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 – avantgarde 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
highpowered 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/pierreboulezdouzenotationsinsidethescore.html)
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 terms 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
clockregulated 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. Nonexperts 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 socalled
‘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 nonrealist
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 closeup 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 nationstates,
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 nonmetricated 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 3d
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.
