ABSTRACT

When we think of elementary logic today, the science of such purely formal properties of inference as validity and soundness come to mind. And elementary metalogic studies the formal properties of systems of logically distinguished items. Aristotle’s logic and its various developments already had these foci, but they were meant to accomplish even more. They were also meant to structure knowledge of the natural world by deducing phenomena from causes. Throughout the early modern period, this venerable logic still had many authoritative, well-placed adherents. A second intellectual tradition that shaped early modern thinking about logic and knowledge came from mathematics. Geometry, especially, stood as a shining example of human learning and ingenuity. In the early modern period, the twin hegemony of these classical influences came to be threatened. Striking advances in algebra and analytic geometry raised questions about the applicability of classical mathematical techniques to newly discovered problems. And the new, mechanical science exposed fatal inadequacies in the framework for empirical science set up two millennia earlier. There was, moreover, some tension between the logic of classical geometry and the logic of science. It had always been fairly obvious that practical mathematical reasoning rarely fit into the formal straightjacket of Aristotelian syllogistic logic. 1