Research
Research
In mathematics, I am interested in two distinct areas: inverse problems and representation theory. I am motivated primarily by the desire for unified mathematical models of abstract formal systems that admit application to diverse areas of mathematics in addition to physics, computer science, philosophy, and beyond. Accordingly, I maintain a significant interest in univalent foundations of mathematics and homotopy type theory.
"On several inverse problems in graph wave networks" (in progress)
In this paper I formulate seven inverse problems in graph wave networks, a spatially discretized version of the wave equation that can be likened to a multi-degree-of-freedom spring-mass system. Given measurements of an output signal at a subset of nodes (the boundary nodes), can any combination of sources, speeds, and graph geometry be recovered given knowledge of the others?
I have exhibited sufficient conditions, an inversion operator, and stability estimate for reconstruction of sources given speeds and graph geometry.
I currently have three sufficient conditions for uniqueness and an inversion algorithm for the recovery of speeds given sources and graph geometry, and am working on a stability estimate.
In philosophy, I think mostly in phenomenological and epistemological terms. I am deeply concerned with the problem of subjectivity and in particular of allowing an (optimistic) epistemological foundationalism for a univocal theory of mind. I have a strong interest in the history of philosophy—in particular the 19th century—with a focus on Fichte, Schelling, Hegel, and Kierkegaard. I am also interested in 20th century continental philosophy, most notably Bataille.
"On the mathematical infinite in Hegel's Science of Logic and the foundations of analysis" (in progress)
In this essay I carefully examine Bertrand Russel's claim that concepts like 'continuity' and 'the infinite' were placed on firm foundations by mathematicians in the late 19th and early 20th century and thereby disentangled from the "metaphysical nonsense" of philosophers like Hegel. I claim on the contrary that Hegel's critique of the differential calculus in the Science of Logic anticipates and preemptively responds to these developments, and that he is rather vindicated by the contributions of Cantor and Lebesgue.
Formal Systems (in progress)
I am currently in the beginning stages of a book-length project about formal systems (considered as abstract philosophical objects).
"On the Faculty of Intuition" (2020). UVM Patrick Leahy Honors College Senior Theses. 377.
In this work I present an account of the faculty of intellectual intuition as analogous to the faculties of empirical sensing (i.e. sight, touch, hearing, proprioception, &c.). I develop an abstract schema into which all of these can be understood as 'presentational' mental activity. I then offer some external criteria that help distinguish between presentational mental activity which can justify belief (like intuition or seeing) and presentational mental activity that cannot justify belief (like dreaming or hallucinating). In the second chapter, I leverage the foregoing to defend epistemological foundationalism. Finally, I leverage my account of intuition specifically to defend against philosophical skepticism.