Negative Größen bei Diophant? Teil I
Identifieur interne : 000A52 ( Main/Exploration ); précédent : 000A51; suivant : 000A53Negative Größen bei Diophant? Teil I
Auteurs : Klaus Barner [Allemagne]Source :
- NTM International Journal of History and Ethics of Natural Sciences, Technology and Medicine [ 0036-6978 ] ; 2007-02-01.
Abstract
In this paper which consists of two parts (Teil I and Teil II) we champion Diophantus of Alexandria and Isabella Bašmakova against Norbert Schappacher. In two publications ([Schappacher 1998a] and [Schappacher 1998b]) he puts forward inter alia two propositions: Questioning Diophantus’ originality he considers affirmatively the possibility that the Arithmetica are the joint work of a team of authors like Bourbaki. And he calls Bašmakova’s claim (in [Bašmakova 1972]) that Diophantus uses negative numbers, a “nonsense”, reproaching her for her “thoughtlessness”. Teil I: First, we disprove Schappacher’s Bourbaki thesis. Second, we investigate the semantic meaning and historical significance of Diophantus’ keywords $$\lambda\varepsilon \tilde{\iota}\psi\iota\zeta$$ and ὕπαρξιζ. Next, we discuss Schappacher’s epistemology of the history of mathematics and defend Bašmakova’s methods. Finally we analyse in detail three problems from Diophantus’ Arithmetica (and their solutions) given by Thomas Heath and Helmuth Gericke as “proof” of the their claim that Diophantus did not use negative numbers. Teil II: In this Part, we give 33 places where Diophantus uses negative quantities as intermediate results; they appear as differences a − b of positive rational numbers, the subtrahend b being bigger than the minuend a; they each represent the (negative) basis ( $$\pi\lambda\varepsilon\upsilon\rho\acute{\alpha}$$ ) of a square number ( $$\tau\varepsilon\tau\rho\acute{\alpha}\gamma\omega \nu o \zeta$$ ), which is afterwards computed by the formula $$(a - b)^2 = a^2 + b^2 - 2ab$$ . Finally, we report how the topic “Diophantus and the negative numbers” has been dealt with by translators and commentators from Maximus Planudes onwards. Und er kommt zu dem Ergebnis: ≪Nur ein Traum war das Erlebnis. Weil≫, so schlieβt er messerscharf, ≪nicht sein k a n n, was nicht sein d a r f.≫ CHRISTIAN MORGENSTERN: Palmström
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DOI: 10.1007/s00048-006-0240-z
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Teil I</title><author><name sortKey="Barner, Klaus" sort="Barner, Klaus" uniqKey="Barner K" first="Klaus" last="Barner">Klaus Barner</name><affiliation wicri:level="3"><country xml:lang="fr">Allemagne</country><wicri:regionArea>Fachbereich Mathematik–Informatik, Universität Kassel, D–34109, Kassel</wicri:regionArea><placeName><region type="land" nuts="1">Hesse (Land)</region><region type="district" nuts="2">District de Kassel</region><settlement type="city">Cassel (Hesse)</settlement></placeName></affiliation><affiliation wicri:level="1"><country wicri:rule="url">Allemagne</country></affiliation></author></analytic><monogr></monogr><series><title level="j">NTM International Journal of History and Ethics of Natural Sciences, Technology and Medicine</title><title level="j" type="sub">NTM Internationale Zeitschrift für Geschichte und Ethik der Naturwissenschaften, Technik und Medizin</title><title level="j" type="abbrev">N.T.M.</title><idno type="ISSN">0036-6978</idno><idno type="eISSN">1420-9144</idno><imprint><publisher>Birkhäuser-Verlag; www.birkhäuser.ch</publisher><pubPlace>Basel</pubPlace><date type="published" when="2007-02-01">2007-02-01</date><biblScope unit="volume">15</biblScope><biblScope unit="issue">1</biblScope><biblScope unit="page" from="18">18</biblScope><biblScope unit="page" to="49">49</biblScope></imprint><idno type="ISSN">0036-6978</idno></series></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0036-6978</idno></seriesStmt></fileDesc><profileDesc><textClass></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract" xml:lang="de">In this paper which consists of two parts (Teil I and Teil II) we champion Diophantus of Alexandria and Isabella Bašmakova against Norbert Schappacher. In two publications ([Schappacher 1998a] and [Schappacher 1998b]) he puts forward inter alia two propositions: Questioning Diophantus’ originality he considers affirmatively the possibility that the Arithmetica are the joint work of a team of authors like Bourbaki. And he calls Bašmakova’s claim (in [Bašmakova 1972]) that Diophantus uses negative numbers, a “nonsense”, reproaching her for her “thoughtlessness”. Teil I: First, we disprove Schappacher’s Bourbaki thesis. Second, we investigate the semantic meaning and historical significance of Diophantus’ keywords $$\lambda\varepsilon \tilde{\iota}\psi\iota\zeta$$ and ὕπαρξιζ. Next, we discuss Schappacher’s epistemology of the history of mathematics and defend Bašmakova’s methods. Finally we analyse in detail three problems from Diophantus’ Arithmetica (and their solutions) given by Thomas Heath and Helmuth Gericke as “proof” of the their claim that Diophantus did not use negative numbers. Teil II: In this Part, we give 33 places where Diophantus uses negative quantities as intermediate results; they appear as differences a − b of positive rational numbers, the subtrahend b being bigger than the minuend a; they each represent the (negative) basis ( $$\pi\lambda\varepsilon\upsilon\rho\acute{\alpha}$$ ) of a square number ( $$\tau\varepsilon\tau\rho\acute{\alpha}\gamma\omega \nu o \zeta$$ ), which is afterwards computed by the formula $$(a - b)^2 = a^2 + b^2 - 2ab$$ . Finally, we report how the topic “Diophantus and the negative numbers” has been dealt with by translators and commentators from Maximus Planudes onwards. Und er kommt zu dem Ergebnis: ≪Nur ein Traum war das Erlebnis. Weil≫, so schlieβt er messerscharf, ≪nicht sein k a n n, was nicht sein d a r f.≫ CHRISTIAN MORGENSTERN: Palmström</div></front></TEI><affiliations><list><country><li>Allemagne</li></country><region><li>District de Kassel</li><li>Hesse (Land)</li></region><settlement><li>Cassel (Hesse)</li></settlement></list><tree><country name="Allemagne"><region name="Hesse (Land)"><name sortKey="Barner, Klaus" sort="Barner, Klaus" uniqKey="Barner K" first="Klaus" last="Barner">Klaus Barner</name></region><name sortKey="Barner, Klaus" sort="Barner, Klaus" uniqKey="Barner K" first="Klaus" last="Barner">Klaus Barner</name></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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