Algebraic Concepts Characterized / Gilles Deleuze / Plotting from History

Articulating quantities – when things depend on whatever can be the case

Paper delivered at the ART OF CONCEPT conference, MaMa in Zagreb (June 2012)

It is a first attempt to speak about what I call the Dedekindian, Boolean, and Deleuzean notion of reasoning as concerning not the totality of what there is, but as the totality of what can be thought rigorously. 

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„Man can think in the sense that he possesses the possibility to do so. This possibility alone, however, is no guarantee to us that we are capable of thinking.“

– Martin Heidegger

„Further still, beyond the world of representation, we suppose that a whole problem of Being is brought into play by these differences between the categories and the nomadic or fantastical notions, the problem of the manner in which being is distributed among beings: is it, in the last instance, by analogy or univocality?“

– Gilles Deleuze

0 Précis

The world is everything that is the case – I would like to take this famous line from Wittgenstein‘s Tractatus as a starting point. The world is not the totality of things, he says, but that of facts. I would like to consider, inversely, the world as the totality for whatever can be the case. After this inversion we can – a small deviation not withstanding – keep with Wittgensteins language game and call the totality of whatever can be the case the totality of artefacts. Artefacts capture and embody acts of concentration by intellect, not things that have happened or are given. What distinguishes them as artefacts from facts is that they conserve an act of concentration by condensating this act into manifest form. (1)

The crucial question, for considering the world as the totality of that which can be the case, regards what I would tentatively suggest to call the auxiliary structure needed for being able to say something about this world at all. My core interest in the following concerns the possibility of a philosophical grammar. I call it an auxiliary structure and not an infrastructure, because referring to it or behaving in it needs thought that considers. Such a grammar allows for conception, of course, but the act of conception it structures is ampliative. I would like to consider the possibility for a philosophical grammar in which conceiving is engendering-by-inference. Ampliative inferences are inferences capable of broadening a terms extension beyond the possibilities that were contained in the premises. Such thinking-as-conceiving involves an aspect of inception, of beginnings. My interest, in short, is to regard artefacts as articulations of such a philosophical grammar, in pursuit of an architectonics that were proper to the city today. Of course this article can be no more than an early step in this pursuit.

The brief sketch I would like to layout in this paper to support such an interest in artefacts as condensations of intellectuality departs from exploring a peculiar proximity between Ludwig Wittgenstein, Martin Heidegger and Gilles Deleuze. A proximity which may seem quite unlikely, at first sight, but for which I would like to argue along the following lines: all three share a common interest in the Kantian insight that reason conditions experience, but more importantly, they explore this insight in relation to acts of learning rather than objects of knowing. This distinction creates, despite all the obvious differences if not incompatibilities, a peculiar proximity among their thinking.

In their own individual ways, Wittgenstein, Heidegger and Deleuze have evoked the ancient sense of mathesis as an art of such conditioned learning. They have embraced the challenge that for learning, the conditions can never be sufficient nor clear and distinct. In such a sense of mathesis, to which I will refer to in the following as mathetical, learning is less concerned with issues of representation or recognition than with an act of appropriation and inhabitation of intellectual capacities and abilities. Learning in a mathetical sense involves a kind of privation which inverses the usual sense of the word: it involves a privation which engages in a relation of giving without depriving.

In the following I would like to extend on this aspect that for learning the conditions can never be sufficient nor clear and distinct. I will suggest to view artefacts in a broad sense – be it software, architecture, film, music, a piece of technology, suggestions for policies, tools for financing, business plans, recipes or theory books – as the manifest instances of acts of learning. I will regard artefacts as cases which are conceived and engendered by ampliative reasoning; this means, in full demand of that gesture, withstanding the temptation to subject thought to the comfort zone provided by artefacts if we assign them the a-conceptual status of incommensurable singularities, or the not-engendered one of generic stem cell like entities. (2)

Considered in their relation to the acts of learning which they manifest, rather than to their status as objects of know-how, artefacts are condensations from the outer space of intellectuality. They are aliens-from-within, if you like. Unlike pieces of art, they are popularized and in that sense de-capitalized acts of concentration. If it makes sense at all to say – with consideration – that they are, we might have to extend the conceptual leap from being to existence towards insistence, and claim that just like things are insofar as they are there (Dasein), artefacts are insofar as they are here. If being corresponds to things-as-they-are-named, and existence to things-in-their-thingness, we can say that insistence corresponds to the prespecific-genericity proper to things considered within the fantasmatics of what can be learnt mathetically.

1 Within the outer space of intellectuality

Wittgenstein had started to sketch out a philosophical grammar for addressing things-as- facts. A philosophical grammar for addressing things-as-artefacts considers things in their pre-specific genericity. It assumes they can be named, in this pre-specificity which they manifest, by mathetical instead of literal names. We can find such mathetical names, I would like to suggest, in algebraic polynomials.

The etymological meaning of polynomials is having many family names. Polynomials name heterogenous things, hybrids that comprehend aspects of many generic lineages. Polynomials name things that have no natural belonging – if natural belonging means that the identity expressed belongs to exactly one genus or genre.

From a pragmatic viewpoint, we can characterize the mathetical context of polynomials today as follows. For all sciences working with methods of probabilistics, factoring polynomials is as ordinary, as elementary and capability-dependent a practice as composing more or less well-formed, more or less well-reasoned arguments in sentences is. Polynomials feature in systems of differential equations, and they are especially useful when describing processes which do not unfold uniformly and steadily in space and time. Polynomials allow to map, probabilistically, they are the building blocks of all sciences involving computation and electronic technology today.

