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2007 | 30 |
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Replacing objects of mathematics by other objects with the same name

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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.
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