Lectures / Plotting from History / Projective Theory of Technology / Thinking as an Algebraic Mechanist

MANUSCRIPT // The creative conservativeness of computation

(Alternative title: The Secretive Conservativeness of Computing)

*this is the manuscript (in draft character) of my paper delivered at the “Within the Domain of the Sun’s Inverse, or: Where are we when we are thinking computationally?” Seminar (cf the documentation of the event).

»The philosophical role of the sun has transformed many times, but it always manifests as that which informs political organization.« (Harold Innis, Empire and Communication)

Because the sun is the icon of the sovereign (ruler), the subject, and capital all at once.


(1) The Computational Cogito

(2) Algebra, or the intimate relation between Law and Economics

(3) The communicational bias, or the nature of contracts

(4) Thinking the Sun’s Sovereignty

(5) The Ruler is Literate

(6) The economic nature of script 



Economic Force, the Power of Formality

(7) Discreting the Many Identities of the Natural Base (natural logarithm)

(8) Keys to Protect the Identity of the Collective Subject (in the Character of its Being: distributiveness)


The Computational Cogito

According to the early theorists of literacy and media, in the Canadian school around Harold Innis, Eric Havelock, Marshall McLuhan, media and communication feature as the linking „item“ between knowledge and power at the largest conceivable scale. Innis, an economist himself, dedicatedly aims with his studies at developing a unifying theory of how forms of political organization crystallize, come into power and dissolve/transform. With this ambitious scope, media theory enters the disciplinary academic scene with the promise of offering a perspective that can integrate, and hence is more abstract and capacious than, the particular „forms“ of „mediacy“ that had been identified around the turn of the century as being capable of providing a foundation and basis for (secular) knowledge: logics, linguistics, geometry, arithmetics. Innis’ point of departure is the affirmation of what he calls „the bias of communication“, as the acknowledgement that media always participate in and shape how we think, and hence play of dispositional role in what can be known. This approach, namely to study the role of media and communication in terms of literacies, appears to me as its great strength, and I would like to explore in my talk how and why.

First to the why: I will try to show that from this approach, we can extract the notion of a new Cogito, a computational rather than a rational or a reasonable Cogito. It is a Cogito – a notion of ‚the subject of which we can say that it knows’ – that is inherently collective without being already general. To put it differently, it is a subject without form, or rather, one whose form it is to be distributive. It is a subject not characterized by properties it masters and owns, or has been gifted with, but one characterized by an activity: to distribute. This is important – not one which is distributed (although it is this, too), but one whose character it is to be distributive. It is a Cogito that is what it is, the subject which can learn to know, in that it has much to give – a muchness which grows vaster the more of it it gives away.

Vaster in what? Not in terms of spatial or temporal extension, but in terms of expansion in value – „value“ as that which mediates the worth that is intrinsic to a thing. 

Algebra, or the intimate relation between Law and Economics

Computation operates with values that are symbolized, according to the categories that pertain to algebra. Algebra expresses identity in terms of richness in distinction, not in terms of adequateness of the expression to what it is supposed to represent. Its form of expression – that of equations – states, in each formulation, particular resolutions of the identity expressed, resolutions that are of greater or lesser capacity and power in affording to accommodate the identity’s distinctive marks. Symbols in algebra are „elements“ taken as „jokers“, they are general equivalents. Like money, their relation to value is a mediate one, what they render intelligible in the sense of discretable (ermessbar) is meaningful only within an economic reality of exchange and substitution. Algebraic symbols, like money, capture, encode and formulate the form that value takes in its circulation.

With a characterization as drastic as this, such a Cogito might, at first, appear like the incorporation of the diabolic! which indeed is what you find throughout all the literature discussing the explosion of abstractness in algebra throughout the 19th century. Our purported understanding is that values pertain to original predication. Law is the only form supposed to state values in un-corrupted form. By decoupling values entirely from the reference to an Originality, Universality, or Divinity of the Laws whose judgements they are to express, the relation between theology and economy appears to collapse, and leave behind a philosophy entirely impotent with regard to supporting political forms of organization. The reality of such a Cogito must appear as sheer tyranny.

The communicational bias, or the nature of contracts

It is at this point that the work of literacy studies by Innis, Havelock and Co on the role played by media and communication in different civilization and their form of organization are capable to expose the short cut and to space out a new distance between the two domains that seem to be collapsing here (God and Money). Innis has  attempted to merge a theory of politics or imperialism with a theory of consciousness and a theory of technology. The main „axiom“ in his work consists in what he called the „communication bias“: every society organizes its processes  through communication, and hence is biased according to the media in which they can communicate. Innis distinguishes two vectors for the cultural development of the civilizations, either towards an emphasis on space and political organization, or towards an emphasis on time and religious organization.

