![]() Although this view is a coherent approach to Zeno’s paradox and is preferable to Skyrms’s (Physics, philosophy and psychoanalysis. In particular, the size of a finite region is the sum of the sizes of its infinitesimal parts. Notably, this view satisfies a version of additivity: for every region that has a size, its size is the sum of the sizes of its disjoint parts. North-Holland, Amsterdam, 1966) nonstandard analysis. According to this view, space is composed of ultimate parts with infinitesimal size, where infinitesimals are understood within the framework of Robinson’s (Non-standard analysis. In this paper, I develop an original view of the structure of space-called infinitesimal atomism-as a reply to Zeno’s paradox of measure. But SIG has an unorthodox mereology, in which the principle of supplementation fails. Against this, I argue that SIG is (part of) such a reformulation. SIA is formulated in intuitionistic logic and is thought to have no classical reformulations (Hellman Journal of Philosophical Logic, 35, 621–651, 2006). (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). (2) It generalizes a standard implementation of spacetime algebraicism (according to which physical fields exist fundamentally without an underlying manifold) called Einstein algebras. A tangent space can be considered an infinitesimal region of space. (1) It provides a simple metaphysics of vector fields and tangent space that are otherwise perplexing. ![]() I argue that SIG has the following utilities. 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). The philosophical concept of thick presentism and the introduction of two dimensional time-physical time and meta-time that are mutually independent on infinitesimal scales-are the the pivot points in these constructions. ![]() result from the principle of microlevel reducibility. I substantiate this statement and demonstrate directly how the formalism of differential equations, the notion of forces in Newtonian mechanics, the concept of phase space and initial conditions, the principle of least actions, etc. This principle assumes, first, that all the properties of physical systems must be determined by their states at the current moment of time, in a slogan form it is "only the present matters to physics." Second, it postulates that any physical system is nothing but an ensemble of structureless particles arranged in some whose interaction obeys the superposition principle. In the present paper I argue that the formalism of Newtonian mechanics stems directly from the general principle to be called the principle of microlevel reducibility which physical systems obey in the realm of classical physics. I argue that Giordano’s nilpotents supply the best answer to Zeno’s paradox. Lawvere’s Smooth Infinitesimal Analysis and the other is inspired by Paolo Giordano’s ring of Fermat Reals. One of these solutions is inspired by F.W. After arguing that any solution to the paradox must satisfy certain theoretical requirements, I examine White’s solution alongside two nilpotent solutions. in relation to the hyperreal infinitesimals of nonstandard analysis. ![]() In this paper, I follow the basic outline of White’s solution but argue that his solution suffers from arbitrariness and a related theoretical artificiality in relation to the system of infinitesimals he invokes, viz. Contra Zeno, this allows the arrow to be moving in the present, rather than frozen in place. In “Zeno’s Arrow, Divisible Infinitesimals, and Chrysippus,” White suggests using an infinitesimal value as the length of the present. Therefore, the arrow is both moving and at rest. However, if the present is a single point in time, then the arrow is frozen in place during that time. Zeno’s Arrow goes like this: during the present, a flying arrow is moving in virtue of its being in flight. Nilpotents are nonzero numbers that yield zero when multiplied by themselves a certain number of times. I offer a novel solution to Zeno’s paradox of The Arrow by introducing nilpotent infinitesimal lengths of time.
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