Notes on: Calamari, M. (2015) RiemannWeyl in
Deleuze's Bergsonism and the constitution
of the contemporary physicomathematical
space. Deleuze Studies 9 (1).
pp.5987. DOI 10.3366/dls.2015.0174
Dave Harris
R was mentioned first in Bergsonism. Weyl's
reading of R was also a major influence
though  we need to extend Plotnitsky on this.
We can explore more implications of R space 
rhizomatic multiplicities, smooth space, links
with 'fibre bundle topology and gauge field
theory'.
The ref to R in Bergsonism relates to the
differences between quantitative and qualitative
multiplicities, as a solution to the old issue of
the One and the multiple. For D, the discrete
multiplicity has its own metric, while the
continuous has an external metrical principle 
phenomena unfolding within, or external forces. D
refers to both R and Weyl's book (Space, Time,
Matter)  the first detailed
exposition of Einstein's general theory and the
first attempt at unified field theory. D says
Husserl was also influenced by R. But W was then
ignored. Calamari says he is crucial  eg in
moving from Leibniz monads to nomads,and to
developing the notion of a plane of immanence
which resembles modern physicomathematical space.
D recognizes that R is both a mathematician and a physicist ,
especially after Einstein. D talks about the
Riemann abstract machine, and says that the notion
of multiplicity is both physical and mathematical
in Riemann. This is a recognition of
mathematics and physics belong together, borne out
by Riemann's own work which included attempts to
elaborate a unified theory of physical
forces. The famed habilitation lecture
established topology but also discussed the
physical applications of the idea of a manifold,
the basis of an infinite number of geometries,
which have the specifically non euclidean ones as
a subset. But this implies a geometry of
physical space, and was a contribution to an
earlier tradition, which included W. D went
on to argue that the debate between Einstein and
Bergson really depended upon different notions of
the Riemann theory of manifolds.
D says that Riemann saw things, any kind of
objects whether empirical or a state of things, as
multiplicities defined by various parameters which
determine them, which makes the manifold exceed
specific domains of application. W also
stresses the many uses of the concept of manifold
in the natural sciences: it can be used to define
geometric entities, phenomena, states of things,
sensible objects, physical systems. The
various coordinates or variables, dimensions,
define these objects, so an ideal gas is two
dimensional because it is determined by two
variables, pressure and temperature, while colours
have three dimensions, adding quality and
intensity [to frequency?]. D is endorsing
this widespread implication when he says
'"everything is a multiplicity insofar as it
incarnates an Idea"' (64) [which seems to be
rather different and suggests some alternate
determination as well as the dimensions?]. Here,
the substantive notion of multiplicity is
connected to ' the transcendental field of virtual
Ideas' (65), making it even more autonomous and
irreducible, but it retains a connection to
empirical determination 'or actualization', as a
matter of 'the differentiation and genesis' of
'"real [spatial temporal] relations and actual
terms"' [citing Difference
and Repetition, 236]. This also
implies a 'fundamental immanence'. In this
way the Idea becomes 'a concrete universal'.
W said the same of Riemann only in the field of
maths and physics.
So the Idea becomes 'a substantive (virtual)
multiplicity', and real actual objects are
produced immanently determined by the elements of
the multiplicity and their differential
[literally] relations or variations
[rendered here as dx and dy , dy/dx, and values of
dy/dx respectively]. A distribution of
singular [only? Not regular?] points
corresponds to these relations. All this is
consonant with W. Both imply the extension
of differential calculus which produces actual
objects, phenomena, phase transitions and states
of things [apparently, singular points play
particular roles in phase transitions].
All this helps us grasp the discussion of the Idea
in g. We are told it is an '"n
dimensional continuous defined multiplicity"'as
when the Idea of colour is a three dimensional
multiplicity as above. Dimensions are to be
understood as variables or coordinates, continuity
as the relations between changes in the variables,
which may take different mathematical forms.
There is a reciprocal determination so that
elements cannot change unless the multiplicity
changes. The example of the colour shows the
debt to W. The particular insistence on the
possibility of a quadratic form of differentials
makes explicit the definition of the Riemann Ian
manifold. This is what is also found in
Bergson ism, where things are actual
multiplicities dependent on the virtual
multiplicities and the differential relations and
points that they embody. There is no other
actual state of things.
