Three Thousand Years of Algorithmic Rituals. The Emergence of AI from the Computation of Space

Di MATTEO PASQUINELLI.

With topographical memory, one could speak of generations of vision and even of visual heredity from one generation to the next. The advent of the logistics of perception and its renewed vectors for delocalizing geometrical optics, on the contrary, ushered in a eugenics of sight, a pre-emptive abortion of the diversity of mental images, of the swarm of image-beings doomed to remain unborn, no longer to see the light of day anywhere.

—Paul Virilio, The Vision Machine[1]

Recomposing a dismembered god

In a fascinating myth of cosmogenesis from the ancient Vedas, it is said that the god Prajapati is shattered into pieces by the act of creating the universe. After the birth of the world, the supreme god is found dismembered, unstrung. In the corresponding Agnicayana ritual, Hindu devotees symbolically recompose the fragmented body of the god by building a fire altar according to an elaborate geometrical plan.[2] The fire altar is laid down by aligning thousands of bricks of precise shape and size that draw the profile of a falcon. Each brick is numbered and placed reciting its dedicated mantra, following step-by-step instructions. Each layer of the altar is built on top of the previous one respecting the same area and shape. Solving a logical riddle that is the key of the ritual, each layer must keep the same shape and area of the contiguous ones but a different configuration of bricks. Finally, the falcon altar must face east, prelude to a symbolic flight of the reconstructed god towards the rising sun, example of divine reincarnation by geometric means.

The Agnicayana ritual is described in the Shulba Sutras, composed around 800 BCE in India to record a much older oral tradition. The Shulba Sutras teach the construction of altars of specific geometric forms to secure gifts from the gods: for instance, they suggest that “those who wish to destroy existing and future enemies should construct a fire-altar in the form of a rhombus.” [3] The complex falcon shape of the Agnicayana evolved gradually from a schematic composition of only seven squares. In the Vedic tradition, it is said that the Rishi vital spirits created seven square-shaped Purusha (cosmic entities, or persons) that together composed a single body, and it was from this form that Prajapati emerged once again. While art historian Wilhelm Worringer argued in 1907 that primordial art was born in the abstract line found in cave graffiti, one may assume that the artistic gesture also emerged by composing segments and fractions, introducing forms and geometrical techniques of growing complexity.[4] In his studies of Vedic mathematics, Italian mathematician Paolo Zellini has discovered that the Agnicayana ritual was used to transmit techniques of geometrical approximation and incremental growth, namely algorithmic techniques, comparable to the modern calculus of Leibniz and Newton.[5] Agnicayana is among the most ancient documented rituals still practiced today in India, and a primordial example of algorithmic culture.

But how can we define a ritual as ancient as the Agnicayana as algorithmic? To the many, it may appear an act of cultural appropriation to read ancient cultures through the paradigm of the latest technologies. Nevertheless, claiming that abstract techniques of knowledge and artificial meta-languages belong uniquely to the modern industrial West is also a forced act and one of implicit epistemic colonialism towards cultures of other places and other times.[6] The French mathematician Jean-Luc Chabert has noted that: “Algorithms have been around since the beginning of time and existed well before a special word had been coined to describe them. Algorithms are simply a set of step by step instructions, to be carried out quite mechanically, so as to achieve some desired result.”[7] Though today some may see algorithms as a recent technological innovation implementing abstract mathematical principles, on the contrary, algorithms are among the most ancient and material practices, predating many human tools and all modern machines:

Algorithms are not confined to mathematics […]. The Babylonians used them for deciding points of law, Latin teachers used them to get the grammar right, and they have been used in all cultures for predicting the future, for deciding medical treatment, or for preparing food. […] We therefore speak of recipes, rules, techniques, processes, procedures, methods, etc., using the same word to apply to different situations. The Chinese, for example, use the word shu (meaning rule, process or stratagem) both for mathematics and in martial arts. […] In the end, the term algorithm has come to mean any process of systematic calculation, that is a process that could be carried out automatically. Today, principally because of the influence of computing, the idea of finiteness has entered into the meaning of algorithm as an essential element, distinguishing it from vaguer notions such as process, method or technique.[8]

Before the consolidation of mathematics and geometry, ancient civilizations were already large machines of social segmentation that marked human bodies and territories with abstractions that remained, and will continue to remain operative for millennia. Drawing also on the work of historian Lewis Mumford, Gilles Deleuze and Félix Guattari offered a list of such old techniques of abstraction and social segmentation: “tattooing, excising, incising, carving, scarifying, mutilating, encircling, and initiating.”[9] Numbers were already components of the primitive abstract machines of social segmentation and territorialization that would make human culture emerge: the first recorded census took place around 3800 BCE in Mesopotamia, for instance. Logical forms that were made out of social ones, numbers materially emerged by labor and rituals, by discipline and power, by marking and repetition.

