Brief notes on: Plotnitsky, A. (2009) . Bernard Riemann's conceptual mathematics and the idea of space. Configurations 17 (1--2), pp.105--130. Retrieved from

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 [non-quantitive] 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 Riemann-function]. 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