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1
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Filozoficznie prowokująca teoria kategorii

100%
Heller M.
Zagadnienia Filozoficzne w Nauce
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2018
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issue 65
PL
Recenzja książki: Elaine Landry (red.), Categories for the Working Philosopher, Oxford University Press, Oxford, 2017, ss. xiv+471.
2

Category Free Category Theory and Its Philosophical Implications

71%
Heller M.
Logic and Logical Philosophy
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2016
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vol. 25
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issue 4
447-459
EN
There exists a dispute in philosophy, going back at least to Leibniz, whether is it possible to view the world as a network of relations and relations between relations with the role of objects, between which these relations hold, entirely eliminated. Category theory seems to be the correct mathematical theory for clarifying conceptual possibilities in this respect. In this theory, objects acquire their identity either by definition, when in defining category we postulate the existence of objects, or formally by the existence of identity morphisms. We show that it is perfectly possible to get rid of the identity of objects by definition, but the formal identity of objects remains as an essential element of the theory. This can be achieved by defining category exclusively in terms of morphisms and identity morphisms (objectless, or object free, category) and, analogously, by defining category theory entirely in terms of functors and identity functors (categoryless, or category free, category theory). With objects and categories eliminated, we focus on the “philosophy of arrows” and the roles various identities play in it (identities as such, identities up to isomorphism, identities up to natural isomorphism ...). This perspective elucidates a contrast between “set ontology” and “categorical ontology”.
3
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Category Theory in the hands of physicists, mathematicians, and philosophers

71%
Stopa M.
Zagadnienia Filozoficzne w Nauce
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2020
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issue 69
283-293
EN
Book review: Category Theory in Physics, Mathematics, and Philosophy, Kuś M., Skowron B. (eds.), Springer Proc. Phys. 235, 2019, pp.xii+134.
4
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“Is logic a physical variable?” Introduction to the Special Issue

71%
Eckstein M., Skowron B.
Zagadnienia Filozoficzne w Nauce
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2020
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issue 69
7-13
EN
“Is logic a physical variable?” This thought-provoking question was put forward by Michael Heller during the public lecture “Category Theory and Mathematical Structures of the Universe” delivered on 30th March 2017 at the National Quantum Information Center in Sopot. It touches upon the intimate relationship between the foundations of physics, mathematics and philosophy. To address this question one needs a conceptual framework, which is on the one hand rigorous and, on the other hand capacious enough to grasp the diversity of modern theoretical physics. Category theory is here a natural choice. It is not only an independent, well-developed and very advanced mathematical theory, but also a holistic, process-oriented way of thinking.
5
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Category Theory and Set Theory as Theories about Complementary Types of Universals

62%
Ellerman D.
Logic and Logical Philosophy
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2017
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vol. 26
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issue 2
145–162
EN
Instead of the half-century old foundational feud between set theory and category theory, this paper argues that they are theories about two different complementary types of universals. The set-theoretic antinomies forced naïve set theory to be reformulated using some iterative notion of a set so that a set would always have higher type or rank than its members. Then the universal u_F = {x | F(x)} for a property F(.) could never be self-predicative in the sense of u_F \in u_F . But the mathematical theory of categories, dating from the mid-twentieth century, includes a theory of always-self-predicative universals - which can be seen as forming the “other bookend” to the never-self-predicative universals of set theory. The self-predicative universals of category theory show that the problem in the antinomies was not self-predication per se, but negated self-predication. They also provide a model (in the Platonic Heaven of mathematics) for the self-predicative strand of Plato’s Theory of Forms as well as for the idea of a “concrete universal” in Hegel and similar ideas of paradigmatic exemplars in ordinary thought.
6
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A critical analysis of the philosophical motivations and development of the concept of the field of rationality as a representation of the fundamental ontology of the physical reality

54%
Grygiel W.
Zagadnienia Filozoficzne w Nauce
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2022
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issue 72
87-108
EN
The unusual applicability of mathematics to the description of the physical reality still remains a major investigative task for philosophers, physicists, mathematicians and cognitive scientists. The presented article offers a critical analysis of the philosophical motivations and development of a major attempt to resolve this task put forward by two prominent Polish philosophers: Józef Życiński and Michał Heller. In order to explain this particular property of mathematics Życiński has first introduced the concept of the field of rationality together with the field of potentiality to be followed by Heller’s formal field and the field of categories. It turns out that these concepts are fully intelligible once located within philosophical stances on the relations between mathematics and physical reality. It will be argued that in order to achieve more extended conceptual clarification of the precise meaning of the field of rationality, further advancements in the understanding of the nature of the human mind are required.
7
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Teoria kategorii i niektóre jej logiczne aspekty

54%
Stopa M.
Zagadnienia Filozoficzne w Nauce
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2018
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issue 64
7-58
PL
This article is intended for philosophers and logicians 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.
8

