卡尔达诺关于四次方程特殊法则的构造原理——兼论数学史的研究范式

Reconstruct Cardano's Four Special Rules of Quartic Equation:A Discussion on the Paradigms of Research on the history of Mathematics

复原了卡尔达诺关于四次方程的4条特殊法则的构造过程,指出这4条形式差异很大的法则所采用的相同的构造方法,由此揭示了这些法则的真正涵义和它们通过命题的形式所表达出的数学内容并不相同,同时也解释了卡尔达诺为什么能得到这些法则。对这一构造过程的复原体现了曲安京所概括的三种数学史研究范式之间的联系。

This paper reconstructs the ways how Girolamo Cardano constituted the four special rules of quartic equation, i. e. , first transforming some simple equation to the quartic equation discussed in the rule, then gaining the suppositions and conclusion of the rule by means of comparing the coefficients of the two equations. The reconstructions discover that the signification of the rules is not the same to the mathematics expressed by the proposition formed rules and explain why Cardano could gain these rules. Moreover, the process of the reconstructions establishes relations among three paradigms of research on the history of mathematics summarized by Professor Qu Anjing as ＂what mathematics was done＂, ＂how mathematics was done＂ and ＂why mathematics was done＂. The paper points out that during the procedure of the reconstructions posing the question of ＂why mathematics was done＂ helps ＂how mathematics was done＂ find the rational way to reconstruct the rules; however, the answer of the problem of ＂how mathematics was done＂ explains the question of ＂why mathematics was done＂, thereby orienting ＂what mathematics was done＂.