abstract
This text inquires how Louis Hjelmslev’s idea of »an algebra immanent to language« can help us to characterize discretized probability densities as a kind of »symbolic alphabeticity«. It is the manuscript for my talk at the 5th metalithikum colloquy in may 2014.
CONTENTS
(1) Introduction: The Materiality of sense, or: “capable of being continued”
(2) Tracing important distinctions regarding the notion of Quantity
(3) Impredicativity, or The mode of insisting existance proper to the circular
(4) The alphabetization of the numerical in computation
(5) The cipher, and its quantum-body of objective neutrality: nothingness
(6) Disparseness, or In the element of neutrality – the many bodies of nothingness
(7) Glossematics: Theory conserves, circulates and differentiates the materiality of sense
(8) Beyond the Apparatus, or: getting over the deadlock of tautology by repetition (alphabeticity)
(9) Signals and codes: the world translates itself in the elementary nature of thought
Introduction
The Materiality of sense, or: “capable of being continued”
In terms of literacy, people can be so more or less, it is not an absolute term. This is different for calculability – we tend to feel that something is either calculable or not. To say that computation may be seen as a literacy means to say that calculation too cannot be regarded as an absolute term. Someone who makes sense in what they say does not necessarily speak the truth. The extension of sense (inclusion of falsity and truth) must be larger than that of knowledge (exclusion of the false from the true). It is from this gap between both that learning is possible, and that new knowledge can be constituted on the basis of sense. Yet how to think about such a notion of “sense”, and why should we dare to reconsider well-established cultural techniques so fundamentally? It seems as if the limits of the representational paradigm for current notions of knowledge is being challenged by another one which we might simply call that of sustainability.
How could we frame this emerging paradigm? We might think of it by conceiving of literacy in a mutually reciprocal relation to calculation. Neither one can be assumed to be absolute, and yet determinable within the constraints they mutually impose upon each other.
What would this entail?
One is dealing with letters of an alphabet, a finite and ordered set of elements each of which has a double make up: a letter embodies a quantity – usually we call it a phoneme, the body of a sound – and the graphemes, the Gestalten of the signs which are capable of expressing the bodies of sound in such an orthogonally governed manner that they form complex units of meaning. The other is dealing with numbers, which are like the signs of the alphabet meant to express bodies of quantities such that they may form, in a well-governed manner, complex units. Yet with calculation, the manner of governing is not orthogonal (vertically oriented) but permeable (horizontally oriented) – everything variably factors in one another, as well as in the whole. From a different angle we are well familiar with these difference, we relate calculation to an economical order, and literacy to an order of political organization.
Let us make this more concrete by thinking with this image about the theme of sustainability. First of all, we link the term up in a pair by coupling it with environment. A logics of the planet‘s oikos is challenged to formulate an identity notion – that of the earth together with its environment – that does not exclude circularity from the manner in which its definition is constituted: the environment is literally that which circum-gives the Earth, and which somehow is thought to be effect of how the planet’s oikos, whose stocks are riches that regenerate themselves in cycles, exchange information between the compartments that together make up, somehow, the whole Earth including what circum-gives and protects it, the environment. Such communication between the compartments, I want to suggest, is a question of how different magnitudes, of which we don’t know how they behave proportionally in relation to each other, factor in together in the environment. Sustainability is all about how the Earth, and its Environment, can manage well together. In order to learn about sustainability in terms of geometry, we have to think of the circle in both its aspects – its perimeter and its area, and this in a temporalized relation that must be characterized as “quick”, “alive” or “self-maintaining”.
Thus, if neither one, literacy or calculation, can be considered as absolute, if both are determinable within the constraints they mutually impose upon each other, how might we tackle this entailment that urges us to engage – somehow – critically with circularity?
We can look to algebra, that part of mathematics which affords the “unification of broken parts”, as its name literally suggests. It is crucial that by characterizing algebra in this manner, we are NOT thinking that “nature IS algebraic”. We are in the realm of SYMBOLICAL STAGINGS. So then, lets see what the Stanford Encyclopedia of Philosophy states, that algebra can do, on this stage for symbolization: algebra provides ways of managing the infinite. Let me clarify how I want to make sense of this: We are on a symbolical stage (in conceptual “reality”) and we have Riches to be harvested and cultivated, by technics and by formal thinking which makes the cultivation possible as something which can be “communicated”. Thus, I would place such “algebraicness”, the stages it provides, as the ARCHITECTONICS and the RICHES OF THE UNIVERSE as the infinite which is being MANAGED by the structures of the architectonics. Thus, I would argue that it is not adequate to think about the symbolic structures and operators from the dipolar level of ontology and epistemology. This dipolar level, I would like to think, is enveloped within a comprehensive ARCHITECTONICS. There can be many such Architectonics, each producing particular epistemologies and ontologies. These Architectonics are how we settle, intellectually, on the world as a place in the universe. With it, we settle intellectually within the vastness of the universe.
In the history of mathematics, there is a long tradition of thinking of the infinite through the figure of the circle. The circle has been thought of as an expression of the infinite because it can never exhaustively be measured. Algebra provides ways of managing the infinite according to methods of measuring the circle either by triangulation or by squaring. With triangulation we are operating algebraically in arithmetics, we partition and differentiate the circle’s area. With squaring, we are operating algebraically in geometry and construct proportionalities that allow to approximate the length of the circles perimeter.
The ways of managing that algebra is capable of providing, capture the data rendered by these ways of operating (triangulating and squaring, and collecting the results in tables) into generic formulations. I call it generic because Algebra expresses its statements in terms of signs that envelop the quantitative bodies in an evaluative manner – as values. We have in algebra, perhaps in algebra alone, an intermediate level of notational code and ciphering between “signs” and what they indicate. This intermediate level is introduced because algebra operates in symmetries: whatever a formula expresses produces a mapping (a function), but not one that stands for something else, a representation. It produces mappings that can stand in for that which has been articulated in a formula. These mappings can stand in for their own “original” so to speak – that is, they can articulate “the original” in a tautological manner – and this needs not be seen as an absurdity, it is rather a manner of how Originality can be conitinuing itself, as originality.
The mappings produced by formula (an EQUATION) are varying expressions of one and the same thing. They can be such if only we affirm that neither expression is ever capable of “expressing all that is to it at once”. In other words: that which is expressed is of a vaster extension that any one of its symbolical expressions can ever be.
In logics and in mathematics such peculiar, self-same mappings are called „impredicative“. Impredicativity concerns self-referencing definitions that build on what they antedate. In a standard and symbolico-technical formulation from analytical philosophy, impredicativity is related to the so-called vicious circle principle: it is impossible to define a collection S by a condition that implicitly refers to S itself. A definition is said to be impredicative if it invokes (mentions or quantifies over) the set being defined, or (more commonly) another set which contains the set being defined.
One can easily see the relevance of this for any earth science, whose point is exactly that all distinguishable „compartments“ that together make up the planet involve such definitions.
These problems with regard to the stating of a definition are intimately related to the representational paradigm. It appears questionable whether they are to be weighted in the same manner within a paradigm that seeks to make sense in terms of sustainablity, meant as “capable of being continued”. It appears as if an embrace of just such circularity is the way of learning to think “sustainably”. Without such an impredicative space of self-same transformability – what I would call a symbolical body of reciprocity, a stage on which we settle into a certain ordinariness in our thinking – no code can operate as a code. (We’d not be aware of the symbolic make up of our “ordinariness”). A code deals with values, within a systematical context that is closed upon itself. This is different for signs. They appear when such self-same symmetry is broken (when something breaks into the ordinariness we got used to in our symbolically set stage). When something factors in that had not been taken into account. When the self-maintaining balance is disturbed.
So how can we think the relation between code and sign while appreciating both?
This is the background context of my talk. It is what I hope to get clearer about with learning to understand what it entails to reciprocally relate literacy to calculation as two mutually constitutive realms of reasoning that is, essentially, circular.
From this point of view, what can be learnt is more extensive a realm than any mapping that is possible at any one point of what can be known. This view sounds quite alien today, but it seems to have been much more natural before, when science and spirituality were not strictly kept apart. It we do not reduce learning to strictly mechanical repetition, it seems, we cannot maintain this categorical break.
Tracing important distinctions regarding the notion of Quantity
One first distinction we should attend to regards the relation between quantity, quality, and equality. We are told by Aristotle (Cat. vi 6a27) that what is most characteristic of quantities is the attribution to them of equality and inequality. for him, to speak of equality in terms of qualitative aspects didn’t make any sense. This is indeed what demarcates his philosophy as a realism – for him there are natural individuals, not natural kinds as for Plato: Aristotle gives primacy to the individual. It alone can be qualified, through specification, and through unifying species into genera. The qualitative was to be specified, and characterized according to how individual entities embody those qualities.
Thus, it was the quantitative alone was referred to the universal, and hence could be treated in terms of “equalities”. In algebraic formula, this separation between a) the universal and quantitative, and b) the individual and qualitative, becomes complicated to sort out. Because here, the distinction between ratio and quantity is crucial, and the notion of ratio, as we have already seen, implies a presumed proportionality between “natural kinds.”
