Notes on: Plotnisky, A. (2005) Manifolds: on the
concept of space in Riemann and Deleuze. In S.
Duffy (Ed) Virtual mathematics.
Chapter 10, pp. 187-- 208. Manchester UK: Clinamen
Press. Retrieved from
http://web.ics.purdue.edu/~plotnits/PDFs/Manifolds.pdf
Dave Harris
[very much a gloss of course]
Riemann developed 'conceptual mathematics'[rather
than formulas or sets] and this matches Deleuze's
notion of philosophy as inventing new concepts,
especially in What is Philosophy? ( WiP.)
The 'conceptual philosophy of [continuous]
spatiality' is what they share (187), and how it
might be developed in phenomenal,mathematical,
physical, cultural or political terms.
Ultimately, there is a 'problematic of
materiality' detectable too. Leibniz also a key
figure for both.
There are differences with set theory, turning on
Riemann's concept of space and manifoldness, a
radical departure from Euclid. There is
connection with the notion of smooth space in
Deleuze—both can be seen as 'conglomerates of
local spaces and multiple transitions between
them'(188). The basic idea is that space as
a concept could well have 'a complex structure or
architecture' and that space itself is a 'primary,
grounding concept', rather than one derived from
some other concept, such as space as a set of
points. This is apparently shared with more
recent developments such as 'category theory' or
'topos theory'. Classic set theory sees a
set as composed of elements with particular
properties and relations among themselves or with
elements of other sets.
Space is even more primary than the point.
There is also a link with Leibniz and the monad.
Actually, points in such a space where they exist
can be seen as monads, 'certain elemental
but structured spaces' (189) [I think the argument
is that the point classically has no
structure]. The implications for
mathematical practice in such a space produced a
radical and minor or nomadic mathematics.
There is even a link between the concept and the
'mode of its production', so that they reflect or
'dedouble' themselves [the concept apparently
belongs to Baudelaire and De Man]. Set
theory by contrast tends to be a major or state
concept, although it might not have started that
way, as D and G suggest.
D&G stresses the need to invent new concepts,
and this implies a rethought notion of a
philosophical concept. It is not a
generalization from particulars or from any
general or abstract idea. It has a 'complex
multi layered structure', components combined in a
multiplicity or manifold, which may include
'figures, metaphors, particular elements, and so
forth, which may or may not form a unity'.
This is also argued particularly in Difference and
Repetition, and 'may indeed be seen as
defining most of Deleuze's philosophical work'
(190).
However, spatial thinking also influences argument
especially in WiP, and
Riemann's influence includes the notion of the
plane of immanence ,originally R's, and the notion
of smooth space. The plane of consistency can be
seen as an avatar or predecessor of this
development, 'topo-philosophy', seen also in their
'geophilosophy' and the 'geo-smooth spaces' which
resist state borders and striations [I didn't see
anywhere where this had much political mileage I
must say].
We have to take care applying maths to other
disciplines, as Deleuze notes himself (in Cinema 2
where he warns about arbitrary metaphors or forced
applications, page 129). However there are
shared dimensions between philosophy and science
and art: not all scientific concepts are precise
and quantitative, some are 'inexact' while still
being rigorous—and here philosophers and artists
make them rigorous [Negotiations?,
Page 29]. [Plotnisky thinks the term
'inexact' might reduce the autonomy of the
qualitative, and collapse useful differences?]
Here, Deleuze might be seeing a positive role for
the quantitative and numerical as producing
'conceptual specificity and significance'(191),
and we see this in the discussion on Bergson too [where
metrication is seen as central to practice].
Bergson makes rigorous non numerical notions of
multiplicity with the concept of duration.
Even Riemann contrasts the manifold with 'metric
manifoldness'. Mathematics must provide
exact numerical features even for things like
smooth spaces, or their specifically mathematical
equivalents—'there is always a number somewhere'.
Philosophy is different—Plato on the c(or k)hora
[useful quick guide in Wikipedia:
'khôra is
neither being nor nonbeing but an interval between
in which the "forms"
were originally held'] is the example of a
philosophical topology that did not develop into
mathematics, which was confined to euclidean
geometry. [We are reminded that topology
disregards measurement and scale and focuses only
on the structure of space and the 'essential
shapes of figures'(192), so we can deform figures
continuously, without separating the points if
they are connected, or connecting them to any
other points,and they will remain the same figure.
There is a brief history of this argument on p.
192].
It is hard to relate these mathematical
conceptions to everyday 'phenomenal ' ones -- but
they are crucial to grasp continuity [the actual
example on p193 is Cantor's notion of the
continuum -- 'highly counterintuitive' ] . Again Bergson's distinctions
between duration and metric time suggest this (via
Weyl). Bergson 'may have a Riemannian
genealogy' as D&G imply and this would
show a 'rhizomatic network' between maths and
other disciplines [denying again a strong division
between exact and inexact].
