History,methodology and early algebra 1

The limits of ‘criterial rationality’ (that is, rationality as rule‐following) have been extensively explored in the philosophy of science by Kuhn and others. In this paper I attempt to extend this line of enquiry into mathematics by means of a pair of case studies in early algebra. The first case is the Ars Magna (Nuremburg 1545) by Jerome Cardan (1501–1576), in which a then recently‐discovered formula for finding the roots of some cubic equations is extended to cover all cubics and proved. The second is the formulation by Albert Girard (1595–1632) of an early version of the fundamental theorem of algebra in his L'invention nouvelle en l'algèbre (Amsterdam, 1629). I conclude that in these cases at least, the questions raised in the philosophy of science debate can also be asked of the history of mathematics, and that a modest methodological anarchism is the appropriate stance.