In the 1970s, the field of ethnomathematics began to foster a break from Platonic loops of elite mathematics, revealing the historical subjects behind computation.[10] The political question at the center of the debate on computation and the politics of algorithms appears very simple in the end, as Diane Nelson has reminded us: Who counts?[11] Who computes? Algorithms and machines do not compute for themselves, they always count for someone else, for institutions and markets, for industries and armies.

 

What is an algorithm?

The term algorithm comes from the Latinization of the name of the Persian scholar al-Khwarizmi. His tractate On the Calculation with Hindu Numerals, written in Baghdad in the 9^th century, is responsible for introducing Hindu numerals into the West and the corresponding new techniques to calculate with them, namely “algorithms.” In fact, the medieval Latin word algorismus referred to the procedures and shortcuts to make the four mathematical operations (addition, subtraction, multiplication, and division) with Hindu numerals. Later, the term algorithm would metaphorically denote any step-by-step logical procedure and become the core of computing logic. In general, we can distinguish three stages in the history of the algorithm: in ancient times the algorithm can be recognised in procedures and codified rituals to achieve a specific goal and transmit rules; in the Middle Ages the algorithm was the name of a procedure to help mathematical operations; in modern times the algorithm qua logical procedure becomes fully mechanized and automated by machines and then digital computers.

Looking at ancient practices such as the Agnicayana ritual and the Hindu rules for calculation, a basic definition of algorithm can be sketched that would be compatible with modern computer science: (1) an algorithm is an abstract diagram that emerges from the repetition of a process, an organization of time, space, labor, and operations: it is not a rule that is invented from above but emerges from below; (2) an algorithm is the division of this process into finite steps in order to perform and control it efficiently; (3) an algorithm is a solution to a problem, an invention that bootstraps beyond the constrains of the situation: any algorithm is a trick; (4) most importantly, an algorithm is an economic process, as it must employ the least amount of resources in terms of space, time and energy, adapting to the limits of the situation.

Today, amidst the expanding capacities of AI, there is a tendency to perceive algorithms as an application or imposition of abstract mathematical ideas upon concrete data. On the contrary, the genealogy of the algorithm shows that its form has emerged from material practices, from a mundane division of space, time, labor, and social relations. Ritual procedures, social routines, and the organization of space and time are the source of algorithms, and in this sense they existed even before the rise of complex cultural systems such as mythology, religion, and especially language. In terms of anthropogenesis, it could be said that algorithmic processes encoded into social practices and rituals were what made numbers and numerical technologies emerge, and not the other way around. Modern computation, jusy looking at its industrial genealogy in the workshops studied by both Charles Babbage and Karl Marx, evolved gradually from concrete towards increasingly abstract forms. 

The rise of machine learning as computational space

In 1957, at the Cornell Aeronautical Laboratory in Buffalo, New York, the cognitive scientist Frank Rosenblatt invented and constructed the Perceptron, the first operative artificial neural network—grandmother of all the matrices of machine learning, at the time classified as a military secret.[12] The first prototype of the Perceptron was an analogue computer composed of an input device of 20×20 photocells (called “retina”) connected through wires to a layer of artificial neurons that resolved into one single output (a light bulb turning on or off, to signify 0 or 1). The “retina” of the Perceptron recorded simple shapes such as letters and triangles and passed electric signals to a multitude of neurons that would compute a result according to a threshold logic. The Perceptron was a sort of photo camera that could be taught to recognize a specific shape, i.e. to take a decision with a margin of error (making it an “intelligent” machine). The Perceptron was the first machine learning algorithm, a basic binary classifier that could determine whether a pattern fell within a specific class or not (whether the input image was a triangle or not, a square or not, etc.). To achieve this, the Perceptron progressively adjusted the values of its nodes in order to resolve a large numerical input (a spatial matrix of 400 numbers) into a simple binary output (0 or 1). The Perceptron gave the result 1 if the input image was recognized within a specific class (a triangle, for instance), otherwise it gave the result 0. Initially a human operator was necessary to train the Perceptron with the correct answers (manually switching to 0 or 1 the output node), hoping that the machine, on the basis of these supervised associations, would recognize correctly similar shapes in the future. The Perceptron was designed not to memorize a specific pattern but to learn how to recognize potentially any pattern.

