Deleuze for the Desperate # 9:  Smooth space

Dave and Maggie Harris

We’ve been trying to show throughout how Deleuze and Guattari in ATP offer specific examples to discuss, and then a more general philosophical notion which emerges. We’ve been suggesting that you operate in a similar way, looking at the examples and then trying to think what they have in common and what some theoretical and political implications might follow. The second stage takes longer, unless you already a philosopher, and much will depend on what you are reading Deleuze and Guattari for, how much time and effort you can devote to them. We are only offering pretty basic summaries here. There is a broader discussion on the website.

In this plateau we are told early on that smooth space rarely appears in a pure form, and is usually mixed with ‘striations’, that is lines that structure and divide up space – boundaries, parallel lines or grids. This will lead to an attempt to get to pure general definition This will be based on philosophical inquiries demonstrated in maths, music and art.

Only proper philosophical notions will be ‘pure’, specifying a space without any striations at all. We have to philosophize because normal everyday conceptions of smooth space are too limited, and will still contain striations, boundaries, lines or channels. To take an easy example, we might think we are engaging in uninhibited action, free of constraint when we are alone, on holiday, unsupervised in the security of our own classroom or wherever. But the point is that none of these activities are really free of striation, direction, control, channeling – even if at an unconscious level. People on holiday still display conventional views of gender, for example, sometimes racism or nationalism, demonstrate their social superiority by heading for the local high culture, or deliberately ignore it to have a laugh with their mates. We know that for Deleuze and Guattari domination is widespread and subtle. We saw this with the comments about subjectification and signifiance (sic) in the video on the BWO, and it is everywhere in the plateaus on language.

Let’s look at the actual examples of smooth and striated space in this plateau. We begin with a reference back to the issue of nomads versus the settled or sedentary, developed in earlier plateaus. Nomads live in relatively unstriated spaces like deserts and unworked land (to plough land is to striate it of course) and develop aggregated forms of social groups on different social principles – not so centralised, not so oriented around work or the State. These less systematically stratified groups can become war machines – more on that in other plateaus, and maybe in the next video.

Striation is associated with settled societies and is an important technique to make scientific progress. It is important for a lot of social and technological development, organizing the land into definite areas, developing agriculture (definite ways to striate land), and subdividing the world into manageable closed systems. So what’s the problem with it? There is an obvious political consequence, that social striation involves constraint, conventional lines to guide thinking, domination and control. Smooth space looks much more free and liberated by comparison. It is a more creative conception because it goes beyond convention, allows qualitative change. If striated space enables progress, smooth space permits becoming as D&G put it. However, we are warned right at the end not to assume that smooth spaces can guarantee free movement: they are always likely to be colonized and striated again

Given the mixed nature of most actual examples, we are going to need some general principle to describe pure smooth space, with no actual or concrete elements. What model best describes space – a regular container as in classical notions with 3 dimensions and objects with regular shapes like triangles and cubes? As we shall see, there are now newer models of space and newer geometries for that matter. If you like pop science programmes, you might have come across models that are even newer than those admired by Deleuze and Guattari – space as a series of multidimensional strings or membranes, a bundle of twisted fibres and so on.

Back to ATP. The discussion involves several ‘models’:

Technological – mostly textiles

Musical – avant-garde music especially in Boulez

Maritime – the sea as a smooth space, including discussion about cities and urban nomads

Mathematical – the important work of Riemann and others on the multiplicity, and more recent mathematicians who have worked on fractals.

Physical – critiques of various attempts to metricate and regularize things, from the objects of Nature like planetary orbits to work. Quantitative and qualitative multiplicities.

Aesthetics – nomad art and whether it really is abstract depicting smooth space

Some of the discussion is unfortunately quite obscure. Perhaps it echoes the discussions between participants in a high-powered Parisian salon Deleuze and Guattari once attended. We can only give a very basic summary of the discussion.

