Deleuze and Guattari see concepts as rhizomes, biological entities endowed with unique properties. They see concepts as spatially representable, where the representation contains principles of connection and heterogeneity: any point of a rhizome must be connected to any other. Deleuze and Guattari list the possible benefits of spatial representation of concepts, including the ability to represent complex multiplicity, the potential to free a concept from foundationalism, and the ability to show both breadth and depth. In this view, geometric interpretations move away from the insidious understanding of the world in terms of dualisms, dichotomies, and lines, to understand conceptual relations in terms of space and shapes. The ontology of concepts is thus, in their view, appropriately geometric, a multiplicity defined not by its elements, nor by a center of unification and comprehension and instead measured by its dimensionality and its heterogeneity. The conceptual multiplicity, is already composed of heterogeneous terms in symbiosis, and is continually transforming itself such that it is possible to follow, and map, not only the relationships between ideas but how they change over time. In fact, the authors claim that there are further benefits to geometric interpretations of understanding concepts which are unavailable in other frames of reference. They outline the unique contribution of geometric models to the understanding of contingent structure:
Principle of cartography and decalcomania: a rhizome is not amenable to any structural or generative model. It is a stranger to any idea of genetic axis or deep structure. A genetic axis is like an objective pivotal unity upon which successive stages are organized; deep structure is more like a base sequence that can be broken down into immediate constituents, while the unity of the product passes into another, transformational and subjective, dimension. (Deleuze and Guattari)
The word that Deleuze and Guattari use for ‘multiplicities’ can also be translated to the topological term ‘manifold.’ If we thought about their multiplicities as manifolds, there are a virtually unlimited number of things one could come to know, in geometric terms, about (and with) our object of study, abstractly speaking. Among those unlimited things we could learn are properties of groups (homological, cohomological, and homeomorphic), complex directionality (maps, morphisms, isomorphisms, and orientability), dimensionality (codimensionality, structure, embeddedness), partiality (differentiation, commutativity, simultaneity), and shifting representation (factorization, ideal classes, reciprocity). Each of these functions allows for a different, creative, and potentially critical representation of global political concepts, events, groupings, and relationships. This is how concepts are to be looked at: as manifolds. With such a dimensional understanding of concept-formation, it is possible to deal with complex interactions of like entities, and interactions of unlike entities. Critical theorists have emphasized the importance of such complexity in representation a number of times, speaking about it in terms compatible with mathematical methods if not mathematically. For example, Foucault’s declaration that: practicing criticism is a matter of making facile gestures difficult both reflects and is reflected in many critical theorists projects of revealing the complexity in (apparently simple) concepts deployed both in global politics. This leads to a shift in the concept of danger as well, where danger is not an objective condition but “an effect of interpretation”. Critical thinking about how-possible questions reveals a complexity to the concept of the state which is often overlooked in traditional analyses, sending a wave of added complexity through other concepts as well. This work seeking complexity serves one of the major underlying functions of critical theorizing: finding invisible injustices in (modernist, linear, structuralist) givens in the operation and analysis of global politics.
In a geometric sense, this complexity could be thought about as multidimensional mapping. In theoretical geometry, the process of mapping conceptual spaces is not primarily empirical, but for the purpose of representing and reading the relationships between information, including identification, similarity, differentiation, and distance. The reason for defining topological spaces in math, the essence of the definition, is that there is no absolute scale for describing the distance or relation between certain points, yet it makes sense to say that an (infinite) sequence of points approaches some other (but again, no way to describe how quickly or from what direction one might be approaching). This seemingly weak relationship, which is defined purely ‘locally’, i.e., in a small locale around each point, is often surprisingly powerful: using only the relationship of approaching parts, one can distinguish between, say, a balloon, a sheet of paper, a circle, and a dot.
To each delineated concept, one should distinguish and associate a topological space, in a (necessarily) non-explicit yet definite manner. Whenever one has a relationship between concepts (here we think of the primary relationship as being that of constitution, but not restrictively, we ‘specify’ a function (or inclusion, or relation) between the topological spaces associated to the concepts). In these terms, a conceptual space is in essence a multidimensional space in which the dimensions represent qualities or features of that which is being represented. Such an approach can be leveraged for thinking about conceptual components, dimensionality, and structure. In these terms, dimensions can be thought of as properties or qualities, each with their own (often-multidimensional) properties or qualities. A key goal of the modeling of conceptual space being representation means that a key (mathematical and theoretical) goal of concept space mapping is
associationism, where associations between different kinds of information elements carry the main burden of representation. (Conceptual_Spaces_as_a_Framework_for_Knowledge_Representation)
To this end,
objects in conceptual space are represented by points, in each domain, that characterize their dimensional values. A concept geometry for conceptual spaces
These dimensional values can be arranged in relation to each other, as Gardenfors explains that
distances represent degrees of similarity between objects represented in space and therefore conceptual spaces are “suitable for representing different kinds of similarity relation. Concept
These similarity relationships can be explored across ideas of a concept and across contexts, but also over time, since “with the aid of a topological structure, we can speak about continuity, e.g., a continuous change” a possibility which can be found only in treating concepts as topological structures and not in linguistic descriptions or set theoretic representations.