The matrix of 20×20 photoreceptors of the first Perceptron was the beginning of a silent revolution in computation that became a hegemonic paradigm with Deep Learning in the early twenty-first century. Although inspired by biological neurons, strictly from the logical point of view, the Perceptron marked a topological turn in computation, not a biomorphic one, that is the rise of the paradigm of computational space or self-computing space. This turn introduced a second spatial dimension into a paradigm of computation that until then maintained a linear dimension (see the Turing machine that reads and writes 0 and 1 along a linear memory tape). This topological turn, that is the core of what people perceive today as “AI” can be described, more modestly, as the passage from a paradigm of passive information to the one of active information. Rather than have a visual matrix processed by a top-down algorithm (like any image edited by a software for computer graphics nowadays), the pixels of the visual matrix dictate the rules for their computation bottom-up according to their spatial disposition. Visual data themselves begin to shape the algorithm for their computation according to their spatial relations.

Because of its spatial logic, the branch of computer science originally dedicated to neural networks was called (nota bene) computational geometry. The paradigm of computational space or self-computing space share common roots in the studies on self-organization principles, that were at the center of post-WWII cybernetics, such as John von Neumann’s cellular automata (1948) and Konrad Zuse’s Rechnender Raum by (1967).[13] The cellular automata by von Neumann are cluster of pixels, perceived as small cells on a grid, that change status and move according to their neighboring cells, composing geometrical figures that resemble evolving forms of life. Cellular automata have been used to simulate evolution and to study complexity in biological systems but they remain finite state algorithms confined to a rather limited universe. Konrad Zuse (who built the first programmable computer in Berlin in 1938) attempted to extend the logic of cellular automata to physics and to the whole universe. His idea of Rechnender Raum, or calculating space, is a universe that is composed of discrete units that behave according to the behavior of neighboring units. Alan Turing’s last essay “The Chemical Basis of Morphogenesis” (published in 1952, two years before his death) also belongs to the tradition of self-computing structures.[14] Turing considered molecules in biological systems as self-computing actors capable of explaining complex bottom-up structures, such as tentacle patterns in the hydra, whorl arrangement in plants, gastrulation in embryos, dappling in animal skin, and phyllotaxis in flowers.[15]

Von Neumann’s cellular automata and Zuse’s computational space are intuitively easy to understand as spatial models, while Rosenblatt’s neural network displays a more complex topology that requires more attention. Indeed, neural networks employ the most complex combinatorial structure, which is probably what makes them the most efficient algorithms of machine learning. Neural networks are said to “solve any problem,” meaning they can approximate the function of any pattern according to the Universal Approximation theorem (given enough layers of neurons and computing resources). All systems of machine learning, including Support Vector Machines, Markov Chains, Hopfield Networks, Boltzmann Machines, and Convolutional Neural Networks, to name just a few, started as models of computational geometry, in this sense as part of the ancient tradition of ars combinatoria.[16]

The automation of visual labor

Even at the end of the twentieth century, no one would have ever thought to call a truck driver a cognitive worker, an intellectual. At the beginning of the twenty-first century, the use of machine learning in developing self-driving vehicles has imposed a new understanding of manual skills such as driving, revealing how the most valuable component of work, generally speaking, has never been merely manual, but also social and cognitive (as well as perceptual, a form of labor still waiting to be located somewhere between the manual and the cognitive). What kind of work do drivers perform? Which human task is AI coming to record with its sensors, imitate with its statistical models, and replace with automation? The best way to answer this question is to look at what technology has successfully automated, as well as what it hasn’t.

