Medieval philosophy a new history of western philosophy volume 2 ( PDFDrive ) 205

PHYSICS revolutions of the moon for every one of the sun Hence, the moon has gone round more often than the sun But the sun has gone round an inWnite number of times; therefore it is possible to Wnd something exceeding what is inWnite in the very respect in which it is inWnite But this is impossible.5 If there were actual inWnities, even if not synchronic, they would be countable, in the way that years and months are countable But if there were countable inWnities, there would be unequal inWnities, and surely this was a scandal Medieval philosophers responded to the scandal in diVerent ways Some denied that ‘equal to’ and ‘greater than’ applied to inWnite numbers at all Others accepted that there could be equal and unequal inWnities, but denied that the axiom ‘the whole is greater than its part’ applied to inWnite numbers The inWnitely divisible continuum, as envisaged by Aristotle, did not raise the problem of unequal inWnities, because the parts of the continuum were only potentially distinct from each other, and potential entities were not countable in the same way as actual entities In the fourteenth century, however, some thinkers began to argue that the continuum was composed of indivisible atoms, which were inWnite in number Notable among these was Henry of Harclay, who was chancellor of Oxford University in 1312 Aristotle had argued that a continuum could not be composed of points that lacked magnitude Since a point has no parts, it cannot have a boundary distinct from itself; two points therefore could not touch each other without becoming a single point But Henry tried to argue that they could touch—they would indeed touch whole to whole, but they could diVer from each other in position, and thus add to each other This theory was diYcult to understand, and Bradwardine was able to show that it made nonsense of Euclidean geometry If you take a square and draw parallel lines from each atom on one side to each atom on the opposite side, these will meet the diagonal in exactly as many atoms as they meet the sides But this is incompatible with the diagonal’s being incommensurable with the sides Ockham took a much more radical stance against Henry As part of his general reductionist programme, he argued that points had no absolute existence Not even God could make a point exist in independence from all other entities So far from a line being constructed out of points, as it was for Henry, a point was nothing other than a limit or cut in a line II Sent 1 2; cited by J Murdoch, ‘InWnity and Continuity’, in CHLMP 570 186