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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 Cahiers de Topologie et Géométrie Différentielle Catégoriques, 21(4), 277–392, 1980). SIA is formulated in intuitionistic logic and is thought to have no classical reformulations (Hellman Journal of Philosophical Logic, 35, 621–651, 2006). Against this, I argue that SIG is (part of) such a reformulation. But SIG has an unorthodox mereology, in which the principle of supplementation fails.

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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.

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Correspondence to Lu Chen.

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Chen, L. Smooth Infinitesimals in the Metaphysical Foundation of Spacetime Theories. J Philos Logic (2022). https://doi.org/10.1007/s10992-022-09653-9

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  • DOI: https://doi.org/10.1007/s10992-022-09653-9

Keywords

  • Continuum
  • Smooth infinitesimal geometry
  • Smooth infinitesimal analysis
  • Vectorial quantity
  • Tangent space
  • Einstein algebras
  • Nonclassical mereology