Smooth Infinitesimals in the Metaphysical Foundation of Spacetime Theories

Abstract

I propose a theory of space with infinitesimal regions called smooth infinitesimal geometry (SIG) based on certain algebraic objects (i.e., rings), which regiments a mode of reasoning heuristically used by geometricists and physicists (e.g., circle is composed of infinitely many straight lines). I argue that SIG has the following utilities. (1) It provides a simple metaphysics of vector fields and tangent space that are otherwise perplexing. A tangent space can be considered an infinitesimal region of space. (2) It generalizes a standard implementation of spacetime algebraicism (according to which physical fields exist fundamentally without an underlying manifold) called Einstein algebras. (3) It solves the long-standing problem of interpreting smooth infinitesimal analysis (SIA) realistically, an alternative foundation of spacetime theories to real analysis (Lawvere 1980). SIA is formulated in intuitionistic logic and is thought to have no classical reformulations (Hellman 2006). Against this, I argue that SIG is (part of) such a reformulation. But SIG has an unorthodox mereology, * Acknowledgments. I want to give special thanks to Jeffrey Russell for the essential discussions and guidance early on and to Tobias Fritz for his very generous technical help, without which the current paper would be impossible. I thank Phillip Bricker and Cian Dorr for their helpful feedback on the early drafts of the paper. Many thanks to the anonymous referee of Journal of Philosophical Logic for their very helpful comments, which have substantially improved the exposition of the paper, among other things. Finally, I would like to thank all the participants at my talks based on various versions of the paper for their stimulating questions and comments, whose names are too numerous to list. 1 in which the principle of supplementation fails.

Stability Problems on D-finite Functions

“…Hellman ([2006]) argued that we cannot interpret these features classically. physically (Chen [2022]; see also Weatherson [2006]). In the standard framework, it is puzzling whether a vector (such as an electric field value) at a spatial point is intrinsic to a point or not.…”

“…Now, as Rynasiewicz pointed out (which is a basic fact in algebraic geometry), such maximal ideals can be conceived as spacetime points because there is a one-to-one correspondence between points of M and maximal ideals of C ∞ (M) that preserves the topology (the collection of all maximal ideals each of which contains an element of C ∞ (M) forms a basis of closed sets for the topology defined of the set of all maximal ideals of C ∞ (M); see Rynasiewicz [1992], pp.583-4 for more details). 6 This shall work as an illustration for how we can construct spacetime structures in manifoldism on the basis of an Einstein algebra, and I will stop short of presenting more technical details.…”

“…To illustrate why algebraicism is not equivalent to manifoldism, I will appeal to three different examples of algebraicism in the literature. The first, which is based on synthetic differential geometry, shows that in the algebraic framework, we have a natural way to define an algebra that can be heuristically understood as an 'infinitesimal region', which is not intrinsically definable in manifoldism or more generally, a geometric-theoretic framework (see also Lawvere [1980], Kock [1981], Chen [2022]). Such an 'infinitesimal region' can facilitate a natural interpretation of vectorial quantities.…”

“…See Footnote 1 5. For simplicity, I will henceforth ignore the algebraic formulation of the metric field, which is not crucial for my arguments 6. A technical caveat: without qualifications, the notion of 'maximal ideal' is more general than that of a point.…”