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Metoda matematyki według B. Bolzano

100%
Dadaczyński J.
Zagadnienia Filozoficzne w Nauce
|
2006
|
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

ІСТОРИЧНІ ВІДОМОСТІ ЯК ЗАСІБ АКТИВІЗАЦІЇ ДІЯЛЬНОСТІ СТУДЕНТІВ ІЗ ЗАСВОЄННЯ НИЗКИ ФУНДАМЕНТАЛЬНИХ ГЕОМЕТРИЧНИХ ІДЕЙ

75%
Антонюк О.
Педагогічні науки: теорія, історія, інноваційні технології
|
2016
|
issue 2(56)
155-162
EN
The purpose of the article is to analyze the possibilities of mathematics’ history for activating students in mastering new geometric concepts, ideas and illustrate them by means of examples. The study used the following methods: theoretical analysis of a number of publications on the subject, synthesizing, deduction. The article refers to the need of increasing students’ cognitive activity for improving the educational process. The work aims to describe a row of aspects of cognitive activity formation by means of history of science. As a result of the analysis of a number of publications on the subject area there were highlighted problematic issues of didactic conditions development of the usage of historical information in the study of geometric disciplines. To illustrate the ability of this tool to increase the cognitive activity of students the article presents examples of biographical information and scientific heritage of R. Dekartes and N. Gulak. Philosophical and mathematical ideas of R. Dekart are as such to interest students and explain them the importance of the ideas of coordinating method of mathematics development. Works of N. Gulak are aimed at facilitating the realization of ideas of non-Euclidean geometry, to describe the role of Lobachevskyi in the appearance of a new hyperbolic geometry. In particular, the biography of N. Gulak with description of his participation in Cyril and Methodius Foundation, his courageous behavior during the criminal and judicial persecution, activities in exile and after, makes considerable emotional impression. Examples of the usage of history of science which are presented in the work eloquently testify to the power of its influence on the educational and training process. Thus, the use of historical material may be the reception that under thorough methodological preparation and systematic use is able to improve the educational process in general, have positive impact on the working atmosphere and students’ capacity for creative activity. And geometry gives many reasons for this, it explains the need of continuation of such research of the various topics that let seamlessly weave historical information in the context of training material program and thus activate the cognitive activity of students.
3
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Strukturalność a dedukcyjność matematyki: współczesny strukturalizm w filozofii matematyki

38%
Czakon M.
Roczniki Filozoficzne
|
2021
|
vol. 69
|
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.
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