Technology – textiles. The examples are familiar enough, although the discussion is pretty obscure. We are told that textiles arose from attempts to striate, literally to weave strands together to produce a solid piece of cloth – a supple one. That’s not the only way to produce cloth, and the nomads, who are much admired elsewhere, use felting techniques instead, compressing fibres together. Other methods of producing textiles are briefly discussed and compared: there is knitting with its horizontal and vertical dimensions versus crochet which emerges from a centre and 'draws an open space in all directions'  (525). Clearly producing textiles is a very important stage in human technology. One interesting example is quilting, commonly a collective activity among women. Quilting produces mixed types of fabric like woven cloth covers filled with looser material like down or kapok, and different types of construction like joining like themed patterns of octagons or the relatively unconstrained patchwork.

There seems to be a connection between types of social order and types of textile, hinting at a more general sociology of technology. Weaving somehow connects with major forms of science and the State as an early demonstration of the power of striation. We thought of the early connection in Britain between automated Jacquard looms and early computing, both using punched cards to convey information (a discussion here) . Textiles also code cultural and symbolic ways of representing important cultural relation like inside/outside. This sociological approach is fairly unexplored and rather simple here, and we are referred to other commentaries. It is pursued rather better, and in connection with a marxist framework in discussions of work in physical models.

Music – Contemporary experimental music can be seen as working in smooth spaces. Conventionally, the sounds accepted as music are constrained by using accepted scales and rhythms or time signatures, as in musical scores – we have a series of lines and spaces between them, with symbols in various shapes to show pitch and time values etc. We use a very limited set of available sounds, of course and experiments soon developed to use other sounds, like bird song or cannon fire, and eventually definitely discordant ones, unfamiliar chords and rhythms. Boulez is the example here with his use of different scales that do not stick to the conventional octaves, distributions of notes, like spirals instead of rigorous sequences, and guidelines to the orchestra rather than strict instructions. A whole musical texture can be created without 'fixed and homogeneous values’ (527).

A particular innovation much admired and used as a metaphor is the ‘diagonal line’ connecting two sections rather than a conventional sequence like a chord progression. It is usually played as an emphatic musical slide, a website on Boulez tells us. (http://www.explorethescore.org/pierre-boulez-douze-notations-inside-the-score.html).

I have since read a short piece on non-pulsed music in Deleuze's Two Regimes... and there is an attached link to an excellent discussion of music as a BwO by Cox

Maritime. The sea is a classic smooth space, although modern navigation has tried to striate it in terns of latitude and longitude, maritime charts and so on. This is an example of a general tendency for modernity/capitalism to striate in order to manage and control space. Some smoothness still persists though in the sense that the points of arrival and departure are the important bits, with the lines only suggesting routes between them. Actual routes rarely follow the lines on the chart, of course. In smooth space generally, lines are like this – vectors, not things that tightly prescribes a route or a measurable distance.

Travel on the sea is also ‘haptic’, a term which crops up later as something also found in nomad art. Haptic events engage more senses than just vision. Other affects are received – a sense of motion, or touch The sea is a space with potentials for a number of affect: It represents 'Spatium instead of Extensio' (527) , a space filled with potentials rather than one with fixed dimensions, just like the body without organs explicitly cited as a comparison.

We also get an important warning here, because the smooth nature of the seas has been weaponised in the form of the modern submarine, which can also follow unpredictable vectors which makes detection more difficult: the smooth characteristics are only for 'the purpose of controlling striated space [like national territories] more completely'(530). This warning serves to remind us that smooth and striated spaces can change into each other and that when comparing impure actual examples, we are usually using only relative terms.

As a further example, cities are striated, at least more than towns are, but we can still stroll around them to some extent, cheerfully ignoring signposts to preferred routes. We can be urban nomads. We used this idea of a wandering path off the beaten track as an example of a rhizomatic path in the second video – and the rhizome is explicitly mentioned here

It might be an idea to pause and consider if any qualities of smooth space are starting to emerge for you. We have the avoidance of lines as determining structures, or at least the ‘subordination’ of lines to points? We have vectors, directions of movement rather than metricated dimensions, following different intensities. We have haptic affects, not just visual representations. We know that empirical examples are mixed, and there is a common tendency to colonize smooth spaces by metric striation especially. However, there are also some ways to smooth out striated space too – rhizomatic wanderings like urban nomadism. Smooth space has a potential to go either way.