The industrial project to automate driving has made clear (more so than a thousand books on political economy) that the labor of driving is a conscious activity following codified rules and spontaneous social conventions. However, if the skill of driving can be translated into an algorithm, it would be because driving has a logical and inferential structure. Driving is a logical activity just as labor is a logical activity more generally. This postulate helps to resolve the trite dispute about the separation between manual labor and intellectual labor.[17] It is a political paradox that the corporate development of AI algorithms for automation made possible to recognize in labor a cognitive component that had long been neglected by critical theory. What is the relation between labor and logic? This becomes a crucial philosophical question for the age of AI.

A self-driving vehicle automates all the micro-decisions that a driver must make on a busy road. Its artificial neural networks learn, that is imitate and copy, the human correlations between the visual perception of the road space and the mechanical actions of vehicle control (steering, accelerating, stopping) as ethical decisions taken in a matter of milliseconds when dangers arise (for the safety of persons inside and outside the vehicle). It becomes clear that the job of driving requires high cognitive skills that cannot be left to improvisation and instinct, but also that quick decision-making and problem-solving are possible thanks to habits and training that are not completely conscious. Driving remains essentially also a social activity, which follows both codified rules (with legal constraints) and spontaneous ones, including a tacit “cultural code” that any driver must subscribe to. Driving in Mumbai—it has been said many times—is not the same as driving in Oslo.

Obviously, driving summons an intense labor of perception. Much labor, in fact, appears mostly perceptive in nature, through continuous acts of decision and cognition implied in any blink of an eye.[18] Cognition cannot be completely disentangled from a spatial logic, and often follows a spatial logic in its more abstract constructions. Both observations (that perception is logical and that cognition is spatial) are empirically proven without fanfare by autonomous driving AI algorithms that construct models to statistically infer visual space (encoded as digital video of a 3D road scenario). Moreover, the driver that AI replaces in self-driving cars and drones is not an individual driver but a collective worker, a social brain that navigates the city and the world.[19] Just looking at the corporate project of self-driving vehicles, it is clear that AI is built on collective data that encode a collective production of space, time, labor and social relations. AI imitates, replaces, and emerges from an organized division of social space (according first to a material algorithm and not the application of mathematical formulas or analysis in the abstract).

The memory and intelligence of space

The French philosopher of speed or dromology, Paul Virilio, was also a theorist of space and topology, for he knew that technology accelerates perception of space as much as it morphs perception of time. Interestingly, the title of Virilio’s book The Vision Machine was inspired by Rosenblatt’s Perceptron. With the classical erudition of a twentieth-century thinker, Virilio drew a sharp line between ancient techniques of memorization based on spatialization, such as the Method of Loci, and modern computer memory as a spatial matrix:

Cicero and the ancient memory-theorists believed you could consolidate natural memory with the right training. They invented a topographical system, the Method of Loci, an imagery-mnemonics which consisted of selecting a sequence of places, locations, that could easily be ordered in time and space. For example, you might imagine wandering through the house, choosing as loci various tables, a chair seen through a doorway, a windowsill, a mark on a wall. Next, the material to be remembered is coded into discreet images and each of the images is inserted in the appropriate order into the various loci. To memorize a speech, you transform the main points into concrete images and mentally “place” each of the points in order at each successive locus. When it is time to deliver the speech, all you have to do is recall the parts of the house in order.

The transformation of space, of topological coordinates and geometrical proportions into a technique of memory should be considered equal to the more recent transformation of collective space into a source of machine intelligence. At the end of the book, Virilio made a reflection of the status of the image in the age of “vision machines” such as the Perceptron, but in fact foreseeing and warning about the upcoming age of artificial intelligence, as “industrialisation of vision”:

“Now objects perceive me,” the painter Paul Klee wrote in his Notebooks. This rather startling assertion has recently become objective fact, the truth. After all, aren’t they talking about producing a “vision machine” in the near future, a machine that would be capable not only of recognizing the contours of shapes, but also of completely interpreting the visual field […]? Aren’t they also talking about the new technology of visionics: the possibility of achieving sightless vision whereby the video camera would be controlled by a computer? […] Such technology would be used in industrial production and stock control; in military robotics, too, perhaps.

