How Many Points Are There in a Line Segment? From Grosseteste to Numerosity

Presentation of the project

"In his commentary on Aristotle’s Physics, Robert Grosseteste (c. 1175–1253), an Oxford theologian and university chancellor, wrote: 'Moreover, [God] created everything according to number, weight, and measure, and He is the first and most exact measurer. With the aid of infinite numbers that are finite for Him, He measured the lines He created. Using an infinite number that is fixed and finite for Him, He measured and numbered the line of one cubit; and with an infinite number twice as large, He measured the line of two cubits; and with an infinite number half as large, He measured the line of half a cubit.'

In Grosseteste's account, the numerosity of points in a finite line segment varies according to the segment's length. This position sparked significant debate in the 13th century, notably following a challenge by the Oxford theologian Richard Fishacre (1205-1248), who established a one-to-one correspondence between points in line segments of different lengths. I will reconstruct aspects of this medieval debate, connect it to later intuitions (Bolzano and Cantor), and then discuss recent results in numerosity theory showing that counting points in a line segment, while preserving the part-whole principle, is compatible with Lebesgue measure. I conclude that Grosseteste’s intuitions can find a proper mathematical realisation."