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1

Set-theoretic mereology

100%

Hamkins J. D., Kikuchi M.

Logic and Logical Philosophy

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2016

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vol. 25

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issue 3 Mereology and Beyond (II)

285-308

EN

We consider a set-theoretic version of mereology based on the inclusion relation ⊆ and analyze how well it might serve as a foundation of mathematics. After establishing the non-definability of ∈ from ⊆, we identify the natural axioms for ⊆-based mereology, which constitute a finitely axiomatizable, complete, decidable theory. Ultimately, for these reasons, we conclude that this form of set-theoretic mereology cannot by itself serve as a foundation of mathematics. Meanwhile, augmented forms of set-theoretic mereology, such as that obtained by adding the singleton operator, are foundationally robust.

2

Georg Cantor i idea jedności nauki

100%

Dadaczyński J.

Zagadnienia Filozoficzne w Nauce

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2009

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issue 44

84-99

PL

G. Cantor presented - in an unpublished paper (1884) - a vision of the unity of science. He argued all sciences can be reduced directly to the set theory. A source of this idea was for Cantor the unity of mathematics (on the basis of set theory). Cantor represented thesis about the unity of science irrespective of the representatives of positivism (E. Mach).

3

Tractarian Ontology: Mereology or Set Theory?

88%

Lando G.

Forum Philosophicum

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2007

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vol. 12

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issue 2

247-266

EN

I analyze the relations of constituency or “being in” that connect different ontological items in Wittgenstein's Tractatus Logico-Philosophicus. A state of affairs is constituted by atoms, atoms are in a state of affairs. Atoms are also in an atomic fact. Moreover, the world is the totality of facts, thus it is in some sense made of facts. Many other kinds of Tractarian notions—such as molecular facts, logical space, reality—seem to be involved in constituency relations. How should these relations be conceived? And how is it possible to formalize them in a convincing way? I draw a comparison between two ways of conceiving and formalizing these relations: through sets and through mereological sums. The comparison shows that the conceptual machinery of set theory is apter to conceive and formalize Tractarian constituency notions than the mereological one.

4

Filling assessment model of strategic cost management system (SCMS)

88%

Petrov O., Minina Y.

Zeszyty Naukowe Wyższej Szkoły Ekonomii i Informatyki w Krakowie

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2018

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issue 14

260-276

EN

The article presents a filling assessment model of SCMS, developed for the reflection of actual content of enterprise strategic cost management system, which can be used in real-time mode. It is proposed to use the mathematical apparatus of set theory and tools for working with sets in hyperspace for practical construction and use of such model. Ref. 7.

PL

Artykuł przedstawia model oceny wypełnienia SCMS, opracowany w celu odzwierciedlenia rzeczywistej zawartości systemu zarządzania kosztami strategicznymi przedsiębiorstwa, który może być wykorzystywany w trybie czasu rzeczywistego. Proponuje się wykorzystanie aparatu matematycznego teorii mnogości i narzędzi do pracy z zestawami w hiperprzestrzeni do praktycznej konstrukcji i wykorzystania takiego modelu.

5

Aksjomat wyboru w pracach Wacława Sierpińskiego

88%

Lewandowska K.

Zagadnienia Filozoficzne w Nauce

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2013

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issue 52

9-33

PL

This paper presents Wacław Sierpiński – the first advocate of the axiom of choice. We focus on the philosophical and mathematical topics related to the axiom of choice which were considered by Sierpiński. We analyze some of his papers to show how his results effected the debate over Zermelo’s axiom. Sierpiński’s impact on this discussion is of particular importance since he was the first who tried to explore consequences of the axiom of choice thoroughly and asserted its undoubted significance to mathematics as a whole.

6

Towards a Formal Ontology of Information. Selected Ideas of K. Turek

75%

Krzanowski R.

Zagadnienia Filozoficzne w Nauce

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2016

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issue 61

23-52

EN

There are many ontologies of the world or of specific phenomena such as time, matter, space, and quantum mechanics1. However, ontologies of information are rather rare. One of the reasons behind this is that information is most frequently associated with communication and computing, and not with ‘the furniture of the world’. But what would be the nature of an ontology of information? For it to be of significant import it should be amenable to formalization in a logico-grammatical formalism. A candidate ontology satisfying such a requirement can be found in some of the ideas of K. Turek, presented in this paper. Turek outlines the ontology of information conceived of as a part of nature, and provides the ‘missing link’ to the Z axiomatic set theory, offering a proposal for developing a formal ontology of information both in its philosophical and logicogrammatical representations.