Ratios figured only in the framework of a formula (an equation), within which they are proportionality established between the terms that are being set “equal”. We can easily see in the schema of a proportion, A : B = C : D, that the distinguished terms are treated as quantities, whereas the sides right and left of the equation sign is treated as ratios. The cunning involved in algebraic reasoning consists in allowing one quantity to be unknown (or more than one, in more complex systems of equations), and to establish procedures of determining it according to (within the confines of) a proportional framework. The incomplete proportion x : B = C : D makes the assumption that x must be a value of A, and on the basis of this assumption, algebraic reasoning proceeds by factoring out the terms into several equations and linking them up into a system of equations that co-constitute a solution space. In other words, the cunning in algebraic reasoning involves dealing with both, quantity and ratio, via recourse to a third and mediating notion, that of value. And values apply to both, qualities and quantities, or rather: they apply to qualities insofar as they are measured in order to expose similarities between the individuals (and thus allow for speciation, generalization).
Values, we can say, arise from the interplay between logics and geometry, ratios (and their logical analog proportion) and quantities (each made up of magnitude and multitude). Values are what mediates this very interplay. Why is this important? Because it is through this mediating notion of “value” that “identity” comes into play, is being temporalized and through that coupled with a level of “individuality”, and through that, “equality” has to be distinguished from “sameness”.
To decipher these differences, we need to recall a further distinction that was crucial for all of mathematics until the advent of conceptual mathematics with set theory, namely that between magnitude and multitude. Magnitudes answer to questions of how much?, and they require a metrical measure that is specific, a means common to the different kinds of magnitudes; multitudes, on the other hand, answer to the question how many?, and they are assumed to reign universally. In other words, multitudes are held to count and govern magnitudes.
In a recent article entitled „Eudoxos and Dedekind: on the ancient Greek theory of ratios and its relation to modern mathematics“ Howard Stein elaborates that the difference between magnitude and multitude „is quite alien to our present way of thinking about such matters, for us, to say that two distinct triangles are equal in area is to say that they have ,the same area‘.“ His argument is that in Antiquity quantity was a category for which an entire philosophical grammar existed. It is this grammar that the thinking in terms of ratio (proportionality, analogicity, logics) would apply – thus, there is an intermediate level between the registered or recorded quantities and their physical referents. Today, this intermediate level is nor usually evoked. It plays in whenever we try to be differentiated with regard to local and global properties and their relations. But overall, we have come to treat both, magnitude and multitude, as particulars that are governed conceptually (as finite sets). And these concepts are not thought of anymore as grammatical (which would place them in literacy), but as strictly formal and logic. What used to be a grammar for articulating quantities has thereby turned into a logics – with that, we have made a step from what may be called natural mathematics in distinction to conceptual mathematics. We can easily see the relevance of this for our interest in literacy: while grammar enables literacy in the comprehensive sense, logics seeks to strip literacy’s extension of that which makes sense from the false, and to purify what is to legitimately make sense from all suggestiveness of eloquence in expression.
According to the philosophical grammar, where no such conceptual logics was available, Stein says: „it would be incorrect to speak of ,the area of this triangle‘” because “a triangle does not have an area, it is an area – that is, a finite surface”. According to the philosophical grammar of quantity, “this area means this figure, and the two distinct triangles are two different, but equal, areas.“ His point is that within such a grammar, entities are universal in their nature, whereas with the shift to conceptual mathematics, entities havea universal nature. So I am nor arguing that the “grammar” is something we have lost and should regain, with out emphasis on “conceptual logics”. Because it is only with the abstraction from words of grammar onto a formal level that we can address the question of how Values factor in, in how “equality” is established. Once we take values into account, we would not seek to identify “equality”, rather we are speaking of “equivalence”. We become aware that in our “grammar of quantity “ we are in a formal space. Our problem today is that we think this formality is Universal, or at least uniformally Existent. It is here that the Sheaf and Category way of thinking in formal concepts appears so interesting to me.
But lets stay with the grammar of quantity more closely and see more how can we imagine it (as something different from quantified formal concepts) – in order to think about how we might think of “quantized” formal concepts.
So, a grammar of quantities.
We can regard multitudes like the VERBS that take nouns or pronouns as their transitive objects, upon which they exert the activity they incorporate. Magnitudes, within this analogy, can be regarded like the transitive OBJECTS of the multitudes(VERBS). Let us take this analogy as a kind of “auxiliary construction”, and put some more stress to it: we can see that multitudes occupy the role of verbs, and magnitudes that of objects, but with regard to the subject position – the voice of mathematical articulation – the two notions that together make up “quantity” is indeterminate. That is, its position is left blank.
My point is that it is the role associated with the subject position that is acted out by the genuinely mediating notion of values. In other words, the values, in their acting out, individualize the subject.
But how can we say they are mediating? Mediation needs at least some sort of “substrate” on which it can act (information, energy, ether or whatever). In the case of our quantitative grammar, the position of the subject is void – subjectivity is acted out in the absence of a unified identifiable actor.
In short, my point is that we take this “absence” as something to work with. We can understand it as “dedicated place for something to actualize”.
We can conceive of this mediation as the peculiar contribution of algebraic forms, that is of equations, and their capacity to express “identity” inversly. And we can conceive of such algebraic identity as the symbolic establishment of a tautological relation that characterizes neutrality, encoded as a cipher. A cipher has a transcendent referent, yet its power consists in “presenting” this transcendent referent symbolically while leaving it absent, just like words are capable of evocating something absent into presence. We can regard a cipher as a symbolical body of a self-referential relation whose identity is being articulated, not represented. I call it a body because we must think of it as a relation that can be organized corresponding to a particular codes, their inventories and rules. If equations are not considered in terms what they might “represent” in a world conceived to be exterior to it, but in terms of what they articulate and express through partitionings extracted out of that formalities own interiority. Such articulated formality cannot be separated from the characters and rules used in the applied notation systems.
Hence, we can think of a cipher, together with a particular code in terms of which we can decipher it, as a virtual body of reciprocity. Such a virtual body cannot be addressed in terms of equality, it must pay tribute to the equivalence that establishes it symbolically. The “identity” is an actor on stage. Hence we have, with this perspective, gained a distinction with regard to how we can think of “identity” – we can speak of equality with regard to the actuality (verbs) and object positions that are affected by such actuality, and we can speak of sameness with regard to the virtuality of the subject position. The “identity” an equation expresses, then, is doubled-up into a complementary (mutually implicative) interplay between its factuality, objectivity, and subjectivity – all three of which must be conceived of as in the infinite tense.
A short interlude, perhaps, to help you keeping track of where I will be going with this. My interest with revisiting these conventions from pre-19th century philosophy and science is that we can gain from them a generic notion of essence, which need not invoke a scholastic and canonical order or natural kinds. But we need the notion of essence when we want to keep a notion of learning from the infinite, in which thought necessarily plunges when it learns something new. A generic notion of essence would allow us to resist the dogmatic decoupling of learning from the infinite that the positivist stances in science and in philosophy have instituted throughout the last 15 decades or so.
In his lecture The Law of Identity (1957) Martin Heidegger drew attention to this distinction between equality and sameness, which has, he suggests, been lost in modern science and philosophy. Heidegger‘s argument is that the law of identity, in logics, has, as its essence, sameness. By insisting on an essence that were proper to a logical law, Heidegger discredits the legitimacy of logics to embody positive foundations for reasoning. Essence is the one concept that has, traditionally, linked up logics within a metaphysical framework – since essence is that which can neither be created nor destroyed, and it persists throughout all forms of becoming, invariantly. But whereas metaphysical discourses have tried to identify essences through representation, and in the sense of stripping identity to equality, Heideggers own stance suggests a different kind of engagement with it. His distinction into ontic and ontologic, Dasein/Existence and Being, suggest, in radical critique of the metaphysical tradition, that nothing of essence can ever be identified through representation in terms of stated equalities. Thereby, what he holds to be essential – a generic essentiality, as I would suggest – is violently muted. Thus, instead of separating learning from an infinite essence we must hold on to the notion of essence (Wesen, Wesentlichkeit) as the spiritual Other that is at work within modern, secular, science and its propositional statements. Heidegger links the question of essence to the question of learning, and hence suggests to not forget, over the huge successes of applied mathematics, about the traditional sense of pure mathematics as “the art of learning”.
By saying that sameness (not equality) be the essence of an equation he suggests that we ought not only apply the knowledge contained in and explicated from formula, in pragmatic vein, but that equations must remain to be the object of contemplation as well. He finds the rituals for such contemplation in the structural ways of proceeding in mathematical proofs. Only mathematical proofs can relate to the purely quantitative without seeking to externalize and objectify (identify through representations) the essence inherent to the symbolical terms that bound a formula. Quantity, that of which we can say that it is characterized by “equality”, is a concept of pure understanding (Verstandesbegriff). It is „[…] nothing else but the synthesis which empowers appearances to appear as specific figurations in space (Raumgestalten)”[1]
Let us rephrase this sentence, to consider actively what it states. Quantity is a concept of pure understanding, that is, its role is with regard to learning, not to knowing (these would be concepts of pure reflection). As such, quantity is not something immediate but constituted by acts of synthesis – and the very act of synthesizing quantity is what “empowers appearances to appear as specific figurations in space”. We can see what this emphasis on learning, with regard to the quantitative, entails: intuition is not a natural sense of sight, innate and common equally to anyone who reasons carefully. What can be intuited is strictly dependent on our powers of synthesis (powers of imagination and fantasy). In Heideggerian idiom: “It is the same precondition which allows the encounter of that which encounters, which arrests what counters into a constellation” (in German: “als Gegen zum Stehen bringen”).