The intention by suggesting such a mathetically grammatical perspective is not to conflate polynomials with names, mathematics with linguistics. The intention is to consider a way in which we can address the peculiar mathetical language-ability triggered by computation and the algebraic symbolicalness proper to it. Such a perspective would allow us to assume, next to the calculating and conceptualizing faculties of reasoning as judging (literally in German Urteilen), a faculty of computation which is concerned with mathesis, with the art of learning to partition (literally in German Teilbarmachen). (3)

If we take the algebraic formula for a circle as an example, we can see that what this formula names is never fully given. Polynomial predication is not directly about the assignation of an object as a thing, nor about the assignation of things as generic objects. The formula for a circle is the formula for any circle, and needs, in order to denotate a particular circle, further determination which relies on input or investment that cannot be deduced from what the formula itself contains. In short, the formula must be placed within a certain problematical domain, it must be ascribed a certain rôle within the act happening in this domain (the problem), and it must, as an actor within a larger play, be equipped, or more precisely: doped, with properties and features. (4) Polynomial predication organizes a dramatical space where things intermingle, in the pure symbolicalness of objects which are regarded in their dissolution into generic pre- specificity. The polynomial space of predication comprehends, it incorporates this intermingling, yet there are no particular instances involved in the non-dramatized purity of it because the polynomial space itself, without a grammar that structures its expressions, is incapable of organizing this intermingling. Strictly speaking, there is nothing to be counted nor judged yet, in the space of polynomial predication, before the drama is staged, before the partitioning is organized.

Such a grammar is a formulaic grammar, and the identities it names are both generic and pre-specific. Hence they are – in a fully determinate yet never exhaustively determinable sense – evoked. They are literally called out, summoned, roused, out of the outer space of intellectuality we all engage with when we learn. These pre-specific identities are only insofar as they are articulated, a bit like daimonions, manifest voices, like those we encounter in erotic recognition. Viewed in their prespecificity, these identities were more adequately called erogenic than generic; they vibrate of erogenous affectivity, these identities are the pure skinning and membraning of the restless intermingling of abundant responsitivity, which is proper to things when dissolved into the symbolicalness of polynomials. Without the evocation of grammatical articulation, the polynomial space of predication comprehends a mere happening of subtle violence.

The attractive promise is that such a grammar may provide us with the ability for systematic and structural thought in domains where reason is not only insufficient but also abundant. In other words: whenever we refer to the probable, the topical, the urban. The space of polynomial predication comprehends virtually anything that can be thought rigorously. As such, it exhausts neither thought nor the outer space of intellectuality – it relies on input or investment that cannot be deduced or induced from within it; but it does provide the virtual consistency of how we can consider this where where all the artefacts that ever were, are or will be here, as manifestations of acts of intellectual concentration, are being conceived. The outer space of intellectuality is vaster than we can imagine or overview, in any one moment or in any summation of moments. But we can be positive that it is not infinite. It is abundant yet not unlimited, because it is only insofar as it is inspired by the thinking of people, finite beings.

The modality of its consistency is virtually real – not actual, and not even necessarily actually possible. It consists in the purity proper to differences of which the parts are potentially set in relation to anything at all, within the boundaries provided by the space of polynomial predication as it is dramatized. Within the outer space of intellectuality, the differences are not yet actually related to a code that joins them; they are not rendered present, they are not symbolized. Or in other words, the erogenous skinning, membraning of this differential space is not yet responding to something. It is not yet organized by the reciprocal liminalization provided by a formulaic identity-relation, i.e. by an algebraic equation. As such, the outer space of intellectuality is the springing origin of the dignity of thinking. Its fertility engenders. It is an erotic, a cultivated fertility, not that of natural reproduction qua multiplication. Thought that strives for articulating the contradictory consistency of this space – it‘s unjoined differences-in-relation – involves the genesis of its actualizations. Thought that claims authority within the outer space of intellectuality is involved in a static genesis, through its acts of intellection. It does nor revolve around a center. Rather it involves condensation, a skinning, a membraning, falling off from its rotating acts of concentration. The products of this fertility are subtle and vulnerable, to the point of their virtual non-existence – if the grammar that expresses these condensations is incapable of receiving them as cases.

In this, thought in such domains of abundant yet insufficient reason is different from the dynamics of dialectial inferential movement. Dialectical inference negates the process of condensation, its mediality and simulacras, and instead of affirming this process it strives for the annihilation of contradictions through normalization. Mathetical inference, on the other hand, strives to articulate the contradictory in myriads of ways. It strives to articulate polynomial names as quantities, not names as generas or singularities. Polynomial names involve diverse ranges of powers, articulating them into formulaic systems of equations means mediating between the involved ranges and orders of powers. Mathetical inference can dissolve the violence involved thereby through a posology of pharmaceutical doping. Such a posology, in the proper sense of logos, does not need to be invented anew, I would like to suggest with the lines of argument presented here. Rather, it can be found in and extracted from algebraic structuralism which attends to the world as the totality of artefacts. Algebraic structuralism is a categorical structuralism, (5) and it looses the dogmatism that usually goes along with categorical thinking if we relate it to domains of abundant and insufficient reason. (6)

What I would like to present in the following is a sketchy proposal of approaching such a way of thinking about the articulate-ability of quantities for polynomial predication.

2 mathesis

There was a time when the theory of the forms of experience and that of the work of art as experimentation had maintained an intimate relation. In a today somewhat outdated sense of the word, the arts were referring to the development of abilities very generally, to a sort of cunning reason and the sophistication in how we can carry out human endeavors in general. As such, the term comprehended a double make-up of the development of such abilities as ars and as techné. The Greek term techné seems to have been applied in a sophistical and pragmatical sense for relating and comparing such developed or cultivated abilities. In its Latin translation as ars, this sophistical, pragmatical dimension was largely reoriented towards a more meditative frame of reference. In both cases, however, techné and ars were meant in a more general sense than any skill or craft in particular. And even more importantly, they both implied an infinite scale: there can be no comprehensive definition, no delineation of how good we can learn to be in something.

