Canonical solutions of domain equations are shown to be final coalgebras, not only in a category of non-standard sets (as already known), but also in categories of metric spaces and partial orders. Coalgebras are simple categorical structures generalizing the notion of post-fixed point. They are also used here for giving a new comprehensive presentation of the (still) non-standard theory of non-well-founded sets (as non-standard sets are usually called). This paper is meant to provide a basis to a more general project aiming at a full exploitation of the finality of the domains in the semantics of programming languages --- concurrent ones among them. Such a final semantics enjoys uniformity and generality. For instance, semantic observational equivalences like bisimulation can be derived as instances of a single `coalgebraic' definition (introduced elsewhere), which is parametric of the functor appearing in the domain equation. Some properties of this general form of equivalence are also studied in this paper.

The present work describes an alternative to the classical partial differential equations-based approach to the discretization of physical field problems. This alternative is based on a preliminary reformulation of the mathematical model in a partially discrete form, which preserves as much as possible the physical and geometrical content of the original problem, and is made possible by the existence and properties of a common mathematical structure of physical field theories. The goal is to maintain the focus, both in the modeling and in the discretizati on step, on the physics of the problem, thinking in terms of numerical methods for physical field problems, and not for a particular mathematical form (for example, a partial differential equation) into which the original physical problem happens to be translated.

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We study some particular examples of quantaloids and corresponding morphisms, originating from primitive physical reasonings on the lattices of properties of physical systems. AMS classification code: 03G10, 18D20, 81P10. Key words: Complete Lattice, Galois Adjoint, Physical System, Quantaloid. 1 Introduction The starting point for our research program is the fact, already observed in Eilenberg and Mac Lane's seminal paper [8], that preordered sets may be considered as small thin categories. One can then not only reformulate a large part of the theory of order structures in categorical terms, but also apply general categorical techniques to specific order theoretic problems. In particular, the notion of an adjunction reduces to that of a residuation [15] 4.5, whereas the notion of a monad reduces to that of a closure operator [15] 6.1-2. Now the above categorical notions have direct physical interpretations in the context of axiomatic quantum theory, the order relation in the prop...

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Aformal proof is a proof written in a precise artificial language that admits only a fixed repertoire of stylized steps. This formal language is usually designed so that there is a purely mechanical process by which the correctness of a proof in the language can be verified. Nowadays, there are numerous computer programs known as proof assistants that can check, or even partially construct, formal proofs written in their preferred proof language. These can be considered as practical, computer-based realizations of the traditional systems of formal symbolic logic and set theory proposed as foundations for mathematics. Why should we wish to create formal proofs?

Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis à vis set theory. It is argued here that these theories require a background logic of relations and substantive assumptions addressing mathematical existence of categories themselves. We propose a synthesis of Bell’s “many-topoi” view and modal-structuralism. Surprisingly, a combination of mereology and plural quantification suffices to describe hypothetical large domains, recovering the Grothendieck method of universes. Both topos theory and set theory can be carried out relative to such domains; puzzles about “large categories ” and “proper classes ” are handled in a

Data Types and Algebraic Semantics The history of programming languages, and to a large extent of software engineering as a whole, can be seen as a succession of ever more powerful abstraction mechanisms. The first stored program computers were programmed in binary, which soon gave way to assembly languages that allowed symbolic codes for operations and addresses. fortran began the spread of "high level" programming languages, though at the time it was strongly opposed by many assembly programmers; important features that developed later include blocks, recursive procedures, flexible types, classes, inheritance, modules, and genericity. Without going into the philosophical problems raised by abstraction (which in view of the discussion of realism in Section 4 may be considerable), it seems clear that the mathematics used to describe programming concepts should in general get more abstract as the programming concepts get more abstract. Nevertheless, there has been great resistance to u...

Examples are discussed of natural statements about irrational numbers that are equivalent, provably in ZFC, to strong set-theoretical hypotheses, and of apparently classical statements provable in ZFC of which the only known proofs use strong set-theoretical concepts. 1.

Along with Frege, Russell maintained an absolutist stance regarding the subject matter of mathematics, revealed rather than imposed, or proposed, by logical analysis. The Fregean definition of cardinal number, for example, is viewed as (essentially) correct, not merely adequate for mathematics. And Dedekind’s “structuralist” views come in for criticism in the Principles. But, on reflection, Russell also flirted with views very close to a (different) version of structuralism. Main varieties of modern structuralism and their challenges are reviewed, taking account of Russell’s insights. Problems of absolutism plague some versions, and, interestingly, Russell’s critique of Dedekind can be extended to one of them, ante rem structuralism. This leaves modal-structuralism and a category theoretic approach as remaining non-absolutist