Natural continuity and the mathematical proofs against indivisibilism in Roger Bacon’s De celestibus (Communia Naturalium, II)

In this paper I discuss the “proof of the square” and other arguments against indivisibilism adduced by Bacon in the De celestibus (Communia Naturalium, part II), as well as in his Opus Majus and Opus tertium. This proof is a reworking of an argument by Al-Ghazali, who assumes that indivisibilist theories (such as the Atomistic and Platonic views of continua as composed of physical or mathematical minima) would imply the geometrical absurdity that side and diagonal are equal. The argument, frequently presented by later philosophers, among whom Duns Scotus, William Ockham, Thomas Bradwardine, is philosophically relevant in the context of Bacon’s natural philosophy not for the sake of accomplishing Aristotle’s view. Rather, his goal is to show that points are nothing but the limits of bodily continuity and that the action of radial species (represented by lines and geometric figures) would not be possible if bodies were not continuous in themselves and contiguous with one another. I argue that this argument is an interesting example of how Bacon maintains a distinction between mathematics and physics by individuating the specific function of the former in providing causal knowledge of natural phenomena. For him, mathematical objects are really in nature, although not as constitutive components of natural things, but of natural action.