Riemann on the manifold and on geometry argues
that some prior
'general concept' is needed to explain
quantitative continuity [OR possibly that
continuity itself suggest some manifoldness underpinning discrete concepts
describing specialist entities, as in
transcendental deducation again?]. Normally,
concepts operate at a general level to include
lots of cases, but we have to think of special
concepts of 'multiply- extended manifoldness'. We
do have the commonsense notion of 'perceived
objects and colours' [reciprocal perceptions and
all that? -- perception and apperception?], but we
really need higher maths. Maths has a special
interest in the simplicity of its concepts --
philosophy or everyday thinking might cope with
more complex ones.
Insisting on concepts means we don't have to see
material objects as 'ontologically pregiven'. Each
concept brings about different 'codes of
determination' [as in discrete vs continuous
manifolds]. The concepts provide the structure,
not ontology or formulae. This is close to D&G
on the concept of the manifold. Thus 'points' is
the appropriate term for continuous manifolds, and
'elements' for discrete ones [conforms to our
perceptions, says Plotnitsky ,where points appear
against continuous space or some constant
background].
For Riemann space is a 'continuous
(three-dimensional) manifold' [although other
possibilities are allowed too]. It is a
conglomoerate of local spaces, each of which can
be considered in Euclidean terms, although there
may not be an overall Euclidian structure.
This is not particularly different from the
existing techniques of understanding euclidean
space starting from the properties of the straight
line.
Weyl's influences included phenomenology as
well as Bergson, and could even be seen as part of
the Kantian tradition.
Topology involves a space described not by points
but by 'open sets'(195), a concept shared by
Riemann. We can think of these in terms of
open intervals of a line 'say, all points between
¼ and ¾', with those two points themselves as
boundaries. Apparently, a closed set would
include those two points themselves. The
intervals involved can be thought of as either
sets or spaces or both. The problem then becomes
one of thinking about a continuum and how it is
constituted by its points. For
mathematicians this becomes whether we can exhaust
a straight line with a set of real numbers
['Cantor's continuum hypothesis'].
Set theory, apparently, sees the continuum as
constituted by real numbers or points, which
Brouwer thought was 'inaccessible to human
intuition'. There is an allied problem of
affirming the equality of two numbers [if we see
them as continua?], which would involve verifying
an infinite number of the qualities among the
decimal digits making up the numbers, 'which is
not possible'. However, we can verify
inequalities between the boundaries of the open
interval [I'm not at all sure why] and this will
be useful in describing a continuous space as a
class of open sub spaces which cover it—and they
may or may not be seen as sets. If we
consider the interval between 1/4 and 1/2 as
overlapping the interval between 1/4 and ¾], we
can get new overlapping open subspaces.
Apparently, this is the 'essential grounding idea
of topology' (196) in its mathematical
sense. We can further think of any open
interval or set which contains a point as the
'neighbourhood of this point'. In the
example above, the intervals 1/4 to 3/4 and 1/4 to
1/2 would be neighbourhoods of a point at 1/3.
Again, neighbourhoods can overlap, and
topologically, they are all equivalent [so one
represents them all].
We can extend this to consider the topology of
curves or other higher dimensional spaces,
manifolds of any dimensions for Riemann. We
do this by defining a curve or other space 'in
terms of its inner properties' rather than having
to relate it to a background euclidean space [this
is an aspect that Deleuze likes, because it helps
us move beyond Cartesian formulae for
curves—Leibniz was on to this too, of
course]. It doesn't stop us thinking of
things like straight lines still as sets of
points, but we now have a more general structure,
a more general foundation [a 'primitive'].
We can see general topological space as a
collection of open spaces acting as sub spaces,
and we can specify algebraic rules for their
relationships [don't include me!].
There has been much development in mathematics,
but it tends to be 'prohibitively difficult', so
we focus on 'essential philosophical ideas'
instead. We might think of a primitive space
as any-space-whatever [Deleuze in Cinema 1], and
this can extend open intervals. We do not
need to specify in any more detail, although we
can think in terms of the relations between sets
as like those between spaces, leading to notions
of mapping or covering. We can render this
as an 'arrow structure', connecting Y and X with a
directional arrow, where X is the main space and
the arrow just means there is some
relationship. We can further understand a
space as structured by relations between subspaces
rather than as a set of points related
together. This has been termed a
sociological notion (197). The related
spaces have to be of the same category, but we do
not need to start with euclidean space, which
becomes just one possibility within a 'large
categorical multiplicity', one where we can
measure the distance between two points
particularly simply. The subspaces need not
be intervals within a particular space, as with
the intervals above, but can be subspaces of the
whole space [maybe]. The same applies to the
classically inexact concept of neighbourhood,
which can now be generalized as 'a relation
between a given point and space associated with
it'. This is particularly important for
Deleuze [I saw its significance much more through
Guattari,
where the neighbourhood seems to be some initial
gathering of components before any semiotic grasp
of their associations].
