(Preprint) Einstein Algebras and Relationalism Reconsidered. [PhilSci-Archive]
This paper reconsiders the metaphysical implication of Einstein algebras, prompted by the recent objections of Chen (2024) on Rosenstock et al. (2015)’s conclusion. Rosenstock et al.’s duality theorem of smooth manifolds and smooth algebras supports a conventional wisdom which states that the Einstein algebra formalism is not more “relationalist” than the standard manifold formalism. Nevertheless, as Chen points out, smooth algebras are different from the relevant algebraic structure of an Einstein algebra. It is therefore questionable if Rosenstock et al.’s duality theorem can support the conventional wisdom. After a re-visit of John Earman’s classic works on the program of Leibniz algebras, I formalize the program in category theory and propose a new formal criterion to determine whether an algebraic formalism is more “relationalist” than the standard manifold formalism or not. Based on the new formal criterion, I show that the conventional wisdom is still true, though supported by a new technical result. I also show that Rosenstock et al. (2015)’s insight can be re-casted as a corollary of the new result. Finally, I provide a justification of the new formal criterion with a discussion of Sikorski algebras and differential spaces. The paper therefore provides a new perspective for formally investigating the metaphysical implication of an algebraic formalism for the theory of space and time.
(2022) The Quasi-Empirical Epistemology of Mathematics, KRITERION – Journal of Philosophy, Special Issue on Lakatos’ Undone Work. [Published version]
This paper clarifies and discusses Imre Lakatos’ claim that mathematics is quasi-empirical in one of his less-discussed papers A Renaissance of Empiricism in the Recent Philosophy of Mathematics. I argue that (1) Lakatos’ motivation for classifying mathematics as a quasi-empirical theory is epistemological; (2) what can be called the quasi-empirical epistemology of mathematics is not correct; (3) analysing where the quasi-empirical epistemology of mathematics goes wrong will bring to light reasons to endorse a pluralist view of mathematics.
(Please email me if you would like to talk about any of these projects or request a draft!)
The Method of Arbitrary Functions, Dynamics, and Coarse-Graining.
Tentative abstract: can there be an objective chance of an event describing a deterministic system? Many of the philosophical discussions on this subject concern the method of arbitrary functions (MAF), a type of mathematical reasoning that was famously employed by Henri Poincare to derive the probabilities of the outcomes of spinning a roulette wheel. In this paper, I provide an analysis of the MAF and the related objective interpretations of probability based on the MAF. I argue that the MAF is a justification for reprensenting the behavior of the roulette wheel system with continuous probability density functions, rather than an evidence for the metaphysical interpretation of probability.
Correlated Equilibria and Rule Utilitarianism.
"Algebraic formalisms and Relationalism"
The Annual Philosophy of Logic, Math, and Physics (LMP) Graduate Student Conference, Western University, May 2025.
"The Method of Arbitrary Functions, Dynamics, and Coarse-Graining"
29th Biennial Meeting of the Philosophy of Science Association Poster Session, Nov 2024.
"Is Mathematics Quasi-Empirical?"
Cambridge Graduate Conference on Philosophy of Mathematics and Logic, Jan 2021.
Lakatos' Undone Work Masterclass & Virtual Conference, Aug 2020.