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The Stone gamut: A coordinatization of mathematics
 In Logic in Computer Science
, 1995
"... We give a uniform representation of the objects of mathematical practice as Chu spaces, forming a concrete selfdual bicomplete closed category and hence a constructive model of linear logic. This representation distributes mathematics over a twodimensional space we call the Stone gamut. The Stone ..."
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Cited by 30 (13 self)
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We give a uniform representation of the objects of mathematical practice as Chu spaces, forming a concrete selfdual bicomplete closed category and hence a constructive model of linear logic. This representation distributes mathematics over a twodimensional space we call the Stone gamut. The Stone gamut is coordinatized horizontally by coherence, ranging from −1 for sets to 1 for complete atomic Boolean algebras (CABA’s), and vertically by complexity of language. Complexity 0 contains only sets, CABA’s, and the inconsistent empty set. Complexity 1 admits noninteracting setCABA pairs. The entire Stone duality menagerie of partial distributive lattices enters at complexity 2. Groups, rings, fields, graphs, and categories have all entered by level 16, and every category of relational structures and their homomorphisms eventually appears. The key is the identification of continuous functions and homomorphisms, which puts StonePontrjagin duality on a uniform basis by merging algebra and topology into a simple common framework. 1 Mathematics from matrices We organize much of mathematics into a single category Chu of Chu spaces, or games as Lafont and Streicher have called them [LS91]. A Chu space is just a matrix that we shall denote =, but unlike the matrices of linear algebra, which serve as representations of linear transformations, Chu spaces serve as representations of the objects of mathematics, and their essence resides in how they transform. This organization permits a general twodimensional classification of mathematical objects that we call the Stone gamut 1, distributed horizontally by ∗This work was supported by ONR under grant number N0001492J1974. 1 “Spectrum, ” the obvious candidate for this appliction, already has a standard meaning in Stone duality, namely the representation of the dual space of a lattice by its prime ideals. “A
THE CHU CONSTRUCTION
, 1996
"... We take another look at the Chu construction and show how to simplify it by looking at ..."
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Cited by 12 (1 self)
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We take another look at the Chu construction and show how to simplify it by looking at