The remaining examples move to more explicit theoretical modelling which appears best of all in the mathematical model. They appear in a different order in the book, but here they are arranged according to how difficult and obscure we found them. We offer a quick gloss only:

We want to discuss first the example of nomad asthetics as an application of the idea of smooth space again. This section offers more on the importance of nomads generally, and goes along with other remarks about the positive nature of nomad culture. Nomads do not take up notions of normal clock-regulated work, for example, but that is not because they are inherently lazy – they want to reject regimented work for positive reasons. Similarly, they can conceive of a modern political system and state but do not want to establish one because it would diminish their local powers.

Here, the issue really is how to understand their paintings and drawings. Should we see them as simple depictions before the great developments of realist representation and perspectival art in the European Renaissance? Or as something more positive – an attempt to depict abstract or smooth space. We can certainly see some modern ‘abstract’ art like that, which is how Deleuze sees Francis Bacon (Deleuze 2014). Some earlier examples might fit too – and they cites other example, including the 'northern Gothic line', referring to the intricate interwoven lines and patterns of Goth art (much of it persists in Celtic designs as well)

A big debate about nomadic art then ensues. The work of two art historians is cited – Worringer and Riegl. Non-experts can’t really contribute here, but according to the Wikipedia entries at least, both believed that a constant and universal human interest in abstract space can be detected there too. So again we have a positive take on so-called ‘primitive’ art. Nomadic artists wanted to avoid the conventional lines of representational art. Instead, such art is 'positively motivated by the smooth space it draws'. The non-realist lines of nomadic art 'is the affect of smooth spaces not a feeling of anxiety that calls forth striation' (548). D&G have some reservations about this, but they basically agree.

Using terms that we shall see later, nomadic art shows us a 'nomadic absolute...a local integration moving from part to part and constituting smooth space in an infinite succession of linkages and changes in direction. It is an absolute that is one with becoming itself, with process' (545) . Nomadic art is not just governed by optical conventions like perspective but is haptic – generating other affects, especially those that relate to our sense of touch. It invites a close-up look, not a distant more unemotional one. It is not centred on conventional human subjects. When depicting humans, D&G say it is unconfined by organism, representing life which organisms only solidify, ' a powerful life without organs, a Body that is all the more alive for having no organs' (550). A deliberate reference to the BWO.

We took the section on physical models next. The argument is that space and solid objects cannot be striated with simple straight lines, and are not easily measured by standard metric scales. Eventually, this will mean that space is not describable in terms of fixed quantities combining in various ways – not adequately represented in formulae or standard equations. Smooth space offers a better account instead 'a continuous variation that exceeds any [regular] distribution of constants and variables, the freeing of a line that does not pass between two points, the formation of a plane that does not proceed by parallel and perpendicular lines' (539).

In one example, Newton’s scheme managed to depict the orbit of the planets according to equations derived from his laws of motion. But the model is OK only for two bodies. Three bodies produce inherent unpredictable variations – a bending of regular lines that become tangents around other bodies. Some stability can be restored in the form of spiral paths. Between the regular lines and the spiral lies smooth space, or maybe chaos according to some definitions. We can make predictions only in the form of probabilities there.

The Greeks were on to this too, apparently providing alternative notions of space to the familiar one developed by Euclid.

Deleuze and Guattari note that there are many common examples where it is impossible to strictly metricate the variables. Sound is one of their examples, where sounds vary in terms of their intensity and frequency, but they cannot just be added together, partly because things like intensity are qualitative differences. Thinking of our own examples, awarding grades for essays is a qualitative matter too, so it is not simply the case for an essay grade of 70 is somehow twice as good as an essay grade of 35. Of course, assessors then employ what is really an illicit procedure to pretend that these numbers are proper quantities, so that we can then add grades, average them and so on. Deleuze and Guattari notice that this tendency is a common one, to impose metrication to manage complexity. Certainly, if you abandon metrication you are left with measures and scales that are far less precise and exact, things like ordinal scales where you only can say that one thing is larger than another, without specifying how great: Deleuze and Guattari use the term ‘anexact’ for these measurements, not exact in metric terms, but still rigorous.

