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

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 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