Papers by Mariusz Stopa
Philosophical Problems in Science (Polish: Zagadnienia Filozoficzne w Nauce, abbrev. ZFN), 2018
This article is intended for philosophers as a short partial introduction to category theory (CT)... more This article is intended for philosophers as a short partial introduction to category theory (CT) and its peculiar connection with logic. First, we consider CT itself. We give a brief insight into its history, introduce some basic definitions and present examples. In the second part, we focus on categorical topos semantics for propositional logic. We give some properties of logic in toposes, which, in general, is an intuitionistic logic. We next present two families of toposes whose tautologies are identical with those of classical propositional logic. The relatively extensive bibliography is given in order to support further studies.
Philosophical Problems in Science (Polish: Zagadnienia Filozoficzne w Nauce, abbrev. ZFN), 2020
Book review: Category Theory in Physics, Mathematics, and Philosophy, Kuś M., Skowron B. (eds.), ... more Book review: Category Theory in Physics, Mathematics, and Philosophy, Kuś M., Skowron B. (eds.), Springer Proc. Phys. 235, 2019, pp.xii+134.
Doctoral Thesis in (theoretical) Physics (written in Polish), 2011
Philosophical Problems in Science (Polish: Zagadnienia Filozoficzne w Nauce, abbrev. ZFN), 2020
It is well-established that topos theory is inherently connected with intuitionistic logic. In re... more It is well-established that topos theory is inherently connected with intuitionistic logic. In recent times several works appeared concerning so-called complement-toposes (co-toposes), which are allegedly connected to the dual to intuitionistic logic. In this paper I present this new notion, some of the motivations for it, and some of its consequences. Then, I argue that, assuming equivalence of certain two definitions of a topos, the concept of a complement-classifier (and thus of a co-topos as well) is, at least in general and within the conceptual framework of category theory, not appropriately defined. For this purpose, I first analyze the standard notion of a subobject classifier, show its connection with the representability of the functor Sub via the Yoneda lemma, recall some other properties of the internal structure of a topos and, based on these, I critically comment on the notion of a complement-classifier (and thus of a co-topos as well).
Philosophical Problems in Science (Polish: Zagadnienia Filozoficzne w Nauce, abbrev. ZFN), 2018
[The paper is in Polish, an English abstract is given only for information.]
This article is in... more [The paper is in Polish, an English abstract is given only for information.]
This article is intended for philosophers as a short partial introduction to category theory (CT) and its peculiar connection with logic. First, we consider CT itself. We give a brief insight into its history, introduce some basic definitions and present examples. In the second part, we focus on categorical topos semantics for propositional logic. We give some properties of logic in toposes, which, in general, is an intuitionistic logic. We next present two families of toposes whose tau-tologies are identical with those of classical propositional logic. The relatively extensive bibliography is given in order to support further studies.
Annales Henri Poincaré, 2010
This paper continues the investigation of the Casimir effect with the use of the algebraic formul... more This paper continues the investigation of the Casimir effect with the use of the algebraic formulation of quantum field theory in the initial value setting. Basing on earlier papers by one of us (AH), we approximate the Dirichlet and Neumann boundary conditions by simple interaction models whose nonlocality in physical space is under strict control, but which at the same time are admissible from the point of view of algebraic restrictions imposed on models in the context of Casimir backreaction. The geometrical setting is that of the original parallel plates. By scaling our models and taking appropriate limit, we approach the sharp boundary conditions in the limit. The global force is analyzed in that limit. One finds in Neumann case that although the sharp boundary interaction is recovered in the norm resolvent sense for each model considered, the total force per area depends substantially on its choice and diverges in the sharp boundary conditions limit. On the other hand the local energy density outside the interaction region, which in the limit includes any compact set outside the strict position of the plates, has a universal limit corresponding to sharp conditions. This is what one should expect in general, and the lack of this discrepancy in Dirichlet case is rather accidental. Our discussion pins down its precise origin: the difference in the order in which scaling limit and integration over the whole space is carried out.
Book Reviews by Mariusz Stopa
Philosophical Problems in Science (Polish: Zagadnienia Filozoficzne w Nauce, abbrev. ZFN), 2020
This is a review of the following book: Kuś M., Skowron B. (eds.), Category Theory in Physics, Ma... more This is a review of the following book: Kuś M., Skowron B. (eds.), Category Theory in Physics, Mathematics, and Philosophy, Springer Proc. Phys. 235, 2019, p.xii+134.
Thesis Chapters by Mariusz Stopa
Doctoral Thesis in Philosophy (in Logic), 2023
The main research goal of the work is to study the notion of co-topos, its correctness, propertie... more The main research goal of the work is to study the notion of co-topos, its correctness, properties and relations with toposes. In particular, the dualization process proposed by proponents of co-toposes has been analyzed, which transforms certain Heyting algebras of toposes into co-Heyting ones, by which a kind of paraconsistent logic may appear in place of intuitionistic logic. It has been shown that if certain two definitions of topos are to be equivalent, then in one of them, in the context of a subobject classifier, there should be a neutral label for the generic subobject, or if the label "true" appears instead, it carries no additional properties or assumptions. It has been shown that a natural isomorphism occurring in one of the definitions of topos need not be an isomorphism of the corresponding algebraic structures on the sets, so that a co-Heyting algebraic structure can be defined on the set of characteristic morphisms, which moreover will also be natural. The analyzes suggest that the notion of co-topos may be considered correct. However, it should be strongly emphasized that although the possibility of defining the notion of co-topos has been shown, its further full consequences should be very carefully examined, which can severely limit its logical applications.
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Papers by Mariusz Stopa
This article is intended for philosophers as a short partial introduction to category theory (CT) and its peculiar connection with logic. First, we consider CT itself. We give a brief insight into its history, introduce some basic definitions and present examples. In the second part, we focus on categorical topos semantics for propositional logic. We give some properties of logic in toposes, which, in general, is an intuitionistic logic. We next present two families of toposes whose tau-tologies are identical with those of classical propositional logic. The relatively extensive bibliography is given in order to support further studies.
Book Reviews by Mariusz Stopa
Thesis Chapters by Mariusz Stopa
This article is intended for philosophers as a short partial introduction to category theory (CT) and its peculiar connection with logic. First, we consider CT itself. We give a brief insight into its history, introduce some basic definitions and present examples. In the second part, we focus on categorical topos semantics for propositional logic. We give some properties of logic in toposes, which, in general, is an intuitionistic logic. We next present two families of toposes whose tau-tologies are identical with those of classical propositional logic. The relatively extensive bibliography is given in order to support further studies.