In the sixth chapter entitled “The End of the Great Bureaucracy,” Bernhard Siegert takes us on a journey from Leibniz to Fourier and Helmholtz, drawing together the “idea of formalism” and mathematical diagrammatics based on infinite series, the “discourse on fluids” and the emerging dynamic mechanics, and the formalization of oscillatory propagations—developments which lead to the rupture (Riss) between representation and the represented and which give passage to the “paralogics” of the theory of signal processing and the digital impulse.
Leibniz’ notion of the “minute perceptions,” which states that the objects we perceive—the sound of a breaking wave, a color—are actually composite phenomena made out of a manifold of small individual elements, corresponds mathematically with the development of infinite series. Such series were thought of as being a “figurative” representation of irrational entities. Siegert refers to the representation of the square root of 2 by making use of continuous fractions (Kettenbrüche): “Kettenbrüche sind ein Verfahren, im Dunklen und Verworrenen des Reellen eine ‘Feinstruktur’ sichtbar zu machen.” (194) Therefore continuous fractions and, more generally, infinite series represent algorithms for the iterative construction of real numbers, which may or may not converge in a distinct value—such as Euler’s number e: “An die Stelle eines finiten Zeichens (das auf eine abwesende Operationsvorschrift verweist) tritt eine infinite, rekursive erzeugende Zeichenserie, die zugleich Repräsentation und erzeugende Regel der Repräsentation ist.” Because they lay open the operative logic of the construction, infinite series represent a visual or “figurative” form of insight (“‘figürliche’ Erkenntnis,” 195). The “dark and intricated” real becomes represented in series of “figures” or “characters” in space, thereby revealing their “operative evidence” (196). It is the same “figurativeness” that causes Leibniz’s enthusiasm about the binary numbers: “Die entscheidende Eigenschaft der Binärziffern für Leibniz wie auch der unendlichen Reihen für Lambert ist, daß sie ‘Figuren’ sind.” (196)
In 18th century physics, Leibniz’ “Analysis” (or calculus) paves the way for a shift from a mechanics of rigid bodies to a mechanics of continua: “Physik—die Wissenschaft von den Körpern—wird im wesentlichen eine Mechanik der Fluide” (197) and hence becomes a theory of the ocean. The description of the world, the “Kosmographie,” ceases to register the exchanges and trajectories of things in order to address oscillatory ruptures in elastic liquids—or media. According to Siegert, this shift explains “the emerging fields of hydraulics, hydrodynamics, elasticity theory in the 18th century, and of wave optics in the 19th century” (198). The discourse on fluids also renders the phenomenon of electricity to be increasingly conceptionalized as a liquid or “current” that could form vortices or shock waves capable to excite both things and nerves. Surfing the wave of this discourse, Benjamin Franklin describes both electricity (“currents”) and money (“currency”) in terms of an “economy of circultations” (204) in closed systems, whereas unequal distributions (of charge or wealth) ultimately lead to processes of compensation or equalization (Ausgleichsprozesse). Franklin’s ideas on both electrical discarge and inflation as a means or medium to control trade, leads Siegert to conclude: “Die entscheidende Leistung von Medien (Elektrizität oder Papiergeld oder Sprache) ist nicht, irgendetwas zu repräsentieren, sondern Fernwirkungen herzustellen, Übertragungen zu bewirken.” (205)
Driven by the discourse on fluids in the 18th century, a dispositive emerges that, according to Siegert, unites “hydrodynamics, circulation, equilibirum and the theory of the oscillatory propagation of disturbances in elastic liquids” (206). In order to describe this new world of movements and propagation of waves—or in other words: to establish cosmography as a theory of fluids (207)—the point-shaped subject needed to become “a staggering, spiraling subject.” Mathematically, this is brought about by new ventures in formalization, such as the introduction of polar coordinates and new operators, such as trigonometric functions, the exponential function and imaginary numbers. Eventually these developments culminate in Eulers famous equation eiθ = cosθ + isinθ, the elements of which Siegert uses to identify what he refers to as the rupture (Riss):
Thereby oscillatory systems (Schwingungssysteme) may be expressed in terms of the exponential function, if frequencies are substituted for θ. But in this algebra, which operates with depictions in terms of trigonometric functions and imaginary powers of e, a rupture between representation and the represented—between the functions and their series expansion (Reihenentwicklung)—occurs. … From it [the rupture], however, no “formal” analysis emerges, which operates with symbols without reference. From it an analysis emerges, whose symbols reference media. (211)
Siegert elaborates this by showing how the deployment of analysis to solve the problem of the vibrating string (from the beginning of harmonic analysis enabled by Bernoulli and Euler to d’Alembert’s wave equation) resulted in Euler’s proposition of “arbitrary curves” or piecewise continuous functions. Discontinuities, however, infringe the common belief (up to Leibniz) that “nature does not make jumps” (natura non facit saltus). Euler’s insistence leads to new functions that count as valid solutions but don’t represent nature or physics anymore, and thus drawing the age of the Great Bureaucracy to a close (216-217).
Instead, Siegert sees the emergence of a theory of signal processing and of electrical media technologies: “Das Ende der Großen Bürokratie ist der Anfang der elektrischen Medien” (218). This logic becomes evident in Euler’s conception of a “digital impulse” or abrupt burst, crack or “spark, stroboscopic flash, which freezes the movement for an instant” (220)—events of which we would describe today using Paul Dirac’s delta function (219). By accounting for arbitrary functions, Euler allows contingent signals to become subject to analysis. “That which is the impossible for d’Alembert—‘that which does not cease to not write itself,’—becomes the contingent for Euler—“that which ceases to not write itself.” (221). This, according to Siegert, directly prompted experimental explorations in signal source production “beyond any representation” (223), such as mechanical “speaking machines:
The media-experiment emerges under the premise of the deconstruction of an analytical order of signs, in which everything that is, must be representable in Leibnizian expressions. Calculating ceases to be the model of an analytical art of inventing truth, and sets out, instead, to design signal generators. (223)
Although I must confess I am not entirely convinced that the shift in the analytical order of signs forms a necessary prerequisite for the epistemic practice of experimental proof, we may conclude with Siegert that mathematical analysis paves the way for signal synthesis—a path that leads from von Kempelen to Helmholtz’ vowel synthesizer as well as to electric transmission media, such as the telegraph and the telephone.