But what is this precondition? It can be grasped, for Heidegger, from within in the mathematical notion of proof. With his emphasis on learning, the very dilemma which mathematical proofs posed to critical Kantian philosophy and science – namely that a proof merely guarantees the consistency of the terms in which a thought is formulated and expressed, but not the existence/truth of the idea that is being formulated thereby (think of the proofs of the existence of God) – this very dilemma is why Heidegger can affirm the mathematical as a spiritual practice that can never turn dogmatic precisely because its relation is inseparable, must be enacted, and dissolves in all attempts to represent it. “Proof is a going in circles [ein Kreisgang]. If we see through and enact this going in circles [diesen Kreisgang], we may receive, as knowledge, the pivot around which everything circles“.
How is this possible? The mathematical is for Heidegger “what we intrinsically already know about things, what we do not have to extract or abstract from things but what we, in a certain way, bring along ourselves“. This understanding holds that all things must be attributed generic and indeterminate essentiality of “sameness”, or in other words, the circle’ infinity. The Universal property of all things, for Heidegger, is not the quantitative, but generic essence. In my emphasis on literacy, I would say that in so far as we are literate, we are – virtually- what everyone else is as far as they are capable of articulating themselves.
In this way, mathematics is giving to a thing what it already has. A thing, in Heidegger, has its being – in this respect, every thing is original, the singular finitude of a thing consists in a thing‘s actual becoming what it virtually already is. This virtuality, and that is why Heidegger‘s position is so radical, is, essentially, not bounded. Heidegger‘s urge to see the essence of mathematics in the infinitary way of proceeding in circular manner – as Kreisgang – identifies all that can be learnt as the object of mathematics. For this view he finds support in the etymology of the term, Gk. mathema means „that which can be learnt“, a mathematikos is „one who is learnt“, in all generality. The understanding of mathematics in its essence attributes infinity as a universal property to all that one can familiarize herself with, in short to all that can be considered, learnt, understood, known.
For Heidegger, the circle as a virtual body of pure indefiniteness – it is the cornucopia that engenders knowledge. The universal itself is this cornucopia! Everything we can articulate mathematically, everything we can learn to understand, springs from attempting to take the point of view of the perfect compass. But this, Heidegger holds, is not possible by merely measuring the circle – it must be appreciated, in a spiritual sense, if its abundant givings are to be received.
And this, as I understand it, is Heideggers main critique on algebra. He sees it merely as providing mechanical recipes, inventories after inventories of how the circle‘s infinity can be managed – and he sees questions of esteem, the humbleness in which finite life must relate to life‘s infinity, recede from attention behind sheer pragmatism. Mathematics is valued only through its applications. Thereby, the universal, as the object of wanton management, is caught up in affairs that are largely trivial, and unrewarding. But what is the structure of the triviality Heidegger perceives?
Impredicativity, or the mode of insistent existence proper to the circular
But more exactly, what is the problem inherent to such trivialization? For Heidegger, it sets in place what he calls a general condition of „facilitation“ (in German: Einrichtung). The critique thereby does not merely regard an emergent general state of comfort societies boredom, due to dulled senses and undertrained intellectual faculties. His critique is more profoundly that of totalitarian control. Facilitation implies, so he argues, to anticipate the future on the grounds of scientific laws alone; the suspension of philosophical principles count to him as constitutive for the age of research in terms of purely operative business. And research in such terms, as purely operative business – strictly applied and case based, we might say today – this, Heidegger stigmatizes as “inauspicious” and “malign” in an “objective” sense, not according to some ill will or intention, but according to the very logics that immanently governs such research.
Heidegger specifies his critique as follows. A scientific law, he says, as opposed to a philosophical principle, draws its legitimation on the inductive basis of experimentation. An experiment, so Heidegger puts it, is this: „To imagine a condition according to which a particular dynamical complex (in German: Bewegungszusammenhang) may be traced in how it necessarily behaves, and this means, such that it may be rendered a subject to proleptic control“_. The malignity Heidegger exposes in his anticipation of technical evolution is not ascribed by him to technics itself – and neither is it ascribed to mathematics per se –, but to logically and circularly stated definitions on grounds that are, necessarily so, elected and exclusively declared to act as a frame of reference. In short, Heidegger‘s critique concerns the impredicative basis of inductive procedures in general. He exposes the inductive procedures of experimental science as presuming a metrical notion of space in order to support the objectivity an experiment seeks to expose; yet the symbolical spaces that even the science that was then contemporary actually works with, Heidegger is well aware, does not, strictly speaking, support such objectivity: vectorial spaces are spaces that may each host a multiplicity of feasible descriptions, of which the possible constellations so described may even diverge and be incommensurable within one and the same space, depending on the level of abstraction.
By „level of abstraction“ here I mean the resolution of the numerical domains, that the coefficients, which are to anchor the variable values in the formula, are considered to range over. Heidegger did not address the problem of impredicativity in these number theoretical terms, but instead in terms of temporality as a dimension that is more comprehensive than spatial extension, and that must even count as constitutive for space. If we think about what adding, subtracting, dividing and multiplying vectors entails, in a common-sensical, non-technical sense, this thought that temporality must be thought of as more extensive than spatiality does not even seem very abstract: vectors are used in probability spaces where each “segment” of a line carries with it a probability cloud of possible inclinations according to different circumstances. Newtons famous arrow of time is multiplied here, when we add vectors we add different arrows of time, so to speak, and we compute in quantum space not only with probabilities, but with probability amplitudes.
Thus we will not follow Heidegger here to where he goes in Being and Time. Because at work in his critique regarding a general condition of facilitation, and its inherent trivialization of knowledge, is, I want to argue, the entire problematical complex surrounding the notions of magnitude, ratio, value, and the role played by algebraic symbolization in how these notions may be related to each other. And in the contemporary practices of computational experimental science, it is this number theoretical complex which makes Heideggers thinking relevant in a new manner that goes beyond Heideggers own discretion of it: Electric facilitation – in his extended sense – is modulated in the realm of quantums proper to electro-magnetism.
The role that mathematics came to play for experimental science largely rests on subjecting experiments to the so-called Induction Principle in Arithmetics. In Turings and Churchs’ notion of “computable numbers”, it is this very principle that governs the suggested class of numbers declared to be “computable”. The principle states that if a property holds for 0, and that whenever it holds for a number n it also holds for n + 1, then the property holds for all numbers. The principle thus guarantees the universal applicability of arithmetics. And yet, it is exactly the uniformity implied thereby which reveals itself to be, more and more, problematical.
Edward Nelson, in his book Predicative Arithmetics (Princeton University Press, 1986), states the reason why the induction principle began to be mistrusted in the following way:
„The reason for mistrusting the induction principle is that it involves an impredicative concept of number. […] That is, the induction principle assumes that the natural number system is given.“
He drastically describes the shift to conceptual mathematics:
„A number is conceived to be an object satisfying every inductive formula; for a particular inductive formula, therefore, the bound variables are conceived to range over objects satisfying every inductive formula, including the one in question.“
In 1986, when Nelson‘s book appeared, he could still observe with some amazement that
„[I]t appears to be universally taken for granted by mathematicians, whatever their views on foundational questions may be, that the impredicativity inherent in the induction principle is harmless – that there is a concept of number given in advance of all mathematical constructions, that discourse within the domain of numbers is meaningful.“
With drastic vehemence he contradicts this view by holding:
„But numbers are symbolic constructions; a construction does not exist until it is made; when something new is made, it is something new and not a selection from a preexisting collection.“
Thereby, he is well aware that the issue is not a strictly scientific one, ; indeed, he concludes his introduction with expressing a sensitivity strikingly similar to Heideggers concerns regarding his diagnosis of our age as the age of world-pictures: „There is no map of the world because the world is coming into being.“
The entire context of science shifting from natural science to a science of technologically facilitated environmentality (our manners of facilitation) which we evoked in the beginning of this talk, makes evident the very pragmatical relevance of this – beyond any “ivory tower interests” as which such philosophical considerations are often discredited.
