We give a uniform representation of the objects of mathematical practice as Chu spaces, forming a concrete self-dual bicomplete closed category and hence a constructive model of linear logic. This representation distributes mathematics over a two-dimensional 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 set-CABA 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 Stone-Pontrjagin 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 two-dimensional classification of mathematical objects that we call the Stone gamut 1, distributed horizontally by ∗This work was supported by ONR under grant number N00014-92-J-1974. 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

Chu spaces have found applications in computer science, mathematics, and physics. They enjoy a useful categorical duality analogous to that of lattice theory and projective geometry. As natural mathematics Chu spaces borrow ideas from the natural sciences, particularly physics, while as rational mechanics they cast Hamiltonian mechanics in terms of the interaction of body and mind. This paper addresses the chief stumbling block for Descartes ’ 17thcentury philosophy of mind-body dualism, how can the fundamentally dissimilar mental and physical planes causally interact with each other? We apply Cartesian logic to reject not only divine intervention, preordained synchronization, and the eventual mass retreat to monism, but also an assumption Descartes himself somehow neglected to reject, that causal interaction within these planes is an easier problem than between. We use Chu spaces and residuation to derive all causal interaction, both between and within the two planes, from a uniform and algebraically rich theory of between-plane interaction alone. Lifting the two-valued Boolean logic of binary relations to the complex-valued fuzzy logic of quantum mechanics transforms residuation into a natural generalization of the inner product operation of a Hilbert space and demonstrates that this account of causal interaction is of essentially the same form as the Heisenberg-Schrödinger quantum-mechanical solution to analogous problems of causal interaction in physics. 1 Cartesian Dualism The Chu construction [Bar79] strikes us as extraordinarily useful, more so with every passing month. Elsewhere we have described the application of Chu spaces to process algebra [GP93], metamathematics [Pra93, Pra94a], and physics [Pra94b]. Here we make a first attempt at applying them to philosophy. It might seem that traditional philosophical questions would be beyond the scope of TAPSOFT. Bear in mind however that Boolean logic as the basis for

. The purpose is to give a simple proof that a category is equivalent to a small category if and only if both it and its presheaf category are locally small. In one of his lectures (University of New South Wales, 1971) on Yoneda structures [SW], the second author conjectured that a category A is essentially small if and only if both A and the presheaf category PA are locally small. The first author was in the audience and at the end of the lecture suggested a proof of the conjecture using some of his own results. This was reported on page 352 of [SW] and used to motivate a definition of "small" in [St]; yet the proof was not published. The proof given in the present paper evolved via correspondence between the authors in 1976-77 while the second author was on sabbatical leave at Wesleyan University (Middletown, Connecticut) but has remained unpublished despite our expectation at various times that it would appear as an exercise in some textbook. In 1979, a longer, but related, proof a...