Now that they are preparing the way for the automation of perception, for the innovation of artificial vision, delegating the analysis of objective reality to a machine, it might be appropriate to have another look at the nature of the virtual image. […] Today it is impossible to talk about the development of the audiovisual […] without pointing to the new industrialization of vision, to the growth of a veritable market in synthetic perception and all the ethical questions this entails. […] Don’t forget that the whole idea behind the Perceptron would be to encourage the emergence of fifth-generation “expert systems,” in other words an artificial intelligence that could be further enriched only by acquiring organs of perception.[20]

Conclusion

If we consider the ancient geometry of the Agnicayana ritual, the computational matrix of the first neural network Perceptron, and the complex navigational system of self-driving vehicles, perhaps these different spatial logics together can clarify the algorithm as an emergent form rather than a technological a priori. The Agnicayana ritual is an example of an emergent algorithm as it encodes the organization of a social and ritual space. The symbolic function of the ritual is the reconstruction of the god through mundane means, also as a practice that symbolizes the expression of the many within the One (or the “computation” of the One through the many). The social function of the ritual is to teach basic geometry skills and to construct solid buildings, that is the production of a common space.[21] The Agnicayana ritual is a form of algorithmic thinking that follows the logic of a primordial and straightforward computational geometry.

The Perceptron is also an emergent algorithm that encodes from a division of space, specifically from a spatial matrix of visual data. The matrix of photoreceptors of the Perceptron defines a closed field and projects an algorithm to compute data according to their spatial relation. Also here the algorithm appears as an emergent process—the codification and crystallization of a procedure into a pattern, after its repetition. All machine learning algorithms are emergent processes, in which the repetition of similar patterns “teach” the machine and cause the pattern to emerge as a statistical distribution.[22]

Self-driving vehicles are an example of complex emergent algorithms as they grow from a sophisticated construction of space that is the road environment as social institution of traffic codes and spontaneous rules. The algorithms of self-driving vehicles register the traffic code of a given locale and spontaneous rules and try to predict unexpected events that may happen on a busy road. In the case of self-driving vehicles, the corporate utopia of automation makes the human driver evaporate and expects that the visual space of the road scenario dictates the map for its own navigation by itself.

The Agnicayana ritual, the Perceptron, and the AI systems of self-driving vehicles are all, in different ways, forms of self-computing space and emergent algorithms (and probably, all of the them, forms of the invisibilisation of labour too).

The idea of computational space or self-computing space stresses, in particular, that the algorithms of machine learning and AI are emergent systems that are based on a mundane and material division of space, time, labor, and social relations. Machine learning emerges from grids that continue ancient abstractions and rituals, of marking territories and bodies, of counting people and goods, basically from an extended division of social labor. Artificial intelligence is not really “artificial” or “alien” as is often advertised: in the usual mystification process of ideology, it appears to be a deus ex machina that descends to the world like in ancient theater to hide the fact that it actually emerges from the intelligence of this world.

What people call AI is actually a long historical process of crystallization of collective behavior, personal data, and individual labor into privatized algorithms that are used for the automation of complex tasks: from driving to translation, from object recognition to music composition. As much as the machines of the industrial ages grew out of experimentation, know-how, and the labor of skilled workers, engineers, and craftsman, the statistical models of AI grow out of the data produced by collective intelligence. Which is to say that AI emerges as an enormous imitation engine of collective intelligence. — What is the relation between artificial intelligence and human intelligence? It is the social division of labor.

 

 

[1] Paul Virilio, La Machine de vision: essai sur les nouvelles techniques de representation (Paris: Galilée, 1988). Print. Translated as: The Vision Machine (Bloomington: Indiana University Press, 1994), 12.

[2] The Dutch indologist and philosopher of language Frits Staal documented the Agnicayana ritual after an expedition in Kerala, India, in 1975. See: Frits Staal, AGNI: The Vedic Ritual of the Fire Altar, 2 volumes, Berkeley: Asian Humanities Press, 1983.

[3] p Dt hte abourst un, Kim Plofker, “Mathematics in India”. In Katz, Victor J (ed.). The Mathematics of Egypt, Mesopotamia, China, India, and Islam. Princeton University Press, 2007.

