Mathematics and Applied Mathematics
April 21, 2020
actual infinity, potential infinity, supertask
It seems reasonable to assume that mathematical infinity was not the objective of Zeno’s Dichotomy (in any of its variants), however, a sort of mathematical infinity was already present in these celebrated arguments. Aristotle proposed a first solution to Zeno’s Dichotomy by introducing what we now call one-to-one correspondences, the key instrument of modern infinitist mathematics. But Aristotle, more naturalist than platonic, finally rejected the method of pairing the elements of two infinite collections (in this case of points and instants) and introduced instead the distinction between actual and potential infinities. Aristotle’s distinction served to define, gross modo, two opposite positions on the nature of infinity for more than twenty centuries. The actual infinity was finally mathematized through set theory in the first years of the XX century and the discussions on its potential or actual nature almost vanished. But, as we will see here, things still remain to be said on this issue.