Shi, Yunjie Ellen
(2025)
Einstein Algebras and Relationalism Reconsidered.
[Preprint]
Abstract
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.
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