Brief notes on: Plotnitsky, A.
(2009) . Bernard Riemann's conceptual mathematics
and the idea of space. Configurations 17
(12), pp.105130. Retrieved from
https://muse.jhu.edu/article/381828
Dave Harris
[ I read this as a bit of background for the one
on Riemann and Deleuze.
The two articles are very similar indeed. I have
made a few points for greater clarification]
Riemann played a major part of a number of
different fields. His notion of conceptual
thinking corresponds closely to that of Deleuze
and Guattari where thought intervenes between
chaos and opinion [an identical section to the
earlier one]. Simple versions of set theory
are extended a bit, with a rather mysterious
example of a sphere defined as a locus of all
points which satisfied particular equations [X X0
squared, where X is the centre of the sphere on
the X axis]. For Riemann, however the sphere
is seen as 'a certain specifically determined
concept'(107), specifically a continuous manifold,
which helps us do more maths with it.
Any kind of space is defined as a conglomerate of
local spaces with networks of relationships among
them, not as a set of points. Each of these
local spaces can be mapped as euclidean, helping
us use conventional geometry, but it would be
wrong to assume the whole space is euclidean,
except when it happens to be so. Each point
has a neighbourhood. Spheres can be seen as
containing small circles on the surface around
each point. Apparently, we can then project
these circles on to the 'tangent plane to this
point to a regular circle on this plane' (108) [in
this way, we see spheres as being made up of
euclidean circles,especially if they are
vanishingly small projected circles?
Technically, we are replacing the idea of points
with that of other spaces, local sub spaces which
in this case can be considered as neighbourhoods
of each point]. This is the sociological
definition of space in Manin's terms. It
leads to philosophical implications for manifolds,
which now have mathematical, philosophical and
other components instead of separating them: this
might be a loss for mathematics but a gain for
other fields.
Mathematical concepts may be a alien to 'general
phenomenal intuition'(109), but might provide a
basis for the exact sciences, especially where
continua are involved. However, there must
always be an interaction between the phenomenal
intuition and the conceptual mathematics, unless
we are to adopt platonic realism. The
obvious differences between phenomenal intuition
and conceptual maths sometimes tempts us to
platonism, but we know that the phenomenal is
always related to any kind of mathematical
intuition. Weyl cites Bergson in Creative Evolution
on the differences between phenomenal time and
quantified mathematical time [does he connect it
to the pragmatic interest?]. Bergson's ideas
may themselves 'have a Riemannian genealogy'(111),
perhaps via Einstein. The whole argument
showed how ideas link with each other and with
different disciplines. Deleuze is another
example. The traffic goes both ways,
although mathematicians like to deny it. Weyl was
particularly perceptive about how any attempt at
definitive elucidation must lead to metaphysics
and therefore to philosophy: such attempts are
'unavoidable in mathematics'(112). Riemann
himself was influenced in the idea of a fold as in
manifold. Apparently Kant was the first to
use the term consistently, and Riemann may have
encountered it in theology which he studied first:
after all the German term for Trinity involves
three folded into one.
What makes maths specific is the project to make
their objects mathematically exact and
specifically numerical, but this does not mean
that philosophy has no influence. We might
grasp mathematical thinking as both heterogeneous
and interactive, involving phenomenal and cultural
fields as well.
There are differences with set theory which
operates with the notion of set as
foundational. Riemann's concepts are not
understood as coming from a single concept,
because each has 'a particular mode of
determination'[and the difference between discrete
and continuous manifolds seems to be the example
which will determine the relation of things like
points]. These differences about the
transcendental nature of the concept divide
Riemann from other mathematicians [his work
preceded set theory, so there is no actual
discussion]. Because so much as happened
since, it is sometimes difficult to follow
Riemann's approach, but we can translate by set
theory as a local version of Riemann, abandoning
the transcendental claims. We will operate
with heterogeneity [and with a less satisfying
pluralism?], But this has been a trend in
philosophy at least since Nietzsche [also taken up
by Derrida, apparently]. The issue is still
whether we see the set or space itself as the
grounding concept [so space itself has become a
foundational concept after all  Plotnitsky says
that this is confined to the mathematics of
spatiality, though,which leaves its status vague]
Geometry and topology have different areas of
application. Topology disregards measurement
or scale and deals with only 'essential shapes of
figures' [risking platonism again?] (115).
Topological figures are seen as continuous
spaces. Sometimes this idea suggests
continuity underneath phenomenally distinct
objects. Such continuity is therefore
difficult to grasp through phenomenal
intuition. The example would be a
topological equivalence of all spheres, however
deformed, even to the extent that they are no
longer geometric spheres. Apparently,
topological equivalence has recently developed
various 'algebraic and numerical properties'(115),
and has thus become a properly mathematical
discipline, compared to the more philosophical
concepts like the khora. Leibniz may
have been on to it with the notion of '"analysis
situs"'.