These commentaries seems to be based on an excerpt
from W, one where he is closely quoting Riemann's
habilitation lecture, which emphasizes this last
connection with the physical world, rendered as
the connection between geometry and physics and
the nature of physical space. W thinks the
distinction between discrete and continuous
manifolds is important. Discrete manifolds
have single isolated elements, determined by the
number of elements belonging to it, with a
metrical principle inherent in it, something
arising from the concept of the number
[apparently, this was part of the problem of
discussing infinitely small space]. The
continuous manifold has a ground or principle
outside it in the form of binding forces: [if
space is a discrete multiplicity it is self
sufficient, and if a continuous one it must have
some external dimension which shapes it].
Apparently, quantum mechanics tends to assume that
space is discrete, but both Riemann and Einstein
imply otherwise. [as in Plotnitsky
on residential flats].
D seems to borrow from W in Bergsonism,
although he inserts a quotation from Riemann,
drawing from the concluding part which they had in
common. This concluding part concerned
applications to space and which geometry is
consistent with the empirical world. This
provides the distinction between discrete and
continuous, or quantitative magnitudes and spatial
topological magnitudes with particular
importance. Riemann is actually arguing that
the geometry of physical space cannot be
determined a priori, and that what is normally
taken as real space is probably a special case of
manifold, one which is three dimensional, but
purely topological, that is devoid of any form,
before the introduction of anything that would
provide metric determination. This means we
must examine the physical forces at work in space
to solve the problem of the geometry of space
instead of presuming any geometry as an axiom—as
euclidean or somehow indifferent to matter which
it merely contains.
D preserves this twofold character in his own
work—the multiplicity can be a purely 'informal
(topological)' (69) and autonomous Idea, and also
something that is necessarily connected to actual
space and physical forces. We see this
twofold nature in the notions of a rhizomatic
multiplicity and a smooth space in ATP.
So smooth space becomes amorphous and non formal,
while rhizomes have a metrical principle acting on
them from outside, or in the physical phenomena
which inhabit them.
We find these laws of nature in the infinitely
small, apparently, and this reveals a further
relationship between Riemann and Leibniz. The
implication is that the physical laws can be
expressed through differential equations and the
differential calculus. Differential geometry
emerges by extending the calculus [well done me 
I suspected this]. Differential equations
will therefore describe natural laws. The
second implication is that natural forces act by
continuous action, 'that is, by local
(infinitesimal) contact' [now I get it] which gets
transmitted through the surrounding space.
We need differential equations because changes
depend on several variables not just one as in
classical newtonian mechanics, which also had an
idea of action at a distance, the instantaneous
transmission of force: Riemann sees mathematical
physics as a matter of fields, which he
anticipated before Faraday and Maxwell.
Deleuze discusses this, mentioning W, in his book on Leibniz, where
he agrees that differential equations explain the
laws of nature [I have always been puzzled by this
interest in the differential equation—now it makes
sense]. He also says that the differential
links physical reality and mathematical
calculus. He has grasped the necessity from
this from Riemann and W.
Riemann's support for Leibniz can also be
understood as an opposition to Newton and the
mechanical model, and more as a support for field
theories. Again, this would imply that the
infinitesimal explains the behaviour of the
external world, 'infinitely near action', implying
the 'local intrinsic construction of space' (71)
as contact interactions propagate along the lines
of force or field lines. Bodies no longer
act biunivocally but are affected by the
continuous space or field between them. It
is this that leads to the
'continuousdiscontinuous dichotomy', and also
challenges scientific reductionism [mechanism
anyway?].
There, however, Riemann and Leibniz also diverge,
Deleuze argues, over the issue of the monad.
[NB the monad can be an isolated particle] Here,
the 'mathematician and philosopher Gilles
Châtelet' is cited. C addressed the
implications of Riemann in the last section of the
habilitation lecture, discussing the implications
for the monad of the continuousdiscontinuous
dichotomy. [It seems that if space is a
continuous manifold, there must be communication
between monads, because signals would be
transmitted from proximate states] (72).
There must be local interactions or communications
in continuous manifolds, apparently as in Maxwell
and Einstein—it is an extension of the idea of
action by contact. There can therefore be no
closure around monads [apparently Leibniz agreed
with the idea of action by contact but saw it as
something representing only extrinsic movement,
but, monads include the totality of the world, so
there can be no genuinely external action, only
interaction between monads]. Thus Riemann
leads to the challenge to closure and a new
openness of communication and coexistence.
For Deleuze, C is useful in arguing that
differentiation propagates step by step in a
community of monads, through tactile relations or
connections.
