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O pewnych modyfikacjach teorii skierowanych liczb rozmytych

100%
Piasecki K.
Optimum. Studia Ekonomiczne
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2017
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vol. 3(87)
PL
Skierowane liczby rozmyte zostały zdefiniowane w doskonały i intuicyjny sposób przez Witolda Kosińskiego. Z tej przyczyny skierowane liczby rozmyte coraz częściej określa się mianem liczb Kosińskiego. W pierwszej części tej pracy zaproponowano w pełni sformalizowaną definicję liczby Kosińskiego. Definicję tę następnie uogólniono do przypadku skierowanej liczby rozmytej z nieciągłą funkcją przynależności. Istotną wadą arytmetyki zaproponowanej przez Kosińskiego był brak zamknięcia przestrzeni skierowanych liczb rozmytych ze względu na podstawowe działania arytmetyczne, takie jak: dodawanie, odejmowanie, mnożenie i dzielenie. Głównym celem prezentowanej pracy jest taka modyfikacja działań arytmetycznych, aby przestrzeń liczb Kosińskiego była zamknięta z racji zmodyfikowanych działań arytmetycznych.
EN
Ordered fuzzy numbers have been defined in an excellent, intuitive way by Witold Kosiński. For this reason, they are increasingly referred to as Kosiński’s numbers. A fully formalized definition of a Kosiński’s number is proposed in the first part of this work. This definition is generalized so as to fit an ordered fuzzy number with an upper semi-continuous membership function. A significant drawback of Kosiński’s arithmetic is that the space of ordered fuzzy numbers is not closed under addition, subtraction, multiplication, or division. The main aim of this paper is to modify the arithmetic in such a way that the space of ordered fuzzy numbers is closed under the modified arithmetic operations.
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Fregovo pojetí aplikace aritmetiky

88%
Sousedík P., Svoboda D.
Filosofický časopis (Philosophical Journal)
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2015
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vol. 63
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issue 3, Special No: Sedmkrát z logiky a metodologie vědy
101-124
EN
The authors believe that the problem of applicability can be approached in two ways. One approach derives from the fact that the empirical world has been the source of many mathematical concepts, and claims that arithmetic captures reality in the same way as common empirical disciplines. Its miraculous applicability can then be explained by the greater universality of the concepts used. Such an approach is designated a poste¬riori. The other approach to the problem of applicability, designated a priori, assumes that arithmetic is not grounded empirically, in fact it is already there before all expe¬rience. Upon analysis, both approaches authors’ view, these merits and shortcomings were already noticed by Frege. Though his conception is to be classified as an a priori approach, he – unlike his predecessors – also learned much from proponents of a posteriori conceptions.
3
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Matematyka nie opisuje świata, lecz wychodzi mu naprzeciw

71%
Mioduszewski J.
Zagadnienia Filozoficzne w Nauce
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2015
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issue 58
7-43
EN
In everyday experience mathematics rarely appears to us as a whole, and certainly never as a system in the sense of David Hilbert’s considerations from early 20th Century. Mathematical disciplines seem to be independent and autonomous. We do not see that specific deduction goes beyond particular convention applicable in given discipline. In the late 19th Century this view was shared by Felix Klein and Richard Dedekind. The latter’s work “What are numbers and what should they be?” (Was Was sind und was sollen die Zahlen?) was the inspiration for writing this article. This essay is an attempt to see mathematics not as a building, but as a living organism seeking its explanation.
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Gerbert a aritmetika : mezi filosofií čísla a počtářským uměním

63%
Otisk M.
Filosofický časopis (Philosophical Journal)
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2019
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vol. 67
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issue 4
503-520
EN
The attitude towards arithmetic in the Middle Ages was closely connected to antiquity’s opinion on the doctrine of the number, in whose framework practical arithmetic (calculating) and theoretical arithmetic (the theory or philosophy of numbers) were differentiated. The latter was traditionally assigned the greater importance in the area of philosophy, but the first had also been, from antiquity, perceived as a means that would lead to an understanding of the nature of numbers and which would cultivate the abstract and philosophical thinking of humans. This study discusses the example of the important, late 10th century thinker Gerbert of Aurillac (Pope Sylvester II) to show how it is possible to use arithmetic for philosophical research. First, the philosophy of numbers is presented, in which Gerbert (in connection with Boethius) reveals an arithmetically based reality, as numbers are also the thoughts of God. Practical arithmetic, for which Gerbert became especially famous (Western Arabic numerals and the abacus), then sharpened the human mind and facilitated the grasping of the nature of numbers.
CS
Středověký postoj k aritmetice úzce navazuje na antické nahlížení na nauku o čísle, v jejímž rámci byla rozlišována praktická (počtářství) a teoretická aritmetika (teorie či filosofie čísla). Druhé uvedené je v rámci filosofie tradičně připisována větší důležitost, ale také první je již od antiky vnímána jako prostředek vedoucí k pochopení povahy čísel, jež rozvíjí lidské abstraktní a filosofické myšlení. Tato studie ukazuje na příkladu významného myslitele konce 10. století, Gerberta z Remeše (papeže Silvestra II.), jak lze aritmetiku užít k filosofickému zkoumání. Nejprve je představena filosofie čísla, která u Gerberta (v návaznosti na Boëthia) odhaluje aritmetické založení reality, neboť čísla jsou zároveň myšlenkami Božími. Praktická aritmetika, v níž se Gerbert zejména proslavil (západoarabské číslice a abakus), pak bystří lidskou mysl a umožňuje pochopení povahy čísel.
DE
Der mittelalterliche Standpunkt zur Arithmetik ist eng mit dem antiken Blick auf die Zahlenlehre verbunden, in deren Rahmen die praktische Arithmetik (Rechenkunst) von der theoretischen Arithmetik (Theorie und Philosophie der Zahl) unterschieden wurde. Der zweitgenannten wird in der Philosophie traditionsgemäß größere Wichtigkeit beigemessen, aber auch die praktische Arithmetik wurde bereits in der Antike als Mittel zum Verständnis der Natur der Zahlen angesehen, was wiederum die Entwicklung des abstrakten und philosophischen Denkens fördert. In der vorliegenden Studie wird anhand des Beispiels eines bedeutenden Denkers vom Ende des 10. Jahrhunderts, Gerbert von Reims (Papst Silvester II.), gezeigt, wie die Arithmetik für die philosophische Erörterung verwendet werden kann. Dabei wird zunächst die Philosophie der Zahl vorgestellt, die bei Gerbert (in Anknüpfung an Boethius) das arithmetische Fundament der Wirklichkeit aufdeckt, da Zahlen gleichzeitig Gedanken Gottes sind. Die praktische Arithmetik wiederum, in der Gerbert insbesondere hervortrat (westarabische Ziffern und der Abakus), schärft den menschlichen Verstand und ermöglicht das Verständnis der Natur der Zahlen.
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Uwagi o arytmetyce Grassmanna

