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Tytuł:
Creating new concepts in mathematics: freedom and limitations. The case of Category Theory
Autorzy:
Semadeni, Zbigniew
Powiązania:
https://bibliotekanauki.pl/articles/1047591.pdf
Data publikacji:
2020-12-28
Wydawca:
Copernicus Center Press
Tematy:
categories
functors
Eilenberg-Mac Lane Program
mathematical cognitive transgressions
phylogeny
platonism
Opis:
In the paper we discuss the problem of limitations of freedom in mathematics and search for criteria which would differentiate the new concepts stemming from the historical ones from the new concepts that have opened unexpected ways of thinking and reasoning. We also investigate the emergence of category theory (CT) and its origins. In particular we explore the origins of the term functor and present the strong evidence that Eilenberg and Carnap could have learned the term from Kotarbiński and Tarski.
Źródło:
Zagadnienia Filozoficzne w Nauce; 2020, 69; 33-65
0867-8286
2451-0602
Pojawia się w:
Zagadnienia Filozoficzne w Nauce
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Replacing objects of mathematics by other objects with the same name
Autorzy:
Semadeni, Zbigniew
Powiązania:
https://bibliotekanauki.pl/articles/748834.pdf
Data publikacji:
2007
Wydawca:
Polskie Towarzystwo Matematyczne
Opis:
The purpose of the paper is to introduce and analyse the following conception: by an ontic shift $X \tilde \rightarrow X''$ we mean an (explicit or hidden) replacement of a mathematical object $X'$ called $X$ by a different object $X''$ which is also called $X$ and is to play the role of $X'$. If $X'$ and $X''$ are sets, then $X' \neq X''$ means that these sets are different in the usual sense. Otherwise $X'$ and $X''$ may stand for basic concepts of mathematics such as, e.g., point, straight line, natural number, addition of natural numbers, (standard) real number, set, ordered pair, and then $X' \neq X''$ is interpreted in terms of the Leibniz principle of indiscernibility. Several examples from secondary and college mathematics are discussed starting with the ontic shift $P \tilde \rightarrow (x, y)$, where a point $P$ of the plane is replaced by a pair $(x, y)$ of real numbers, also called a point; an angle $\varphi$ (thought of as a geometric figure) is replaced by its measure which is also denoted by $\varphi$; a number (natural, integer, etc.) is replaced by a sophisticated set; a real number a is replaced by the complex number $(a, 0)$, and so on. We distinguish three types of ontic shifts: (α) object $X'$ is replaced by $X''$, and $X'$ is discarded; (β) object $X'$ is replaced by a conglomerate $X' \coprod X''$ of two alternatively and exibly used objects $X'$ and $X''$; (γ) object $X'$ is replaced by a new object which is a mental synthesis $X' \& X''$ integrating features of $X'$ and $X''$. What is crucial in ontic shifts is the change of approach: $X'$ may be identified with $X''$ by declaring that: $X'$ is the same as $X''$. For example, "a function may be identified with a set of pairs" is replaced by "a function is the same as its graph", i.e., a function becomes a set of pairs. In turn, a set of pairs of real numbers may be identified with a geometric figure. If such ontic shifts were composed, a function such as, say, $x \mapsto \sin x$ would be the same as a geometric figure. Such arguments show that ontic shifts are often local in the sense that they apply only to a certain part of mathematics and composition of two identifications may be unacceptable. A point in $\mathbb{R}^3$ may be replaced by the corresponding vector and one may say: "A point is the same as a vector". Also a vector may be replaced by the corresponding translation and then we may say that "A vector is the same as a translation". This identification, however, is not transitive. To see that the statement "point is the same as translation" is unacceptable imagine a student who is asked "How do we define a translation of 3D-space?" and answers: "A translation is an arbitrary point of the space". The paper also deals with distinguishing between ontic shifts and metonymies and with the delicate border between two phenomena: (a) a change of the linguistic form, and (b) replacing one mathematical object by another. Although labelling different objects with the same name does not fit the stereotypical image of mathematics, ontic shifts are its significant feature. They should be used in an elastic way, with proper understanding of the concepts involved. However, if such shifts are introduced prematurely, they cause serious didactical troubles.
