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Metoda matematyki według B. Bolzano

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Dadaczyński J.
Zagadnienia Filozoficzne w Nauce
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2006
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issue 38
76-113
PL
The matter under discussion is the methodology of mathematics presented by Bernard Bolzano (1782-1848) in his early pamphlet 'Beitraege zu einer begruendeteren Darstellung der Mathematik' (Prague 1810). Bolzano built, with success, the classical axiomatic-deductive method of nonspacial and atemporal concepts (Begriffe). He abandoned the traditional custom of formulating primitive concepts of deductive theories. Bolzano opposed the traditional conviction that the axioms of mathematical theories should be clear and distinct sentences. He divided the domain of nonspacial and atemporal sentences into the subdomains of objectively provable and objectively nonprovable sentences. In his view, the axioms of mathematical (deductive) theories are only the objectively nonprovable sentences, and each of the objective nonprovable sentences is an axiom of a certain deductive theory. He postulated, at the time when only the (Euclidean) geometry was axiomatized, the axiomatization of all mathematical theories.
2
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Niejasności związane z Bernarda Bolzana 'definicją' zbioru nieskończonego

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Dadaczyński J.
Zagadnienia Filozoficzne w Nauce
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2009
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issue 44
100-115
PL
The aim of the paper is to remove the ambiguities contained in the 20th Section Bernard Bolzano's 'Paradoxien des Unendlichen'. Conclusion is that Bolzano originally wanted only to present in the 20th Section one of the (many) paradoxes of infinite sets: an infinite set may be equivalent with its subset. The proof in the 20th Section applies only to this property. Later added Bolzano to the text of the 20th Section a generalizing remark. Only this remark suggests that Bolzano anticipated the reflexive (Dedekindian) definition of the infinity set.
3
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Bernard Bolzano on colours, shapes, and non-figurative painting

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Hlobil T., Stephens P.
Umění (Art)
|
2023
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vol. 71
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issue 4
347-355
EN
In the analyses that have appeared so far of Bernard Bolzano’s posthumously published treatise On the Classification of the Fine Arts [Über die Eintheilung der schönen Künste], a salient opinion is that the Prague philosopher was an original writer who anticipated twentieth-century modernist non-figurative painting. A progressive updating of Bolzano’s reflections, relating them to twentieth-century non-figurative painting, significantly contributed to establishing the place of his long-forgotten aesthetic treatises in the history of European aesthetics. At the same time, however it tended to obscure the original contents of his writings. Working from these “updating” attempts, it is not possible to gain a sufficiently clear picture of what Bolzano really thought about colour and shape on the one hand, and about drawing, painting, and the painter’s art on the other. The study which follows presents these views of Bolzano on the basis of his two published aesthetic treatises, On the Classification of the Fine Arts and On the Concept of the Beautiful [Über den Begriff des Schönen, 1843], with the aim of showing that Bolzano never did elaborate a theory of non-figurative painting, and indeed that in his reflections on drawing, painting, and the painter’s art, he did not discuss at all monochrome, the autonomy of colours and shapes, or their connection with the sublime.
CS
Z dosavadních výkladů posmrtně publikovaného pojednání Bernarda Bolzana O dělení krásných umění (1849) vynikají postřehy představující pražského filozofa jako originálního autora předjímajícího modernistické nefigurativní malířství dvacátého století. Progresivní aktualizace Bolzanových úvah směrem k nefigurativnímu malířství dvacátého století významně napomohly proniknutí jeho dlouhou dobu opomíjených estetických pojednání do dějin evropské estetiky. Zároveň ale zastřely jeho samotný výkon. Jaké byly skutečné Bolzanovy názory na barvu a tvar na straně jedné a na kresbu, malbu a malířství na straně druhé, se z nich nelze dostatečně dozvědět. Následující studie představuje tyto názory na základě obou publikovaných estetických pojednání, O dělení krásných umění a O pojmu krásna (1843), s cílem ukázat, že Bolzano nedospěl k teorii nefigurativního malířství, ba že v úvahách o kresbě, malbě a malířství o monochromii, autonomii barev a tvarů či jejich sepětí se vznešenem vůbec nepojednal.
4
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Bernard Bolzano: pierwsze (historycznie) matematyczne ujęcie pojęcia kontinuum

88%
Dadaczyński J.
Zagadnienia Filozoficzne w Nauce
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2018
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issue 64
195-199
EN
Book review: Lukas Benedikt Kraus, Der Begriff des Kontinuums bei Bernard Bolzano, Beiträge zur Bolzano-Forschung, vol. 25, Academia Verlag, Sankt Augustin 2014, pp. 112.
PL
Recenzja książki: Lukas Benedikt Kraus, Der Begriff des Kontinuums bei Bernard Bolzano, Beiträge zur Bolzano-Forschung, vol. 25, Academia Verlag, Sankt Augustin 2014, ss. 112.
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Zamyšlení Petra Vopěnky nad Bolzanem

63%
Šebestík J.
Filosofický časopis (Philosophical Journal)
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2016
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vol. 64
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issue 4
493-511
EN
For Petr Vopěnka, Bolzano was not only an important figure in the history of mathe­matics, but also a source of inspiration and a locus of confrontation. Almost in every one of his works, Bolzano takes an essential place, especially when Vopěnka talks about set theory. He dedicated a whole book Podivuhodný květ českého baroka (The Wonderful Flower of the Czech Baroque, Praha, Karolinum 1998) to Bolzano’s posthumous Paradoxy nekonečna (Paradoxes of the Infinite, 1851). There he returns to the native grounds of modern science, namely the discussions and quarrels between different Christian sects in the first four centuries, followed after more than a thousand years by Protestantism. His pilgrimage leads us to Cervantes, the Spanish mystics and the Spanish baroque. All this goes together with a discussion of the infinite according to Augustine, Thomas Aquinas, Giordano Bruno, Galilei, Rodrigo de Arriaga and others. While Medieval scholasticism showed reticence towards actual infinity, modern science (Gauss, Cauchy, etc.) has refused it without hesitation and has returned to Aristo­telian potential infinity. Bolzano, educated in the 18th century neo-scholastic tradition in Bohemia, was the first important mathematician to have introduced actual infinite collections and sets into mathematics. In section 20 of his Paradoxes, he stated a characteristic property of infinite sets: their reflexivity (the possibility to put into 1-1 correspondence a set with some of its infinite subsets). Vopěnka interprets this text as a general collapse, namely the possibility to reduce all Cantorian cardinalities of infinity to one: that of the natural numbers. He also shows, why Bolzano sought his infinite numbers not among Cantor´s cardinalities, but among infinite sums of real numbers.
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