Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Mathématiques --- Philosophie --- Histoire --- 17e siècle. --- Mathematics --- Mathematics. --- Filosofia da matemática --- Matemática (história) --- Philosophy. --- History --- 1800-1899. --- Europe. --- 17e siècle --- Europe

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Mancosu offers an original investigation of key notions in mathematics: abstraction and infinity, and their interaction. He gives a historical analysis of the theorizing of definitions by abstraction, and explores a novel approach to measuring the size of infinite sets, showing how this leads to deep mathematical and philosophical problems.

Abstraction. --- Infinite. --- Infinity --- Finite, The --- Abstract thought --- Cognition --- Logic --- Thought and thinking

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Paolo Mancosu provides an original investigation of historical and systematic aspects of the notions of abstraction and infinity and their interaction. A familiar way of introducing concepts in mathematics rests on so-called definitions by abstraction. An example of this is Hume's Principle, which introduces the concept of number by stating that two concepts have the same number if and only if the objects falling under each one of them can be put in one-one correspondence. This principle is at the core of neo-logicism. In the first two chapters of the book, Mancosu provides a historical analysis of the mathematical uses and foundational discussion of definitions by abstraction up to Frege, Peano, and Russell. Chapter one shows that abstraction principles were quite widespread in the mathematical practice that preceded Frege's discussion of them and the second chapter provides the first contextual analysis of Frege's discussion of abstraction principles in section 64 of the Grundlagen. In the second part of the book, Mancosu discusses a novel approach to measuring the size of infinite sets known as the theory of numerosities and shows how this new development leads to deep mathematical, historical, and philosophical problems. The final chapter of the book explore how this theory of numerosities can be exploited to provide surprisingly novel perspectives on neo-logicism.--

Mathematical logic --- Philosophy of science --- Mathematics --- Infinite --- Philosophy --- Mathematics - Philosophy

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

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.

Syllogism. --- Syllogisme. --- Preuve (philosophie) --- Evidence.