63%
Hanusek J.
Diametros
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2015
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issue Diametros 45
107-121
EN
Hermann Grassmann’s 1861 work [2] was probably the first attempt at an axiomatic approach to arithmetic (of integers with a distinguished subset of positive ones). The historical significance of this work is enormous, even though the set of axioms has proven to be incomplete. Basing on the interpretation of Grassmann’s theory provided by Hao Wang in [4], I present its detailed discussion, define the class of models of Grassmann’s arithmetic and discuss a certain axiom system for integers, modeled on Grassmann’s theory. At the end I propose to modify the set of axioms of Grassmann’s arithmetic, which consists in adding an elementary sentence and removing a non-elementary one. I prove that after this modification the only model of the theory up to isomorphism is the standard model.
PL
Praca Hermanna Grassmanna z roku 1861 była pierwszą próbą aksjomatycznego ujęcia arytmetyki (liczb całkowitych z wyróżnionym podzbiorem liczb dodatnich). Znaczenie historyczne tej pracy jest ogromne, choć sama aksjomatyka okazała się niepełna. Opierając się na interpretacji teorii Grassmanna dokonanej przez HaoWanga [1957], przedstawiam szczegółowe jej omówienie i definiuję klasę modeli tej teorii. Na koniec podaję propozycję modyfikacji aksjomatyki arytmetyki Grassmanna, która polega na dodaniu pewnego zdania elementarnego i usunięciu zdania nieelementarnego. Przedstawiam dowód że po takiej modyfikacji teorii jej jedynym modelem z dokładnością˛ do izomorfizmu jest model standardowy.
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On Writing and Handwriting

51%
Kučera M.
Journal of Pedagogy
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2010
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vol. 1
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issue 2
11-28
EN
Writing is often considered secondary to the spoken language, as it is only coded sound-by-sound. But other scholars have demonstrated that writing is similar to ‘arithmetic’: a cognitive structuring, a shift to the meta-level (‘for the eye’). Handwriting (referred to here as the cursive writing in the sense of joined up handwriting, of ‘écriture liée’) differs from writing (in the first analysis): it has its own grammar composed of paradigmatic gestemes and tracemes and its own syntagmatic rules that connect them. In emotional terms, handwriting is designed to provide a special pleasure by its own drive (instinct, ‘Trieb’). But there is also cognitive aspect to it: the rapidity and fluidity of a cursive writing could be (in professional writing, for instance) more important (at the climax of the creative process) than it being legible for all eternity. The project of the new handwriting reform for Czech schools, abolishing the liaison between letters, is shown to be a modern and technically simplified form of calligraphy.
7
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Does Science Progress Towards Ever Higher Solvability Through Feedbacks Between Insights and Routines?

45%
Marciszewski W.
Studia Semiotyczne
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2018
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vol. 32
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issue 2
153-185
EN
The affirmative answer to the title question is justified in two ways: logical and empirical. (1) The logical justification is due to Gödel’s discovery (1931) that in any axiomatic formalized theory, having at least the expressive power of PA (Peano Arithmetic), at any stage of development there must appear unsolvable problems. However, some of them become solvable in a further development of the theory in question, owing to subsequent investigations. These lead to new concepts, expressed with additional axioms or rules. Owing to the so-amplified axiomatic basis, new routine procedures like algorithms, can be reached. Those, in turn, help to gain new insights which lead to still more powerful axioms, and consequently again to ampler algorithmic resources. Thus scientific progress proceeds to an ever higher scope of solvability. (2) The existence of such feedback cycles – in a formal way rendered with Turing’s systems of logic based on ordinal (1939) – gets empirically supported by the history of mathematics and other exact sciences. An instructive instance of such a process is found in the history of the number zero. Without that insight due to some ancient Hindu mathematicians there could not arise such an axiomatic theory as PA. It defines the algorithms of arithmetical operations, which in turn help intuitions; those, in turn, give rise to algorithmic routines, not available in any of the previous phases of the process in question. While the logical substantiation of the point of this essay is a well-established result of logico-semantic inquiries, its empirical claim, based on historical evidences, remains open for discussion. Hence the author’s intention to address philosophers and historians of science, and to encourage their critical responses.
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