Abilities as abilities, both in ars and techné, cannot be mastered, strictly speaking. Developing them means learning, in a non-transitive sense. Today we have largely dismissed such a notion of learning, in favor of orientating thought to an objective dimension of knowledge which we can learn to cultivate by what is today called literacy. Yet different from the old notion of mathesis, attending to the literal assumes a given naturalness of meaning, to be received and represented. In this respect, it leaves us with an insuperable helplessness within the apparent phenomenality peculiar to the fertility and autonomy of thoughts thought, which they acquire within the outer space of intellectuality.

Heidegger has paid attention to the mathetical alternative to the notion literacy. He referred to it with cautious consideration as the mathematical, and meant by it, in an open sense, that which can be learnt. In Die Frage nach dem Ding (1935/36)7, Heidegger comprehends genuinely philosophical thought as thought revolving around the notion of the thing. The mathematical is concerned with things, he says, insofar as we can learn about things. Not simply how to use them, name them, or master them, but rather how we can learn about things in their thingness. About bodies as bodiliness, plants as plantness, etc. With these abstract terms Heidegger does not refer to an idea of a thing, but to a certain kind of intellectual experience of an object as a thing in a certain appearance. This experience is conditioned by a sort of intellectual intuition, yet it only concerns the possible awareness of our interiority. Such experience is not real, for Heidegger, it is purely projective.8 He introduces the mathematical as that which at one and the same time gives things to us, and allows us to learn about them: „The mathematical, this is 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“. (9)

Learning, he continues, is a giving to oneself what one already has. It contains an element of a-substantiality which for Heidegger is, in this certain mathematical way, strictly personal.

3 There is a naturalness proper to reasoning

Different from Heidegger, Gilles Deleuze has suggested to consider a possible generalization of this peculiar element of a-substantiality that is involved in mathetical learning.10 He conceives of purity not as an attribute, but as an elementarity, as a transcendental quasi-naturalness proper to reasoning. This elementarity, for Deleuze, is conditioned by three inseparable principles: that of pure quantitability, complemented by those of pure qualitability and pure potentiality. Raising these terms, the quantitative, the qualitative, and the potential, to the level of –abilities is crucial for understanding Deleuze. He calls them principles, but – by raising them to the level of abilities – he manages to call them so in such a way that they do not presuppose anything given. These principles do not allow us to recognize, imagine or picture ideas by thought. Rather, ideas bath in this pureness as a natural elementarity, and this pureness grants that thought is natural in a different way from assuming its Good Nature. Within such a setting, ideas need to be indexed before they can be treated analytically or synthetically. They need to be actively coded. Thought can engender thinking, within this elementarity of pureness, because thought is mobilized by what Deleuze in a later work calls the surplus value of code. (11) Deleuze conceives of ideas as the differentials of thought, and thinking for him involves determining – reciprocally – the differential relations contained within them. (12) Thus, Deleuze presupposes a naturalness for reasoning which precedes the assignability of truth or falseness to any act of thought in particular. This naturalness itself provides conditions, yet neither sufficiency nor well- foundedness for emerging thoughts.

Considered together, Deleuze‘s ideas and this elementarity of pureness make up for considering reason not from the point of view of its conditioning, but from the point of view of inception or genesis, as Deleuze called it. Precisely because he assumes a naturalness to reason, Deleuze can hold, in a mathetical sense, that reasoning depends upon learning. (13)

Deleuze inverts the analytical assumption of an objectivity of problems. There is an objectivity of problems, for him, but it is given to thought only as ideas – of which he conceives, in turn, as differentials of thought. Within such an elementarity of naturalness proper to reason, we cannot have representations of problems/ideas, we can only attend to them by formulating them. Reasoning, for Deleuze, is the faculty capable of formulating problems in ways that allow for Critique, and this means: formulating problems-in-general. This way of formulating problems-in-general, I would strongly like to argue, can only be considered as algebraic and symbolic – not as literal or numeral in any direct sense.

But let us look more closely at this articulate-ability of quantities within such a transcendentally-empirical setup. A differential takes the fractional form of a ratio. If ideas are not what is represented or mapped in reasoning, if they are differentials which need to be formulated – in order to pose the problem whose objectivity they embody, once they are formulated – we cannot deal with a differential‘s fractional form as a ratio directly. We have to empirically-experimentally investigate the ratio (think: the idea, the differential of thought, the being of a problem) by expressing it in a variety of forms. This is what polynomials allow for. Polynomials are algebraic ways of how to index ratios such that they can be put into symbolical terms that allow for a variety of ways of how to express the ratio‘s quantity. As algebraic expressions, ratios are put into an arrangement of terms which involve indeterminate variables and constant values. The sum of these terms either needs to equal zero, or another version of the same quantity articulated differently, i.e. factored differently. Like this, ratios can be algebraically expressed such that they can be determined strictly reciprocally. The abstract identity postulated by algebraic equations is expressed as symmetry relations that can only be unified according to a mapping that involves elective symbols, as George Boole had called it. (14) Algebraic expressions with polynomials allow for all the arithmetic operations except division – and division is precisely what is expressed in the form of ratios. Like this, Deleuze can maintain that ideas can be tested – by algebraically articulating their form symbolically, as a differential, i.e. as a form which comprehends a fully determinable ratio.

