GABRIELE GRAMELSBERGER
On the occasion of an invitation to a lecture on Leibniz as a forerunner of today’s artificial intelligence at the Leibniz Library in Hannover, where most of his manuscripts are kept and edited, I had the opportunity to see some excerpts from his vast oeuvre. Prof. Michael Kempe, head of the research department of the Leibniz Edition, gave me some insights into the practice of editing Leibniz’s writings. Leibniz literally wrote on every piece of paper he could get his hands on. Hundreds of thousands of notes, because Leibniz wrote various notes on a large sheet of handmade paper and then cut it up himself to sort the individual notes thematically. A kind of early note box. However, he did not actually sort many of his notes and left behind a jumble of snippets.
How do you deal with the jumble of 100,000 snippets?
Nowadays, Artificial Intelligence (AI) technology is used to put together the “puzzle”, as Michael Kemper calls it. Supported by MusterFabrik Berlin, which specializes in such material cultural heritage puzzles, the snippets are reassembled again and again and reveal many surprises. For example, a snippet of Leibniz’s idea on “Motum non esse absolutum quiddam, sed relativum …” (Fig. 2 front/back side) showed a fragment of a geometric drawing. However, the snippet 22 preserved in box LH35, 12, 2 was not completed by any other snippet in this box. The notes were sorted by hand in the late 19th century by the historian Paul Ritter (Ritter catalog) as a basis for a later edition. Ritter’s catalog was a first attempt to bring some order to the scattered notes. Now, more than a hundred years later, AI technology is bringing new connections and affiliations to light. Snippet 43, shown in Figure 3 (front/back side), completed this part of the puzzle. It was located in box LH35, 10, 7 and had never before been connected to snippet 22.
Trains of thought made visible
“What these recombined snippets tell us,” says Michael Kempe, “is how Leibniz’s thinking worked. He used writing to organize and clarify his thoughts. He wrote all the time, from morning, just after waking up, until late at night. And he often used drawings to illustrate, but also to test his ideas. He changed the sketches and thus further developed his train of thought.” Combined snippets 22/43 are such an example. While writing about the relativity of motion, Leibniz made some geometric sketches of the motion of the planets and added some calculations (fig. 4b).
Leibniz’ contributions to AI
An interesting side aspect is that the AI technology, used for solving the Leibniz puzzle, is based on a modern version of Leonhard Euler’s polyhedron equation, which was inspired by Leibniz’s De Analysi situs. De Analysi situs, in turn, was the topic of my talk the day before on the influence of Leibniz’s ideas on AI technology. So, it all fitted together very well. However, Leibniz’ contributions to AI were manifold. Already his contribution to computation were outstanding—he had developed a dyadic calculation system, an arithmetic mechanism (Leibniz wheel), which was in use until the beginning of the 20th century, and directed the construction of a four species arithmetic machine. However, his contribution to a calculus of logic was even more significant, because he had to overcome sensory intuition and to develop an abstract intuition solely based on symbolic data. De Analysi situs was precisely about this abstract stance, which came into use only in 19th century’s symbolic logic. Furthermore, De Analysi situs is considered a precursor of topology, which inspired Euler’s polyhedron equation, which expresses topological forms with graphs. Graphs, in turn, play a crucial role in AI for network analysis of all types of data points and relationships. This closes the circle from Leibniz to AI.
De Analysi situs (1693)
How did Leibniz overcome sensory intuition and develop an abstract intuition solely based on data points and relationships? The text begins with the following sentences: “The commonly known mathematical analysis is one of quantities, not of position, and is thus directly and immediately related to arithmetic, but can only be applied to geometry in a roundabout way. Hence it is that from the consideration of position much results with ease which can be shown by algebraic calculation only in a laborious manner” (Leibniz, 1693, p. 69). Leibniz criticized the limited arithmetic operativity of algebraic analysis (addition, subtraction, multiplication, division, square root) and called for the expansion of operations through the analytical method for geometry and geometrical positions.
This expansion was the following: “The figure generally contains, in addition to quantity, a certain quality or form, and just as that which has the same quantity is equal, so that which has the same form is similar. The theory of similarity or of forms extends further than mathematics and is derived from metaphysics, although it is also used in mathematics in many ways and is even useful in algebraic calculus. Above all, however, similarity comes into consideration in the relations of position or the figures of geometry. A truly geometrical analysis must therefore apply not only equality and proportion […] but also similarity and congruence, which arise from the combination of equality and similarity” (p. 71).
Leibniz criticized that it was the fault of the philosophers, who were content with vague definitions. And now comes the decisive step: he proposed an exact definition for the concept of similarity. He writes: “I have now, by an explanation of the quality or form which I have established, arrived at the determination that similar is that which cannot be distinguished from one another when observed by itself” (pp. 71-72). Thus, he replaced similarity by indistinguishability and argued that indistinguishability only requires the comparison of data “salva veritate.” He thus established a concept of indistinguishability which can “be derived from the symbols by means of a secure computations and proof procedure” (p. 76), which is the basis of all data operations to this day.
With this algorithm, Leibniz hoped, that “all the questions for which the faculty of perception is no longer sufficient can be pursued further, so that the calculus of position described here represents the complement of sensory perception and, as it were, its completion. Furthermore, in addition to geometry, it will also permit hitherto unknown applications in the invention of machines and in the description of the mechanisms of nature” (p. 76). It is an algorithm that is intended to help recognize similarities purely on the basis of data. Today, we call this clustering and it is the central strategy of unsupervised learning, i.e. a method for discovering similarity structures in large data sets.
References and further readings
De Risi, Vincenzo: The Analysis Situs 1712-1716, Geometry and Philosophy of Space in the Late Leibniz, Basel: Birkhäuser 2006.
Gramelsberger, Gabriele: Operative Epistemologie. (Re-)Organisation von Anschauung und Erfahrung durch die Formkraft der Mathematik, Hamburg: Meiner 2020. Open access URL: https://meiner.de/operative-epistemologie-15229.html
Gramelsberger, Gabriele: Philosophie des Digitalen zur Einführung, Hamburg: Junius 2023.
Kempe, Michael: Die beste aller möglichen Welten: Gottfried Wilhelm Leibniz in seiner Zeit, S. Fischer: München 2022.
Leibniz, Gottfried W.: De analysi situs (1693), in: Philosophische Werke (ed. by Artur Buchenau and Ernst Cassirer), vol. 1, Meiner: Hamburg 1996, pp. 69–76. (All quotes translated by DeepL).
Ziegler, Günter M., Blatter, Christian: Euler’s polyhedron formula — a starting point of today’s polytope theory, Write-up of a lecture given by GMZ at the International Euler Symposium in Basel, May 31/June 1, 2007. URL: https://www.mi.fu-berlin.de/math/groups/discgeom/ziegler/Preprintfiles/108PREPRINT.pdf