Riemann has been developed since in analysis of
the topos by a certaun Grothendieck. This
does replace set as a transcendental primary
concept. Apparently it helped avoid
some of the paradoxes have set theory 'such as
that of the concept of the set of all sets' which
can never be consistently defined because sets
cannot be members of themselves [this is the old
regimental barber paradox]. Apparently, in
topos theory, we can have different sorts of
'esoteric constructions', including spaces with a
single point or even spaces without points [hence
the jokey term 'pointless topology']. The
idea again is that space or something space like
is a primary concept.
Riemann defined continuous vs. discrete
manifoldness in his habilitation lecture
(117). Individual 'specialisations' in the
manifold are called points or elements. It
is possible to derive concepts from the elements
in discrete manifolds that occur frequently in
everyday life—indeed, anything normal or frequent
cannot be conceptualized in this way.
However, conceptualizing the continuous manifold
is much more difficult—the closest we get is in
thinking about objects and colours. Mathematics is
required for a more systematic
conceptualizing. Thus concepts are crucial,
especially for continuous manifolds and the
relations between elements inside them. The
continuous manifold is seen as a conglomerate of
local spaces. The original conceptual
division between discrete and continuous manifolds
lead to further kinds of determinations within
each one—so the concepts continuous or discrete
are what determine the relations [close to
idealism then?]
This also means a different 'concept of "concept"'
(118) [any discussion of this paradox? It is
rather like the way in which heterogeneity is a
concept in Deleuze in the sense that it is formed
on the basis of generalizations or from some other
notion like repetition, and it implies consistency
and homogeneity]. As with D and G, a
mathematical concept is not just a generalization,
but is 'defined by a specific architecture' with
definite components. In Riemann's case, we
move beyond just calculation or the manipulation
of formulas, and this helps laypersons grasp the
underlying ideas. The concepts enable a
grasp of otherwise unobtainable mathematical
objects and relationships [idealism is denied in
this way, but the implication is that mathematical
objects and relationships have an independent
existence? Unless it is all a language
game?]. Thus seeing a sphere as a manifold
will lead to more information, including
topological differences with other objects like
the torus: a practical application might turn on
the flows of liquid on the two structures
[apparently, flow on the torus can be turbulence
free. I can see the point if we might be
working with deformed objects where it is not
clear if they are spheres or tori?].
Conceptual determinations might lead to formulas,
but the trick is to look first for 'certain deeper
properties reflected in the formula'(119) [a
transcendental deduction maneuver? Some
subsequent abductive confirmation?]
Points have a phenomenal dimension, because we can
only detect them against some continuous
space. Again, apparently, set theory had
difficulties explaining the connections between
points and continuous spaces, and there are now
new ideas of spaces as we saw above. Riemann
works with a three dimensional notion of space as
a continuous manifold, but even so he could
understand space as curved and geometrically
complex, again a series of sub spaces which we can
treat as if they were euclidean. The key
term for topology is the notion of the open set, a
matter of open intervals of a line as in the
earlier article. We can think of these
intervals as spaces or sets. In set theory
the continuum has to be understood as a set of
points, or real numbers [note 20 on page 120
explains that the problem is to ask whether there
can be a set whose 'power (a number of elements)'
is larger than the set of natural numbers.
It must also be smaller than the set of real
numbers which is infinite. Apparently, the
problem was thought to be solved, although others
think it is undecidable, incapable of proof from a
set of axioms]. This led to the notion of
space being covered by open spaces and there might
be algebraic rules for their relationships [note
21 suggest that the procedure can also be used to
understand the topology of curves in multi
dimensional space—again we use the internal
properties rather than the relation to an
underlying space, presumably, just as Leibniz did
with calculus, exploring local regions of curves
in terms of the relation between distance on X and
rise on Y, only, somehow, for n dimensions].
Again we find this developed in topos theory:
basically the characteristic of a given space can
be left unspecified and instead we might consider
relationships between this space and other spaces
of the same type which might be seen to cover
it. The general structure is the arrow
structure as in the earlier article, also known as
a morphism. A particular example involves 'A
"fiber bundle" or "sheaf"' (121)—fibres connect a
subspace to those other sub spaces which cover it,
or are projected on to it. Sheafs themselves
might be conglomerated [still seems circular to
me, as a total non mathematician. The
covering sub spaces are assumed to be related to
the sub space in the first place as in the phrase
'of the same type'?]. As with Leibniz, flat
spaces.
Thus external relationships with other objects is
the crucial thing rather than any intrinsic
structure, the sociological relationship.
[Here, neighbouring spaces have to belong to the
same category rather than type, e.g. discrete or
continuous manifold]. This is where we get
the term [nonquantitive] multiplicity—a
'("society") of other spaces rather than...a
multiplicity of its points' (122). Even
pointless spaces relate to other spaces.
