Gerloff: Relations between the analog and the digital (Session 5)

As we approach the rupture in the history of Analysis we work our way through Siegert’s rendition of Leibniz’s version of and contribution to the Analysis epistème. In several aspects it becomes clear why Leibniz is a step towards the rupture, laying the ground for what will nevertheless be a distinct step into the digital.

His invention of a procedure of the infinitesimal calculus, his idea of a ‚universale Charakteristik’, the tables of the state, and the parable ‚Palais des destinées’ all actualize forms of representation. If we view Leibniz’s imaginations of universal knowledge systems in a series with Ovando’s and Hooke’s we see a development towards an increased operativity of signs. Where Ovando’s operations had to ascertain the true place and numbers of things in the world and Hooke and Wilkins focused on experimentation and a universal code for the acquired data Leibniz aims at calculating the truth within this code without having to think about the signified things:

“Im Unterschied zur Algebra des Royal Society-Sekretärs wird Analysis – die nun auch unter diesem Namen auftritt – Voraussetzung für eine mit Zeichen operierende Erfindungskunst, die tatsächlich rechnet, anstatt nur zu zählen und zu klassifizieren. Wobei Rechnen seit Vieta und Descartes heißt, mit Charakteren auf Papier zu hantieren.” (p. 167)


“Mittel zum Erfinden zu erfinden, war die Aufgabe, die Hooke der Wilkinsschen Universalsprache zugedacht hatte und unter deren Perspektive er sie als ‘Algebra’ im Sinne Vietas bezeichnet hatte. Leibniz’ Projekt hingegen geht weiter: Es strebt an, Wahrheitsbeweise auf Richtigkeitsbeweise zu reduzieren. Die logischen Operationen der Begriffsanalyse werden damit arithmetische bzw. algebraische.” (p. 168)

This is the move from a correspondence theory to a coherence theory of truth. What is interesting to me then is the connection of Leibniz’s characters and the world. If the primary relation is true and undisputed a proof can veritably depend on the validity of sign-operations in a coherent system. Representation, differentiation, classification, order – these principles of Analysis all rely on distinct entities, elements, that can be coded, related and accounted. This leads to some crucial questions: How are these elements even identified? Is it self-evident where they have their margins? How are they linked to the characters?

The epistemic operations to gather the facts don’t seem to be relevant in the context of this chapter. The connection between characters and things is addressed, though. Surprisingly, there shouldn’t be any figurative or mimetic relation between the two:

“Die mimetischen Zeichen der Malerei, Plastik oder Wissenschaft sind jedoch nicht nur deswegen ungeeignet, weil sie zu sperrig sind (die Staatstafeln müssen transportabel sein) oder weil sie bloß mimetisch sind, sondern weil die Dinge, von denen die Staatstafeln den Landesherrn benachrichtigen, von einem Typ sind, die eine Repräsentation erfordern, die eher die Verbindung der Dinge mimetisch vor Augen stellt als die Dinge selber.” (p. 172)

So the representational function of the ‘universale Charakeristik’ tries to capture the relations between the things of the world rather than the things themselves. These relations apparently are represented by the operators of the calculus. On a side note, this reminds me of Foucaults explanation that it isn’t important to him whether the elements of a dispositive are discursive or non-discursive, but which relation exists between them. Is this another hint to support my intuition that basic principles of analysis – explain the world by isolating and ordering elements – is still in play especially in styles of thought in the humanities?

When it comes to the question whether Leibniz’s world is continuous or made up of particular, distinct things and events I think the latter is the case. At a first glance the metaphor of the ocean that is not to be resolved by human perception gives the impression that the world is continuous. But this is just the mode of human perception: With a higher resolution of our senses we could discern all the particular noises of the singular waves. Continuous representations and imaginations are a function of his own infinitesimal calculus. On the other hand, as Siegert states, the principle of continuity is a condition of this function. Only if there are relations of similarity between the instances there can be a finite expression or term in a concrete, melded perception or thought. So maybe the continuous and analog is to be found on a deeper level:

“Die analoge Schrift, die die Kette der Ordnungen in den göttlichen Staats-Tabellen (oder im Palais des destinées) repräsentiert, ist infolgedessen ein Funktionsgraph. ‘Ich meine also gute Gründe zu der Annahme zu haben, daß alle die verschiedenen Klassen von Wesen, deren Gesamtheit das Universum ausmacht, in den Ideen Gottes, der ihre wesentlichen Abstufungen distinkt erkennt, nur ebenso viele Ordinaten ein und derselben Kurve sind.’(129)” (p. 189)

The world becomes a graph in this version, a function of God’s ideas. What better example of the thought-informing qualities of material media structures could you find? In my opinion, adding the handling of the continuous to the modus operandi of Analysis seems to be Leibniz’s groundbreaking contribution here. The way to handle the big data of the world and induce its truth then could be, again, the infinitesimal calculus. This makes me think of another grandiose science fiction of holistic knowledge transgressing space and time: Isaac Asimov’s notion of ‘psychohistory’ – the idea, that history itself can be calculable if we just zoom out in time enough to handle the unpredictability of the human psyche with mass data and probabilities.