Deleuze has extracted the philosophical consequences of this when he writes that quantitas, the Kantian concept of the understanding capable of grasping the quantum, i.e. things-in-their-extension, cannot be regarded as powerful enough for dealing with the different kinds of generality at stake: „The zeros involved in dx and dy express the annihilation of the quantum and the quantitas, of the general as well as the particular, in favour of ‘the universal and its appearance’.“ (15)

The assumption of abundant yet insufficient conditions for reasoning allows for an empirical science of investigating the universal and its appearances. The particular and the general come to be, therein, the toolbox of such experimental measuring – concepts are representing the objectivity of something problematical only insofar as they are tools for learning to think. Deleuze degrades Kantian concepts of the understanding quite plainly in favor of such learning: „As a concept of the understanding, quantitas has a general value; generality here referring to an infinity of possible particular values: as many as the variable can assume.“ So far so good, but he continues: “However, there must always be a particular value charged with representing the others, and with standing for them: this is the case with the algebraic equation for the circle, x2 + x2 – R2 = 0. The same does not hold for ydy + xdx =0, which signifies ‘the universal of the circumference or of the corresponding function’.“ (16) The algebraic formula for a circle needs a symbolic investment in order to become apparent as a particular circle. The particular, hence, is not a given concrete but an evoked appearance. An appearance engendered through a kind of abstraction which renders symmetries within the purely asymmetrical, it creates consistencies by testing the reciprocal determinations of differential relations. We have to dramatize ideas, as Deleuze calls it. (17) The generalities are what can be extracted from abstract thought, not the other way around. Abstract thought does not presuppose the General Forms as given. Thus, the validity of General Forms can only be empirically grounded. Concepts can be created mathetically, they are grounded in what we have learnt to conceive rigorously.

Just like in the case of Heidegger, for whom such mathematical learning as „giving to oneself what one already has“ is strictly personal, also Deleuze‘s notion of reasoning as learning is enacted by Personas. But for Deleuze, attending to the thingness of things means attending to ideas within the outer space of intellectuality – and this is only possible if we actively dramatize them. Both, Heidegger and Deleuze assume a dynamism which allows such attending or dramatization. For Heidegger, this dynamism takes the mechanical and in that sense self-sufficient form of a proof which pivots around the given axis of time. (18) This self-sufficiency is opened up by Deleuze. He allows the mechanical, linearly circular dynamism – Heidegger calls his notion of proofing Kreisgang – to follow lines of flight which always depart from what has just been learnt. (19)

3  A locus in quo of imaginary points and figures

Let us raise some of the background issues to algebraic numbers and symmetrical quantities.

In 1883 Arthur Cayley, a British algebraist working on variational calculus and invariance-theory, gave his presidential address of the British Association for the Advancement of Science in London with the following endeavor. There is a notion, he told his fellow intellectuals, which is „really the fundamental one (and I cannot too strongly emphasize the assertion) underlying and pervading the whole imaginary space in geometry.“ (20) It is hard to see at first what this statement implies, and why he holds it of such importance to devote his entire speech to it and this with such a tone of gravity in his voice. Has not geometry, at least since its analytic turn to the Cartesian Space of abstract representation, lost its cosmologically ordered elementarity in favor of merely providing an imaginary plane for experimental science? (21) So what exactly is Cayley referring to with this imaginary space in geometry – what had happened?

The crucial sentence is the following specification Cayley gives: „I use in each case the word imaginary as including real.“ (22) Both terms, imaginary and real, are meant in their number theoretical sense, but nevertheless, the issue Cayley wants to address is not one dedicatedly for mathematicians. Quite to the contrary, his concern is: „This has not been, so far as I am aware, a subject of philosophical discussion or enquiry“. (23)

The issue raised in this address concerns the grand question of whether and in what sense a notion of space is relying on experience and subjectivity. Yet the extraordinary take it presents, for philosophers, is that this question is raised out of the field of number theory. This is an unusual perspective. How can we, philosophically, conceive of space such that it features „as a locus in quo of imaginary points and figures“, or in other words: as the scene of the event of a peculiar kind of „elementarity“ (hence, geometry) where figures are articulated out of a numerical domain of which we must, somewhat paradoxically, try to understand that it literally „includes the real“.

By „including the real“ is meant that the numerical domain at stake is said to extend beyond the infinite number line of the real numbers. In their continuity, the domain of the real numbers comprehends all the positive and negative integers, zero, as well as all the rationals and the irrationals. It is indeed difficult to picture, mentally, what could be left out by the real numbers, but this is precisely the point of Cayley‘s address. From the perspective of number theory, Cayley‘s question considers the possibility of a kind of intellectual intuition, and it considers that the quantitative may host something like forms of construction which might hive off such a notion of intuition out of the threatening swamps of unconditioned revelation in a mystical or theological sense. (24)

The imaginary numerical domain Cayley is referring to is that of the Complex Numbers, and what this domain allows for – as we could perhaps put it – is operations on real infinities. The crucial point about them is that their conditioning cannot be thought of as natural (if we understand natural by its more conventional notion, not the Deleuzean one we have put forward above) namely that the quantities describing it need to be factorizable in a unique and necessary way, according to an assumedly universal and unique order of primes.

This may seem like a fancy question for number-crunchers, not for political and intellectual realists, materialists, or idealists, but just consider that none of our electronically maintained infrastructures today would be working without those quantities. And yet, their usage is still today commonly put into rhetorical brackets which claim that only the „real“ part of these operations was of importance, philosophically, whereas the imaginary part is called „but a technical trick“ which we can apply when dealing with symbols. Contrary to this view, Cayley raised the question concerning the „nature“ of such tricks.

