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Full Description
Does syllogistic logic have the resources to capture mathematical proof? This volume provides the first unified account of the history of attempts to answer this question, the reasoning behind the different positions taken, and their far-reaching implications. Aristotle had claimed that scientific knowledge, which includes mathematics, is provided by syllogisms of a special sort: 'scientific' ('demonstrative') syllogisms. In ancient Greece and in the Middle Ages, the claim that Euclid's theorems could be recast syllogistically was accepted without further scrutiny. Nevertheless, as early as Galen, the importance of relational reasoning for mathematics had already been recognized. Further critical voices emerged in the Renaissance and the question of whether mathematical proofs could be recast syllogistically attracted more sustained attention over the following three centuries. Supported by more detailed analyses of Euclidean theorems, this led to attempts to extend logical theory to include relational reasoning, and to arguments purporting to reduce relational reasoning to a syllogistic form. Philosophical proposals to the effect that mathematical reasoning is heterogenous with respect to logical proofs were famously defended by Kant, and the implications of the debate about the adequacy of syllogistic logic for mathematics are at the very core of Kant's account of synthetic a priori judgments. While it is now widely accepted that syllogistic logic is not sufficient to account for the logic of mathematical proof, the history and the analysis of this debate, running from Aristotle to de Morgan and beyond, is a fascinating and crucial insight into the relationship between philosophy and mathematics.
Contents
Introduction
1: Aristotelian Syllogism and Mathematics in Antiquity and the Medieval Period
2: Extensions of the Syllogism in Medieval Logic
3: Syllogistic and Mathematics: The Case of Piccolomini
4: Obliquities and Mathematics in the 17th and 18th Centuries: From Jungius to Wolff
5: The Extent of Syllogistic Reasoning: From Rüdiger to Wolff
6: Lambert and Kant
7: Bernard Bolzano on Non-Syllogistic Reasoning
8: Thomas Reid, William Hamilton and Augustus De Morgan
Conclusion
