There is an urgent need in philosophy of mathematics for new approaches which pay closer attention to mathematical practice. This book will blaze the trail: it offers philosophical analyses of important characteristics of contemporary mathematics and of many aspects of mathematical activity which escape purely formal logical treatment.

Neoplatonism is a term used to designate the form of Platonic philosophy that developed in the Roman Empire from the third to the fifth century AD and that based itself on the corpus of Plato's dialogues. Sara Rappe's challenging study analyses Neoplatonic texts themselves using contemporary philosophy of language. It covers the whole tradition of Neoplatonic writing from Plotinus through Proclus to Damascius. Addressing the strain of mysticism in these works, the author shows how these texts reflect actual meditational practices, (...) methods of concentrating the mind, and other mental disciplines that informed the tradition as a whole. In providing such a broad survey of Neoplatonic writing, the book will appeal to classical philosophers classicists as well as students of religious studies. (shrink)

The ArgumentTo highlight speculative trends specific to the mathematical tradition that developed in China, the paper analyzes an excerpt of a third-century commentary on a mathematical classic, which arguably contains a proof. The paper shows that the following three tasks cannot be dissociated one from the other: (1) to discuss how the ancient text should be read; (2) to describe the practice of mathematical proof to which this text bears witness; (3) to bring to light connections between philosophy and mathematics (...) that it demonstrates were established in China. To this end the paper defines its use of the word “proof” and outlines a program for an international history of mathematical proof. It describes the sense in which the text conveys a proof and shows how it simultaneously fulfills algorithmic ends, bringing to light a formal pattern that appears to be fundamental both for mathematics and for other domains of reality. The interest in transformations that mathematical writings demonstrate in China at that time seems to have been influenced by philosophical developments based onThe Book of Changes (Yi-jing), which the excerpt quotes. This quotation within a mathematical context makes it possible to suggest an interpretation for a rather difficult philosophical statement. (shrink)

In sixteenth century Paris a circle of mathematicians produced texts of the most advanced kind of algebra. This "French algebraic tradition" will be the context for Viete's symbolic algebra. Comparing French algebraic texts with Italian and German, and examining the publishing context , I establish a periodization in two phases. ;Jacques Peletier stands for the introduction of the abacus tradition and algebra at the court. Peletier's algebraic program is connected to his theory of rhetoric. He establishes a genre of texts (...) devoted to algebra in vernacular, promoting French as a scientific language. Rhetorical criteria order L'Algebre, emphasizing structure and theory . He innovates on Cardano's and Stifel's treatments of equations in several unknowns. ;Phase two changes style and content; Guillaume Gosselin is representative. Manuals shift focus from problems and questions to equations and classification. Solution of equations becomes the purpose of algebra. Problems are no longer addressed as such, but conceived in their most general form as corresponding equations. Algebrists connect with a milieu of jurists important in politics and scholarship. This erudite milieu favored the recovery of Diophantus' Arithmetic, which provides a set of theoretical problems on numbers. Diophantus solved problems by transforming them into particular cases; Gosselin reaches a general solution by transforming them into equations. Gosselin's notation makes the difference. The algebrists create an illustrious genealogy for algebra, deriving it from Greece rather than abacus schools. ;Three features of this tradition can be traced back to Italian humanism: Cardano's and Tartaglia's algebra, imitatio , and the construction of a history for the discipline. The French algebrists radically transformed all three. Their translatio authorized them to abandon links to the medieval tradition and to build a new discipline that they could see as national. Preparing the adoption of this discipline by the legal elite was the rhetorical interpretation of logic developed in Paris at the time. This provided a theoretical frame in which generalized algebraic problems were seen as Cicero's quaestiones infinitae, i.e. scientific questions. (shrink)

Philosophy of mathematics today has transformed into a very complex network of diverse ideas, viewpoints, and theories. Sometimes the emphasis is on the "classical" foundational work (often connected with the use of formal logical methods), sometimes on the sociological dimension of the mathematical research community and the "products" it produces, then again on the education of future mathematicians and the problem of how knowledge is or should be transmitted from one generation to the next. The editors of this book felt (...) the urge, first of all, to bring together the widest variety of authors from these different domains and, secondly, to show that this diversity does not exclude a sufficient number of common elements to be present. In the eyes of the editors, this book will be considered a success if it can convince its readers of the following: that it is warranted to dream of a realistic and full-fledged theory of mathematical practices, in the plural. If such a theory is possible, it would mean that a number of presently existing fierce oppositions between philosophers, sociologists, educators, and other parties involved, are in fact illusory. (shrink)

Sara Rappe has given us a stimulating book full of interesting suggestions concerning philosophers hardly known, in some cases, in the English-speaking world. She raises a question concerning these philosophers that has not previously been discussed on this scale. The question arises from the comparison of two features of Neoplatonism. For the Neoplatonist philosopher, discursive thinking does not yield knowledge. By discursive thought is meant the kind of thinking we normally practice. It has to do with objects external to thought, (...) objects mediated by images derived from sense perception or also from a higher form of thought. From these images discursive thought elaborates further ideas derived by methods of reasoning such as those codified in Aristotelian logic. This kind of thinking is open, however, to the attacks of Skepticism, against which the Neoplatonist argues by appealing to another form of thinking and to the knowledge it constitutes, a form of thinking free of all mediation separating subject and object of thought, where subject and object of thought are identical. This is not discursive thought; dis- cursive thought does not yield knowledge that is immune to skeptical attack and that is obtained by a form of thought beyond discursive thinking. To the extent that language is the expression of discursive thought, this knowledge is not expressed in language. Yet the Neoplatonists of Late Antiquity were plunged in discursivity. In particular, they did philosophy by interpreting texts, the authoritative texts of their tradition, not only Plato’s dialogues, but also the Aristotelian corpus and what they supposed were the more ancient sources of their philosophy, Pythagorean and Orphic texts and the Chaldaean Oracles. How are and to be reconciled? The thesis explored by Rappe in this book is that the Neoplatonists had recourse to nondiscursive textual strategies, images, symbols, myths, ritual formulae, so as to convey nondiscursive thought and knowledge through their reading of discursive texts. (shrink)