Wittgenstein's Conception of Space and the Modernist Transformation of Geometry via Duality

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概要

Wittgenstein's disagreement with the set-theoretical view of mathematics led him to the idea
that space is not an extensional collection of points, but the intensional realisation of a law.
Brouwer's theory of the continuum is arguably
based upon the point-free conception of space qua law;
this would exhibit yet another case of Brouwer's influence on Wittgenstein's philosophy (in particular of space).
I consider the conception of space qua law to represent epistemology of space,
and the conception of space qua points to represent ontology of space.
From this perspective, modern geometry, such as Topos Theory, Algebraic and Non-Commutative Geometry, and Formal Topology,
some of which are conceptual ramifications of Brouwer's intuitionism,
yields rich instances of duality between epistemology and ontology of space.
Links with Husserl, Whitehead, Cassirer, Granger, Lawvere, and Japanese philosophers are briefly touched upon as well.

From Operational Chu Duality to Coalgebraic Quantum Symmetry

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We investigate into duality and symmetry building upon Pratt's idea of the Stone Gamut and Abramsky's representations of quantum systems. In the first part of the paper, we first observe that the Chu space representation of quantum systems leads us to an operational form of state-observable duality, and then show via the Chu space formalism enriched with a generic concept of closure conditions that such operational dualities (which we call ``-type" as opposed to ``sober-type") actually exist in fairly diverse contexts (topology, measurable spaces, and domain theory, to name but a few); the universal form of -type dualities between point-set and point-free spaces is described in terms of Chu spaces and closure conditions. From the duality-theoretical perspective, in the second part, we improve upon Abramsky's ``fibred" coalgebraic representation of quantum symmetries, thereby obtaining a finer, ``purely" coalgebraic representation: our representing category is properly smaller than Abramsky's, but still large enough to accommodate the quantum symmetry groupoid. Among several features, our representation reduces Abramsky's two-step construction of his representing category to a simpler one-step one, thus rendering the Grothendieck construction redundant. Our purely coalgebraic representation stems from replacing the category of sets in Abramsky's representation with the category of closure spaces in the light of the state-observable duality telling us that closure is a right perspective on quantum state spaces.

From an algebraic point of view, we consider that,
while propositional logic is a single algebra,
predicate logic is a fibred algebra; in the context of intuitionistic logic,
such an idea goes back to Lawvere, in particular his concept of hyperdoctrine.
Here, we aim at demonstrating that
the notion of monad-relativised hyperdoctrines,
which are what we call fibred algebras,
yields algebraisations of a wide variety of predicate logics.
More specifically,
we discuss a typed, first-order version of the Full Lambek calculus,
which has extensively been studied in the past few decades,
functioning as a unifying language for different sorts of logical systems
(classical, intuitionistic, linear, fuzzy, relevant, etc.).
Through the concept of Full Lambek hyperdoctrines,
we establish both generic and set-theoretical completeness results
for any extension of the base system; the latter arises from a dual adjunction,
and is relevant to the tripos-to-topos construction and quantale-valued sets.
Furthermore, we give a hyperdoctrinal account of Girard's and G¥"odel's translation.

The theory of natural dualities is a general theory of Stone-Priestley-type dualities based on the machinery of universal algebra. In this paper, by introducing the new notion of ISPM, we attempt to extend the theory of natural dualities so that it encompasses J´onsson-Tarski duality and Kupke-Kurz-Venema coalgebraic duality for the class of all modal algebras. Our main results are topological and coalgebraic dualities for ISPM(L) where L is a quasi-primal algebra with a bounded lattice reduct, which encompass both J´onsson-Tarski and Kupke-Kurz-Venema dualities, and give new coalgebraic dualities for algebras of many-valued modal logics and some insights into equivalence of categories of algebras involved. It also follows from our dualities that the category of relevant coalgebras has a final coalgebra and cofree coalgebras.

Dualities for algebras of Fitting's many-valued modal logics

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Stone-type duality connects logic, algebra, and topology in both conceptual and technical senses. This paper is intended to be a demonstration of this slogan. In this paper we focus on some versions of Fitting’s L-valued logic and L-valued modal logic for a finite distributive lattice L. Using the theory of natural dualities, we first obtain a duality for algebras of L-valued logic. Based on this duality, we develop a Jonsson-Tarski-style duality for algebras of L-valued modal logic, which encompasses Jonsson-Tarski duality for modal algebras as the case L = 2. We also discuss how the dualities change when the algebras are enriched by truth constants. Topological perspectives following from the dualities provide compactness theorems for the logics and the effective classification of categories of algebras involved, which tells us that Stone-type duality makes it possible to use topology for logic and algebra in significant ways.

Inspired by locale theory, we propose “pointfree convex geometry.” We introduce the notion of convexity algebra as pointfree convexity space. There are two notions of point for convexity algebra: one is chain-prime meet-complete filter and the other is maximal meet-complete filter. In this paper we show the following: (1) the former notion of point induces a dual equivalence between the category of “spatial” convexity algebras and the category of “sober” convexity spaces as well as a dual adjunction between the category of convexity algebras and the category of convexity spaces; (2) the latter notion of point induces a dual equivalence between the category of “m-spatial” convexity algebras and the category of “m-sober” convexity spaces. We finally argue that the former notion of point is more useful than the latter one from a category theoretic point of view and that the former notion of point actually represents polytope (or generic point) and the latter notion of point properly represents point. We also remark about the close relationships between pointfree convex geometry and domain theory.

Inferentialism about logic contradicts the thesis of meaning as use

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In this article I argue that inferentialism about logic, which is roughly the same as what is called proof-theoretic semantics, contradicts the thesis of meaning as use, although many proponents of inferentialism about logic such as Belnap (1962) and Dummett (1991) have considered that it is in harmony with the thesis. Inferentialists about logic such as Dummett (1991), Read (2004), and Murji and Hjortland (2009) claim that the meaning of a logical constant in a logic can be given by some or all of the rule(s) of inference governing the logical constant in a deductive system for the logic (usually, its introduction rule(s) in the standard system of natural deduction). I show that the inferentialist claim leads to a contradiction via the thesis of meaning as use.

Sets and Categories as Foundations of Mathematical Practice

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概要

Kreisel distinguishes between foundations and organization of mathematics. Broadly speaking, however, both could be seen as foundational studies on mathematics; the former is concerned with epistemological and/or ontological basis of mathematics in an ideal sense, while the latter is involved in conceptual, methodological machinery for organizing actual mathematics (as both problem solving and theory building) in an effective manner, which shall be called foundations of mathematical practice. The issue of sets versus categories have been discussed mainly in the context of the former, and they are usually supposed to be in conflict as a foundational enterprise. I thus aim to shed light on the issue from the second perspective, and clarify how sets and categories have worked as foundations of mathematical practice through the modernization of mathematics, with an emphasis on the vital role of sets in category theory, and on the way sets and categories interact in mathematical practice. They are not really in conflictwhen regarded as foundations of mathematical practice.hey are not really in conflict when regarded as foundations of mathematical practice.

Categorical Universal Logic relativizes the logic of topos to monads, with the purpose of reaching a universal conception of logic, and of providing foundations of categorical semantics for various logical systems, including intuitionistic logic, substructural logics, quantum logics, (topological and convex) geometric logics, and many others.