The Logic of God

45%
Heller M.
Edukacja Filozoficzna
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2019
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issue 68
227-244
EN
When speaking on the “logic of God”, we can understand the logic of our reasoning about God or the logic as it is supposedly employed by God. It is rather obvious, at least for believers in God, that we can infer something about God’s logic in the latter meaning, from how the logic operates in the world created by Him. In the present essay, my strategy is to use this narrow window through which we can grasp some glimpses of “God’s ways of thinking”. There are strong reasons to believe that it is category theory that best displays the role of logic in the system of our mathematical and physical knowledge. It gives us a refreshingly new perspective on logic and its various applications, and could be a good starting point for our speculations concerning the “logic of God”. A quick look at category theory and its applications to physics shows that logic can change from theory to theory, or from level to meta-level. This poses the question of the existence of “superlogic” to which all other logics would somehow be subordinated. The fact that this question remains unanswered forces us to face the problem of plurality of logics. Usually, it is tacitly assumed that the role of “superlogic” is played by classical logic with its non-contradiction law as the most obvious tautology. We briefly discuss paraconsistent logic as an example of a logical system in which contradictions are allowed, albeit under the condition that they do not make the system to explode, i.e. that they do not spill over the whole system. Such logic is an internal logic in categories called cotopoi (or complement topoi). I refer to some theological discussions, both present and from the past, that associate “God’s logic” with classical logic, in particular with the non-contradiction principle. However, we argue that this principle should not be absolutized. The only thing we can, with some certainty, assert on “God’s logic” is that it is not an exploding logic, i.e. that it is not an “anything goes logic”. God is a Source-of-All-Rationality but His rationality need not to conform to our standards of what is rational. This “principle of logical apophaticism” is formulated and briefly discussed. In the history of theology at least one attempt is known to reconstruct the “process of God’s thinking”, namely Leibniz’s idea of God’s selecting the best world to be created from among all possible worlds. Some modifications are suggested which we believe Leibniz would have introduced in his reconstruction, if he knew present developments in categorical logic.
PL
Mówiąc o „logice Boga”, możemy mieć na myśli logikę naszego myślenia o Bogu albo logikę, jaką w naszym wyobrażeniu posługuje się Bóg. Jest dość oczywiste, przynajmniej dla wierzących, że to i owo na temat logiki Boga w tym drugim znaczeniu możemy wywnioskować z logiki obowiązującej w stworzonym przez Niego świecie. W tym eseju stosuję właśnie tę strategię, by zidentyfikować niektóre przebłyski „Boskiego sposobu myślenia”. Istnieją dobre powody, by przyjąć, że rolę logiki w systemie naszej matematycznej i fizycznej wiedzy najlepiej obrazuje teoria kategorii. Daje nam ona nowe, świeże spojrzenie na logikę i jej różne zastosowania, i może być dobrym punktem wyjścia dla badań nad „logiką Boga”. Pobieżne nawet przyjrzenie się zastosowaniom teorii kategorii w fizyce pozwala się przekonać, że logika może się zmieniać przy przejściu od teorii do teorii i z jednego poziomu ogólności na drugi. Nasuwa się pytanie o istnienie „superlogiki”, której w jakimś sensie podlegałyby wszystkie logiki niższego rzędu. Fakt, że nie potrafimy na to pytanie odpowiedzieć, stawia nas w obliczu problemu logicznego pluralizmu. Zwykle przyjmuje się milcząco, że rolę „superlogiki” odgrywa logika klasyczna z zasadą niesprzeczności jako najbardziej oczywistą tautologią. Artykuł omawia krótko logikę parakonsystentną jako przykład logiki, w której sprzeczności są dozwolone, choć jedynie pod warunkiem, że nie eksplodują, tzn., nie rozlewają się na całość systemu. Taka logika jest wewnętrzną logiką w kategoriach zwanych ko-toposami (complement topoi). Przywołuję niektóre teologiczne dyskusje, zarówno dawne, jak i współczesne, w których „logikę Boga” utożsamiano z logiką klasyczną, a zwłaszcza z zasadą niesprzeczności, starając się pokazać, że zasady tej jednak nie powinno się absolutyzować. Jedyne, co z jakąś pewnością możemy powiedzieć o „logice Boga”, to że nie jest to logika bez żadnych reguł, w której wszystko jest dozwolone. Bóg jest Źródłem-Wszelkiej-Racjonalności, ale Jego racjonalność nie musi spełniać naszych standardów. Formułuję zatem i krótko omawiam tę „zasadę logicznej apofatyczności”. Historia teologii zna przynajmniej jedną próbę zrekonstruowania „procesu Boskiego namysłu” – wizję Leibniza, u którego Bóg ze wszystkich światów, które mógłby stworzyć, wybiera najlepszy. Sugeruję na koniec kilka zmian, które moim zdaniem Leibniz wprowadziłby do swojej rekonstrukcji, gdyby znał współczesną logikę kategorialną.
9

Special Issue on point-free geometry and topology. An introduction

45%
Coppola C., Gerla G.
Logic and Logical Philosophy
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2013
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vol. 22
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issue 2
139-143
EN
In the first section we briefly describe the methodological assumptions of point-free geometry and topology. We also outline the history of geometrical theories based on the notion of a region. The second section is devoted to a concise presentation of the content of the LLP special issue on point-free theories of space.
10
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On the validity of the definition of a complement-classifier

45%
Stopa M.
Zagadnienia Filozoficzne w Nauce
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2020
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issue 69
111-128
EN
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).
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