C understands connection in the modern
mathematical and physical sense, involving a
theory of fibre bundles for maths, or gauge field
theories for physics. Again, W extending
Riemann and linking with Einstein's general theory
suggested this [although so did other
mathematicians, cited on page 74] [much seems to
turn on debates about incorporating work on
parallel transport of vectors into Riemann notion
of space as a manifold with its own autonomous
properties.] W did this by thinking of parallelism
as a matter of connection between manifolds as a
'projective' or 'conformal' nature [aka 'affine
connection'  wikipedia
defines this as 'an
affine connection is a geometric object on
a smooth
manifold which connects nearby tangent spaces, and so
permits tangent vector fields
to be differentiated as if they
were functions on the
manifold
with values in a fixed vector space', and
there is a diagram:
An affine connection on the sphere rolls
the affine tangent plane from one point to
another. As it does so, the point of contact
traces out a curve in the plane: the development. .
The term was originally used to describe the
displacement of vectors in 'affine' [tangent]
spaces, but W. saw that there might be a physical
process involved in field theory, perhaps
explaining interactions between gravity and
electromagnetism. First Riemann had to be
read as a thorough rejection of geometry at a
distance, 'the possibility of a direct comparison
of lengths (specifically, the length of vectors)
located at distant points'(745). W argued
instead that spatial and temporal lengths were
path dependent, with local changes in the scale:
length therefore became relative, and 'variable
scale factors' (75) became important. This
would lead to 'local gauge theory'[gauge seems to
be something invariant, irrespective of any
particular choice of unit length or gauge {sic}].
Einstein apparently argued that physical
observations contradicted this idea, leading to
further research by W. 'two new fundamental
interactions' had been identified by then, the
weak and the strong, which opened possibilities
for application to quantum mechanics, and
apparently this is the basis of modern gauge
theory. That led to possible unifications
between the interactions in quantum field theory
and it now constitutes the standard model in
particle physics.
Back at the mathematical level, the idea of
connections produce the theory of 'topological
fibred spaces', again with global implications for
understanding connections. We can think of a
fibre bundle as an enlarged topological
space. There is a base space, an
ndimensional manifold and a total space above it
which has a group structure [pass]. Fibers
correspond to points on the base space [and they
also connect the two levels of space to produce a
total space]. In the simplest example, a
product space explains a total space as a product
of the base space and the fibres  as in a
cylinder, with a circle as base space and a line
segment as a fibre. As an additional
complication, each fibre itself has m dimensions,
so a fibre bundle is not a simple multi
dimensional space but an M plus N dimensional
space. The way the fibers work is through
'bundle projection (a continuous map)' (76), a
relation that can be described as extrinsic but
not external. More grippingly, the total
space does not just have the basis space embedded
in it and is not reducible to base space.
Each fibre adds extra dimensions, so a bundle
connection is a geometric structure in its own
right, and [as continuous space?] helps us deal
with the relations between infinitesimally
neighbouring fibers. The connections with Riemann
and Einstein were not realized at first, but
eventually, gauge theory could be related to fibre
bundle topology as a further deep connection
between physics and maths. There is also a
link with general relativity theory.
In quantum field theory, quantum particles must
interact with a field in order to exist at all, so
those particles themselves can be grasped as
interactions between matter and force. If a
quantum particle interacts with a gauge field
[some additional force field?] we can see
the effects in terms of changes in the internal
states of the particle , such as the 'phase change
in the wave function'. Originally, it was
thought that the intensity of the electromagnetic
field caused this change. Mathematics was
much more interested in the potential, produced by
differences in particular electrical or magnetic
vectors which produced a '"tension"': this is a
gauge potential.
However potentials have always been seen as
ambiguous in physics, even arbitrary. It is
not sure if they are physically entities or just
convenient mathematical devices ['conventional and
arbitrary']. It seems that modern thinking
sees potentials as having physical realities
because they can produce effects that are not
entirely attributable to the field strength.
It is now thought that the phase differences of
quantum particles are physically determined by the
gauge potentials, that field strength is a
quantity derived from these potentials, and that
the potentials cause the changes. Later
heroes demonstrated that gauge fields can be
better understood as 'principle fibre
bundles'. The base space is the spacetime
manifold, and the total space shows the dynamics
of interacting matter and force quantum fields
[matter fields are now seen as 'associated vector
fields.]. Each fibre can then be seen to
correspond to the 'internal symmetry space' of
quantum fields [this is where it gets its extra
dimension from]. We can also understand that
total space of the fibre bundle as a space of
phases of a quantum particle as it moves through
space and time and interacts with a quantum
field. These phase differences are
determined by gauge potentials, so gauge
potentials are identified with bundle connections,
or changes of phase along a path. The
strength of the gauge field is connected to the
curvature of the connection [apparently 'phase
changes imply either curvature of the connection
to be nonzero']. We can think of this as
twists in the fibre bundle, producing more than
just a simple or trivial connection.