Can there be, in short, something like an intuitive rendering-present by intellect, such that we can learn to say something reasonable about the conditions of this rendering- present – even though we cannot assume any necessity for it to appear as it appears? (25) What was preoccupying Cayley, and many others in the second half of the 19th century, was the unsettling suspicion that we cannot exhaustively address reality by investigations following the Cartesian doctrine verum et factum convertuntur. The status of numbers has grown problematic in a new way, with this newly developed capacity to render-present, symbolically and insofar intersubjectively, by acts of intellection.

4  Considering the symbolicalness of symbols

The troubling question can be put like this: how can we conceive of the symbolicalness of symbols in Universal Algebra? For Whitehead it was an open question. For Russell just as for Husserl, it was clear that assuming for symbols a status of their own – one that is not grounded in geometry nor in arithmetics nor in language – would be profoundly mislead; they both held firm – albeit in different versions – that symbols need to regard necessary facts. (26)

Yet with algebraic expressions, there is an objectivity proper to symbolic encodings that allows the encoded to be referred to and represented in purely general terms. This generality is not gained by strictly deductive reasoning, and it nevertheless does not depend upon psychological subjective experience. (27)

Conceiving of a genuine symbolicalness of symbols means tackling with the primacy of abstract algebra as the means for formulating symbolic constitutions. These constitutions provide the structures for what can be expressed as the cases of this peculiar algebraic generality. (28) Strictly speaking, the fundamental theorem of algebra leaves the general applicability of arithmetics problematic. If algebra is granted a universal status, applying arithmetics turns into a practice of engendering solutions as cases, i.e. of calculating solutions which are not, strictly speaking, necessary solutions. (29)

For the majority of philosophers, an affirmation of this would be a straight forward capitulation of enlightenment philosophy at large, because it means that the strong link between calculability and necessity were broken, and along with that, the distinction between philosophy as metaphysics and philosophy capable of critique.

Yet if Algebra‘s universal status is considered as complementing a probabilistic element, into which the formula – i.e. the algebraic identity-as-relation-to-be-established – is seeked to be integrated, all that the fundamental theorem of algebra asks for, philosophically, is to ascribe a different modality to the abstract objects of mathematics and logics than that of either necessity or contingency. (30) I read Deleuze‘s concept of the virtual in these terms, as the modality for the experienciability of things which are not merely actually possible but virtually real. Virtually real means in principle fully determinate (and hence conceptually exact) yet never actually exhaustively determinable. We can consider the virtual as the modality of the things engendered by abstract thought. The symbolicness of symbols encodes forms of structure for determining unknown quantities, and is itself neither form nor content. Such algebraic quantity-expressions can be considered „pure“ in a quasi-Kantian sense: They make reference to no specific magnitudes at all and work only with conceptual definitions. „It sets before the mind by an act of imagination a set of things with fully defined self- consistent types of relations“, writes Whitehead about such vectors of imaginary verticality. (31)

5  Coda

Aristotle had performed a bold move when he appropriated from the Olympian Gods their mark of distinction, their family names as a sign of belonging to different generations and genera, and claimed this divinely distinctive mark to be applicable to all there is. All there is is things that can be named, he set out to consider.

Once people had started to conceive of the mythical Happenings in terms of philosophical consequences and inferences between things that can be named, a structure was needed to receive and conserve the voices of the Mythical Personas. Words originally simply meant verbs, abstract acts in infinitive form. Energeia was Aristotles term such an abstract Principle of Actuality. With the verbs, grammar was providing a structure to receive and conserve the mythical voices by distinguishing cases, as a sort of a negative form, in which we can encounter things-that-can-be-named as affected by energeia, as involved into the actuality at play with the activities expressed by verbs.

We still commonly say today – albeit we mean it, undoubtedly, in a largely technical and sterile sense – that grammatical cases are the structures provided to receive and express what is decadent, what is falling or declining. This is where the term casus comes from. Language and its grammar solves the threat of decadence for community by turning it into a problem to be articulated. As such, it needs not be solved anymore. The effect of expressing the threat in language literally dissolves it, by probabilizing the forms in which it might appear.

These articulated expressions have led a fertile live, within the space of intellectuality. Entire populations of words have been conceived, engendered, and raised, which allow for this enormeous richness in articulating what may be the case. The real question today is not the purely metaphysical one about Being‘s analogy or univocity, as Deleuze suggests in the quote which I aligned, together with the one by Heidegger about the ability to think, as a kind of entrance to this article. The real philosophical questions today asks us to actively value and esteem artefacts as the conditions for everything that can be the case.

The interesting aspect about raising the question of whether and how the consideration of an art of concepts is meaningful, I would like to suggest, is how we can account for polynominality and the spatio-temporal dynamisms they engender. And for this, I have attempted to argue, it is crucial to include techné alongside ars, in their correlation within mathesis as the art of learning.



(1)  In giving this twist to Wittgenstein‘s interest on cases, casuistics and on a philosophical grammar, my perspective is much influenced by the writings of Louis Hjelmslev, cf. especially: Catégorie des cas (1 vol.). Acta Jutlandica VII, IX. (1935/37).