The alphabetization of the numerical in computation
Today, Nelsons insistence on the symbolical make-up of number concepts, and on the impredicativity of an inductively verified formula, must appear much more urgent and important to the eyes of anyone who is slightly familiar with state of the art computer science. Here, impredicativity is welcomed with an open mind as „parametric polymorphism“. We can learn the basic interest of this affirmation from the wikipedia article which characterizes parametric polymorphism as „a way to make a language more expressive“. What can we understand by this? „Using parametric polymorphism, a function or a data type can be written generically so that it can handle values identically without depending on their type.“ This may sound very technical, but such functions and data types are called generic functions and generic datatypes respectively, and they form the basis of what is called generic programming. As we can learn again from wikipedia: „In the simplest definition, generic programming is a style of computer programming in which algorithms are written in terms of to-be-specified-later types that are then instantiated when needed for specific types provided as parameters.“ Important for our context is that by regarding the metrical diagonal as the scalar product in a notion of space whose constitution is vectorial, confronts us with an understanding of space that is neither static nor dynamic, but rather vibrant and burstingly full of possible and probable extensions.
What I would like to suggest is that such a notion of space can most adequately, perhaps, be illustrated by relating it to the infinity of the space that is contained, immanently, in the characters of an alphabet. Wilhelm von Humboldt, the great empirical linguist in the 19th century, has characterized such a notion of literal space with great sensitivity:
„Language faces a truly boundless realm“, he held, „that of the thinkable. It must make an infinitary use of a finite stock of means, and it can achieve this through the identity of the power which engenders thought as one and the same with that which engenders language. Language is not to be treated as a dead something, engendered. It is not an ouevre (ergon), but itself active (energeia).“_
In similar manner, we ought to conceive of a notion of intellegibility that is to bound what can be computed. Indeed, it seems as if the respective developments have engendered a veritable alphabetization of number through programming languages. The authors of that book which most prominently raised interest in parametric polymorphism, Design Patterns. Elements of Reusable Object-Oriented Software (1994), Erich Gamma, Richard Helm, Ralph Johnson and John Vlissides note in dry understatement that this technique is very powerful, but that „[dynamic], highly parameterized software is harder to understand than more static software.“_
An alphabetization of number means to decouple numbers from the apparent sequentiality that constitutes the Induction Principle in Arithmetics, and instead to treat them like the letters of an alphabet. In a text, the appearance of a B does not necessarily follow an A, as it would according to the positional system within the alphabet. Instead it can just as well precede, lets say, an E in order to form the word BE. Of course, with this analogy I do not mean that numbers might be compassed by the alphabets of languages as we treat them in (mainstream, structural and transformational) linguistics. Rather, we might say that the elements of language and the elements of arithmetics belong together within a third, and this third is symbolicity as a quasi-material elementarity which is governed by Laws, in distinction to the logicist idea of it as a symbolic Order that is stateable by Laws.
It is here that Louis Hjelmslevs quest for a general linguistics that builds around “an algebra that were immanent to language” becomes to relevant. We will come back to this in a moment.
In any case, we would be mistaken, or at least acting prematurely, if we reduced our speaking of the alphabetization of number to a merely metaphorical usage; indeed, it is the essence of coding to establish finite and symbolical alphabets, within which the place-value logics of natural numbers (as we have it for example in the decimal number system) is combined with the mathematical notion of information in terms of quantities that can thus be positioned within a numerical space. In information science information is not, like we would expect from linguistics, semantically or materially determined (e.g. nuclei of meaning, or the phoneme as the basic element of sound). Instead it is a strictly symbolical notion, which is determined by taking a symbolization of chance as its quasi-material, or quasi-semantic „magnitude“ – magnitude because in such symbolizations, chance is counted by an indefinitely stated, yet finite variable (the so-called „chance-variable“), whose values sum up all the combinations possible for the elements of the coding-alphabet. This is what renders chance measurable. All probabilistic procedures depend upon such alphabets.
It is with their help that algebra manages the infinite today in manners as yet unseen._ Yet from a philosophical point of view, the meaning of this characterization of algebra entirely depends upon our notion of the infinite – do we mean, by the infinite, only what we can account for by enumeration, i.e. in regard to number, or do we mean, by the infinite, also what we can learn to account for by qualitative specification of that which we can, symbolically, enumerate as entities, i.e. in regard to the modularization form, and that which gives measure. From the literature on mathematical practice, the common assumption today seems to be the former. On this assumption, the status of algebra in what we can know is subordinated and instrumental. Algebra, according to this understanding, can only be applied to what is assumed to be identical in its kind by nature, independent of how we think about it. Algebra in this understanding can help us to find out how what is identical in kind, and hence potentially determinable, can actually be determined through reason. According to the latter characterization of the infinite, on the other hand, identity itself is the nature that is engendered by the contracts worked out with symbolical procedures. A thing‘s determinability, here, is not independent from the thinking that appropriates and develops the skills of actually determining it on reasonable grounds. The status of algebra according to this latter view is not only instrumental in how we can learn to know, but also constitutive for what we can learn to know.
The cipher, and its quantum-body of objective neutrality: nothingness
So let us sharpen this line of thought: our contemporary form of technics is algebraically constituted – electricity and information are both concepts with real and physical manifestations, even though they are formalized in terms of symbolical quantities. Both of these formalizations work by symbolizing their systematicity in terms of entropy. The concept of entropy originates in the science of dynamical systems, i.e. in thermodynamics._ Here, entropy is a measure of the number of specific ways in which a thermodynamic system may be arranged. Thermodynamic systems are systems constituted by an operative measure that acts upon a temporal happening, not by a particular design of a constructive order that would apply to a static constellation. Yet in order to still count as a system, i.e. in terms of a finite number of specific ways in which it can be arranged, thermodynamics must assume the state of an ideal balance for each system – what is called their state of maximum entropy. This state is an ideal state because it describes a state in which, paradoxically, nothing actually happens. As long as such a system can count as real, it never is in this ideal state. This ideal state is commonly called „disorderly“. Yet what is referred to in terms of „disorder“ here is ill understood if we conceive of it as chaotic in any genuine sense: such a state is not confused in the sense that it lacks determinability; quite oppositely, maximum entropy in fact denotes a state of too much determinability – namely the state where any way of arranging the system‘s course is equally likely to happen._
So entropy is a measure, and dynamical systems are dynamical because they are constituted not by one particular order, but a structure that supports the full operability of this measure. To picture what this means, we could say that the metrical constitution of these systems extends as the totality of all the orders that can be constituted by applying the measure. Maximum entropy, hence, would be the state of a system in which its constitutive measure measures nothing but itself.
The ideal state of maximal entropy is like an impredicative definition in logics, i.e. a definition that is self-referentially defined. Hence, the thermodynamic idea inverses the scheme of how philosophy has traditionally thought about order and disorder._ Here, ideality is not linked to pure order, and reality is not linked to deficient and confused order, but the other way around. Ideality is disorder because it is a too much in possibilities, a system is the more real the less ideality characterizes it. In other words, a system whose dynamics unfolds in approximatively uniform manner counts as a “well-ordered” system, whereas one which remains difficult to predict in its behavior counts as a “disordered” system. Noise is actually the amphibolic milieu in which a signal will trigger effects. There are interesting links here to Simondon’s theory of individuation – in a way that would lift his “emphasis” on physical instruments and apparatus (their “mode of existence”) to the level of the symbolic schemes and forms that constitute their mechanical workings.
The concept of entropy has been introduced early in the rise of thermodynamics (it is already stated in the first law), but at first it merely played a theoretical and postulational role, its assumption was necessary for accounting how a thermodynamic system can conserve energy (however asymmetrically). It was only with a formula that assumed the number of ways in which the energy of a system may be distributed to its parts can be identified globally, for all thermodynamic systems, as a constant. With this formula, Ludwig Bolzmann introduced a way of how the particular entropy of a system could actually be determined in predicative (not in self-referential and impredicative) terms._
This inverted set-up in thinking about order and disorder is constitutive for how the entropy concept has been appropriated in information science. In the mathematical theory of communication, a message is considered within a probabilistic framework, analogous to how thermodynamical systems are regarded in nature. In communication technology, the technological pendant to “nature” in thermodynamics is the “channel”. Here, entropy also counts as a measure of disorder, i.e. as an excess in ways of how the structure of the message (conceived as a dynamical system) might be arranged. Thus, from the point of view of communication technology, this line of reasoning results in the following proportionality: The higher the entropy with which a message arrives at the destination of its transmission through a channel, the higher the amount of information involved in the act of communication. It was the genius of Claude Shannon to find a formula for statistically determining the value of entropy in informational systems analog to that found by Bolzmann. With recourse to George Boole‘s binary algebra, and through applying Boole‘s notational code not to the consistency of logical arguments (this is what Boole himself applied his algebra to_) but to the conductivity of electricity, he was able to introduce a binary unit of information, the BIT, and hence think about the transmission of messages „mathematically“.
