diagrammatic possibilities of chaos or catastrophe are banished from the canvas. The danger of abstract expressionism is that the diagram may cover the whole canvas and result in nothing but an undifferentiated mess. Bacon’s strategy is to paint portraits and studies of human figures, and hence to remain in a certain sense within the confines of representation, but to allow the diagram in each painting to deterritorialize the human subject, to introduce ‘a zone of Sahara into the head’, to split ‘the head into two parts with an ocean’ (FB 65), to make a leg melt into a puddle of purple or a body start to turn into a piece of meat. One finds resemblances between the configurations of paint and human figures, deserts, oceans, puddles, and rolled roasts, yet such resemblances are no longer productive, but simply produced. A resemblance may be said to be produced rather than productive ‘when it appears suddenly as the result of entirely different relations than those which it is charged with representing: resemblance then surges forth as the brutal product of non-resembling means’ (FB 75).1
An abstract machine is characterized by its matter – its hecceities, or relations of speeds and affects – but also by its function. The abstract machine of panopticism, for example, consists of a ‘pure matter’, a human multiplicity, and a ‘pure function’, that of seeing without being seen. What is important to note is that this function is neither semiotic nor physical, neither expression nor content, but an abstract function that informs both the expression-form of the discourse on delinquency and the content-form of the prison. Such an abstract function, characteristic of every abstract machine, Deleuze and Guattari call a ‘diagram’. Semioticians generally classify diagrams as simplified images, or icons, of things. But as Guattari points out, the image represents both more and less than a diagram; the image reproduces numerous aspects which a diagram does not retain in its representation, whereas the diagram brings together the functional articulations of a system with much greater exactitude and efficacy than the image. (Bogue, p. 135)
Visual graphs and charts are diagrams, but so are mathematical formulae, musical scores, and models in particle physics; and the more abstract the diagram is, the less it represents any particular thing, and the less it can be conceived of in terms of expression and content. Mathematical equations articulate a self-referential system of relations which may be embodied in diverse contexts. Musical scores, although heavily ‘coded’ in traditional music (specific designations of instruments, tempi, and so on), in much electronic music function as abstract diagrams of differential speeds and intensities which a synthesizer embodies in various sounds. Models in particle physics fuse mathematical theories and experimental particles (theories isolating particles and particles generating theories) to such an extent that one may speak no longer of particles or signs, but of ‘particle-signs’, units in a self-referential experimental-theoretical complex. The function of an abstract machine is a diagram of this sort, a function ‘which has only “traits”, of content and expression, whose connection it assumes: one can no longer even say whether a trait is a particle or a sign’ (MP 176). Thus, in an abstract machine, content and expression yield to ‘a content-matter which presents only degrees of intensity, resistance, conductibility, heatability, stretchability, speed or slowness; an expression-function which presents only “tensors”, as in a mathematical or musical notation’ (MP 176-7). (Bogue, p. 135)
Indices and Data
So in the above when Bogue speaks of deterritorializeingthe human subject we should thinkg ‘decoding’ which is at the heart of Landian non-dialectical materialism. Land eschews the orthodox philosophical reception of Gödel as the mathematician who put an end to Hilbert’s dream of absolute formal consistency, thus opening up a space for meta-mathematical speculation. More important, for Land, are the implications of Gödel’s ‘decoded’ approach to number, which builds on the Richard Paradox, generated by the insight that numbers are, at once, indices and data. (Land, Nick. Fanged Noumena: Collected Writings 1987 – 2007, ed. Robin Mackay and Ray Brassier).
This notion of numbers as ‘indices and data’ underlies the diagrammatic a-signifying theories of information of our digital age, and go to the heart of Deleuze’s conceptions of Societies of Control that modulate both individual and dividual by way of both the older form of discipline (Foucault) and newer forms of control (Deleuze). Such works as Ronald E. Day’s ‘Indexing It All: The Subject in the Age of Documentation, Information, and Data’ and others support as shift in the production of subjectivity showing the transition as indexes went from being explicit professional structures that mediated users and documents to being implicit infrastructural devices used in everyday information and communication acts. Doing so, he also traces three epistemic eras in the representation of individuals and groups, first in the forms of documents, then information, then data. Day investigates five cases from the modern tradition of documentation. He considers the socio-technical instrumentalism of Paul Otlet, “the father of European documentation” (contrasting it to the hermeneutic perspective of Martin Heidegger); the shift from documentation to information science and the accompanying transformation of persons and texts into users and information; social media’s use of algorithms, further subsuming persons and texts; attempts to build android robots — to embody human agency within an information system that resembles a human being; and social “big data” as a technique of neoliberal governance that employs indexing and analytics for purposes of surveillance. Finally, Day considers the status of critique and judgment at a time when people and their rights of judgment are increasingly mediated, displaced, and replaced by modern documentary techniques.
- Bogue, Ronald (2008-03-07). Deleuze and Guattari (Critics of the Twentieth Century) (p. 120-122). Taylor and Francis. Kindle Edition.
- Land, Nick (2013-07-01). Fanged Noumena: Collected Writings 1987 – 2007 (Kindle Locations 620-627). Urbanomic/Sequence Press. Kindle Edition.