Syllogistic Logic and Mathematical Proof

Abstract Syllogistic Logic and Mathematical Proof chronicles and analyzes a debate centered on the following question: does syllogistic logic have the resources to capture mathematical proofs? The history of the attempts to answer this question, the rationales for the different positions, their far-reaching implications, and the description of the cast of major and minor mathematicians and philosophers who made contributions to it, has hitherto never been the subject of a unified account. Aristotle had claimed that scientific knowledge, which includes mathematics, is given by syllogisms of a special sort, “scientific” (“demonstrative”) syllogisms. Thus, it is puzzling that in ancient Greece and in the Middle Ages the claim that Euclid’s theorems could be recast syllogistically was accepted without further scrutiny. And yet Galen had early on already noticed the importance of relational reasoning for mathematics. More critical voices will emerge in the Renaissance and the topic will attract more sustained attention in the next three centuries when, supported by detailed analyses of Euclidean theorems, one finally encounters attempts at extending logical theory to include relational reasonings, arguments purporting to reduce relational reasoning to syllogisms, and philosophical proposals to the effect that mathematical reasoning is heterogeneous with respect to logical proofs. The latter position was famously defended by Kant and the philosophical implications of the debate discussed in the book are at the very core of Kant’s account of synthetic a priori judgments. We know today that syllogistic logic is not sufficient to account for the logic of mathematical proof. Yet, the history and the analysis of this debate, spanning from Aristotle to de Morgan and beyond, is a fascinating and crucial aspect of the relationship between philosophy and mathematics.