Euclidean spaces are not privileged but is rather
just one object in 'a large categorical
multiplicity', one which happens to make it easy
to measure the distance between two points.
[Note 22 describes a particular mathematical
definition of a category which involves a
'multiplicities of mathematical objects endowed
with Givens structures and of relations among
them' such as arrow structures. By
considering objects as consisting of relations, we
can often 'learn more than we could by considering
them only or primarily as individual' We can amend
the idea of a neighbourhood slightly, since its
relation with its point can be by fibre
bundles or sheaths. Categories can be
considered as objects like this too and so they
become functors not simple morphisms: hence functor
connects categories of topological and geometrical
objects with categories of algebraic objects 'such
as groups'].
Riemann built on earlier discoveries of non
euclidean geometry, including
the work on geodesics as in the earlier article,
but he attempted to encompass these others as
special cases. Nevertheless, such rethinking
had an impact in developing the concept of
manifold. As well as
continuous and discrete manifolds , they can also
have infinite dimensions. Continuous
manifold has been the most significant concept,
offering a doctrine of space itself rather than a
limited geometry. Such geometry is
independent of euclidean space and turns on the
internal properties of curved surfaces.
Riemann's use of the tensor calculus extended an
earlier idea of internal curvature. Einstein
used the procedure. Curvature of space can
vary from point to point, so the concept of
manifold needs to allow for such variations—this
is the breakthrough compared to the earlier non
euclidean geometries which assumed constant
curvature. Spatiality is seen as relational
rather than as something given or absolute
containing geometrical figures or material
objects. One internal determination can be
gravity, but such 'empty space' can also be
defined mathematically or philosophically.
We need to investigate each space in its own
terms, with no privileged terms, and in relation
to its other spaces rather than to one primary
space. The structure of a space can be
determined 'sociologically', that is by these
relations. We can start with euclidean maps
[and presumably conventional physics? Cause
and effect, resonance and all that?].
The specific implications for this article are
those that affected Einstein. Apparently he
prioritized Leibniz against Newton [on absolute
space?] and saw that space and time are also
effects of our own measuring instruments and our
conceptual apparatus—these provide the conditions
of possibility of space and time, in a
Kantian gloss. Our perceptual
machinery depends on the materiality of our
bodies. Einstein was only interested in the
material determinants of physical space, although
there are some general implications. Riemann
anticipates some of Einstein, at least in
suggesting that metric relations depend on the
concept of discrete manifold, and may not apply to
continuous ones. We should start with
phenomena rather than from general concepts which
risk a contamination of 'traditional
prejudices'. The implication is that the
nature or structure of a discrete manifold can be
defined by a specific mathematical concept,
although most conceptions assume that space is the
continuous manifold, at least until recently with
new thinking about quantum gravity [of all
things]. Certainly Einstein replaced Newton
on the instant effect of gravity [in fixed space]
with the notion of curved space itself.
Riemann had already suggested this, adding that
curved space is usually occupied by actual
material content which adds flatness [hints that
jokey term 'residential flats']. This might
mean that the universe is flat one the average,
despite local regions of curvature. The physical
universe given phenomenally can be coextensive
with matter  but Weyl says we needed to add time
as a fourth dimension to fill out the picture of
'binding forces' [there is a weasel about whether
metricable behaviour is really produced by
gravity, including the characte4ristsics of
measuring rods etc  all our phenomenal
perceptions.
In R's terms, this means that a manifold
explains space, matter and our [intuitive]
philosophies of it, Space and time have an
'efficacy' (128). Plotnitsky suggests that
Derrida has a similar idea with différance,
producing all sorts of proximities and
interactions from the one process rather than
seeing everything as unconditionally separate—différance
is also 'the material efficacy of both spatiality
and temporality' [note 32 refers us to Margins
of Philosophy]. For Derrida, writing
is material, and when combined with technology
produces things ('neither terms nor
concepts'] like trace, supplement etc. Here,
Einstein on technology has been extended to
cultural production [Bergson said it first, maybe
even Husserl?], including scientific theories 
material dynamics produce différance. Some
versions of social constructivism say the same
[Plotnitsky likes Latour].
Materiality presupposes everything else  our
material bodies, their history and their
technology. Bodies arise from the universe  and
there are many connotations of the term 'body'. We
might never grasp this deep materiality
phenomenally, or even conceptually but not for
Kantian or theological reasons [the universe
prevents understanding of itself?] . We might be
able to study affects and effects  quantum
objects show the possibilities of a new
technology, although its relation with the
manifold is still unclear  it is discrete at
present although there is a notion of a continuous
manifold implied.
Other contributions have included developments in
number theory [the distribution of prime numbers
according to a Riemannfunction]. This could also
impact on quantum theory. The ideas are
still having impact  they relate to many fields.
There may be no single term for his maths  but
then there is no separation of maths from other
modes of thought.
Deleuze page

