Mathematical Logic: An Oxymoron

Extensionalism revolutionized logic. Revolutions seem somewhat implausible when first conceived and plainly inevitable in hindsight, and usually the hindsight comes naturally: the antecedents require reconstruction. Extensionalism is no exception. Fascinating, detailed studies of the hindsight perspective comprise investigations into 18th and early 19th century mathematics. They emphasize every precursor to extensionalism. They often show that in hindsight attitudes clearly identifiable as extensional slowly crept into the agenda of mathematics raising rigor. To this end mathematicians found it fruitful to consider certain mathematical objects and operations in the abstract, disregarding their particular features, concentrating, instead, on formal characteristics shared by these objects and operations. This cleared the road to the study of certain formal mathematical laws in the abstract. These were the early steps in the study of formal (“universal”, or “symbolical” as it was labeled then) Algebra. Studies of the history of the formalization of algebra are thus of the background to the history of the extensionalist revolution. In hindsight they are clearly right: indeed, we now know that the very notion of a formal law anticipates extensionalism. Such studies, then, are vital, interesting, and rarely contestable. Rather than summing them up here, however, I have focused instead on the obstacles on the way to the revolution, for these are rarely discussed.