Returns of geometry: From the Pythagoreans to mathematical modernism and beyond

Abstract

This article considers the persistence of geometrical thinking in mathematics amidst the twentieth-century transformation of mathematics associated with “mathematical modernism,” which may be characterized, as it was by this author previously, by an algebraization of mathematics, even in such fields as geometry and topology. This article, by contrast, discusses the role of geometry or, more generally (so as to include topology and related fields), the mathematics of spatiality, its “returns” to the scene of mathematics, in its interaction with algebra during this period. Thus, it also argues for a continuity, along with a break, between mathematical modernism with the preceding history of mathematics. As part of this history, I shall consider a form of thinking, in mathematics and beyond, defined by the role of the unthinkable in thought, or in the language of the ancient Greeks, who introduced this architecture of thought with the discovery of the incommensurable magnitudes in Pythagorean mathematics, the alogon within a logos. I shall argue that this architecture, which takes a more radical form, designated here as radical Pythagorean thinking (in mathematics itself and in physics), underlies the interplay of algebra and geometry in mathematical modernism and beyond.