This distinction is powerful, and Innis’ accounts cast focus especially on the development of writing and its relation to material carrier and tools. For example the hardness of stone comes to play the role of emphasizing time and endurability, expressing a privilege with respect to originality, while he shows the turn towards the softness of clay to support an emphasis on space and the building of imperiums and city states linked by trade rather than cult. Especially McLuhan has pointed out certain limits of Innis approach with regard to post-19th century technology in the form of electronic media, and has suggested to generalize from Innis emphasis on space and time towards a new emphasis on the aesthetic organization of sensory data at work within the intuition of space and time, we can still gain extremely valuable insights from Innis studies on writing in relation to the discrete character of computations. Because he departs from value in relation to Law, that is in abstraction from values’ concrete expression in mathematics or language.

Thinking the Sun’s Sovereignty

There is one assumed premise on the basis of which Innis thinks about law: that civilizations have developed forms of political organization through the identification of the position in power with the symbolization of the sun – this is the equation (the postulated identity) which is articulated and spelt out in every form in which politics and economics find their manners of organization, such that it can be legitimated.

He relates the discovery of forms of government in ancient cultures to the idea which sees in the Sun the enactor of divine Law, law which is imposed upon all things, while simultaneously also seeing in it a subject to this same, divine law. From this gap, the relationships of Sun-Kings and priestly Cults could crystallize in different manners. The person in power was endowed with attributes according to the particular relation an idea of government postulates between the representation of power and the Sun-God(s): in different civilizations there were networks of relations on either side, many representatives that together make up the Ruling Power and families of Gods, or there were figurations of absolute monarchy and dynasties by either identifying the Ruler on Earth as descending from the Sun-God, or as its administrator, bookkeeper, or scribe in general.

The Ruler is Literate 

Hieroglyphs as they began to appear from about 4000 BC was the Greek name for sacred engraved writing. It recorded the names of kings, wars, political events, accounts of things in inventories. The knowledge of these writings was crucial for counting time and the bonding of a society by imposition of a shared calendar. It is through calendars that the rulers were assessed as good or bad – if they foretold events like the flood of the rivers wrongly, their legitimation of power was inevitably called into question. Many of the major developments in forms of political organization were hence related with more powerful systems of counting time. Innis describes the development of both, calendars and writing systems as one from strictly tabular and formulaic recordings to less and less formulaic and more articulate and differentiated ones. Important is the common origin of mathematical notions and what evolved eventually into alphabetic ones, because the question of the nature of the values expressed is exposed thereby as irreducibly entangled.

Given his comparative interest in the role of media in political forms of organization, Innis’ concern is not one in the „truth“ of these values, but an operational one in how the question of their reference has been treated, and triggered different approaches that manifest in the political, economic and religious forms of civilization. He acknowledges that the background for relating articulated expression, both verbal and written, to the creation of facts, and hence the assumption of their divine nature, arises from the legitimation narratives in support of the political/religious power and the social support and control of the community afforded thereby. He treats „value“ that express divine Law and its Judgements as a cipher, and studies how the role played by communication and its media provided manners of deciphering and introduces additional changes in the encryption of value’s essentially cryptic character. This, he traces in how written records develop towards phonocentrism, towards documenting utterances of values (judgements) with greater and greater degrees of mediation – which means, distance to and decoupling from the assumed original act of judgement/creation. The first hieroglyphic scripts knew words for things, and they rapidly gathered immense stocks of different signs for words. It was an economic necessity to develop a script that expresses how words can be represented phonetically, in order to decrease the amount of signs that are to be mastered. Different systems hence began to distinguish vowels and consonants in order to measure the sound of words. The fewer signs were used, the more abstract the script in the sense that it provides a relatively easy to handle formal framework within which any word can be expressed such that it can be deciphered with precision. Innis studies the degrees of mediation that were gradually introduced through how articulate sounds were discreted, measured, and characterized. The key distinction thereby is that between vowels and consonants. Many early scripts did not include vowels into their sets of elements for phonetic representation of words. The reason for this, Innis argues, is that consonants were thought to express the concept of the original value’s/judgement’s roots, they were the closest one could get in reconstructing the Divine Laws whose judgements are expressed in the values articulated in speech. The vowels, on the other hand, were thought to express the variable form of those roots, and hence the changes in meaning related to the changes in the form of the root’s conception – that is, changes in meaning that arise from mediation, and that distance one from the original statements that express the Laws.

The entire discourse of derivatives, copies, simulacra seems to root in this distinction between vowels and consonants. Vowels were regarded like profane „glue“, necessary but potentially a source of confusion, as they are related to the rhetorical power of expression and hence to the political instrumentalization of power that introduces arbitrariness into how the Laws are interpreted and formulated. (Innis, p. 50ff.). This indeed, is the entire link for Innis to study forms of political organization by studying communication and its conditions of possibility.