Źródło:
Didactica Mathematicae; 2007, 30
2353-0960
Pojawia się w:
Didactica Mathematicae
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Stany i działania na stanach jako aspekty znaczeniowe pojęć matematycznych
States and actions on states as meaning-based aspects of mathematical concepts
Autorzy:
Semadeni, Zbigniew
Powiązania:
https://bibliotekanauki.pl/articles/749535.pdf
Data publikacji:
2004
Wydawca:
Polskie Towarzystwo Matematyczne
Źródło:
Didactica Mathematicae; 2004, 27, 01; 169-192
0208-8916
2353-0960
Pojawia się w:
Didactica Mathematicae
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Aspekty znaczeniowe i aspekty strukturalne pojęć matematycznych
Meaning-based aspects and structural aspects of mathematical concepts
Autorzy:
Semadeni, Zbigniew
Powiązania:
https://bibliotekanauki.pl/articles/749541.pdf
Data publikacji:
2004
Wydawca:
Polskie Towarzystwo Matematyczne
Źródło:
Didactica Mathematicae; 2004, 27, 01; 151-168
0208-8916
2353-0960
Pojawia się w:
Didactica Mathematicae
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Koncepcja sieci wzajemnych powiązań idei głębokich i powiązań ich modeli formalnych
A conception of a web of ties between deep ideas a rid ties between their formal models
Autorzy:
Semadeni, Zbigniew
Powiązania:
https://bibliotekanauki.pl/articles/749551.pdf
Data publikacji:
2005
Wydawca:
Polskie Towarzystwo Matematyczne
Źródło:
Didactica Mathematicae; 2005, 28, 01; 311-338
0208-8916
2353-0960
Pojawia się w:
Didactica Mathematicae
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Koncepcja idei głębokich epistemicznych i idei głębokich indywidualnych w matematyce
Concept of Epistemic Deep Ideas and Individual Deep Ideas in Mathematics
Autorzy:
Semadeni, Zbigniew
Powiązania:
https://bibliotekanauki.pl/articles/967216.pdf
Data publikacji:
2012-12-01
Wydawca:
Uniwersytet Warszawski. Wydział Filozofii
Opis:
The aim of this paper is to present a conception of the triple nature of mathematics. It is argued that the nature of mathematics is best served by distinguishing deep ideas (of concepts or propositions), their surface representations (signs which can be perceived by senses) and their formal models (in axiomatic theories). For instance, the concept „number π” has several different models in set theory (those based on Dedekind cuts and on Cantor's equivalence classes of Cauchy sequences) and yet all working mathematicians in the world have the same object π in mind. They have a common deep idea of π. Generally, the deep idea of a concept X is a well-formed mental construction of X which controls reasoning. It manifests itself in a characteristic, definite feeling of purpose, in firm certainty of the meaning of X in various contexts, and in robustness of understanding of X in cases of typical cognitive conflicts. Epistemic deep ideas are intersubjective and have been formed in phylogeny whereas individual deep ideas (or deep intuitions) are formed in ontogeny. In certain situations a deep idea may be described in terms of intuition, of meaning or sense, or of understanding, but none of these terms can provide a satisfactory description fitting all cases. Deep ideas of certain concepts are identified in authoritative texts where the actual use of the concept formally, although unnoticeably, conflicts with the declared definition. Specific examples, discussed in the paper, include: vertex of a straight angle; switching the meaning of fraction from a single number to a pair numeratordenominator; identifying vector with point and with translation. A peculiar anomaly is known in axiomatic set theory. The standard definitions are: an ordered pair is (a1,a2)={{a1}, {a1,a2}}; a function f : X›Y is a set of pairs; a sequence (a1,…,an) is a function on the set {1,…,n}; an ordered pair is the same as a two-term sequence (a1,a2), which is different from {{a1}, {a1,a2}}. This is an unavoidable definitional loop; however, it does not affect reasoning, for mathematicians use the deep idea of a pair, and not the definition. An example is given to show that two geometric phrases with analogous surface grammatical structures (with adjectives linked by „and”) may be interpreted differently (as the union or the intersection of the relevant sets), depending on their deep linguistic structures. The transitional mechanisms in the history of science and psychological development described by J. Piaget and R. Garcia, in particular those leading from the intra level (object analysis), to the inter level (analysing relations or transformations), and then to the trans level (building structures), may be used to outline the formation processes of epistemic deep ideas and those of individual ones; in each progression what gets surpassed is always integrated with the new (transcending) structure. Metaphorically, the deep idea of a concept is built on layers of earlier constructions and meanings.
Źródło:
Filozofia Nauki; 2012, 20, 4; 119-138
1230-6894
2657-5868
Pojawia się w:
Filozofia Nauki
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Reformy programów i podręczników szkolnych sprzed pół wieku inspirowane prądami tzw. nowej matematyki
Reforms of curricula and school textbooks inspired by the so-called new math half a century ago
Autorzy:
Semadeni, Zbigniew
Powiązania:
https://bibliotekanauki.pl/articles/1789250.pdf
Data publikacji:
2020-12-31
Wydawca:
Uniwersytet Pedagogiczny im. Komisji Edukacji Narodowej w Krakowie
Tematy:
New Math
sets
axiomatic geometry
curricula
reforms
teachers
Polska
Opis:
The purpose of this paper is to outline the reforms of mathematics education in the spirit of “New Math” in USA, France and to document their features in Poland. Activities and achievements of Zofia Krygowska are listed. Particular attention is paid to two radical reforms: the 1967 textbook on geometry based on set-theory for grade IX and far-reaching changes in primary math education in the 1970s. Excerpts from articles, curricula and textbooks are included.
Źródło:
Annales Universitatis Paedagogicae Cracoviensis. Studia ad Didacticam Mathematicae Pertinentia; 2020, 12; 211-248
2080-9751
2450-341X
Pojawia się w:
Annales Universitatis Paedagogicae Cracoviensis. Studia ad Didacticam Mathematicae Pertinentia
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Enactive and iconic representations in Bruner’s sense exemplified by representation of set-theoretical notions
Autorzy:
Semadeni, Zbigniew
Powiązania:
https://bibliotekanauki.pl/articles/748924.pdf
Data publikacji:
1982
Wydawca:
Polskie Towarzystwo Matematyczne
Opis:
The article contains no abstract
Źródło:
Didactica Mathematicae; 1982, 1, 01
2353-0960
Pojawia się w:
Didactica Mathematicae
Dostawca treści:
Biblioteka Nauki
Artykuł
Wyświetlanie 1-10 z 23

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