3 thoughts on “Gerloff: Relations between the analog and the digital (Session 5)”

  1. “When it comes to the question whether Leibniz’s world is continuous or made up of particular, distinct things and events I think the latter is the case.”

    This is an intriguing point. I wonder whether we might state it differently: that Leibnizian analysis is interested in rendering the continuous discrete? It seems like the premise of Leibniz’s approach to calculus is precisely about blurring the space between these two categories, so that a wave is actually calculable as having many distinct parts, and those distinct parts can be summed as infinitesimally close to the wave. At least in the world of sound, these are the great examples of analog vs digital: analog is a continuous wave, digital is the bite-size sample. But I sense that Siegert is not merely looking for a pre-history of the digital, but rather setting us up to see the digital and analog as part of the same Zeichenpraktik. Is that a possible way forward or just a cop-out? Is it visible already in Leibniz?

    Also, I’m not sure I understand the noise-of-waves analogy that Leibniz proposes. (I’ll partially take this up soon with the second part of my post on speech/noise.) But just for now, what do you think he means by an individual wave in the ocean? Is that one point of a wave? Is it the entire length of a wave–say, running the whole length of a beach and cresting at different times? (When is an ocean wave sounding–is it only when it breaks? And regardless, doesn’t it break–and thus make sound–in an ongoing way, as the break extends transversely along the wave?) I’m embarrassed to say my knowledge of wave noise here is more based on sitting at the beach than actually knowing how waves (are supposed to) function!


    1. In light of the 6th chapter it seems to be clear that the shift of the epistème towards a world of fluids, processes, exchange and equilibrium is of the greatest significance here. Especially the question of a fitting scheme of representation – or whether this just folds into operativity – emerges in this context. How can a world that is conceived of not as distinct, fixed object but as a sea of fluids, currents, and energies be turned into signs? I guess this leads directly into Eulers functions and graphs. I think this fits to your point of “blurring the space between these two categories, so that a wave is actually calculable as having many distinct parts, and those distinct parts can be summed as infinitesimally close to the wave”. I would also be interested in other ways to represent and mobilize such a model…


  2. I think these issues touch at the heart of what the book is about. With Leibniz, the diffuse border between the continuous and the discrete, which will grow into the “Riss” of the second half of the book starts to emerge.

    The curious contradiction with Leibniz, as I understand it, is the seeming contradiction between his ‘Law of Continuity’ presupposing a world that “makes no jumps,” and the logic of the ‘Monads,’ which are discreet, completely self contained elements. In Leibniz’s view, however, everything from the (infinitesimally) smallest to the largest is continuously and proportionally connected, hence the importance of the ‘Grenzwert’ of threshold. The difficulty of combining these two positions (which is Leibniz’s difficulty and the difficulty of all analysis afterwards) becomes apparent with his ‘solution’ through the idea of ‘small perceptions,’ as well as in the infinitesimal calculus. This is where the noise of the sea comes in.

    Whether or not there is such a thing as a ‘sound’ of a ‘single’ wave is not really relevant, what his example tries to show is that our senses only perceive one single “roar” of the sea, whereas this roar must be, according to Leibniz’s logic, made up of many single sounds (as “a hundred nothings can’t make something”). Each of these small sounds, or small perceptions is a single self-contained ‘thing’ and there are an infinite, but in principle countable or knowable, amount of these things. Only god, however, is able to oversee the entirety of these perceptions. We can only perceive one, inextricable mess. Similarly, only God can choose at each given moment the best possible world from an infinite, but nonetheless countable, number of possible worlds.

    In my view, it is exactly this problem of infinity, which already became apparent with the bureaucratic and cartographic issues that we discussed earlier and which was so nicely addressed in John Durham Peter’s essay we posted last session, that rears its head and that causes the very fundamental problems of representation that enter into the modern analysis from Leibniz onward. As the next chapter, and the second part of the book, begins: “Until Leibniz, there was the silence of the library in the rooms of the Mathesis universalis that was managed by analysis. From Leibniz onward this silence is ruled by an unconscious noise.”


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