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Recenzja z Fulvia Furinghetti, Alexander Karp (eds), Researching the History of Mathematics Education. An International Overview, Springer, 2018, ss. 314 +XV.

100%

Błaszczyk P.

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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2019

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tom 11

167-171

EN

Recenzja

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From Euclid's Elements to the methodology of mathematics. Two ways of viewing mathematical theory

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Błaszczyk P.

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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2018

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tom 10

5-15

EN

We present two sets of lessons on the history of mathematics designed for prospective teachers: (1) Euclid's Theory of Area, and (2) Euclid's Theory of Similar Figures. They aim to encourage students to think of mathematics by way of analysis of historical texts. Their historical content includes Euclid's Elements, Books I, II, and VI. The mathematical meaning of the discussed propositions is simple enough that we can focus on specific methodological questions, such as (a) what makes a set of propositions a theory, (b) what are the specific objectives of the discussed theories, (c) what are their common features. In spite of many years' experience in teaching Euclid's geometry combined with methodological investigations, we cannot offer any empirical findings on how these lectures have affected the students' views on what a mathematical theory is. Therefore, we can only speculate on the hypothetical impact of these lectures on students.

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Nota o rozprawie Richarda Dedekinda Stetigkeit und irrationale Zahlen

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Błaszczyk P.

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

|

2017

|

tom 9

153-168

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A Purely Algebraic Proof of the Fundamental Theorem of Algebra

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Błaszczyk P.

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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2016

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tom 8

7-23

PL

Proofs of the fundamental theorem of algebra can be divided upinto three groups according to the techniques involved: proofs that rely onreal or complex analysis, algebraic proofs, and topological proofs. Algebraicproofs make use of the fact that odd-degree real polynomials have real roots.This assumption, however, requires analytic methods, namely, the intermediatevalue theorem for real continuous functions. In this paper, we developthe idea of algebraic proof further towards a purely algebraic proof of theintermediate value theorem for real polynomials. In our proof, we neither usethe notion of continuous function nor refer to any theorem of real and complexanalysis. Instead, we apply techniques of modern algebra: we extend thefield of real numbers to the non-Archimedean field of hyperreals via an ultraproductconstruction and explore some relationships between the subringof limited hyperreals, its maximal ideal of infinitesimals, and real numbers.

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Nota o Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen Georga Cantora

100%

Błaszczyk P.

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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2016

|

tom 8

159-168

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Nota o rozprawie Eduarda Heinego Elemente der Functionenlehre

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Błaszczyk P.

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

|

2014

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tom 6

139-152

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Nota o rozprawie Otto Höldera Die Axiome der Quantität und die Lehre vom Mass

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Błaszczyk P.

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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2013

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tom 5

129-142

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Nota o Lehrbuch der Algebra. Einlaitung Heinricha Webera

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Błaszczyk P.

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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2015

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tom 7

129-139

PL

Historia matematyki

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XXX Konferencja z Historii Matematyki, Bedlewo, 5–8 V 2016

100%

Błaszczyk P.

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

|

2015

|

tom 7

155-163

PL

Sprawozdanie z konferencji

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Nota o Über den Zahlbegriff Davida Hilberta

100%

Błaszczyk P.

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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2012

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tom 4

195-197

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O ciałach uporządkowanych

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Błaszczyk P.

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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2012

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tom 4

15-30

PL

In this paper, we present some basic facts concerning ordered fields. We review definitions of an ordered field, give an example of a field that admits many orderings, and present equivalent definitions of the axiom of Archimedes and the continuity axiom. We show how to extend an ordered field by means of an ultrapower construction and formal power series.

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Euclid’s theory of proportion revised

64%

Błaszczyk P., Petiurenko A.

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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2019

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tom 11

37-62

PL

Teoria pola po raz pierwszy została opisana w pracy Chou, Gao, Zhang w 1994 roku. W kolejnej pracy (Janicic, Narboux, Quaresma 2012) zaprezentowano nowy system aksjomatów teorii pola i program przeznaczony do automatycznego dowodzenia twierdzen. W artykule chcemy przedstawić interpretację teorii pola w geometrii analitycznej na płaszczyznie kartezjanskiej R×R z porządkiem leksykograficznym. Również pokażemy nową metodę dowodzenia twierdzeń geometrycznych (szczególnie twierdzeń z ksiegi VI Elementów Euklidesa), w której pole trójkąta wystepuje w dowodach (szczególnie w proporcji) jako element pierwotny (wzór na pole trójkąta wprowadza się, jako aksjomat). Podobną metodę stosował Euklides na objektach geometrycznych bez użycia liczb. W omawianej teorii pole trójkąta jest liczbą, a twierdzenie VI.1 Elementów, podstawowe dla teorii Euklidesa, jest przyjmowane jako aksjomat. W artykule również omówimy mało znaną własność, która jest modyfikacją twierdzenia VI.1: w miejsce proporcji trójkątów o wspólnej wysokości, wykorzystuje proporcje trójkątów o wspólnej podstawie.

