Arguments on motivation in the rise and decline of a mathematical theory; the ''Construction of equations'', 1637 - ca. 1750
Die folgende originale Einleitung des Autors gibt anhand des Beispiels der Konstruktion gewisser Gleichungen eine Darstellung der Motive für den geschichtlichen Aufstieg und Abstieg mathematischer Theorien: ''Between 1650 and 1750 the 'construction of equations' was a recognized area of mathematical research and a standard subject in books on algebra. The equations were polynomial equations in one unknown; 'constructing' such an equation meant performing a geometrical construction yielding line segments with lengths equal to the roots of the equation. - The subject is now forgotten; it fell into oblivion shortly after 1750. But in the seventeenth century it was considered a crucial technique within analytic geometry, and it was developed by top-ranking mathematicians. In the present study I shall describe the rise and decline of the subject. I shall devote particular attention to the arguments which various mathematicians used to motivate their studies about the construction of equations or to argue against the work of others. From these arguments it will appear that the rise and decline of the construction of equations fits into a general line of development in mathematics in the period, namely the receding of the geometrical conception of mathematical entities and operations, and the emergence of an analytic style in which the main interest focused on formulas and the manipulations performed on them. A discussion of the arguments on motivation will also shed light on the particular process of decline that occurred in the case of the construction of equations, and thus it may contribute to the understanding of processes of development of separate sectors of science. The story of the construction of equations provides an interesting test case for ideas about 'degenerating' research programs which have been put forward by Lakatos and other philosophers and historians of science.''
geometrical conception of mathematical entities
scientific article; zbMATH DE number 3878877