Grosseteste’s and William of Auvergne’s Legacies in the Third Quarter of the Thirteenth Century

Abstract Chapter 4 explores the legacies of Grosseteste and William of Auvergne in the period 1245–1275. In the case of William of Auvergne, it is shown how his discussion of paradoxes of infinity might have influenced several thirteenth-century Franciscan scholars (Bacon, Olivi, Peckham) who exploit the paradoxes to give arguments for the finitude of time or of the world. In addition, the chapter explores the remarkable similarities between William of Auvergne’s and Giles of Rome’s accounts of infinity. In the case of Grosseteste’s legacy the focus is on the discussion, mostly of English origin, concerning Grosseteste’s idea that segments of different lengths contain unequal infinities of points. A powerful objection by Richard Fishacre, based on a one-to-one correspondence between the points of segments of different lengths, brought to the fore the opposition between two criteria for measuring infinities, one based on a part-whole intuition and the other on one-to-one correspondence.