58 research outputs found

    Smooth Infinitesimals in the Metaphysical Foundation of Spacetime Theories

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    I propose a theory of space with infinitesimal regions called \textit{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 \textit{Einstein algebras}. (3) It solves the long-standing problem of interpreting \textit{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, in which the principle of supplementation fails

    Rethinking inconsistent mathematics

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    This dissertation has two main goals. The first is to provide a practice-based analysis of the field of inconsistent mathematics: what motivates it? what role does logic have in it? what distinguishes it from classical mathematics? is it alternative or revolutionary? The second goal is to introduce and defend a new conception of inconsistent mathematics - queer incomaths - as a particularly effective answer to feminist critiques of classical logic and mathematics. This sets the stage for a genuine revolution in mathematics, insofar as it suggests the need for a shift in mainstream attitudes about the rolee of logic and ethics in the practice of mathematics

    Metaphysics of Ersatzism about Possible Worlds

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    According to actualism about possible worlds everything that exists is actual. Possible worlds and individuals are actually existing abstract parts of the actual world. Aristotelian actualism is a view that there are only actual individuals but no possible ones, nor their individual abstract representatives. Because of that, our actualist account of modality should differ depending on whether it concerns actual individuals or possible ones. The main goal of the dissertation is to develop a metaphysical framework for Aristotelian actualism. Chapter 1 explains basic issues associated with the possible world approach to modality. I overview modal realist and actualist views on possible worlds and explain why I support the actualist approach. Subsequently, I introduce a distinction between Platonic and Aristotelian actualism, and discuss some semantic issues associated with actualism as such. In Chapter 2 I argue that Aristotelian actualism, modeled on linguistic ersatzism, is preferable over its Platonic counterpart. Subsequently, I propose a metaphysical framework for Aristotelian ersatzism which is based on a claim that our modal concepts work differently for actual and possible individuals. In order to explain that claim I introduce three specific differences concerning modal features of actual and possible individuals: (a) Representational Difference, according to which actual and possible individuals are represented differently by possible worlds; (b) Metaphysical Difference, according to which actual and possible individuals are represented by possible worlds as having different metaphysical nature; (c) Modal Difference, which says while there are singular and contingent possibilities involving actual individuals, all possibilities about possible individuals are general and necessary. I propose to interpret those differences in terms of the doctrines of haecceitism, antihaecceitism and existentialism. There is however no consensus on how those views should be characterized. Chapters 3, 4, and 5 focus on providing a precise characterization of those doctrines. Chapter 3 focuses on the doctrines of modal haecceitism and antihaecceitism, which I view as opposite accounts of how possible worlds represent possibilities. According to modal haecceitism what possible worlds say about particular individuals does not supervene on what they say qualitatively. Modal antihaecceitism is a denial of such a claim. Chapter 4 concerns metaphysical haecceitism and antihaecceitism, which I take to be alternative accounts of the fundamental structure of reality. For the metaphysical haecceitist reality contains irreducible singular facts, while for the metaphysical antihaecceitist reality is purely qualitative and general. Chapter 5 focuses on an argument between existentialists and antiexistentialists. Existentialists claim that there are contingent singular propositions, while antiexistentialists deny that. I defend existentialism against antiexistentialist counterarguments, as well as criticize some of the antiexistentialist accounts of singular propositions modeled on the notion of individual essence. In Chapter 6, by appealing to the results of investigations conducted in Chapters 3, 4 and 5, I reconsider Representational, Metaphysical and Modal Differences. According to a view that I propose: (a) Representational Difference entails (extreme) modal haecceitism for actual individuals but (extreme) modal antihaecceitism for possible individuals; (b) Metaphysical Difference entails metaphysical haecceitism (individualism) for actual individuals, but metaphysical antihaecceitism (generalism) for possible individuals; finally (c) Modal Difference entails existentialism: while there are singular and contingent possibilities involving actual individuals, all possibilities about possible individuals are general and necessary. In Chapter 6, I also explain the implications of those views for the various issues, including transworld identity, essentialism, or the modal status of modal space. Lastly, Chapter 7 overviews some semantic and metaphysical applications of Aristotelian ersatzism. I explain how it manages to accommodate Kripkean semantics and how it is able to account for the possibilities of indiscernibles, alien individuals and iterated modalities. I also address some possible objections to my proposal, including an issue of implicit representation and the Humphrey objection