One particularly interesting section deals with attempts to quantify, measure and evaluate human work. A physical conception of work as a matter of weights, heights, forces and displacements became allied to a socio economic concept of labour power or abstract labour, something homogenous and abstract which could be applied to all work and which could be metricated. As a result, every human activity could be translated into work and free action could be disciplined, or at least relegated to mere leisure, defined against work. This work model became a fundamental part of the state apparatuses. The concept of labour that developed was also always associated with surplus and stockpiling, and the disciplining of free action, 'the nullification of smooth spaces'. What we lost was 'the continuous variation of free action', passing from action to song to speech to enterprise: only 'rare peak moments' will resemble work. (541)

There is the usual warning: capitalist calculation no longer requires just calculating quantities of labour, but is more complex and qualitative and includes infrastructure, the media, 'every semiotic system' (543). In this, it has 'reconstituted a sort of smooth space' The multinationals produce this deterritorialized smooth space, breaking with the classical striations, so the real struggle these days is between striated capital, still tied to territories like nation-states, and smooth capital, able to operate across these boundaries. New forms might be able to reverse colonization and create more liberating smooth spaces. Guattari’s politics explore these possibilities best with his interest in Italian autonomism or various anarchic cultural and political ventures like French free radio. [Italian autonomism is referenced briefly at the end of Plateau 13,and the politics generally are outlined well in Guattari and Rolnik on the Workers Party in  Brazil].

The mathematical model is the final one for us. It begins with the work of Riemann on the multiplicity. There are a couple of readable articles on Riemann included on the transcript (Plotnitsky and Calamari). To simplify a great deal Riemann was interested in trying to imagine and define what a multi dimensional space would look like, and for various reasons, he thought it would consist of a number of more recognizable spaces folded together in complex ways. One helpful analogy on a website http://www.mu6.com/riemann_space.html] suggests you take an ordinary flat sheet of paper and then crumple it in your hand. The resulting object would give some idea of what Riemann meant by n dimensional space. It is a many folded structure, a manifold, or to use another term a multiplicity.

The analogy is helpful in another way as well, in that you can see that the crumpled sheet of paper features lots of smaller conventional spaces joined together in diverse ways, like a sort of three dimensional patchwork. And this apparently was another argument developed by Riemann, that there is no overall structure for space, but rather a patchwork of local spaces joined together by local relations, in infinitely complex ways in n dimensions. It is also clear that there are no simple linear relations between points on the local spaces – if we want to explain the connections between them we will have to talk about lines and curves and angles altering distance and direction. Some of the connections will involve maybe qualities, like intensity, frequency – none of them really metric. To pursue the simplistic grasp of this, space itself contains potentials forces or vectors and these are actualized, made concrete in the local spaces that are joined together. We can follow this process of non-metricated intensive forces taking actual forms as a result of some simple basic operators at work, bending, stretching or deforming forces. These are indeed the only forces we can specify in the geometry or topology of n dimensional space. DeLanda (2002, ch.1) offers the best description of how further operations like operations that reduce symmetry are required to produce the familiar solid 3-d bodies we know.

All sorts of implications follow, including the end of dialectical reasoning, which, classically, had suggested that forces and tendencies come together only in opposing pairs, which interact and produce a third term. Here, there are not just opposing pairs but a multiplicity of forces and tendencies, and they can interact in all sorts of ways, not just by opposing each other.

