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

Piotr Błaszczyk — 2018

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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 Their historical content includes Euclid's Elements, Books I, II, and VI.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...

O ciałach uporządkowanych

Piotr Błaszczyk — 2012

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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.

A Purely Algebraic Proof of the Fundamental Theorem of Algebra

Piotr Błaszczyk — 2016

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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...

Euclid’s theory of proportion revised

Piotr BłaszczykAnna Petiurenko — 2019

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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...

Calculus without the concept of limit

Piotr BłaszczykJoanna Major — 2014

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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...

O liczbach ujemnych z perspektywy historycznej i dydaktycznej

Piotr BłaszczykMirosława Sajka — 2017

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

We identify two ways of introducing negative numbers. In the first one, a totally ordered set (L, ) 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...

Cantor on Infinitesimals. Historical and Modern Perspective

Piotr BłaszczykMarlena Fila — 2020

Bulletin of the Section of Logic

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.

19th-century real analysis, forward and backward

Jacques BairPiotr BlaszczykPeter HeinigVladimir KanoveiMikhail Katz — 2019

Antiquitates Mathematicae

19th-century real analysis received a major impetus from Cauchy's work. Cauchy mentions variable quantities, limits, and infinitesimals, but the meaning he attached to these terms is not identical to their modern meaning. Some Cauchy historians work in a conceptual scheme dominated by an assumption of a teleological nature of the evolution of real analysis toward a preordained outcome. Thus, Gilain and Siegmund-Schultze assume that references to limite in Cauchy's work necessarily imply that Cauchy...

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