As a unit, the BIT is conceived such that it is thought to count a magnitude; yet this magnitude is entirely pre-specific. It is Chance rendered measurable by having it articulatable by a particular alphabet (whose possible articulations sum up to the symbolical totality of a chance variable). Such a magnitude is pre-specific not only because it represents the continuity of a magnitude in discrete manner, but also because the kinds of the magnitudes which the BIT discretizes are relative to a finite alphabet of coding, the sum of whose elements constitutes the „horizon of events“ which operates, in probabilistics, as the particular magnitude‘s form. This horizon of events is a so-called „chance variable“. Hence, from the point of view of communication technology, anything at all can count as a „message“ as long as it can be encoded within the finite characters of an alphabet (morse code, color code, musical scores, the laws of kinetics, those of dynamics, those of quantum states, the table of chemical elements, elements of DNA, etc)._
We can see where the initial and ingenious analogy between nature and a particular probabilistic framework gives rise, meanwhile, to all the confusions regarding the not-strictly technical status of the concept of information: Information technology renders plainly evident that the „nature“ of any probabilistic framework is a symbolically contracted nature. Therefore, the particular status of the symbolical in probabilistic reasoning has turned problematical: It is with regard to this peculiar status of the symbolic that our initial question regarding infinity matters so much: symbolical procedures are not only instrumental in how we can learn to know, but also constitutive for what we can learn to know.
Disparseness, or In the element of neutrality – the many bodies of nothingness
Strictly speaking we are, with probablistic reasoning, not in the realm of nature any more. We are in a duplication of the natural in a manner similar to how cities were always conceived to be duplications of the cosmos. We are in a symbolically contracted element of naturalness, we are in a civic nature. Ludwig Wittgenstein’s theory of rule following is highly aware of this, and placed by him within a philosophical grammar once more: Ask yourself whether our language is complete, he challenged us. Whether it was so before the symbolism of chemistry and the notation of the infinitesimal calculus were incorporated in it. »Our language can be seen as an ancient city: a maze of little streets and squares, of old and new houses, and of houses with additions from various periods; and this surrounded by a multitude of new boroughs with straight regular streets and uniform houses«, he suggested. Rule following and learning need not exclude each other, if we let go of the idea that that which we have learnt mimics nature. But if we are in such a civic nature, a technically mannered duplication of the natural, the creativity involved in thinking it can only be approached by a philosophy for which origin and originality is placed in the world, and thus, a philosophy which has overcome the distinction between the natural and the cultural, and which instead embraces – as its very element – “formality”. As an element, we are speaking here about “formality” as “stuffness”, not as “pure form” or something like that. It is a philosophy which does not seek complicity with an assumed “pure interiority” of the natural, by recollecting ones own participation in it (in the Platonic manner of Plato, continued by Husserl, Heidegger, Derrida, Stiegler today) but one which enjoys encountering it with grace, as something to be praised and appreciated, as something which will remain, essentially, arcane. We take this to mean that thought which encounters the locus of originality (every philosophy departs from this) must necessarily behave discretely. It can encounter it in a space of homothesis, a formal space where thought can be precise, witty, responsible, caressing, and self-confident, because it knows itself in an abstract space (where both time and space are indefinite) it engenders specifically for its encounters with the locus of originality.
But is this not a contradiction with the very possibility of learning? If we can behave always already only within a very element we have, ourselves, contracted? No one can learn without repetition. How could we possibly learn form repeating within a space where no real events, no real limitations, bare of all deliberations, would by our guide? This question is at the very core of Gilles Deleuzes philosophy, which discredits the dialectic circle of identity and contradiction in favor of a virtualized dialectic circle which revolved around difference and repetition. If we are, in thinking, in an element of pure formality – on a stage of abstraction on which we participate in the drama of an encounter with originality – than that which “stands out against” is not a contradiction but an indication of the naturally dispars (unequal and vanishing) materiality of sense in its formlessness:
»Repetition is […] the formless power of the ground which carries every object to that extreme ‘form’ in which its representation comes undone. The ultimate element of repetition is the disparate [dispars], which stands opposed to the identity of representation. Thus, the circle of eternal return, difference and repetition (which undoes that of the identical and the contradictory) is a tortuous circle in which Sameness is said only of that which differs.«[2]
For our interest in literacy, sense must condition knowledge, not the other way around. The dialectical circle Deleuze is proposing makes this its point of departure. Much of the unease that rises against Deleuzian philosophy stems from this characterization of the dialectical circle (which is circle of learning, in Deleuze no less than in Plato or Hegel) – because a virtualized dialectic revolving around difference and repetition is, in a certain sense, painful as learning too is painful. Deleuze calls it a “tortuous” circle.
If we forget that this dialectic circle relates to individual historicity, not to history in any general sense, then this this seems to suggest that we simply have to settle and accept a violent wielding of the powers of the real, and live with the irrevocability of unequality. In its political implications, this seems completely unacceptable – for well understandable reasons! It seems to strip all faith in intellectuality from its hope of being salvational. But “tortuous” has these connotations only in a sense that derives from our familiarity with the dialectical circle of identity and contradiction, not in the Deleuzian virtualized form as one revolving around difference and repetition. “Tortuous” means, primarily, as the etymologoy of the notion tells us: “sinous, winded, twisted, spiral”. Now, if we acknowledge that (1) we can never form complicity with an interiority of nature, but (2) that we also can not avoid duplicating it in a contractual, symbolical sense, then there can be no talk of “having to settle and accept a violent wielding of the powers of the real”. Because the virtuality ascribed to this circular and dialectical thinking has, as the very element, in the sense of “stuff”, the unequal and ceaselessly vanishing materiality of sense. The characterization of “tortuous” appears in a different light.
We can comprehend of the virtuality proper to the materiality of sense if we assume an atomism of the power to imagine that is at work in sense-making. This “atomism” must be providing symmetry structures for sub-atomic particles, and it can be conceived as equally electrified like the sub-atomic particles are conceived in quantum physic. Virtuality, then, is not another name for “potentials” – it is potentials considered in the totality of the articulatable probability spaces that are thinkable. The virtual, then, is a force, but one which does not harvest positive resources, but which gains strength and develops through the tortuous character of the disparse: its sinuous, winding, entropic (!) character.
The disparate is the very positivity of the unequal: if we place this determination in what we have discussed about entropy, as the virtual state of the impredicative amount of specific ways in which a dynamic or communicational system may be arranged, it is clear that we are in state of pure in-sinuosness where possibilities are vibrantly co-existing. To regard the element of sense as such an entropic materiality, then this suggests that learning derives its force from its discretionary power over nothingness, or, less threateningly, over the cipher. Learning derives its force from inventing codes that allow thought to appropriate virtual bodies of reciprocity, as I have called them earlier, that is a cipher together with a particular code in terms of which we can decipher what can be rendered out of it (the application of mathematics to physics, or any other domain, does nothing else!
With this, we are back to the distinction we have gained earlier regarding two different manners of thinking “identity”. In the pre-modern philosophical grammar of quantity, we saw that equality is only suitable for addressing the actuality position (verbs) and the object position, but not the subject position. It is in this sense, the sense as a subject, that a symbolic body of reciprocity cannot be addressed in terms of equality. It is a “subject” because it must pay tribute to the codes that establish and contract the relation of equivalence which it states formally. We have in such bodies a “subjectivity” which is genuinely “neutral” – in all implications of this word. To put it drastically: we have symbolical corporealities of pure neutrality. They are bodies of entropy.
But, after all, what does this have to do with the energetics on a sub-atomic level, as which I have suggested to characterize this peculiarly in-sinous, unequal and vanishing materiality of sense?
Let us attend to quantum physicist and feminist scholar Karen Barads remarks in her booklet “What is the measure of nothingness”. She writes:
»The electron is a structureless point particle “dressed” with its intra-actions with virtual particles: it intra-acts with itself (and with other particles) through the mediated exchange of virtual particles. (For example, an electron may intra-act with itself through the exchange of a virtual photon, or some other virtual particle, and that virtual particle may further engage in other virtual intra-actions, and so on.) Not every intra-action is possible, but the number of possibilities is infinite. In fact, the energy-mass of this infinite number of virtual intra-actions makes an infinite contribution to the mass of the electron«.[3]
But what is a »virtual particle« supposed to be? In quantum field theory, and this is her major thesis, the so-called virtual particles do not stand for an epistemological uncertainty, but for an objective and ontological indeterminacy. [4] »Virtual particles: virtual particles are quanta of the vacuum fluctuations. That is, virtual particles are quantized indeterminacies-in-action.« The difficult to swallow implication of this is that »Virtual particles are not in the void but of the void«. Nothingness is not some sort of receptacle, which supposedly can be endowed with particles. Rather, it is contractually bound up with the particles it is endowed with. Nothingness is of a contractual nature which must count as positive and material, even though this regards as symbolic and purely formal sense – that’s why she insists on the positivity of an ontological indeterminateness. Only in this “nature” can particles be virtual. Nothingness, the void, the vacuous is ascribed, thereby, a material elementarity of sheer neutrality. Such neutrality, I would like to suggest, is civic nature. Every “chance variable” we compute with is of this nature.
Before turning to Hjelmslev, who indeed invented an utterly general theory of pure/stuffy/elemental “formality,” where everything is engendered out of the measurement of nothingness, I would like to add a lenghty quote from how Barad illustrates her point about ontological indeterminacy:
»They are on the razor edge of non/being. The void is a lively tension, a desiring orientation toward being/becoming. The vacuum is flush with yearning, bursting with innumerable imaginings of what could be. The quiet cacophony of different frequencies, pitches, tempos, melodies, noises, pentatonic scales, cries, blasts, sirens, sighs, syncopations, quarter tones, allegros, ragas, bebops, hip hops, whimpers, whines, screams, are threaded through the silence, ready to erupt, but simultaneously crosscut by a disruption, dissipating, dispersing the would-be sound into non/being, an indeterminate symphony of voices. The blank page teeming with the desires of would be traces of every symbol, equation, word, book, library, punctuation mark, vowel, diagram, scribble, inscription, graphic, letter, inkblot, as they yearn toward expression. A jubilation of emptiness.
