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1

O definicji 7 z Księgi V „Elementów” Euklidesa

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Błaszczyk P.

Zagadnienia Filozoficzne w Nauce

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2010

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issue 46

118-140

PL

Euclid's 'Elements' Book V develops theory of proportion of 'geometric magnitudes'. Definition V,5 is the definition of proportion, A:B::C:D, definition V,7 is the definition of the order of ratios, A:B - C:D. In commentaries on Book V it is usually supposed, and sometimes even proved, that the order of ratios is a total order, while it is also supposed that 'magnitudes of the same kind' obey the Archimedean axiom only, i.e. Euclid's definition V,4. The purpose of this paper is to show that the linearity of the order of ratios cannot be deduced from the Archimedean axiom; to this end we define a structure of magnitudes that obeys the Archimedean axiom and show that the conjunction of negations is satisfied (for mathematical symbols applied see the original paper).

2

On the Mathematical Object

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Błaszczyk P.

Filozofia Nauki

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2004

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vol. 12

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issue 2

5-19

PL

In this paper we show that the field of the real numbers is an intentional object in the sense specified by Roman Ingarden in his Das literarische Kunstwer and Der Streit um die Existenz der Welt. An ontological characteristics of a classic example of an intentional object, i.e. a literary character, is developed. There are three principal elements of such an object: the author, the text and the entity in which the literary character forms the content. In the case of the real numbers the triad consists of Richard Dedekind, his work Steitigkeit und irratinale Zahlen, and the intentional object determined by this work. Showing that we are indeed faced with the inten-tional object we analyse three moments: two-sidedness of the formal structure of the intentional object, the moment of its existentional derivation, and schematism of the intentional object. Since there are many constructions of the real numbers known in mathematics, we show what relation obtains between Dedekind's and Cantor's constructions of the real numbers as these are taken as intentional objects. Moreover, we show what relation obtains between Dedekind's construction and the axiomatic characteristics of the real numbers.

3

Fragments of Ingarden's Ontology. On Schematism of a Purely Intentional Object

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Błaszczyk P.

Filozofia Nauki

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2009

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vol. 17

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issue 4

71-93

PL

In this paper, we present a reinterpretation of Roman Ingarden's theory of intentional objects. There are four types of intentional objects in Ingarden's ontology, we offer a detailed analyses of an intentional object that is a correlate of a text. Such an object is characterised by Ingarden as a two-sided and schematised formation. We focus on the notion of schematism. We classify different interpretations of schematism and propose our own definition of schematism of a purely intentional object.

4

Eudoxos versus Dedekind

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Błaszczyk P.

Filozofia Nauki

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2007

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vol. 15

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issue 2

95-113

PL

All through the XXth century it has been repeated that "there is an exact correspondence, almost coincidence between Euclid's definition of equal ratios and the modern theory of irrational numbers due to Dedekind". Since the idea was presented as early as in 1908 in Thomas Heath's translation of Euclid's Elements as a comment to Book V, def. 5, we call it in the paper Heath's thesis. Heath's thesis finds different justifications so it is accepted yet in different versions. In the paper its historical and mathematical version is reconstructed. We next reconstruct Eudoxos' theory of proportions in an axiomatic fashion. Finally, we show that Heath's thesis both in the historical and mathematical version is false. To this end a counterexample is given; it is based upon a specific interpretation of the uniform distribution theorem.

5

Galileo’s paradox and numerosities

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Błaszczyk P.