(2)  Rem Koolhaas has recently given two highly interesting accounts of the contemporary urban conditions, viewed from the perspective of infrastructures‘ new factual primary rôle in architecture and in urbanism: The Generic City, he holds, is pure abundance and availability of potentials, with no earthening in either identity, history, or God. The Generic City, in his description, is less a utopia/dystopia (as any utopia/dystopia keeps within the idea that city‘s order represents the possible/impossible path to salvation) but more radically, it fabulates a kind of paradise generalized, and thus paradise decoupled from man‘s (always individual) banishment and expulsion. There is no need, in this paradise, for engendering and conceiving, nor for learning. In an origin generalized, there is knowledge, but not recognition; there is no sexuality within the generic. The other text is called Junkspace, and describes the same reality of infrastructures‘ primacy, yet in the light of the decoupled other: Junkspace is the space of objectivity provided by language, once the modernizing movement of globalization has taken possession of it. As such, it is a pure place of guilt, Koolhaas says, not of beauty. There is not only no sexuality anymore, if we ascribe infrastructures the primary rôle for how we understand our living spaces, there is also no erotics anymore. While the Generic City draws an account of paradise without the prior loss of it, Junkspace is banishment without origin and without the possibility for salvation. Cf. Rem Koolhaas: „The Generic City“ (1994) in: X, M, L, XL , Monacelli Press 1998, pp. 1248-1257; and „Junkspace“, in: october, vol. 100, obsolescence (spring 2002), p. 172-190.

(3)  This idea draws much from the suggestions made by Jules Vuillemin in his seminal books La Philosophie de l‘ Algèbre (1962, no engl. translation), What are Philosophical Systems ([1986] / 2009 ) and Necessity and Contingency ([1984] / 1996).

(4)  The relevance of doping for quantum-mechanical technology and its artefacts has been the topic of a recent conference the contributions to which are forthcoming as a book: Vera Bühlmann, Ludger Hovestadt (Eds.): Printed Physics, Springer, Vienna/New York 2012; cf. especially the article by Ludger Hovestadt: „A Fantastic Genealogy of the Printable“, pp. 18-70.

(5)  There is currently, and somewhat symptomatically, a great debate emerging within philosophy interested in metaphysics and the foundational issues of mathematics. It concerns the relation between set theory and category theory. Cf. for a condensed and rich introduction to many of the issues involved: Mary Tiles‘ review of Penelope Maddy‘s book Defending the Axioms: On the Philosophical Foundations of Set Theory, Oxford University Press, 2011, accessible online at Notre Dames Philosophic Journal, Sept.13 2012: http:// (accessed: July 15th 2012). Also, with many further references, the article on Category Theory at Stanford Encyclopedia: Marquis, Jean-Pierre, “Category Theory”, The Stanford Encyclopedia of Philosophy (Spring 2011 Edition), Edward N. Zalta (ed.), (accessed: July 15th 2012).

(6)  If we conceive of such genesis in terms of a grammar, we can avoid subscribing to what Hegel had wisely stigmatized as „the bad infinity“ – which would mean ascribing artefacts the status of an absolute self- or auto-conditioning. Though infinitary in regard to its engagement with an element of the probable, such a grammatical structuralism is nevertheless finitary in the actualizations it conditions. We can express that and only that which the grammatical structure allows to be the case. The powerful move of articulating the polynomials as quantities, symbolically, instead of names as generas or individuals, lies in its capacity to deal with what linguists call „grammaticalization“, the becoming of grammar. Cf. for the relevance of grammaticalization and IT technology especially Bernhart Stiegler: „Nanomutations, hypomnemata and grammatisation“, online at: (accessed July 15th 2012). Cf. also footnote [13].

(7)  Heidegger, Martin: Die Frage nach dem Ding. Zu Kants Lehre von den transzendentalen Grundsätzen, in: Martin Heidegger, Gesamtausgabe, 2. Abteilung: Vorlesungen 1923-1944, Band 41, Frankfurt am Main, Vittorio Klostermann 1984.

(8)  It is purely projective in a specific way, which Heidegger clarifies as follows: „Where the casting of the mathematical design is ventured, the pitcher of this cast places herself on a ground which comes to be itself projected itself only in the design ventured. There is not only a liberation proper to mathematical design, but also a new kind of experience and designing of freedom itself, i.e. of the actively resumed attachment. Within a mathematical design, an attachment to the conditions claimed is taking place.“ My own translation, cf. the original German: „Wo der Wurf des mathematischen Entwurfs gewagt wird, stellt sich der Werfer dieses Wurfes auf einen Boden, der allererst im Entwurf erworfen wird. Im mathematischen Entwurf liegt nicht nur eine Befreiung, sonder zugleich eine neue Erfahrung und Gestaltung der Freiheit selbst, d.h. der selbstübernommenen Bindung. Im mathematischen Entwurf vollzieht sich die Bindung an die in ihm selbst geforderten Grundsätze. Befreiung zu einer neuen Freiheit.“ Heidegger (1984), p. 97.

(9)  Heidegger 1984: „Das Mathematische, das ist jenes an den Dingen was wir eigentlich schon kennen, was wir demnach nicht erst aus den Dingen herholen, sondern in gewisser Weise selbst schon mitbringen“, p. 74.

(10) cf Deleuze, Difference and Repetition [1968], transl. by Paul Patton, Columbia University Press 1994. In what follows I refer mainly to the chapter entitled „Ideas and the Synthesis of Difference“, p. 168-221.

(11)  Gilles Deleuze and Félix Guattari, Mille Plateaux [1980], transl. by Brian Massumi, University of Minnesota Press 1993; cf. especially „The Geology of Morals“, pp. 53ff.

(12)  I elaborated on this Deleuzean notion of ideas as the differentials of thought in Vera Bühlmann, inhabiting media. Annäherungen an Herkünfte und Topoi medialer Architektonik, PhD University of Basle 2009 / 2011, published online:ühlmann_Juli2011.pdf (accessed July 15th 2012). Cf. especially the subchapters comprehended in the section “Funktion, Sinn und Form”: pp. 120-189.