    Beyond the Wigner’s friend dilemma: A new indeterminacy-based quantum theory

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    I propose a novel (interpretation of) quantum theory, which I will call Environmental Determinacy-based or EnD Quantum Theory (EnDQT). In contrast to the well-known interpretations of quantum theory, EnDQT doesn’t modify its equations or add hidden variables, is not in tension with relativity, and provides a local causal explanation of Bell-type correlations without measurement outcomes varying according to perspectives or worlds. Unlike collapse theories, in principle, arbitrary systems can be placed in a superposition for an arbitrary amount of time, and no modifications of the equations of quantum theory are required. Furthermore, it provides a series of novel empirical posits that may distinguish it from other interpretations of quantum theory. According to EnDQT, some systems acquire determinate values at some point, and the capacity to give rise to determinate values through interactions propagates to other systems in spacetime via local interactions. This process can be represented via certain networks. When there is isolation from the rest of the systems that belong to these networks, such as inside the friend’s isolated lab in the extended Wigner’s friend scenarios, indeterminate values non-relationally arise inside

    Algebraic Models for Qualified Aggregation in General Rough Sets, and Reasoning Bias Discovery

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    In the context of general rough sets, the act of combining two things to form another is not straightforward. The situation is similar for other theories that concern uncertainty and vagueness. Such acts can be endowed with additional meaning that go beyond structural conjunction and disjunction as in the theory of ∗*-norms and associated implications over LL-fuzzy sets. In the present research, algebraic models of acts of combining things in generalized rough sets over lattices with approximation operators (called rough convenience lattices) is invented. The investigation is strongly motivated by the desire to model skeptical or pessimistic, and optimistic or possibilistic aggregation in human reasoning, and the choice of operations is constrained by the perspective. Fundamental results on the weak negations and implications afforded by the minimal models are proved. In addition, the model is suitable for the study of discriminatory/toxic behavior in human reasoning, and of ML algorithms learning such behavior.Comment: 15 Pages. Accepted. IJCRS-202

    Beyond the Wigner’s friend dilemma: A new indeterminacy-based quantum theory

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    I propose a novel (interpretation of) quantum theory, which I will call Environmental Determinacy-based or EnD Quantum Theory (EnDQT). In contrast to all the other current interpretations of quantum theory, EnDQT doesn’t modify its equations or add hidden variables, is not in tension with relativity, and provides a local causal explanation of Bell-type correlations without measurement outcomes varying according to perspectives or worlds. Unlike collapse theories, in principle, arbitrary systems can be placed in a superposition for an arbitrary amount of time, and no modifications of the equations of quantum theory are required. Furthermore, it provides a series of novel empirical posits that may distinguish it from other interpretations of quantum theory. According to EnDQT, some systems acquire determinate values at some point, and the capacity to give rise to determinate values through interactions propagates to other systems in spacetime via local interactions. This process can be represented via certain networks. When there is isolation from the rest of the systems that belong to these networks, such as inside the friend’s isolated lab in the extended Wigner’s friend scenarios, indeterminate values non-relationally arise inside

    The Role of Reconstruction in the Elucidation of Quantum Theory

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    The quantum reconstruction program seeks to uncover the deeper physical meaning of the quantum formalism by distilling its content into a set of physically-graspable postulates. Once reconstructed, one can further elucidate the theory by reflecting philosophically upon the postulates rather than the inscrutable formalism in which the theory is couched. Quantum reconstruction thereby opens up a two-step strategy for developing a quantum conceptual framework which reflects the full physical content of quantum theory and can be used to establish precisely what aspects of the classical conceptual framework are retained, modified, or abandoned. The paper is organized as follows. Sec. II describes the tripartite structure of classical physics, whose coherency I argue underlies its intelligibility. Sec. III discusses the lack of intelligibility of quantum physics, analyses the limitations of the main traditional elucidative approaches, and describes how reconstruction can remove the key interpretative bottleneck. Sec. IV discusses the methodology of reconstruction per se, showing that it is part of the natural life-cycle of physical theories, and summarizes its use in the elucidation of the theories of classical physics. Finally, circling back to quantum theory, Sec. V summarizes the reconstruction of quantum theory, and Sec. VI describes some of the interpretative insights thus far obtained through interpretation of reconstructions of the abstract quantum formalism and the quantum symmetrization postulate

    Truthmaker Semantics for Relevant Logic

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    I develop and defend a truthmaker semantics for the relevant logic R. The approach begins with a simple philosophical idea and develops it in various directions, so as to build a technically adequate relevant semantics. The central philosophical idea is that truths are true in virtue of speciic states. Developing the idea formally results in a semantics on which truthmakers are relevant to what they make true. A very natural notion of conditionality is added, giving us relevant implication. I then investigate ways to add conjunction, disjunction, and negation; and I discuss how to justify contraposition and excluded middle within a truthmaker semantics
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