Don’t for a minute think that there are no material effects of yearning and imagining. Virtual particles are experimenting with the im/possibilities of non/being, but that doesn’t mean they aren’t real, on the contrary. Consider this recent headline: “It’s Confirmed: MatterIs Merely Vacuum Fluctuations.” The article explains that most of the mass of protons and neutrons (which constitute the nucleus and therefore the bulk of an atom) is due not to its constituent particles (the quarks), which only account for 1 percent of its mass, but rather to contributions from virtual particles. Let’s try to understand this better«.[5]
The measurement of nothingness involves turn nothing into a ciphered body of neutrality (reciprocity is another word for it). A cipher-turned nothingness includes the code that establishes it as a cipher. It is in the characteristics of this code that the cipher is the sign for nothingness. Perhaps we can say that “Nothingness” stands for all that which has not yet been learnt by anyone ??
Whatever its transcendent referent might be, the cipher represents literally nothing but itself. That is, it poses an enigma – the enigma that there is something rather than nothing. The enigma of thinking, learning, understanding. It poses the enigma in a self-engendering manner: operatively, characterizable, such that it can be rendered intuitable and discretable. The cipher is the symbolic “degree zero”, the being of the void. It affords the neutral space for any determination.
If I’d try to put this in suggestive terms: the cipher, together with its lofty skeleton (the code) is a symbolical body which insists in all determinability as a distributed and hence desiring quantum-body. This quantum-body is what it is only because it is dressed up as cipher. Its ‘disguise’ is no concealment, it is Heidegger’s sameness of the infinite’s identity, invariant and insistent generic essence. The Parmenid’ean “impossibility” it incorporates is that it cannot exist barely, nakedly. Yet this impossibility is not a lack, but a cornucopia of abundant sense that can be made, of a knowledge of a nature which is mannered, civic.
It is the body of the intelligible Earth as it engenders and gives birth to itself. It is the Earth’s natural body, whose nakedness is the neutrality of the cipher – seeking to glance at its bareness is nothing but obscene. It insists on being paid respect through discreetness – like the circle, the elusive figure of infinity – while at the same time it will always resist being the object of discretion. It places us in a position of modesty that is not in conflict with the pursuit of intellectuality – quite opposite. The main requirement for this modesty not to be a false modesty is to pursue distinctiveness, formality, eloquence, power, sophistication – in the awareness that these abilities are conditioned and grounded in an abstract space of symbolical contracts, in the being of the Earth’s body of neutrality, the cipher. Symbolical contracts constitute stages for abstract thought – such stages can be thought of as cities that desire to accommodate nature in its universality. Cities where no one is native, and where everyone who wants to learn is welcome.
Glossematics: Theory conserves and circulates and differentiates the materiality of sense
The first inception of such a space is usually associated with Thales, whose theorem states how quantities might be extended or diminished to infinity while keeping their ratio as well as their proportionality. The theorem postulates a Logos, as “a manner of speech” which articulates an identical relation, the invariance of one and the same form. It is a logos of similarity whose operative metrics is not external to itself, such speech bears it immanently. It is the logos of speaking mathematically, and that is why its constitutive metrics needs not be external to it. Conceived of as a logos, as a linguistic characteristics, mathematics is a cornucopia which donates infinitely much out of almost nothing at all, as Michel Serres has put it.
Electricity and information technology reminds us that the most formal manner of thinking we know, mathematics, is not the silent, non-conversing Other to natural (or “original”) language’s eloquence, but that it too is communicational. It is caught up in the conversation about originality which is, even though it is a conversation, not idle talk but the springing source of intellectual mastership which we associate with “Knowledge”.
The Danish linguist Louis Hjelmslev has begun to link back to the space of homothesis and he has come up with a „General theory of linguistics“ which is set up, as I would like to argue, as a kind of theoretical geometry (this is the meanig of the Thales moment in geometry, the inception of theoretical geometry) of anything that can be uttered in a manner that makes sense – now or at anytime. He called it Glossematics, from the Greek glossa for “obscure word, a word that requires explanation, a foreign word, a word by a foreign tongue”. His ambition is to come up with a theory of language which must take nothing for granted, a theory which carries its own principle of metricality within itself just like theoretical geometry after the theorization by Thales and others. “Such a linguistics” he writes, is distinguished from conventional linguistics, and “would be one whose science of the expression is not a phonetics and whose science of the content is not a semantics. Such a science would be an algebra of language, operating with unnamed entities […].” These entities are the glossemes around which he constitutes his theory. Glossematics “[…] aims to produce just such an immanent algebra of language. To mark its difference from previous kinds of linguistics and its basic independence of non-linguistically defined substance, we have given it a special name, which has been used in preparatory works since 1936: we call it glossematics (from ‘Yhwuua ‘a language’), and we use glossemes to mean the minimal forms which the theory leads us to establish as bases of explanation, the irreducible invariants.”
Hjelmslevs theory is availabe in his “prolegomena to a general theory of language” as well as in his posthumeously published “Résumé of a theory of language”. Both texts are utterly formalized, and form a system. In that, they are not unlike Ludwig Wittgenstein’s Tractatus. Thus, it would be vain if I tried to introduce it on a technical level as a basis for you to relate to in my discussion of it. Instead, I have tried to prepare the stage for my consideration of glossematics with regard to CHARACTERIZING DISCRETIZED PROBABILITY DENSITIES AS A KIND OF ALPHABETICITY .
Hjelmslev continues the structural approach of Saussure, with perhaps no more than one crucial divergence. It concerns the manners of dealing with negative values. As is well known, Saussure‘s key stroke was to assume negative values and thus to think about language as a system, with a proper structure, the elaboration of which was his key interest. Saussure marks the beginnings of engaging with language systematically and constructively. This was only possible because he did not seek to characterize instances of text descriptively, according to what features in a particular instance of text, but by placing each and every instance within one horizon of possible features. Like this, he hoped to find a way of formalizing linguistic substance (the meaning of a word or a grammatical, morphological etc feature) without having to assume an essential positivity for it – instead, as he saw it, we need to analyze text differentially, and in absence of the very movement one tries to trace, just like distributed distinctive points are integrated by calculus in mathematical analysis. Signification, for him, is the relation between Signifier and Signified, and yet, this relation is never congruent with a sign itself. Signs are what counts as “distinctive points” in the analysis of dynamics, and Saussure assumed that no sign exists outside of its relationality with all the other signs. Harald Weinrich describes as follows:
“Every linguistic sign has a determinate value insofar as it differs from all the other signs within one and the same linguistic system. With regard to this, Saussure conceives of signs as ‘terms’ within a relational complex. The relation between one term and all the other terms of language is, by principle, a differential relation, oppositional and negative. All that matters is that a linguistic sign be not all the other signs of the same language.” Saussure himself puts this plain and simple: “Dans la langue il n’y a que des différences sans termes positifs” or: “tout est négatif dans la langue.” Thus, linguistic analysis which aims at positif results would have to go through all the negatives to all the signs of a linguistic inventory – a pragmatical burden which has lead to the oblivion of Saussures “theory of values” aspect. Instead, structural linguistics was pursued for example on the basis of a particular notion of the phoneme as a “nucleus of positivity” such that linguistic analysis may provide an empirical and inductive basis for logical foundations of meaning, semantics, syntax, etc.
Well, this is exactly the point where Hjelmslev continues the Saussurean legacy. The pragmatical issues regarding inventories of signs to be ‘overcomplex’ to be treated according to the value theory, rather than as variations from a particular magnitude (the phoneme) did not bother him – the dedicated goals of his theory was not to provide an inductive basis for systems of logical foundation, but an economical one: to conserve and cultivate the “inexhaustibel abundance of manifold treasures” through which language is “mankind’s patent of nobility”.
As disturbing as this sounds to our ears – a patent of mankind’s nobility – it bears the aspect under which I would like to raise the question of my title: is “an algebra immanent to language” suitable for characterizing discretized probability densities as a kind of mathematical alphabeticity.
With this twist it is clear that I don’t treat this question as a question which could be “verified” or “falsified” empirically. Rather, I treat it as the question of “under which considerations could we hold it to be suitable, or non-suitable”.
Beyond the Apparatus, or: getting over the deadlock of tautology by repetition (alphabeticity)
This means that we have to look closer at what such an “alphabeticity” would entail, from what it would distinguish itself, and what would be its promise. Let’s begin then by contrasting it to what is the common practice today – namely to affirm that the characterization of probability densities must enable us to deal with them “mechanically”.
