Mereological summation and the question of unique fusion

Forrest, Peter

doi:10.1111/j.1467-8284.2007.00680.x

Cited by 2 publications

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“…The statue and its lump of clay are spatially coextensive and geometrically indiscernible, while a nilpotent region and a point are not spatially or geometrically identical. The idea that the mereology of space could violate Supplementation has rarely been discussed (with the exception of Forrest 2003Forrest , 2007). 11 While the loss of supplementation is counterintuitive, I believe that we should not reject the theory outright on this ground.…”

Section: Nonclassical Mereologymentioning

confidence: 99%

“…Not all quotient rings of C ∞ (M ) are isomorphic to the rings (or algebras) of all smooth functions on some manifolds. 18 In particular, not all quotient rings of C ∞ (R) are isomorphic to the rings of all smooth functions on some subspaces of R. 19 For example, there are no (non-zero-dimensional) subspaces on which all smooth functions are affine. So the ring of affine functions L does not correspond to any manifold.…”

Section: Manifold-algebra Dualitymentioning

confidence: 99%

“…(Insofar as spacetime algebraicism aims at dealing away with manifolds and substantive spacetime, the restriction cannot 18 For our purposes, the difference between rings and algebras is not important. 19 As will become apparent soon, this is not entirely due to that n-manifolds are standardly defined as being locally like R n and therefore a closed interval is not a manifold (since its boundary points do not have line-like neighborhoods), which entails that rings like C ∞ ([0, 1]) are not smooth algebras. The discussion in the main text won't be affected even if we use "manifold" in the broad sense that includes those with boundaries.…”

Section: Manifold-algebra Dualitymentioning

confidence: 99%

“…Also suppose that the fusion of all nilpotent regions, each of which includes a point, is the whole line. Then, it follows that the fusion of all points is also the whole line, even though intuitively they do not cover any nilpotent region Forrest (2007). discusses alternative definitions of "fusion" when Supplementation is violated.…”

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confidence: 99%

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Smooth Infinitesimals in the Metaphysical Foundation of Spacetime Theories

Chen

2022

J Philos Logic

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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.

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Section: Nonclassical Mereologymentioning

confidence: 99%

Section: Manifold-algebra Dualitymentioning

confidence: 99%

Section: Manifold-algebra Dualitymentioning

confidence: 99%

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confidence: 99%

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Smooth Infinitesimals in the Metaphysical Foundation of Spacetime Theories

Chen

2022

J Philos Logic

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“…This argument uses a similar idea to the Cantor-Forrest-Arntzenius measure argument discussed inRussell (2008). (See alsoForrest 1996;Forrest 2007;Arntzenius 2008. ) In fact, though I won't take this up here, I believe that the indefinite extensibility approach also offers a promising alternative response to the measure argument.11 Here's a sketch of how Global Tukey's Lemma can be derived from Global Choice.…”

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confidence: 99%

Indefinite Divisibility

Russell

2015

Inquiry

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Some hold that the lesson of Russell's paradox and its relatives is that mathematical reality does not form a "definite totality" but rather is "indefinitely extensible". There can always be more sets than there ever are. I argue that certain contact puzzles are analogous to Russell's paradox this way: they similarly motivate a vision of physical reality as iteratively generated. In this picture, the divisions of the continuum into smaller parts are "potential" rather than "actual". Besides the intrinsic interest of this metaphysical picture, it has important consequences for the debate over absolute generality. It is often thought that "indefinite extensibility" arguments at best make trouble for mathematical platonists; but the contact arguments show that nominalists face the same kind of difficulty, if they recognize even the metaphysical possibility of the picture I sketch. … endure not yetA breach, but an expansion.

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Mereological summation and the question of unique fusion

Cited by 2 publications

References 1 publication

Smooth Infinitesimals in the Metaphysical Foundation of Spacetime Theories

Smooth Infinitesimals in the Metaphysical Foundation of Spacetime Theories

Indefinite Divisibility

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