One interest of Riemann was in noneuclidean
geometry, especially the geometry of 'positive
curvature'. In euclidean geometry, a
geodesic [shortest possible?] line crossing a
point external to it is straight and has one
[concept of?] parallel [?]. In other
geometries [including a 'geometry of negative
curvature or hyperbolic geometry', the first to be
discovered] there are infinite numbers of such
lines. Riemannian geometry encompasses both
examples as special cases and allows for still
more, so these discoveries were really only a
point of departure for his radical thoughts on
spatiality.
Riemannian geometry involves the study of space
defined as a manifold, specifically a continuous
one. Such manifolds may have a variety of
dimensions, from one to infinity. There are
also discrete manifolds 'which mathematically have
the dimension zero', formed by isolated points or
elements. We need this concept to grasp
space in physics [and in Deleuzian politics?—
Plotnisky says D and G are at least aware of the
difference between continuous and discrete
manifolds, and gives a reference to ATP
p.32]. Mostly it though we think of manifold
as a continuous manifold, and this 'provides the
primary mathematical model of smooth space for
Deleuze and Guattari'(198), while being aware of
other possibilities such as porous spaces which
may be discrete manifolds.
Deleuze and Guattari agree that Riemann's geometry
refers to space itself, not just particular
configurations of it, and see it [in the plateau on smooth space
in ATP, especially page 485] a
decisive event, when the manifold became a noun
[they render it is a multiplicity] in its own
right. They see the enveloping nature of
smooth space as incorporating metric
multiplicities, but also relating other kinds of
nonhomogenous space, which may appear Euclidean to
observers in each subspace, but which cannot be
related to each other directly [I dunno though --
what of the Lorentz transformations -- see Wikipeda].
For them, the subspaces are juxtaposed but not
attached, a matter of an accumulation of sets of
neighbourhoods, not at all like metric space, but
rather like '"pure patchwork"' with the pieces
connected via '"rhythmic values"', providing
continuous variation of heterogeneous
elements. The '"determinations"' involved
should be understood as part of one another,
relating to '"enveloped distances or ordered
differences"', whatever the magnitude. Other
determinations may not be part of one another, but
they are still connected '"by processes of
frequency or accumulation"'. This is clearly
an 'underlying mathematical conception'.
To pursue the idea of an internal geometry, we can
consider Riemann building on Gauss to develop
'tensor calculus'(199), used to measure in curved
spaces of three or more dimensions [Guattari uses
the notion of a tensor quite a lot. It can
be briefly defined, via Wikipedia,
roughly, as a geometric object, a multi
dimensional vector]. This takes full account
of the curvature of space itself. Riemann's
manifold also allows for variations in the
curvature of space, a more general conception than
earlier ones which assumed universal homogenous
curvature.
There is also a link to Leibniz and the relational
nature of spatiality, again with internally
determined structures, either mathematical or
material [the latter case would involve gravity as
in the general relativity theory]. Again
there need be no relation to an ambient
space. We need to investigate all spaces 'in
their own terms and, essentially, on equal
footing' (200): there is no 'uniquely primary
space'. We then replace a depth or vertical
model of space with what D&G call '" a
typology and topology of manifolds"': they also
say that this will replace dialectic ontologies
[all apparent opposites are states of the same
manifold?]. There will be a sociological
relation between neighbouring spaces, or indeed
between any spaces whatever, local structures of
neighbourhoods, including euclidean ones, which
are only special cases, a patchwork requiring
local striations of a given smooth space.
There will be no 'homogenous global
striations'(201), and this 'cartographic or
terminology and conceptuality' is crucial to
Deleuze, and to Deleuze's Foucault.
Mathematicians can extend the notion of a manifold
to even more general topological spaces with open
neighbourhoods—'"any (open) spaces whatever"', or
relations as general as those defined by the arrow
structure above. These may not be accessible
to ordinary intuition, so it might require a new
name. It might even be the case that it is
these general structures, more general even than
manifolds, which inform the [politicized?] notion
of smooth space in D and G. [I think this
could also arise because of the infinite regress
of transcendental deduction?]. These might
forbid any kind of striation including
metrication, and so might not be suitable for
state mathematics. This could provide a
general understanding of how smooth space develops
all the others, having the capacity to disable all
striation. Smoothness could also refer to
connectives between neighbourhoods which will
provide a constant continuity even if there are
striations.