(13)  Deleuze sais for example: „Just as the right angle and the circle are duplicated by ruler and compass, so each dialectical problem is duplicated by a symbolic field in which it is expressed.“ The capacity of such symbolic fields considered as tools he defines as follows: „Instead of seeking to find out by trial and error whether a given equation is solvable in general, we must determine the conditions of the problem which progressively specify the fields of solvability in such a way that ‘the statement contains the seeds of the solution’. This is a radical reversal in the problem-solution relation, a more considerable revolution than the Copernican.“ Deleuze [1968], pp. 179/180.

(14)  In his Algebra of Logic George Boole has introduced three laws according to which logical identity relations can be established algebraically: the distributive law, the commutative law, and the idempotency law (which Boole also called the index law). All three regulate the establishment of symmetry relations within not fully determined givenness (hence Boole‘s crucial involvement of probabilistics into logics). It is crucial to understand the abstract move involved in considering an algebra of logics, as this is directly inverse to the consideration of logical foundations for mathematics (i.e. Frege‘s and Russell‘s Logicism). While the latter seeks in logics a formal representation of abstract identities, the latter seeks in mathematics the means for learning to think abstract identities in ever more differentiated manners. For algebraists, abstract identity is treated as a difference relation, while for logicists abstract identity is treated as the representation of a unified relation. While logicists seek, ultimately, to determine a unified universe (of discourse), and proceed by seeking to analyze elementary sets or atomic units from which to build up by generalization, algebraists like Boole and Dedekind proceed inversly. They do not begin with assuming ultimately fully analyzable elementary units to work with. They begin with assuming a never exhaustively representable abstract unity as the „universe of thinkable objects“. Hence the former approaches are referred to as „finitary approaches“, and the algebraic approaches as „infinitary approaches“. By this, algebraists can experimentally test logical structures. Thereby, the values of mathematical rigor and exactness are no less important for algebraists than for logisists, yet they are seeked for in the exact definition of concepts to be experimentally applied within the infinite universe of the thinkable. Logicists, on the other hand, seek exactness for the definition of the whole universe of discourse, in order to legitimate the application of one defined concept as opposed to another definition of that concept – this features most prominently in Frege‘s „metaphysics“ of Three Empires (the empire of physical/empirical „objects“, that of psychological „objects“, and that of logical „objects“), as well as in Popper‘s many world theory. I would strongly like to suggest, without being able to go into details here, that Booles laws of thought are misunderstood profoundly, if considered as an early version along the same lines as sets of axioms are proposed in the foundational discourse on mathematics. Boole‘s laws of thought are less like logical axioms than like experimental laws. They are like postulated laws of nature that allow to regulate and orientate experiments, within the „nature“ of the intellectual Universe of Thought (rather than that of the Physical Universe). Obviously, some of the most diversely and all too often overheatedly disputed issues are at stake here. Introductory articles on George Boole and on his Algebra of Logics that are rather uncorrupted by taking a specific stance on the foundational question in philosophy of mathematics can be found at Stanford Encyclopedia: Burris, Stanley, “George Boole”, The Stanford Encyclopedia of Philosophy (Summer 2010 Edition), Edward N. Zalta (ed.), entries/boole/; and Burris, Stanley, “The Algebra of Logic Tradition”, The Stanford Encyclopedia of Philosophy (Summer 2009 Edition), Edward N. Zalta (ed.), entries/algebra-logic-tradition/ (accessed July 15th 2012). Cf. also footnote [6].

(15)  Deleuze (1968), p. 171.

(16)  Deleuze (1968), p. 171

(17) cf Deleuze (1968), pp. 216ff. Also his article „The Method of Dramatization“, in Desert Islands and other texts, 1953–1974, semiotexte 1994, pp. 94-116.

(18)  cf. Heidegger (1984), „This concept of pure understanding, quantity, is nothing else but the synthesis which empowers appearances to appear as specific figurations in space. Hence, all appearances, insofar as they are intuited, are quantities, extensive quantities (space). It is the same precondition which allows the encounter of that which encounters, which brings what counters into constellation. 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“. My own translation, cf. the original German: „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“.“ p. 252.

(19)  Deleuze, with this characterization of elementary pureness of reason, in which thought dramatizes ideas through involving them into spatio-temporal dynamism by specifying those dynamisms, opens up sights onto a philosophical domain that is conditioned by abundant and insufficient reason. I would like to suggest that this is the domain of symbolic algebra, and that the cases engendered by it – the artefacts – make up all the spatio-temporal dynamisms conceivable. Every single artefact embodies a multitude of acts of concentration which was necessary for dramatizing an idea. It is in this sense that they conserve intellectual energy. What for Heidegger is „the power of synthesis which brings appearances into extensive form“ (cf. footnote 18), the dynamisms conceived through their extension as artefacts make this „power of synthesis“ accessible in terms of intellectual energy: artefacts allow us to apply, store and construct with symbolically encapsulated acts of Heidegger‘s „power of synthesis“. Intellectual energy can be conceived as encapsulatable, storable, transmittable, and even transformable in artefacts of any kind, i.e. of any symbolic constitution. The power of synthesis thereby acquires a qualifiable quantification. For the energy conserved, by such acts of concentration articulated into the manifest form of artefacts, depends upon our ability to understand it. If we naturalize those artefacts, as autolog or self-sufficient objects, without esteem for the acts of concentration they encapsulate, they don‘t conserve but consume energy, through annihilation of their mediality. Because then, there are always too many artifacts, and too many which are not optimally configured, or not to the right optimization configured, etc. The wealth of artefacts then inevitable appears like a waste of resources, intellectual as well as material resources. Cf. footnote [2] on Koolhaas‘ generalization of the concept of paradise in what he calls The Generic City and Junkspace – without understanding and esteem, artefacts must be seen, within the context of Western history at least, as establishing a realm of pure guilt.

