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            <title xml:lang="en">The institutional relativity of mathematical organizations.</title>
            <title xml:lang="fr">La relativité institutionnelle des organisations mathématiques</title>
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                <forename type="first">Yves</forename>
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            <idno type="halRefHtml">&lt;i&gt;Caminhos da Educação Matemática em Revista 2023,&lt;/i&gt;, 13 (4), pp.135-158, 2023, ISSN 1983-7399</idno>
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                <title xml:lang="en">The institutional relativity of mathematical organizations.</title>
                <title xml:lang="fr">La relativité institutionnelle des organisations mathématiques</title>
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                <term xml:lang="en">relativity</term>
                <term xml:lang="en">Instititution</term>
                <term xml:lang="fr">relativité</term>
                <term xml:lang="fr">Institution</term>
                <term xml:lang="fr">transposition</term>
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              <p>How is mathematics organized at each stage of its transposition, didactic or, more generally, institutional? How do they satisfy the needs associated with the defining tasks of a given institution, whatever it may be? This article shows, from an example, the necessity to give up the point of view according to which mathematical objects would be intangible, identical to themselves, regardless of places, times and institutions. It shows, on the contrary, the variability of mathematical organizations concerning the same mathematical object. It is the theorem which, in some countries, bears the name of Thales. The article is interested, in various works, in the mathematical organizations around what can be considered as the same theorem. These works reveal the subjection of their authors to various institutions, carrying various questions. They are either producers of mathematics, or transpositive and noospherical, or transpositive producing teaching proposals. The chosen methodology is of a clinical type. The evidence from observation is confronted with theoretical frameworks provided by mathematics and anthropological theory of the didactic. The works analyzed range from Euclid's Elements to "modern" textbooks, using the notions of affine and vector spaces and elements of linear algebra.</p>
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              <p>Comment les mathématiques sont-elles organisées à chaque étape de leur transposition, didactique ou, plus généralement, institutionnelle ? Comment satisfont-elles les besoins associés aux tâches définitoires d'une institution donnée, quelle qu'elle soit ? Cet article montre, à partir d'un exemple, la nécessité de renoncer au point de vue selon lequel les objets mathématiques seraient intangibles, identiques à euxmêmes, quels que soient les lieux, les époques et les institutions. Il montre, au contraire, la variabilité des organisations mathématiques concernant un même objet mathématique. Il s'agit du théorème qui, dans certains pays, porte le nom de Thalès. L'article s'intéresse, dans divers ouvrages, aux organisations mathématiques autour de ce qui peut être considéré comme le même théorème. Ces ouvrages révèlent l'assujettissement de leurs auteurs à diverses institutions, portant diverses questions. Ces institutions sont soit productrices de mathématiques, soit transpositives et noosphériennes, soit transpositives produisant des propositions d'enseignement. La méthodologie choisie est de type clinique. Les indices issus de l'observation sont confrontés aux cadres théoriques fournis par les mathématiques et la théorie anthropologique du didactique. Les ouvrages analysés vont des Éléments d'Euclide aux manuels « modernes », utilisant les notions d'espaces affines et vectoriels ainsi que des éléments d'algèbre linéaire.</p>
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          <orgName>Département IRES Aix-Marseille Université</orgName>
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