Materiality is also involved in shaping and
architecture and making it possible. Leibniz
will be important here for both Riemann and
Deleuze. For him, space was never a
'primordial ambient given', a mere container of
material bodies, and an arena for physical
processes. We have to stick with the notion
of the phenomena of space here, since there may be
no general concept at all other than what we can
perceive. Einstein develops this notion by
suggesting that space or time can be understood
best as the affects of instruments like rods or
clocks, and also 'our perceptual and conceptual
interactions with those instruments' [so it wasn't
just Bohr as Barad
seems to argue]. We can connect this with
the idea of the monad in Leibniz. Space
requires there to be matter and technology both,
and also our own 'perceptual phenomenal machinery'
[which gives a link to Kant if we want to see this
as primary, or condition of possibility of the
material including space].
Riemann anticipated some of these and Einstein had
built on his geometry. A Riemann lecture
(listed on 203) discussed the difference between
discrete and continuous manifolds and thought that
physical space might be a discrete manifold, which
could still be the case although mostly in physics
space is assumed to be a continuous
manifold. Space is certainly a continuous
phenomenon, and and Reimann thought this was
a matter for physics not mathematics—the physical
content does not just take possession of space,
but rather that physical matter gives space its
form '" filling it and determining its metric
relations"' [quoting Weyl]. Ordinary, phenomenal
understanding can conceive of space only as a
three dimensional manifold, smooth within those
limits. It might be co-extensive with
matter, considered either as bodies or as
fields. This phenomenal understanding could
only be extended to modern findings by adding time
as another dimension.
The gravitational field does determined the
discrete manifold of physical space and the
general properties of its variable curvature. But it also shapes space as a
smooth space. The normal conception of
materiality provides the phenomenal qualities of
space. It consists of bodies and their
material history, and also technology enabled by
these bodies [mental operations can be considered
as part of the body]. Technology enables us
to deal with the universe as 'the ultimate body
without organs' (204) through the desiring
machines which bodies provide us with them,
including the ability to perceive and think.
The body can be thought of as involving concepts
relating objects from quantum constituents to the
universe itself, as well as to political, textual
or other bodies.
We may not be able to access it in principle, so
actually terms like matter or even body without
organs 'may be inapplicable'. It might be
inaccessible in practice or even in
principle. Not only Kant but Leibniz argued
that a grasp of the ultimate nature of the world
is not available, never to any monads. We
can operate only by examining affects and their
effects. We have still been able to build
technologies that 'establish' [for practical
purposes anyway?] the existence of material
objects even in quantum mechanics—that is they
obey 'materialist epistemology'. This is why
the body has been so important, even for Kant, and
why D and G prefer the notion of the nomad to the
'"unitary subject of euclidean space"'[citing note
27 page 574, of ATP].
We can extend the notion of the nomad though -
make monadology into nomadology, in a 'new post-
Riemannian Baroque' (205). Here monads do
interact with each other,not only through the
world, within an overall architecture known only
to God, as in Leibniz. The fold becomes the
manifold. We see the move in Boulez (who coined
the term smooth space) and Cezanne ( discussed in
ATP 477-8, 4593--4),although earlier forms
indicate it too. The quilt is the metaphor [there
is a connection with cubism, says Plotnitsky].
Generally, specific paintings becomes allegories
of underlying smooth space. So does the new
baroque music from Boulez on, and minor literary
practices in Kafka (and Woolf). But it is all
attributed to Riemann - monads have local
connections, and eventually become nomads (
defined in terms of '" the identity of striated
spaces versus the realism of smooth spaces"' (ATP
573).
The conclusion of The Fold
describes the new baroque 'linked to divergent
series on the one hand and Joycean chaosmos on the
other'. Apparently infinite series [same as
divergent ones?] were studied by both
Leibniz and Riemann. The example of music
illustrates the idea best. ATP
(p.137) argues that divergent series that
make up the world, the chaosmos, means that monads
can no longer operate by containing the world
first and then projecting their
understandings. We now have a spiral of
expansion. Vertical harmonics can't be
distinguished from horizontal ones, so that the
themes of a dominant monadic need no longer
correspond to a crowd of monads following their
own melodies. The vertical and horizontal
dimensions can fuse and produce '"groups of
prehension"'. We still have to live in the
world but we have a changed conception of it, more
like the habitat of Stockhausen or the plastic
world of Dubuffet, where lines are blurred between
inside and outside, public and private. Individual
monadic conceptions no longer accord with
the world: we need to discover '"new ways of
folding ...new envelopments"'. But Leibniz
is right to insist it is still always a matter of
'"folding, unfolding,refolding"'.
In other words, all these processes now take place
in smooth spaces, manifolds. Because they
are smooth, new and speedy movements of transition
and convergence can occur, from point to point,
from concept to concept, and from mathematics to
philosophy and vice versa. These rapid
movements are still not easy, but they are very
worthwhile.
Deleuze page
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