[4] See: Wilhelm Worringer, Abstraction and Empathy: A contribution to the psychology of style. (Chicago: Ivan R. Dee, 1997 [Abstraktion und Einfühlung, 1907]). collective machine intelligence.t hte abourst un,

[5] For an account of the Agnicayana mathematical implications see: Paolo Zellini, La matematica degli dèi e gli algoritmi degli uomini. Milano: Adelphi, 2016. Translated as: The Mathematics of the Gods and the Algorithms of Men. London: Penguin, forthcoming, 2019.

[6] Frits Staal, “Artificial languages across sciences and civilizations.” Journal of Indian Philosophy 34.1-2 (2006).

[7] Jean-Luc Chabert (ed.) A History of Algorithms: From the Pebble to the Microchip. Berlin: 1999, 1.

[8] Jean-Luc Chabert, cit, p. 1.

[9] Gilles Deleuze and Felix Guattari. Anti-Oedipus. Vol. 1 of Capitalism and Schizophrenia, New York: Viking, 1977, p. 145.

[10] See: Ubiratàn D’Ambrosio, “Ethno Mathematics. Challenging Eurocentrism”, in Arthur B. Powell, Marilyn Frankenstein (eds.) Mathematics Education, State University of New York Press, Albany 1997.

[11] Diane M. Nelson, Who Counts?: The Mathematics of Death and Life After Genocide. Duke University Press, 2015.

[12] Frank Rosenblatt, “The Perceptron a Perceiving and Recognizing Automaton”, Technical Report 85-460-1, Buffalo: Cornell Aeronautical Laboratory, 1957.

[13] John von Neumann and Arthur W. Burks, Theory of Self-Reproducing Automata. Urbana, University of Illinois Press: 1966. Konrad Zuse. “Rechnender Raum”. In: Elektronische Datenverarbeitung, vol. 8, 1967. As book: Rechnender Raum, Friedrich Vieweg & Sohn: Braunschweig, 1969. Translated as: Calculating Space, Cambridge, MA: MIT Technical Translation, 1970.

[14] Alan Turing, “The Chemical Basis of Morphogenesis”, Philosophical Transactions of the Royal Society B. 237 (641), 1952.

[15] It must be noted that Marvin Minsky and Seymour Papert’s 1969 book Perceptrons (which attacked superficially the idea of neural networks and nevertheless caused the so-called first “winter of AI” by stopping all research funds to neural networks) claimed to provide “an introduction to computational geometry”. Marvin Minsky and Seymour Papert, Perceptrons: An Introduction to Computational Geometry. Cambridge, MA: MIT Press, 1969.

[16] See the work of 12^th century Catalan monk Ramon Llull and his rotating wheels. In the ars combinatoria an element of computation follows a logical instruction according to its relation with other elements and not according to instructions from outside the system. See also: Amador Vega and Siegfried Zielinski, DIA-LOGOS: Ramon Llull’s Method of Thought and Artistic Practice, Minnesota Press: 2018.

[17] Specifically, a logical or inferential activity does not necessarily need to be conscious or cognitive to be effective (this is a crucial point in the project of computation as mechanization of “mental labor”). See the work of Simon Schaffer and Lorraine Daston on this point. More recently, Katherine Hayles has stressed the domain of extended nonconscious cognition in which we are all implicated. Simon Schaffer, “Babbage’s intelligence: Calculating engines and the factory system,” Critical inquiry 21.1 (1994). Lorraine Daston, “Calculation and the division of labor, 1750-1950”. Bulletin of the German Historical Institute, 62 (Spring), 2018. Katherine Hayles. Unthought: The power of the cognitive nonconscious. University of Chicago Press, 2017.

[18] As both Gestalttheorie and the semiotician Charles Sander Peirce remarked, vision always entails cognition and even a small act of perception is inferential, it has the form of an hypothesis.

[19] School bus drivers will never achieve the same academic glamour of drone pilots and airplane cockpits with their adventurous “cognition in the wild”, nonetheless we should acknowledge their help and the contribution of the microhistory of labour in marking crucial insights on the ontology of AI.

[20] Virilio, The Vision Machine, cit., p. 76.

[21] As Stall and Zellini noted among others, these skills include also the so-called Pythagorean theorem, that is helpful in the design and elevation of buildings, demonstrating that it was known in ancient India probably transmitted via Mesopotamian civilisations.

[22] In fact, more than machine ‘learning’ it is data and their spatial relations ‘teaching’.