7

Nieskończoność w matematyce. Zmagania z potrzebnym, acz kłopotliwym pojęciem

75%

Murawski R.

Zagadnienia Filozoficzne w Nauce

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2014

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issue 55

5-42

PL

Infinity has appeared in mathematics since the very beginning. Moreover the mathematical concept of infinity was and is connected with philosophical and theological concepts. The aim of the paper is to show how mathematicians struggled with this concept and how they tried to bring it under control.

8

Kilka uwag na temat koncepcji mnogościowych Richarda Dedekinda i Stanisława Leśniewskiego

75%

Obojska L.

Kwartalnik Historii Nauki i Techniki

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2014

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vol. 59

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issue 1

95-107

EN

Mereology, is a part-whole theory, also called the theory of collective sets. It was founded in1916 by Stanislaw Lesniewski and this is an alternative theory versus the classical set theory by Georg Cantor. These two theories are usually teamed up together as Leśniewski himself was referring to the concept of the set by Cantor and Cantor is considered the "main" ideologist of the set theory. However, when analyzing the original texts of various authors, it seems that the very concept of a collective set is closer to the idea of Richard Dedekind rather than that of Georg Cantor. It is known that Cantor borrowed some concepts on the notion of set from Dedekind, whose ideas were also known to Leśniewski, however, there is no study on this topic. This work is therefore an attempt to compare some set-theoretical concepts of both of these authors, i.e. S. Leśniewski and R. Dedekind and the presentation of their convergence.

9

Kryzys podstaw matematyki przełomu XIX i XX wieku

71%

Fila M.

Semina Scientiarum

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2014

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vol. 13

EN

By the end of the 19th century, mathematics had become very intensively developed. Mathematical logic became an independent discipline, and in the 1880s Cantor published his work on set theory. All this led to questions about the consistency of mathematical theories and decidability theorems. Therefore, for the second time in the history of mathematics, there emerged a crisis of the basis of mathematics.There were a few ideas for overcoming the crisis. In this paper, there will be described three trends in the philosophy of mathematics in the late 19th and early 20th centuries: logicism (Frege), intuitionism (Brouwer) and formalism (Hilbert). These three trends were described from the philosophical point of view and in the context of the crisis. Moreover, for each of them there will be present the most important methodological assumptions, and I will briefly describe attempts to achieve them. This will describe the problem in such a way that allows for the grasping of important differences and similarities between logicism, intuitionism and formalism and better understand their causes.

10

How to define a mereological (collective) set

63%

Gruszczyński R., Pietruszczak A.

Logic and Logical Philosophy

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2010

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vol. 19

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issue 4

309–328

EN

As it is indicated in the title, this paper is devoted to the problem of defining mereological (collective) sets. Starting from basic properties of sets in mathematics and differences between them and so called conglomerates in Section 1, we go on to explicate informally in Section 2 what it means to join many objects into a single entity from point of view of mereology, the theory of part of (parthood) relation. In Section 3 we present and motivate basic axioms for part of relation and we point to their most fundamental consequences. Next three sections are devoted to formal explication of the notion of mereological set (collective set) in terms of sums, fusions and aggregates. We do not give proofs of all theorems. Some of them are complicated and their presentation would divert the reader’s attention from the main topic of the paper. In such cases we indicate where the proofs can be found and analyzed by those who are interested.

11

Conceptual Classifications versus Collections of Objects in Biology

63%

Stuchliński J. A.

Dialogue and Universalism

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2014

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issue 2

149-153

EN

In logic the concepts of set and class are understood in two senses: distributively (in set theory) or collectively (in mereology). This paper discusses problems of the usefulness of both the concepts in evolutionary biology, particularly in cross–researches of the biological taxonomy, evolutionism and genetics.

12

Pierwszy krok w chmurach: o teorii siedliska wydarzeniowego Alaina Badiou

45%

Kuźniarz B.

