Negative Größen bei Diophant? Teil I

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