A New Look at Geometry

Abstract

We close Part II with an examination of two of the most revolutionary developments in modern mathematics—both directly related to infinity. The first of these, the creation of projective geometry, takes us back to the Renaissance, and it has its roots not in science but in art. During the Middle Ages, both science and art were subordinated to the religious and mythological beliefs of the time. Nature was depicted not as she really was, but as the observer’s fantasies and religious beliefs wanted her to be. Thus, the world believed in a sun that moved around the earth, not because the available evidence, based on an objective observation of the heavens, made such a conclusion inevitable, but because the Roman Catholic church decreed that it must be so. The earth itself was flat—despite mounting evidence to the contrary—because to believe in a round earth meant to let the poor creatures on the “other side” plunge into the abyss of infinite space. And a painter depicted his saints and heroes not in their natural perspective—that is, faraway figures appearing smaller than nearby ones—but according to their status in the Church hierarchy.

All finite things reveal infinitude.

— Theodore Roethke (1908–1963), The Far Field, IV