The economic nature of script 


It is striking how „algebraic“ in character this genealogy of writing comes along! Triggered by records of events sorted into tabular manner, the lawfulness of how to link this data is established by algorithmic procedures: if this and this, then that and that can be followed. In Michel Serres’ Book Elements of a History of Science there are two chapters on Babylonian Mathematics and on the Origin of Geometry which give a lively account of this. The form of the tables organized the form of experience in the manner of analogies. The data recorded was classified into typical form, and accordingly it was to be filled in the provided spots of the tables where it was called „givens“. Like in computer programs today, these givens were used as inputs for a series of steps to be taken (a procedure) in order to yield an output (the results of predictive and economically driven calculations on which planning was based). Such reasoning was called „divination“, and it was applied in medicine, astrology and astronomy, as well as in economic regards.


Reading as the deciphering of values, and writing as their encryption was essentially algebraic (albeit, we must note, this is avant la letter – as the resolution of equations (what we today call algebra) was not distinguished from arithmetics as the calculation by method of completion and balancing (Diophantus 3 BC). The arabic word al-ǧabr means „completion“/„balancing“ and was introduced from arabic scholars to Europe not before Renaissance. The entire method of balancing and completion required three components: a tabular form (later turning into systems of categorization), the identification of givens (later turning into systems of classification) and the identification of variables – the results that could be calculated. And, of course, the rules/habits/customs for identifying what can count as givens and variables.

Economic Force, the Power of Formality

We can easily see how this structure was followed as well in the development of the scripts that are to represent spoken words. The consonants appear to have been identified as the ‚givens‘, and the variables had to be the expression of the original values. The role of the vowels therein was inherently ambiguous: it allowed for a variable ‚power of expression’ a power which at the same time also manifests as a rhetorical power that arises out of the mastership of the formality, decoupled from the lawfullness that is supposedly expressed in this formality. Innis describes the emergence of literature that pertains to largely secular concerns of life, and to a secularization of religion and politics, in relation to this rhetorical power. And he explains the artistic explosion in Greek culture after the 800 BC with their alphabet as a finite and ordered (alpha beta …) set of elements that represent phonetically the utterance of words as a system that included into its formalization vowels as well as consonants. It is plausible to assume, as Innis does, that in the case of Greek cities that were increasingly governed by tyrants, leaders that were not entangled in priestly or aristocratic power structures but that came to power out of successful trade relationships, had less religious reservations and greater pragmatical need to make compromises and just begin to standardize the role played by vowels – and hence create an artificial language that is not one naturally spoken dialect, but, peculiarly so, a phonetic script that came to be spoken while having been written first.

This is the background to Derrida’s famous argument about the primacy of writing in the phonetic language, and the deconstruction of metaphysics as a metaphysics of presence. I will try to link these dots very sketchily here, by departing from Innis and Havelock emphasis on how a new kind of concepts was introduced in Greek Antiquity – concepts that are genuinely abstract, and that introduce a notion of Universality which breaks the immediacy of Values, Divine Law, and the particular religious set-ups for how such Divinity is thought.

The Abstract Form of the Universal

The big claim of metaphysics was that the Universal as the origin of abstract concepts, was outside of time and eternal. The subsequent categorical distinction of systems of writing from manners of computation bases upon this claim – it is why mathematics counted as the art of learning, and as an art, it kept is entanglement with spirituality that cannot properly be positivized exhaustively in formal terms. While words name the things pertaining to the finite domain of all there is, numbers name the relations between all things finite in a manner that is, in its procedurality, infinite, in the sense that it cannot be exhausted. Concepts subsequent to Greek philosophy did not merely express the roots of original values according to forms however derived and impure through mediation, instead they now come to occupy the mediate level in between the assumed original act of judgement (predication of an identity) and its uttered or written formulation. Concepts now are thought to express the eternal (divine) predication of finite things in their abstract and universal identity. The deconstruction of such identity-based metaphysics, which characterizes 20th century philosophy, points out this „forgetfulness“ with regard to the character of language itself. This character shares the same origin like that of computation, and hence calculation and mathematics. The so-called linguistic turn that followed out of this deconstruction is more familiar than the corresponding turn towards a certain alphabeticity in number theory itself, hence I would shortly like to outline the latter.

With the identification of numerical domains like the Integers, the rational and the real numbers etc, was possible because if follows the same gesture introduced with the Greek phonetic alphabet – it is clear that this is an assertion in contest, but let me point out how its argument might work while being aware of its hypothetical character.