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O liczbach ujemnych z perspektywy historycznej i dydaktycznej

64%

Błaszczyk P., Sajka M.

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

|

2017

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tom 9

5-36

EN

We identify two ways of introducing negative numbers. In the first one, a totally ordered set (L, $\prec $) is presupposed, an element 0 in L is arbitrarily taken, and a number a is negative when a 0. In the second one, a negative number is defined by the formula a + (−a) = 0. From a mathematical perspective, the first method involves the idea of a totally ordered group (G,+, 0,<), while the second one considers the idea of the algebraic group (G,+, 0) alone. Through the analysis of source texts, we show that the first model originates in John Wallis’ 1685 Treatise of Algebra, while the second one comes to form the theory of polynomials, as developed by Descartes in his 1637 La Géométrie. In mathematical education, the first model is applied in the overwhelming majority. Still, we identify a theory that applies to the second model. We show how to develop it further and simplify the representation of the operation a + (−a) = 0 by turning the second model into a tablet game.

PL

We identify two ways of introducing negative numbers. In the first one, a totally ordered set (L, $\prec $) is presupposed, an element 0 in L is arbitrarily taken, and a number a is negative when a 0. In the second one, a negative number is defined by the formula a + (−a) = 0. From a mathematical perspective, the first method involves the idea of a totally ordered group (G,+, 0,<), while the second one considers the idea of the algebraic group (G,+, 0) alone. Through the analysis of source texts, we show that the first model originates in John Wallis’ 1685 Treatise of Algebra, while the second one comes form the theory of polynomials, as developed by Descartesin his 1637 La Géométrie. In mathematical education, the first model is applied in the overwhelming majority. Still, we identify a theory that applies to the second model. We show how to develop it further and simplify the representation of the operation a + (−a) = 0 by turning the second modelin to a tablet game.

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Calculus without the concept of limit

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Błaszczyk P., Major J.

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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2014

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tom 6

19-40

PL

There are two different approaches to nonstandard analysis: semantic(model-theoretic) and syntactic (axiomatic). Both of these approachesrequire some knowledge of mathematical logic. We present a method basedon an ultrapower construction which does not require any mathematical logicprerequisites. On the one hand, it is a complementary course to a standardcalculus course. On the other hand, since it relies on a different intuitivebackground, it provides an alternative approach. While in standard analysisan intuition of being close is represented by the notion of limit, in nonstandardanalysis it finds its expression in the relation is infinitely close. Asa result, while standard courses focus on the " − technique, we explorean algebra of infinitesimals. In this paper, we offer a proof of the theoremon the equivalency of limits and infinitesimals, showing that calculus can bedeveloped without the concept of limit.

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Cantor on Infinitesimals. Historical and Modern Perspective

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Błaszczyk P., Fila M.

Bulletin of the Section of Logic

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2020

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tom 49

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nr 2

EN

In his 1887's Mitteilungen zur Lehre von Transfiniten, Cantor seeks to prove inconsistency of infinitesimals. We provide a detailed analysis of his argument from both historical and mathematical perspective. We show that while his historical analysis are questionable, the mathematical part of the argument is false.

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Wprowadzenie do rozprawy Bernarda Bolzano Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwei Werthen, die ein entgegengesetzetes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege

52%

Błaszczyk P., Fila M., Mrówka K.

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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2017

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tom 9

185-195

Ograniczanie wyników

15 Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

1 Bulletin of the Section of Logic

16 Błaszczyk P.

2 Fila M.

1 Major J.

1 Mrówka K.

1 Petiurenko A.

1 Sajka M.

1 2020

2 2019

1 2018

3 2017

2 2016

2 2015

2 2014

1 2013

2 2012

Recenzja z Fulvia Furinghetti, Alexander Karp (eds), Researching the History of Mathematics Education. An International Overview, Springer, 2018, ss. 314 +XV.

From Euclid's Elements to the methodology of mathematics. Two ways of viewing mathematical theory

Nota o rozprawie Richarda Dedekinda Stetigkeit und irrationale Zahlen

A Purely Algebraic Proof of the Fundamental Theorem of Algebra

Nota o Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen Georga Cantora

Nota o rozprawie Eduarda Heinego Elemente der Functionenlehre

Nota o rozprawie Otto Höldera Die Axiome der Quantität und die Lehre vom Mass

Nota o Lehrbuch der Algebra. Einlaitung Heinricha Webera

XXX Konferencja z Historii Matematyki, Bedlewo, 5–8 V 2016

Nota o Über den Zahlbegriff Davida Hilberta

O ciałach uporządkowanych

Euclid’s theory of proportion revised

O liczbach ujemnych z perspektywy historycznej i dydaktycznej

Calculus without the concept of limit

Cantor on Infinitesimals. Historical and Modern Perspective

Wprowadzenie do rozprawy Bernarda Bolzano Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwei Werthen, die ein entgegengesetzetes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege