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The Finite Volume, Finite Element, and Finite Difference Methods as Numerical Methods for Physical Field Problems
 Journal of Computational Physics
, 2000
"... The present work describes an alternative to the classical partial differential equationsbased 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 possi ..."
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Cited by 47 (1 self)
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The present work describes an alternative to the classical partial differential equationsbased 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.
On the Foundations of Final Semantics: NonStandard Sets, Metric Spaces, Partial Orders
 PROCEEDINGS OF THE REX WORKSHOP ON SEMANTICS: FOUNDATIONS AND APPLICATIONS, VOLUME 666 OF LECTURE NOTES IN COMPUTER SCIENCE
, 1998
"... Canonical solutions of domain equations are shown to be final coalgebras, not only in a category of nonstandard sets (as already known), but also in categories of metric spaces and partial orders. Coalgebras are simple categorical structures generalizing the notion of postfixed point. They are ..."
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Cited by 47 (10 self)
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Canonical solutions of domain equations are shown to be final coalgebras, not only in a category of nonstandard sets (as already known), but also in categories of metric spaces and partial orders. Coalgebras are simple categorical structures generalizing the notion of postfixed point. They are also used here for giving a new comprehensive presentation of the (still) nonstandard theory of nonwellfounded sets (as nonstandard 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.
On the Foundations of Final Coalgebra Semantics: nonwellfounded sets, partial orders, metric spaces
, 1998
"... ..."
Quantaloids describing causation and propagation for physical properties
 Foundations of Physics Letters
, 2001
"... We study some particular examples of quantaloids and corresponding morphisms, originating from primitive physical reasonings on the lattices of properties of physical systems. ..."
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Cited by 14 (7 self)
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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.
Finitary Spacetime Sheaves
 International Journal of Theoretical Physics
, 2000
"... The notion of finitary spacetime sheaves is introduced based on locally finite approximations of the continuous topology of a bounded region of a spacetime manifold. Finitary spacetime sheaves are seen to be sound mathematical models of approximations of continuous spacetime observables. 1 ..."
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Cited by 12 (8 self)
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The notion of finitary spacetime sheaves is introduced based on locally finite approximations of the continuous topology of a bounded region of a spacetime manifold. Finitary spacetime sheaves are seen to be sound mathematical models of approximations of continuous spacetime observables. 1
Formal proof—theory and practice
 Notices AMS
, 2008
"... 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 nume ..."
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Cited by 12 (1 self)
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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, computerbased 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?
Does category theory provide a framework for mathematical structuralism?
 PHILOSOPHIA MATHEMATICA
, 2003
"... 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 ..."
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Cited by 10 (3 self)
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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 “manytopoi” view and modalstructuralism. 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
C∞Smooth Singularities
 Chimeras of the Spacetime Manifold, in preparation
, 2001
"... Abstract. We present herewith certain thoughts on the important subject of nowadays physics, pertaining to the socalled “singularities”, that emanated from looking at the theme, in terms of ADG (: abstract differential geometry). Thus, according to the latter perspective, we can involve “singularit ..."
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Cited by 6 (3 self)
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Abstract. We present herewith certain thoughts on the important subject of nowadays physics, pertaining to the socalled “singularities”, that emanated from looking at the theme, in terms of ADG (: abstract differential geometry). Thus, according to the latter perspective, we can involve “singularities ” in our arguments, while still employing fundamental differentialgeometric notions, as connections, curvature, metric and the like, retaining also the form of standard important relations of the classical theory (e.g. Einstein and/or YangMills equations, in vacuum), even within that generalized context of ADG. To wind up, we can extend (in point of fact, calculate) over singularities classical differentialgeometric relations/equations, without altering their forms and/or changing the standard arguments; the change concerns thus only the way, we employ the usual differential geometry of smooth manifolds, so that the base “space ” acquires now a quite secondary rôle, not contributing, at all (!), to the differentialgeometric technique, we apply, the latter being thus, by definition, directly referred to the objects involved, that “live on the space”, not being, of course, i p s o f a c t o “singular”!
The growth of mathematical knowledge: an open world view
 The growth of mathematical knowledge, Kluwer, Dordrecht 2000
"... mathematical knowledge: “The advance of science is not comparable to the changes of a city, where old edifices are pitilessly torn down to give place to new ones, but to the continuous evolution of zoological types which develop ceaselessly and end by becoming unrecognizable to the common sight, but ..."
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Cited by 5 (5 self)
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mathematical knowledge: “The advance of science is not comparable to the changes of a city, where old edifices are pitilessly torn down to give place to new ones, but to the continuous evolution of zoological types which develop ceaselessly and end by becoming unrecognizable to the common sight, but where an expert eye finds always traces of the prior work of the centuries past ” (Poincaré 1958, p. 14). The view criticized by Poincaré corresponds to Frege’s idea that the development of mathematics can be described as an activity of system building, where each system is supposed to provide a complete representation for a certain mathematical field and must be pitilessly torn down whenever it fails to achieve such an aim. All facts concerning any mathematical field must be fully organized in a given system because “in mathematics we must always strive after a system that is complete in itself ” (Frege 1979, p. 279). Frege is aware that systems introduce rigidity and are in conflict with the actual development of mathematics because “in history we have development; a system is static”, but he sticks