An introduction to the basics of graph theory and its applications aimed at philosophers. We introduce graph-theoretic concepts and definitions working only from a familiarity with predicate logic and set theory in enough depth to understand the main results and open questions in contemporary graph theory – with sections devoted to explaining a few celebrated theorems (e.g. the four colour theorem).
In addition to drawing out various connections to logic and philosophy, the book will devote space to some philosophical “applications” — uses of graph-theoretic tools for modelling in a mathematical approach to philosophy (e.g. reference graphs, grounding structures, Kripke frames, etc.). For just two examples of applications see:
“Dangerous reference graphs and semantic paradoxes” (2013, Journal of Philosophical Logic, with L. Rabern and M. Macauley)
We used resources from graph theory in an attempt to determine which relations of reference afford the structure necessary for paradoxicality. We developed the relevant mathematical and philosophical groundwork required to tackle this question, and we made some significant progress towards an answer but a full characterisation of paradox supporting “reference structures” remains an open question
“Well-founding grounding grounding” (2016, Journal of Philosophical Logic, with G. Rabin)
We used resources from graph theory to develops and utilize a formal framework based on the notion of a “grounding structure” in order to settle some questions about the logic of ground and the operative notion of “well-foundedness” for the metaphysical notion of grounding.
"In addition to that branch of geometry which is concerned with magnitudes and which has always received the greatest attention, there is another branch, previously almost unknown, which Leibniz first mentioned, calling it the geometry of position [geometria situs]. This branch is concerned only with the determination of position and its properties; it does not involve measurements, nor calculations made with them [...] Hence, when a problem was recently mentioned, which seemed geometrical but was so constructed that it did not require the measurement of distances, nor did calculation help at all, I had no doubt that it was concerned with the geometry of position.” - L. Euler (1736) ‘Solutio problematis ad geometriam situs pertinentis’, Commentarii Academiae Scientiarum Imperialis Petropolitanae, 8: 28-140.
The problem Euler is referring to is, of course, the Bridges of Königsberg problem. The allusion to the “the geometry of position” concerns Leibniz’s attempt to build a new foundation for, and devise a new formalism for, geometry. In a series of manuscripts, papers, and notes (unpublished in his lifetime) Leibniz tried to reduce geometry to a system of basic relations between points — the latter is connected to his theory of relational space. Leibniz’s analysis situs (or sometimes called geometria situs) affords a way of expressing various “situations” of points relative to one another, and promised a calculus so that all geometrical theorems could be deduced from propositions concerning the points and relations (see especially volumes 5 and 7 of Gerhardt’s edition of the Mathematische Schriften). But although Euler was clearly inspired by Leibniz's analysis situs, it is a misattribution to credit Leibniz with the key graph-theoretical ideas that Euler goes on to develop in the paper. (See discussion in Benson Mates (1986) The Philosophy of Leibniz: Metaphysics and Language, Oxford University Press, p. 235-240.)
This project was a joint project with my brother Landon Rabern but I won't finish it without him. Someone else should...