Zagadnienia Filozoficzne w Nauce

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2021

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issue 70

73-107

EN

Galileo's paradox of infinity involves comparing the set of natural numbers, N, and the set of squares, {n2 : n ∈ N}. Galileo (1638) sets up a one-to-one correspondence between these sets; on this basis, the number of the elements of N is considered to be equal to the number of the elements of {n2 : n ∈ N}. It also characterizes the set of squares as smaller than the set of natural numbers, since ``there are many more numbers than squares". As a result, it concludes that infinities cannot be compared in terms of greater--lesser and the law of trichotomy does not apply to them. Cantor's cardinal numbers provide a measure for sets. Cantor (1897) gives a definition of the relation greater–lesser between cardinal numbers and establishes the law of trichotomy for these numbers. Yet, when Cantor's theory is applied to subsets of N, it gives that any set can be either finite or of the power ℵ0. Thus, although the set of squares is the subset of N, they are of the same cardinality. Benci, Di Nasso (2019) introduces specific numbers to measure sets called numerosities. With numerosities, the following claim is true: numerosity of A < numerosity of B, whenever A ⊈ B. In this paper, we present a simplified version of the theory of numerosities that applies to subsets of N. This theory complies with Galileo's presupposition that when A ⊈ B, then the number of elements in A is smaller than the number of elements in B. Specifically, we show that as the numerosity of N is the number α, the numerosity of the set of squares is the integer part of the number √α, that is ⌊√α⌋, and the inequality ⌊√α⌋ < α holds.

6

O definicji 5 z Księgi V Elementów Euklidesa

100%

Błaszczyk P.

Investigationes Linguisticae

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2006

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vol. 14

120-146

EN

It is well known fact that there are two definitions of proportion in Euclid's Elements: Book V, def. 5 and Book VII, def. 20. In the present paper we show that three different interpretations of definition V.5 can be given as modern notation is used: two of them arise from different readings of the definition itself, the third is a negation of disproportion (V, def. 7).

7

Miedzy oczywistością a dedukcją. Platon i Euklides o równości

63%

Błaszczyk P., Mrówka K.

Zagadnienia Filozoficzne w Nauce

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2011

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issue 48

127-147

PL

We confront Plato's understanding of equality in geometry with that of Euclid. We comment on Phaedo, 74b-c, Meno, 81e-85d and Elements, Book I. We distinguish between two meanings of equality, congruence and equality of the area, and show that in Plato equality means congruence. In Euclid, starting with the first definitions until Proposition I.34, equality means congruence. In the proof of Proposition I.35 equality gains a new meaning and two figures that are not congruent, and in this sense unequal, are considered to be equal. While Plato's geometry is based on self-evident facts, Euclid's geometry rests on deduction and the axioms that are by no means self-evident. However, the shift of meaning from congruence to equality of the area can be substantiated by reference to Euclid's axioms of equality. Finally, we present an ontological interpretation of the two attitudes to equality that we find in Plato's and Euclid's writings.

8

Euclid and Aristotle about Continuity. Part I. Euclid

63%

Błaszczyk P., Mrówka K.

Filozofia Nauki

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2013

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vol. 21

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issue 4

91-115

PL

Line segment is a kind of ancient Greek μέγεθος. It is described mathematically in Euclid’s Elements and in a philosophical way in Aristotle’s Physics. In this first part of our paper we present Euclid’s twofold attitude toward a line segment: the first one developed in his theory of proportion of magnitudes (book V), the second in his plain geometry (books I-IV). Euclid’s magnitudes are of several different kinds: lines segments, triangles, convex polygons, arcs, angles. Magnitudes of the same kind can be added to one another and compared as greater–lesser. We provide a set of axioms for the line segments system (M, +, <) and show that the total order of segments < is compatible with the addition operation +. The positive part of an Archimedean field is a model of these axioms. Next, we present an interpretation of Euclid’s proposition I.10 and show that Aristotle’s famous saying “everything continuous is divisible into divisibles that are infinitely divisible” applies to a single line segment. Our study is based on Heiberg’s Euclidis Elementa.

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O definicji 7 z Księgi V „Elementów” Euklidesa

On the Mathematical Object

Fragments of Ingarden's Ontology. On Schematism of a Purely Intentional Object

Eudoxos versus Dedekind

Galileo’s paradox and numerosities

O definicji 5 z Księgi V Elementów Euklidesa

Miedzy oczywistością a dedukcją. Platon i Euklides o równości

Euclid and Aristotle about Continuity. Part I. Euclid