John’s nice post has reminded me of the importance of repetitive series for Deleuze (an issue I also discuss here). Picking up on John’s discussion of the perception of colors, series play an important role in attempting to account for our use of predicates: in short, Deleuze will often place predicates within the context of a series of predicates – e.g., shades of blue. This pattern is most obvious in Deleuze’s Logic of Sense, where each chapter is titled “First Series of…” “Second Series of…” etc.… But why series?
Two short answers, which I’ll expand on below the fold: 1) a series of differences is precisely what provides, in good Spinozist fashion, the principle of sufficient reason for determinate phenomena; and 2) series in turn provide the metaphysics science needs.
Let us take a simple series of phenomena, E, E, E, E, etc. … Let us also assume there is no difference between the elements of the series. For Deleuze, however, each element, as an identifiable, determinate phenomenon, refers “to an inequality by which it is conditioned” (Difference and Repetition, 222). Thus every extrinsically distinct element presupposes an intensive difference, what Deleuze will call an “intensive quantum” (Desert Islands, 88), and thus in the series E, E, E, E …, E itself presupposes the intensive quantum e-e’, and the element e presupposes ε-ε’, and so on ad infinitum. Deleuze will call this “state of infinitely doubled difference which resonates to infinity," "disparity,” which he then adds is “the sufficient reason of all phenomena, the condition of that which appears.” (DR 222; emphasis mine).
One can picture this “state of infinitely doubled difference” by way of the graphs of Feigenbaum’s constant, where functions approach chaos through period doubling:
Returning to the use of predicates, this difference or disparity that is the “sufficient reason of all phenomena” does not inhere in phenomena as their predicate, nor even as a separate or separable element, but it is rather the particularity of each element taken to the limit, or what Deleuze will call a “concrete universal.”
For example, if we take a series of colors such as the several shades of blue one might find on a paint sample card at a hardware store, our tendency is indeed to consider each of these individual colors as a “shade of blue.” In other words, each particular color is differentiated from one another by a matter of degrees from one general color, blue, with these degrees running the spectrum from high to low saturation.
Deleuze, however, argues that this is wrong. Following through on Bergson’s discussion of Ravaisson (see DI 43), the universal is not an abstract concept distinct from each particular shade of blue, in which case we have an external difference, or a difference between the shades made possible by virtue of a universal that is external to them and of which they are varying degrees or shades. To the contrary, the concrete universal is the infinitely doubled difference that resonates and inheres within each appearing shade. In the case of a particular shade of blue, this concrete universal is “white light,” or it is the infinitely doubled difference (the far right of the above graph) that “makes the difference come out between the shades”; or,
...the different colors are no longer objects under a concept, but nuances or degrees of the concept itself. Degrees of difference itself, and not differences of degree. White light is still a universal, but a concrete universal, which gives us an understanding of the particular because it is the far end of the particular… (DI 43)
One can find further evidence for Deleuze’s metaphysics in Mark Wilson’s essay “Theory Façades,” (which can be gotten here) and in his subsequent book Wandering Significance. Wilson provides numerous examples, and with dizzying detail, to argue that throughout “scientific work” one finds that what is put to work in the effort to provide theoretical directives are “sheets of doctrine that do not truly cohere into unified doctrine in their own rights, but can merely appear as if they do if the qualities of their adjoining edges are not scrutinized scrupulously” ("TF" p. 273). What may work well at one level and scale may begin to fail at a more detailed and enhanced level of description. As Wilson puts this in Wandering Significance,
…as our everyday descriptive terms become pressed to higher standards of accuracy or performance, as commonly occurs within industry or science, a finer and more perplexing grain of conflicting opinion begins to display itself within our applications of “hardness,” “force” and even “red.” (p. 7)
Put in the Deleuzian terms discussed above, the effort to produce accurate descriptions of phenomena encounters, with the increasing demands of more detailed and nuanced analysis, the substantive multiplicity or concrete universal that is the sufficient reason for the phenomena being described. The result is the failure of descriptive terms as these terms get pushed towards increasing particularity of detail; in short, as Deleuze says, as they are pushed toward the concrete universal that is “the far end of the particular” (DI 43).
What happens, Wilson argues, in our attempts to maintain “inferential headway” in the face of the difficulties that arise as the level of particularity increases, is that we often find it easier “to decompose the system’s overall behavior into descriptive fragments where the intractable complexities of the full problem become locally reduced to more tractable terms” (TF pp. 273-4).
Wilson offers the example of what happens when we attempt to use applied mathematics to understand the formation of spray on “the surface of a choppy ocean” (WS p. 210, as are subsequent quotes). If the ocean is modeled as a continuous fluid, the partial differential equations will provide accurate descriptions to a point, but then it fails to track the phenomena, for the equations continue “to plot an attached blob that never relinquishes its absurdly elongated umbilical tie to the mother ocean.” To offset this poor description, one solution is to run the model with an already detached blob that then separates from the ocean. This provides for a good description where the continuous model does not, but then the description is poor where the continuous model’s was good. If we combine the two models together, we can overlap them such that it provides a good description for the entire process.
While this may be effective at providing an accurate description, Wilson argues that what is going on here is an exercise in “physics avoidance in that we do not directly describe the molecular processes that lead to drop separation, but merely cover the relevant region with an interpolating patch.” In other words, there is a repressed difference or boundary between the two patches that is then mistakenly held to be a unified account of water separation when it is not.
Wilson is not arguing that no account of the water separation is possible. His argument is that an adequate account of the boundary where the different patches converge may well entail a complex mathematics beyond our ken at this point. As a result, and due largely to impatience, we are often tempted, Wilson claims, “to pretend as if our façade patchwork provides a wholly adequate descriptive framework solely on its own terms…” (TF 275)
Wilson’s point, however, and this is just what one would expect given the Deleuzian metaphysics and its use of the PSR, is that however detailed and nuanced the theoretical and mathematical description might be, there are underlying differences that subvert them as the level of description pushes to the “far end of the particular.” In other words, substantive multiplicity (or concrete universal) may be the sufficient reason for every phenomena, but it is also the reason our mathematical equations and theories which track phenomena will forever flirt with, and be challenged by the intensive differences that fail to be explicated and hence modeled by their equations.
Wilson makes a very similar point in the early pages of Wandering Significance, and one quite in line with reading of Deleuze’s metaphysics offered here. Wilson argues that,
The main consideration that drives the argument of the book is the thesis that the often quirky behaviors of ordinary descriptive predicates derive, not merely from controllable human inattention or carelesseness, but from a basic unwillingness of the physical universe to sit still while we frame its descriptive picture. (WS 11)