(20)  Cayley, Arthur (1996) [1883],“Presidential address to the British Association“, in Ewald, William, From Kant to Hilbert: a source book in the foundations of mathematics. Vol. I, II, Oxford Science Publications, The Clarendon Press Oxford University Press, pp. 542–573, reprinted in collected mathematical papers volume 11.

(21)  Cayley‘s is not a solitary voice at that time. Within the last decades of the 19th century, Gustav Lejeune Dirichlet, Ernst Kummer, Leopold Kronecker and Karl Weierstrass all wrote on the theory of numbers involving algebraic quantities; Husserl published his Habilitationsschrift entitled Über den Begriff der Zahl (1987); Dedekind wrote Über die Theorie der ganzen algebraischen Zahlen (1871) and Was sind und was sollen die Zahlen (1888), Russell wrote his PhD on the Foundations of Geometry (1897) and Whitehead published a comprehensive volume entitled Universal Algebra (1898). All of this before the writing and appearance of Russell‘s and Whitehead‘s Principia Mathematica in 1910/1913.

(22)  Cayley [(1996) [1883]], as reprinted in collected mathematical papers volume 11, p. 784.

(23)  Cayley [(1996) [1883]], as reprinted in collected mathematical papers volume 11, p. 784.

(24)  Cf. how Deleuze [1968] speaks about the possibility of intellectual intuition, by reference to an „ideal cause of continuity“: „the limit must be conceived not as the limit of a function but as a genuine cut [coupure], a border between the changeable and the unchangeable within the function itself. […] the limit no longer presupposes the ideas of a continuous variable and infinite approximation. On the contrary, the notion of limit grounds a new, static and purely ideal definition of continuity, while its own definition implies no more than number, or rather, the universal in number. Modern mathematics then specifies the nature of this universal of number as consisting in the ‘cut’ (in the sense of Dedekind): in this sense, it is the cut which constitutes the next genus of number, the ideal cause of continuity or the pure element of quantitability.“ p. 171.

(25)  This is indeed a very old meaning of symbols – symbols evoke an immaterial presence in our thinking of something which lacks manifest presence. Symbols are place-holders, indexes, and they enforce a certain immediacy upon us. Hence our associations with symbols tend to center around mystical or sacral contexts. Or around contexts of undisputable control, when we think of our passports as symbols of our identity, for example.

(26)  Husserl referred to them as „Anschauungsthatsachen“, as intuitive facts (although he means it in a different sense than the intuitionist schools of Herman Weyl or Luitzen Egburtus Jan Brouwer), and Russell‘s main preoccupation remained the quest for how logical quantification is possible.

(27)  It is in this sense that Deleuze speaks of „the universal in number“ as „the next genus in number, the ideal cause of continuity or the pure element of quantitability“ ([1968] p. 171); and in which Jules Vuillemin proposes an „Ontologie Formelle“ to complement a „Critique Générale de la Raison“ in the end of his book La Philosophie de l‘Algèbre, Epiméthée, Paris 1962, pp. 465ff. The crucial point is that the notion of the universal is related to the learning made possible by mathematics, and not to a realm of logical representation of the achievements of such learning. Hence, instead of universal logical and ontological quantification, these philosophers suggest to consider the quantification of the general in relation to a formal ontology of the general, by viewing in it a kind of philosophical auxiliary structures for learning within an experimental empiricism in the realm of abstraction. Cf. footnote [28].

(28)  In physics and Engineering we more or less casually calculate spaces with algebraic numbers which relativize the assumed unproblematical rootedness of all numerical values in positive natural integers or a homogenous spatiality; and with regard to the formalization of language in logics, we very well know meanwhile about all the problems related both, to universal and to existential quantification. For a discussion of this issue in analytical philosophy after Kant and Frege, cf. Michael Potter, Reason’s Nearest Kin. Philosophies of Arithmetic from Kant to Carnap. Oxford University Press, Oxford, 2000. Potters book is a great overview and introduction over the issues involved and approaches presented, but its limits are within the author‘s decision to exclude any discussion concerning the challenges introduced by algebraic quantities. The framing question in his book is: „Can we give an account of arithmetic that does not make it depend for its truth on the way the world is? And if so, what constrains the world to conform to arithmetic?“ (p.1). Potter takes it as a given that such an account is possible, and he assumes that all of the figures he discusses do as well. Certainly for Wittgenstein and for Dedekind, this seems to me like a crucial misreading. Cf. for a critique in a similar direction also the review by Richard Zach, „Critical study of Michael Potter’s Reason’s Nearest Kin“, Notre Dame Journal of Formal Logic 46 (2005) 503-513 (online: http://

(29)  Algebraic terms are polynomials, they embody unequal potentials. The liberty of engendering solutions as cases comes in because every algebraic solution requires a depotentialization of its terms, such that an order can be established which is shared by all the components. If number spaces that may extend into the imaginary and algebraic ideality are allowed for solving polynomial equations, there are many different ways of achieving this depotentialization; hence the vastness of the possible solution spaces.

(30)  Algebraic reasoning means specifying what outcome you will want to have, and producing it by taking an infinitary approach to deal with this probabilisitc element. Infinitary means, by eliminating from the vast and non-controllable possibility space of your solutions anything you do not want to feature in the result. This elimination procedure works by injecting into equations, as a kind of doping, whatever is necessary and sufficient for common denomination and factorization of the terms involved, such that the two sides of an equation may be transformed and balanced in ways that are neither fully necessary nor arbitrary.

(31)  Alfred North Whitehead, Treatise on Universal Algebra with applications, Cambridge University Press, Cambridge 1910, p. vii.


3 thoughts on “Articulating quantities – when things depend on whatever can be the case

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