To treat them mechanically involves considering them as potentials in the thermodynamic, energetic sense. But when natural kinds are symbolized by categories, categories explicate the notion of forms, and properties do not belong to things, neither individually, specifically or generally, but extend into a qualitative space whose principle is logical, then probability densities are caught up in a virtuality that results from their status as ciphers, as entropic bodies of neutrality. We can characterize this status as the mutual implicativity of “epistemological uncertainty” and “ontological indeterminacy”, but it is not enough. This virtuality introduces to quantum physics the entire range of expression that is proper to literacy: grammatical correctness and logical consistency as well as poetic voice, varying degrees of mastership in the sophistication and eloquence in expression and articulation, instrumental and strategic use of “quantitative” speech and writing, involving dramatization, narration, humor and lies. A mechanism, we are used to thinking, liberates us from all the difficult interpretations of sorting out the reliability of a statement in this manner – mechanisms never lie. They never instrumentalize you. They never try to trick you into a position you don’t want to find yourself in. They are purely general and objective. Isn’t this what we are used to thinking? Is this not why we trust academic language if it is as in-eloquent as possible, as conforming and general as is established in and demanded by the communities of expertise?
And yet – don’t we experience in all those information-based infrastructures that condition our current manner of “being facilitated” the almost exact opposite of this? Do not all the technologies impose “imperatives” on their users – especially if they are supposedly straight forwardly mechanical?[6] Well exactly. The idea that mechanism are straightforwardly what they are corresponds to mechanism applied to sets of particles where each one is replaceable with any other one without change in the smoothness of how the mechanism works. But in information technology, the (classical, non-virtual, non-quantum) physical “particles” are expressed as “values” in a “variable” which ranges over “magnitudes”. This is what makes “mechanisms” in information technology parametrizable according to different “weightings”. Operating with “particles” expressed as “variables”, mechanisms turn “generic”. Generic mechanisms do not directly transform “forces” (as we would say for mechanisms that operate on particles that are not expressed as values) – they do so within an environment of power within which they modulate how they transform forces. The interplay of such value-based mechanisms are more adequately conceived as “instruments”, not unlike “instruments” in Music.
Michel Serres has given a characterization of language which very well summarizes Hjelmslevs concerns with his insistence on “nobility”: „La langue n‘est pas, elle peut“, and further „une langue vivante, c‘est une puissance“. The symbolical techniqes of discretion in language are alive, and as long as they are alive, they are capable of saying all: „Une langue a la capacité de dire ce qu‘elle ne dit pas encore, elle est creative, elle invente“. The same holds for “mechanisms” in technics based on information/quantum physics.
Thus, it is Saussures “lost beginning” of developing a theory of language based on values in all their algebraic complexity (which cannot be captured by “sign-based representations of signifying relations) that is continued by Hjelmslev. Half a decade later, the resistance towards it is still the same as in Saussures time: too abstract, difficult and demanding, if viewed proportional to what it can yield. Similar concerns are raised against category and sheaf theory today.
But the singular “benefit” of these approaches is that value-based theories, not representation-based theories, can be conceived as “instruments” rather than as “inventories” or “logical grids that are grounding explanation”. They can be conceived as being capable of saying even that which it cannot say yet – a theory’s virtual possibilities. The music that can be made with a violin can never be exhausted. And so is the case, if we follow Hjelmslev and Serres, for theories. The influence between a theory and the object that it makes understandable are not one-way (inductive, the object informing the theory, or deductive the theory informing the object), but reciprocal. That is why Serres, as well as Hjelmslev, maintain that a characteristics, an alphabet that is capable of symbolizing something, maintains relations to theories as instruments like musical notations maintain to mustical instruments: „a characteristics peut dire tout ce qui est écrit, et on peut inventer avec un piano.“
Theories, then, can be as much or as little “verified” as the compositional notations for a musical piano piece. They do not themselves “explain” something – they can be “played” (more or less well) to produce “articulations” of a perspective (theoria) offered and grounded in the object around which it revolves. This refrains from a “realist” position – language does not describe objects independent of it. But it also does not follow a correlational point of view, where the “real things” were always already constituted by a formal language – rather it maintains that we ought to treat “objects” as ciphered “invariants”, which cannot be comprehended without articulating them in their totality. This totality, and that is the crucial point, is itself an articulation of the totality’s infinity. They ways we can to think about that is not that the articulation provides a representation or description of it, but rather that it sets in place within language the homothetical realm of theory where the object can be staged and saturated with its own objectivity. Hence a theory does give us the real object, but it does so in lesser or greater richness. That’s where how Hjelmslev gives priority to literacy rather than to logics or grammatical correctness. He declares:
“Naive realism would probably suppose that analysis consisted merely in dividing a given object into parts, i.e., into other objects, then those again into parts, i.e., into still other objects, and so on. But even naive realism would be faced with the choice between several possible ways of dividing. It soon becomes apparent that the important thing is not the division of an object into parts, but the conduct of the analysis so that it conforms to the mutual dependences between these parts, and permits us to give an adequate account of them. In this way alone the analysis becomes adequate and, from the point of view of a metaphysical theory of knowledge, can be said to reflect the “nature” of the object and its parts.”
and he continues:
“When we draw the full consequences from this, we reach a conclusion which is most important for an understanding of the principle of analysis: both the object under examination and its parts have existence only by virtue of these dependences; the whole of the object under examination can be defined only by their sum total; and each of its parts can be defined only by the dependences joining it to other coordinated parts, to the whole, and to its parts of the next degree, and by the sum of the dependences that these parts of the next degree contract with each other. After we have recognized this, the “objects” of naive realism are, from our point of view, nothing but intersections of bundles of such dependences. That is to say, objects can be described only with their help and can be defined and grasped scientifically only in this way. The dependences, which naive realism regards as secondary, presupposing the objects, become from this point of view primary, presupposed by their intersections.”
Totality, on which his entire approach to pursue a value-based theory of language hinges, must not be thought of as consisting of “elements”, “parts”, “things” – but of relationships. Of course these relationships contract “substance” that is being totalized – but “substance” must be dealt with like “energy” in physics, that is entirely without a positive or a negative, (a semantically determined) definition of it. We can operate with the concept of energy precisely because its sole scientific relevance stems from the internal and external relations that can be examined by the Laws of Conservation: we can articulate structures of reciprocal transformation on the premise that the quantity of “energy” – whatever it may be, it is dealt with as an unknown quantity – remain invariant throughout the transformations. This is how Hjelmslev suggests to think about substance (of real things) that is expressed in language. To do linguistics systematically for him is like establishing a network for the circuit of “substance” not unlike the networks for “circuiting” energy – which lies at the very basis of chemistry, physics, biology, etc. System, if it is not to give a representation, affords logistic infrastructures. In the case of linguistics, hence, systematical theories provide logistical infrastructures to make sense of the world in its abundant richness. This is indeed how he begins his treaties:
“Language-human speech-is an inexhaustible abundance of manifold treasures. Language is inseparable from man and follows him in all his works. Language is the instrument with which man forms thought and feeling, mood, aspiration, will and act, the instrument by whose means he influences and is influenced, the ultimate and deepest foundation of human society.”
[[“But it is also the ultimate, indispensable sustainer of the human individual, his refuge in hours of loneliness, when the mind wrestles with existence and the conflict is resolved in the monologue of the poet and the thinker. Before the first awakening of our consciousness language was echoing about us, ready to close around our first tender seed of thought and to accompany us inseparably through life, from the simple activities of everyday living to our most sublime and intimate moments – those moments from which we borrow warmth and strength for our daily life through that hold of memory that language itself gives us. But language is no external accompaniment. It lies deep in the mind of man, a wealth of memories inherited by the individual and the tribe, a vigilant conscience that reminds and warns. And speech is the distinctive mark of the personality, for good and ill, the distinctive mark of home and of nation, mankind’s patent of nobility.”]]
Obviously, this raises the theme of “knowledge” and “anthropocentrism”. My whole point is that such an anthropocentric approach to literacy, and “rooting” formal value-based theories therein, is exactly the way to keep “knowledge” free from anthropocentric dogmatism in science. It we forget that we are speaking – if neurons are believed to be “immediately the thing itself” – well, then we are back in the rich legacies of our past, when science was pursuing to lay bare an Original Language, an Adamitic Language. The language in which we express science is never natural – it is highly ciphered, symbolized, and formalized.
This feels very counter intuitive and even paradoxical today. Just like the insistence that systematical thought must needs hold on to “totality” – in fact, that only thinking in terms of totalizations, that is, insisting on finite forms in science – can prevent totalitarianism. But only if thought plunges in an infinite element of abundant plenty of things to be known.
So, in all brevity, how is the set up of such a conservational theory of language?
It presumes the “infinite element of abundant plenty of things to be learnt and known” as “purport”. Purport literally means that which is contained, conveyed, carried forth. Purport itself is the “real body of linguistical substance” – that is, not its invariant and infinite substance but the bodies that incorparate this substance in finite manners. In an analogy we could say that like all things existing incorporate “energy”, or all animate beings incorporate “live”, thus all that is purported incorporates “linguistic substance” – spirituality? breath? meaning? It doesn’t matter how one sympathizes to characterize this substance in semantic terms. As an invariant which can only be “trafficked” it can be treated entirely formal and void of attributed characteristics. Purport, to Hjelmslev, is the body of language that is alive. Hence, purport itself cannot be represented or described – it can only be articulated. What Hjelmslev introduces into the immediacy of articulation (and the metaphysics of presence it seems to be entwined with) is a level of mediacy by making it constitutive for all analysis and systematical theorization of the alive bodies of language (literacy) that they must be dramatized on an abstract stage of homothesis, where the equivalence of an articulation with what it purports, must be established.