The alphabet means that a finite stock of elements is introduced in terms of which all words that can be uttered can be expressed phonetically – in a manner decoupled from a distinction between literal and figurative speech, truth and fiction, argument and rhetorics. In that sense, the formal character of the alphabet is that of a code system. Now, we might object, with numbers this is exactly what we don’t have – every one knows that the amount of numbers cannot itself be counted, that they name the infinite. But the notational systems for numbers are treating them as figures or ciphers, Ziffern in German, because they express numbers according to positional value systems that take a certain value as its base – 60 in the hexadezimal, 10 in the decimal, or more recently now, 2 in the binary system. Such positional systems are organized in what was called „logarithmic tables“ – a term introduces by John Napier in the 17th century, expressing „ratio-numbers, or „numbers put in proportionate notation“ from logos, proportion, and arithmos, number. The decision with regard to which base characterizes the notion of numbers as a particular code. It is within algebraic number theory that the positional logics of such notational systems itself is being thematized, in a manner which up to the 19th century has usually taken the form of placing numbers on one infinite line – the so-called number continuum. Dedekind and Peano have introduced a general procedure of how to identify numerical domains as number classes embedded and nested within each other and within that continuum (the rationals, reals, integers etc). The application of this procedure requires further and further levels of relative abstraction attributed to the algebraic symbols – symbols used as jokers – at work in identifying the positional logics of these domains, up to the situation we have today where number theory is understood by many as the object of cryptology/cryptography/cryptoanalysis. We have entirely abstract numerical bodies called „fields“ in english, and constructs that build upon these – with beautiful names such as rings, latices, sheafs, and so on. We need not go further into this, for the point I want to highlight – and with which I want to link back to my interest in the beginning, namely to think about a Cogitio of what I called „computational thought“ – regards the peculiar character of the binary applied in digital computation.

Discreting the Many Identities of the Natural Base (natural logarithm)

It is clear that we can compute electronically with different basis than the binary as well, and for what is often distinguished from digital computing  as analog computing we can even more easily work if we take a different base. But the peculiar character of the binary is that it does not simply introduce a different base, and hence a positional system, but that it can discretize any positional system conceivable. This is, as I understand it, why the algebra behind it is called „universal algebra“ – because it produces systems of symbolic reasoning that are universal, as Whitehead called them, without expressing a contradiction by emphasizing systems in the plural. In algebra, Laws are economic in nature.

Let us try to understand more closely how this works in terms of operations, by bracketing out the unsettling aspects of its implications – like Innis bracketed out the irreducibility of politics, economics and religion in the treatments of how value has been articulated and communicated in different civilizations.

The crucial moment is the introduction/invention/finding of a particular number that constitutes the so-called „natural logarithm“. This number is the inverse to the exponent: this means that a² can be written as x times e raised to a certain power. In other words, every single number can be expressed as a certain power of e, e being one common base to all numbers. There is a nature of numbers that can be developed out of one value-constant that factors in all numbers. This constant itself is irrational (it can only be approximated) and transcendental. The crucial point about this is that in principle, the question of how to determine commensurability can thereby be attempted in inverse manner: there is no need to find the smallest common denominator in order to sort „what fits and what remains as a rest“ and hence must be contracted as contingent. Commensurability can actually be established between any number – but only symbolically so. And there are different manners of how we can rationalize differences systematically into commensurability. Before this background, this seems to be the distinctive mark of the binary – it is the sole „base-system“ which operates entirely decisional, and does not necessitate the identification of a smallest common denominator (decimals or the like), or a common alphabet (number class, like that introduced by Turing and Church as the class of „computable numbers“).

This is what the digital, and the universal algebra that constitutes it, is all about. The digital leaves in-determined whether computations can be treated as logics, linguistics, of physics. Every emphasis on „analog computing“ is committed to foreclosing this indeterminacy and imposing upon it certain symmetry structures – the proportionality given by the form of an analogy (A : B like C : D) rendered into the arbitrarily decided formulation of a particular and (as they claim) „concrete“ foundation.

Keys to Protect the Identity of the Collective Subject (in the Character of its Being: distributiveness)

It is against this arbitrarily attributed ideality to information that we should insist on a materialist stance in regard to information. by materialist stance i mean one where information and energy are not strictly kept apart, such that a Universal Cogito (that computes algorithmically) can be said to have a nature that evolves on the basis of literacies – in the expansive sense this term gains from Innis study, literacy as the forms political organization crystallizes, relative to the Character of the „how much“ the collective subject can mobilize to give in its particular modality of being distributive.

The Collective Subject, the Cogito of Computation, is characterized only by the contracts it is capable to sign regarding its own nature. That might be a scope, perhaps, in which we can speak of a Nature of Information, and on which one can build up a science that informs not a political economy, but a politics of judgement – as Lyotard suggested, and as Michel Serres seems to be picking up in his Natural Contract book. Science produces keys, and keys are passwords that protect the identity of the Collective Subject. The dignity for which this subject finds recognition in the contracts that articulate its universal rights depends entirely upon the sophistication of these keys. The Dignity established by keys energetizes this virtual Universality to which the Collective Cogito that computes is subject. They govern in constitutive manner the Domain of Values upon which algorithmic computation operates.

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