There is a reminder of Bergson too. We have discussed him a bit in the video on images in film. He insisted that time and space are more complex than the basic scientific definitions suggest – both have qualitative dimensions as well. This is seen most clearly when considering human notions of duration – time passes and we change from childhood to old age, but these are qualitative changes not quantitative ones as we add more years to our age. Space is discussed through some famous dilemmas in describing motion as progress through space. Deleuze and Guattari like to revisit Bergson on ancient Greek paradoxes discussed by Zeno, but the clearest example is that the motion of a horse involves qualitative change, not just a quantitative increase in velocity as it moves from walk to trot to canter to gallop

There is a delightfully baffling aside on number as well. Strictly speaking we cannot use conventional metric numbers to map smooth space, but we retain a sense of number before it got fully metricated, almost an original notion of number. We use ordinal numbers which deal with magnitudes but only in that we can say that a size 7 shoes is bigger than a size 5.

Other mathematical explorations of complex space are perhaps more familiar. Fractals, for example, actually discussed quite clearly in this plateau. The length of a fractal line cannot be established definitively because it is always branching off to add extra shapes. We can examine fractal lines from far away and then they look like simple straight lines – but closer up they have more dimensions (that is more lines and points). As a result fractal lines have more dimensions than a normal straight line between two points (1) but fewer than a regular surface (2). So they can only be represented by a fraction or decimal between I and 2.

The plateau discusses other equally odd objects like Sierpinski [their spelling] sponges. These have a different ratio between surface area and volumes from fully solid objects, because they have holes in them. They have a volume than they should, but they are not just surfaces either. [There is a helpful explanation of the Sierpintski triangle at least]

After this section on mathematical models and objects, we finally have a pure definition of smooth space as opposed to striated (538):

(1) a striated or metric aggregate has a whole number of dimensions and constant directions;

(2) non metric or smooth space involves the 'construction of a line with a fractional number of dimensions greater than one', or a 'surface with a fractional number of dimensions greater than two'; (3) the fractional dimensions indicates 'a properly directional space (with continuous variation in direction…

(4) a smooth space therefore has dimensions lower than [any conventional 2 or 3 dimensional space] 'which moves through it or is inscribed in it; in this sense it is a flat multiplicity, for example, a line that fills a plane without ceasing to be a line' [like a really elaborate fractal]

(5) smooth space tends to be identified with that which occupies it, with the same power, with the same 'anexact yet rigorous form of the numbering or nonwhole number

(6) a smooth space of this kind can be seen as 'an accumulation of proximities' constituting zones of indiscernibility -- this is why smooth spaces are proper to becoming .[Indiscernibility is a concept Deleuze discusses in his book on Leibniz . To cut a long argument short it depends on a view that two, usually adjacent, objects can be distinct while having the same properties. What we have described is a mathematical notion of a patchwork]

So the overall implications are theoretical and political again (combined as usual)

Theoretical implications first . We are told there is no need to multiply models in different areas since the distinction between smooth striated space summarises everything else. It turns on the idea that there are different kinds of multiplicities 'metric and nonmetric; extensive and qualitative; centered and acentered; arborescent and rhizomatic; numerical and flat; dimensional and directional; of masses and of packs; of magnitude and of distance; of breaks and of frequency; striated and smooth' (534) This underpins much of the specific discussions in their book – differences between packs and organized groups, rhizomes and trees, BWO and actual bodies, war machines and states.

It is still worth focusing on specific aspects of smooth space though. We can cut through them to simplify and organize our investigations. To be literal about this, take your sheet of crumpled paper and a pair of scissors and cut through it. The flat surface you have produced is a plane, a common term in Deleuze and Guattari ,and it offers a basic way for philosophy to grasp the intricacies .

There are political implications here too. Perhaps politics should focus on the basics, not the multiple surface issues like multiple identities and different models of division and solidarity. Marxists might want to argue that class divisions are the most fundamental, but for Deleuze and Guattari the issue is the notion of space and constraint.

There are of course political warnings as usual – smooth spaces offers no guarantee of freedom because they are always liable to striation by powerful political groups.

References

Delanda, M.  (2002) Intensive Science and Virtual Philosophy, London: Continuum
Deleuze, G. (2014) Francis Bacon London: Bloomsbury Academic.
Deleuze G and Guattari F (2004) [1987] A Thousand Plateaus, London: Continuum.