It is the set up of such a stage that is formal apparatus must provide. The following ingredients are crucial: the stage of homothesis is made up of two coordinated planes, each of which unfolds around an axis: a) between form and content, and b) between substance and expression. We have a fourfold set-up, to planes that are per declaration to be congruent. Congruence is a property that must old between a content that is articulated, and the form in which it is articulated. The measure according to which it must be congruent must be derived – by the sole criteria of adequacy – from the other plane, that between substance and expression. For Hjelmslev these are the two respective procedures which constitute an articulation as mediate, and hence as the possible object of scientific treatment: that is, an articulation as being irreducibly double, corresponding to the homothetical space: the consistency of an articulation is relative to a particular manner of partitioning.
So what is being staged, in such a set-up? How are “values” dealt with purely formally, by an algebra immanent to language? Values are treated as the “Unkonwn Quantities” that are being determined by the procedures to manage the infinite which every algebra produces. Language, the totality of what is being purported, serves as the alive and empirical “world” with which calculated designations of the values must be brought into contractual agreement.
How precisely does this work: Signs are central to glossematics – and signs are conceived as functions. The crucial distinction vis-a-vis propositional calculus is in the insistence that the form of functions itself cannot be left transparent:
„Avoid the ambiguity that lies in the conventional use made of it in science, where „function“ designates both the dependence between two terminals and one or both of these terminals.“
For Hjelmslev, “A dependence that fulfills the conditions for an analysis we shall call a function. Thus we say that there is a function between a class and its components (a chain and its parts, or a paradigm and its members) and between the components (parts or members) mutually. The terminals of a function we shall call its functives, understanding by a functive an object that has function to other objects. A functive is said to contract its function. From the definitions it follows that functions can be functives since there can be a function between functions. Thus there is a function between the function contracted by the parts of a chain with each other and the function contracted by the chain with its parts.”
The functives, as the terminals of a function, must be regarded in a manner decoupled form the relation captured in the one function, are the “bodies” of the values. These bodies are indeterminate but determinable. They are thought to be the “subjects” whose integrity is granted the dignity of an “active” being, an “agent”, but in a “noble” sense, not in a “general” sense. These bodies of values are the “subjects” between which the “contracts” of the formal theory are worked out.
Signals and Codes: The world translates itself in in the nature of thought
This is more than a metaphorical way of speaking, Hjelmslevs’ glossematics is a veritable law-based politics – yet the laws are natural, not political, in the same sense as Newton’s laws are “natural”: understanding them affords mastership and understanding of natural processes – only here, it the nature of thought itself that can be learnt to be mastered. The nature of thought is civilized by stating it not into propositions, that hold true in a referential sense, but by bringing it into the terms of contracts that need to be worked out continuously. The subjects, as integral bodies of values, are themselves not being formalized – here lies the crucial difference to “agent-based” theories of networks like Bruno Latours ANT or Manuel deLandas assemblages. And it is in this sense that Hjelmslves framework of a “politics of the nature of thought” links up so helpfully with Gilbert Simondon’s theory of individuation. As bodies of values, these “impersonal” subjects are truly capable of learning through the communicational environments which they “inhabit”. Because these environments are made up of disparate magnitudes only – the kind of magnitudes we ascribed to the bodies of neutrality earlier. These communicational environments are “entropic”, entirely symmetrical and reciprocal, yet not in the sense of disorderly but in the sense of impredicativity.
I would like to end my talk by characterizing such communication more precisely through the work of Gilles Deleuze. Like in Hjelmslev, sign-systems are a contractual settlements within a nature that is law-based. There is no causality in Signs themselves, causality resides in the laws. Inbetween the two we have the play of signals. Signals are neither signs, and neither are they physical materials. They have a materiality that is symbolic.
“[…] it is not the elements of symmetry present which matter for artistic or natural causality, but those which are missing and are not in the cause; what matters is the possibility of the cause having less symmetry than the effect.” As Deleuze puts it. The effect, this is what is being contracted on the level of signification, and the inversion that is so difficult to get is that signification introduces symmetry through differentiation. Signals are the translators between the physical and the significant. He continues:
“For this reason, the logical relation of causality is inseparable from a physical process of signaling, without which it would not be translated into action.” He specifies:
(1) By ‘signal’ we mean a system with orders of disparate size, endowed with elements of dissymmetry;
(2) by ‘sign’ we mean what happens within such a system, what flashes across the intervals when a communication takes place between disparates. The sign is indeed an effect, but an effect with two aspects: in one of these it expresses, qua sign, the productive dissymmetry; in the other it tends to cancel it. The sign is not entirely of the order of the symbol; nevertheless, it makes way for it by implying an internal difference (while leaving the conditions of its reproduction still external). (Deleuze DW p.x)
But how can it do that – not entirely be of the order of the symbol, and yet implying an internal difference while leaving the conditions for its reproduction external? Learning takes place not in the relation between a representation and an action (reproduction of the Same) but in the relation between a sign and a response (encounter with the Other). Signs involve heterogeneity in at least three ways:
(1) first, in the object which bears or emits them, and is necessarily on a different level, as though there were two orders of size or disparate realities between which the sign flashes;
(2) secondly, in themselves, since a sign envelops another ‘object’ within the limits of the object which bears it, and incarnates a natural or spiritual power (an Idea);
(3) finally, in the response they elicit, since the movement of the response does not ‘resemble’ that of the sign.
“The negative expression ‘lack of symmetry’ should not mislead us: it indicates the origin and positivity of the causal process. It is positivity itself.” (Deleuze)
With this I would like to end, thank you.
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[1]„Dieser reine Verstandesbegriff Quantität ist aber nichts anderes als jene Synthesis, kraft deren Erscheinungen als bestimmte Raumgestalten erscheinen können. Also sind alle Erscheinungen als Anschauungen Quantitäten, und zwar extensive (Raum). Es ist dieselbe Bedingung, die das Begegnende begegnen lässt und die es als Gegen zum Stehen bringt. Der Beweis ist ein Kreisgang. Wenn wir diesen Kreisgang als solchen durchschauen und vollziehen, gehen, bekommen wir eigentlich zu wissen, worum sich alles „dreht“.“
[2] Gilles Deleuze, Difference and Repetition S. 57.
[3] Ebd., S. 30.
[4] Sie konturiert den Hintergrund dessen wie folgt: »Die so häufig erzählte Geschichte von der Existenz virtueller Teilchen Iautet, dass sie ein unmittelbares Resultat der Heisenberg’schen Unschärferelation sei. Doch die ›Unschärfe‹- Relation (sic) von Energie und Zeit ist weit davon entfernt, ein beständiger Sachverhalt zu sein. Vor allem die neuere Forschung stützt die Deutung dieser Relation eher unter dem Aspekt der Unbestimmtheit als der Unschärfe. Zur Debatte steht mithin eine ›objektive [ontologische] Unbestimmtheit‹(Paul Busch), nicht jedoch eine epistemologische Unschärfe. Siehe zum Beispiel Paul Busch, ›The Time-Energy Uncertainty Relation‹, in: Time in Quamum Mechanics, hrsg. v. Juan Gonzalo Muga, Rafael Saia Mayato und Inigo L. Egusquiza, 2. Auf., Berlin: Springer 2008 [Orig. 2002]. Für eine ausführliche Darstellung der Deutungsunterschiede, die durch Fragen der ›Unschärfe‹(Heisenberg) vs. ›Unbestimmtheit‹(Bohr) gekennzeichnet sind, siehe auch Barad, Meeting the Universe Halfway (Anm. 3).«
[5] Ebd., S. 30.
[6]do not all the discussions around systems-optimization and efficiency, around securing the smooth control of networks on which many sub-structures depend, scare us with the threat of systems-breakdown, corruption (what exactly is meant by this??), into willingly subjecting to the dictates of those imperatives? Do not our discussions around the “intolerability” of privacy and the demand for “transparency” touch upon exactly the insufficiency of thinking that “mechanisms” spare us from critical evaluation, interpretation, and our exhausting efforts in striving to be convincing? Critical evaluation and interpretation requires the possibility for taking a distance to demands that make themselves noticed as only apparently real and immediate necessities.
cf in relation to this text also my other articles on articulation, computability, entropy, especially:
Atomism, alphabet and idiosyncrasy (the amorous nature of intellectual conception)
Arché, arcanum, articulation. The universal and its characteristics.
Articulating quantities. If things depend on whatever can be the case.
The idea of a characteristica universalis between Leibniz and Russell, and its relevancy today.
Quantum City: from the cosmological city to the articulated city I: A psycho-political essay.
Quantum City: from the cosmological city to the articulated city II
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