Diametros

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2013

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issue 37

69-84

PL

Zgodnie z tytułem najważniejszej książki Alaina Badiou, L’être et l’événement, jego myśl rozpada się na dwie części: tę dotyczącą bytu oraz tę, w której na pierwszy plan wysuwa się to, co wobec bytu inne, czyli kategoria wydarzenia. Pierwsza część ogranicza się do ustaleń nauki, przepisując na język ontologii aksjomaty teorii zbiorów, by na pewnym etapie, wyczerpawszy swój koncepcyjny potencjał, ustąpić miejsca filozofii. Zadanie tej ostatniej polega na metodycznym opisie możliwości zawartych w wykraczającym poza ontologię „cudzie” wydarzenia. Na pograniczu ontologii i filozofii Badiou umieszcza siedlisko wydarzeniowe. Stanowi ono rodzaj matematycznego odpowiednika kartezjańskiej „szyszynki”, a zarazem ostatnie pojęcie systemu francuskiego filozofa, którego zrozumienie nie wymaga skoku w metafizyczne zaświaty, jakim jest w swej istocie kategoria wydarzenia. Siedlisko wydarzeniowe nie przesądza o możliwości pojawienia się wydarzenia, lecz pozwala określić miejsce, w którym można sensownie oczekiwać jego nadejścia. W swoim artykule podejmuję się rekonstrukcji tego kluczowego pojęcia systemu teoretycznego Badiou.

EN

In line with the title of Alain Badiou’s main book, Being and Event, his thought breaks down into two components: one that deals with being, and another where pride of place is given to the being’s other, namely the category of event. The former involves scientific findings, which rewrite the axioms of the set theory for the purposes of ontology. Only the second part is properly philosophical, describing the possibilities provided by the otherworldly “miracle” of the event. On the very border between ontology and philosophy Badiou places the evental site: the last concept of Badiou’s system that is fully intelligible without resorting to the metaphysical leap known as the event. The evental site does not settle the matter of the event’s actual occurrence but it allows us to sensibly determine the place or site of its coming. In my paper I undertake the task of reconstructing this crucial concept of Badiou’s system.

13

Strukturalność a dedukcyjność matematyki: współczesny strukturalizm w filozofii matematyki

38%

Czakon M.

Roczniki Filozoficzne

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2021

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vol. 69

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issue 2

241-268

EN

It is common for different types of mathematical structuralism that the conjunction of two statements ( a) mathematics is science about structures and b) mathematics is deductive science) is true, Distinct arguments for this two features of mathematics are exanimated therefore the main concepts (structurality and deductivity) are understood differently, the results are various types of structuralism. We claim that it is possible to establish the way of understood of this two concepts in witeh they are equivalent. We argue that can interpret mathematical structuralism as equivalence: a) mathematics is science about structures if and only, if b) mathematics is deductive science

PL

Wspólne dla różnego typu strukturalizmów matematycznych jest stwierdzenie, że dla matematyki jako nauki prawdziwa jest koniunkcja: a) matematyka jest nauką o strukturach oraz b) matematyka jest nauką dedukcyjną. Przedstawiane są odmienne argumenty na rzecz tych dwóch własności matematyki i różnie rozumiane są pojęcia strukturalności i dedukcyjności, co skutkuje powstawaniem różnego rodzaju strukturalizmów. Twierdzimy, że przy pewnym ustalonym sposobie rozumienia tych pojęć możliwa jest ich równoważność. Argumentujemy na rzecz takiego rozumienia strukturalizmu, które streszcza się w stwierdzeniu: a) matematyka jest nauką o strukturach wtedy i tylko wtedy, gdy b) matematyka jest nauką dedukcyjną.

Refine search results

Set-theoretic mereology

Georg Cantor i idea jedności nauki

Tractarian Ontology: Mereology or Set Theory?

Filling assessment model of strategic cost management system (SCMS)

Aksjomat wyboru w pracach Wacława Sierpińskiego

Towards a Formal Ontology of Information. Selected Ideas of K. Turek

Nieskończoność w matematyce. Zmagania z potrzebnym, acz kłopotliwym pojęciem

Kilka uwag na temat koncepcji mnogościowych Richarda Dedekinda i Stanisława Leśniewskiego

Kryzys podstaw matematyki przełomu XIX i XX wieku

How to define a mereological (collective) set

Conceptual Classifications versus Collections of Objects in Biology

Pierwszy krok w chmurach: o teorii siedliska wydarzeniowego Alaina Badiou

Strukturalność a dedukcyjność matematyki: współczesny strukturalizm w filozofii matematyki