So the paths of the monads in space can be
seen like this, related to the fibers constituting
the space of phases. These will synthesise
the dynamics of fundamental interactions.
The potentials arising from connection create the
vertical particles of the interaction field, and
its curvature extends the notion of captures, from
the trivial and from Cartesian grids. Space
therefore becomes heterogeneous, according to C,
an open space of virtual potentials and
intensities of matter, a necessary connection
between monadic particles and fibers or quantum
fields, all of which are affected by others
outside them. Interactions of matter and
force will affect each fibre [and
particle?]. Apparently everything depends on
seeing a (mathematical) connection as involving a
(physical) potential. The idea of open space
in Riemann is retained, and monadic perspectives
overcome. [Fascinating, I wish I could
understand this more fully].
Deleuze apparently summarizes all this in a short
passage in the Leibniz
book, discussing the new monadology in
mathematics, which he describes apparently [on
page 81] as a matter of monads testing their paths
and entering into synthesis with other paths
through a process of capture. Thus modern
developments in science help overcome the
important restriction of absolute closure in
Leibniz, and confirms the progress made by'
neo leibnizianism'. The open world in
constant becoming is the chaosmos, where beings
are pushed apart and connections kept open through
divergent series and '"incompossible totalities"'
[citing page 81 again]. The world is open
not to something external and supplementary but to
a field of interactions. Thus we can develop
a nomadological perspective instead, as Plotnitsky puts it,
the ideality of striated space vs. the realism of
smooth space.
The notion of rhizomatic multiplicity also draws
upon contemporary notions of interactions and
connections of heterogeneous elements. The
idea of Riemannian space as a patchwork indicates
this. Deleuze sees interactions as in gauge
theory, with space as open, a matter of '"
extrinsic coexistence"', or a 'plane or "milieu of
exteriority"', extrinsic [but not
external]. Nor do they have any
'supplementary dimensions of embedding'(81).
Technically this means that open spaces are
matters of accumulation, not at all like euclidean
conjunction, which implies overlaying each point
with a tangent euclidean space [I thought this was
what Riemann argued at one stage] —for Deleuze,
this would still imply that smooth space was still
embedded in homogeneous and striated space.
This apparently is an intervention with the
earlier debate about Riemann and parallelism
summarized above. The best way to understand
Deleuzian fields and multiplicities is as a non
trivial fibre bundle, with the base space
corresponding to the multiplicities, and an
additional total space composed of internal but
extra dimensions of interactions and connections,
a field of '"heterogeneous potentials and
intensities"' [quoting Difference and
Repetition, p.50].
This field of interaction and connections is not
an external space for the multiplicities, but
rather a necessary milieu. Deleuze thinks
that even Leibniz was forced to talk about a
minimal outside for the monads, although he saw it
as complementary. Deleuze goes on to
describe complementarity here in terms of the
'topological property of "unilaterality"' (82),
that is '"a torsion of the world, an infinite fold
that can be unwrapped' but in a particular way:
the other side of the fold should not be seen as
exterior to the monads but '"as the exterior or
outside of its own interiortiy"', a construction
like that of the Mobius strip, already described
as a non trivial fibre bundle [above, but I did
not mention that]. Deleuze had already, in
the Logic of Sense,
argued the 'outside' of sense is like a Mobius
strip that we need to understand by splitting it,
unfolding in its length, untwisting it. This
would then agree that unilateratality would become
an extrinsic or relational property [something to
do with in extrinsic action, introducing no
external forces]: the dimension of sense seen from
the outside is folded like this. Since the
characteristics of the Mobius strip are also
intrinsic, this shows that the extrinsic is
strictly complementary [apparently the extrinsic
refers to unilaterality, and the intrinsic to
nonorientability, which might be another term
referring to folding or twisting]. This is
not to say that the outside is not different to
the monad.
The same might be said about the relations between
the multiplicities and the field—there is an
extrinsic outside dimension of openness, but also
'a strict complementarity that prevents any kind
of transcendence'(83). The field of
interactions and connections can be seen as the
'"plane of consistency" of multiplicities, which
defines precisely their internal exteriority; that
is, of pure immanence'. The plane of
consistency is actually the outside of all
multiplicities. Its dimensions increase
every time connections are made on it, but it is
at the same time 'a single, pure plane of
immanence'. It is neither an external
outside, nor an internal inside: it is instead a
twisted fibre bundle, correlated to the
contemporary idea in physical science of a
physicomathematical space.
Deleuze page

