Les articles de ce recueil discutent les principes de la pensee mathematique qui ont fait le partage entre les trois fameuses options philosophiques: logicisme, formalisme et intuitionnisme. Leur auteur, Paul Bernays, fut un des plus proches collaborateurs de David Hilbert, qui a si profondement marque de son empreinte les mathematiques du XXe siecle et la philosophie construite a leur sujet. Defenseur de l'infini, de la methode axiomatique, des structures generales, des raisonnements abstraits, de la formalisation logique, Hilbert a investi (...) une grande energie pour ancrer ces audacieuses conquetes sur le terrain balise et controlable du fini, de l'intuitif, du concret, du constructible. Les essais de Bernays sont un precieux guide pour mieux s'orienter dans la pensee logique et philosophique de Hilbert et se debarasser une bonne fois des prejuges hatifs qui jalonnent les opinions ressassees sur le formalisme, le finitisme, le platonisme, etc. Mais ces essais nous decouvrent aussi un penseur libre, parlant de sa propre voix. Sensible a la pluralite des approches possibles, assumant le risque de se rectifier sans cesse, mesure dans sa visee d'une certitude pratique plutot que d'une verite absolue, Bernays defend l'idees que les mathematiques ne sont pas un reservoir de connaissance, mais une activite de l'esprit reagissant a des situations. (shrink)

English summary: This volume offers French readers important texts of modern philosophical mathematics, addressing in particular ontological questions and others on mathematical objects, how to demonstrate the need for mathematical truths, and how to explain that mathematics apply to the real world. French description: Le compagnonnage entre la philosophie et les mathematiques ne date pas d'hier. Mais l'emergence des nouvelles logiques, au debut du XXe siecle, a profondement modifie la forme des interactions entre les deux disciplines, suscitant de nouvelles (...) interrogations et modifiant la formulation des problemes herites de la tradition. Ce premier volume vise a rendre accessible aux lecteurs francophones des textes importants de la philosophie contemporaine des mathematiques. Il est plus particulierement consacre aux questions ontologiques et a celles liees aux fondements: Qu'est-ce qu'un objet mathematique? Comment rendre compte de la necessite des verites mathematiques? Comment expliquer que les mathematiques s'appliquent au monde reel? (shrink)

L’œuvre mathématique de La Ramée est constituée d’ouvrages publiés tout au long de sa carrière. Les éditions abrégées des Éléments d'Euclide, ses manuels d’arithmétique et de géométrie plusieurs fois remaniés, un court traité d’algèbre ainsi que les Scholae mathematicae de 1569 sont autant d’ouvrages dans lequels il met en œuvre une réforme des mathématiques héritées et de leur enseignement. Des justifications de cette réforme apparaissent dans les trois livres d’un Prooemium mathematicum portant sur l’histoire des mathématiques, la défense (...) de leur « utilité » et une critique sévère de l’exposé euclidien. Pour autant, l’œuvre mathématique de La Ramée ne doit pas être mise à part de l’œuvre philosophique qu’il élabore par ailleurs. Dans cet article, je m’efforce de le montrer en m’interrogeant d’abord sur les raisons qui ont poussé La Ramée à se tourner, au début de sa carrière, vers les mathématiques ; l’examen du livre I du Prooemium mathematicum me permet ensuite d’examiner le rôle que jouent Platon, Aristote et Euclide dans son histoire des mathématiques. Finalement, je suis amené, en abordant la critique raméenne de l’ « abstraction » pythagoricienne, à chercher dans les éditions de la Dialectica et les Scholae in liberales artes les éléments d’une « philosophie des mathématiques » – conception originale de la nature des objets mathématiques et de la façon dont on accède à leur connaissance – qui offre une certaine lumière sur la réforme des mathématiques qu’entreprend La Ramée. (shrink)

C’est, beaucoup plus que l’essai annoncé, une remarquable synthèse sur la philosophie des mathématiques que nous procure Jean-Michel Salanskis (J-M. S), synthèse qu’il faut placer dans la continuité de ses autres ouvrages, L’herméneutique formelle, Paris, Éditions du CNRS, 1991, Le constructivisme non standard, Lille, Presses universitaires du Septentrion, 1999, Sens et philosophie du sens, Paris, Desclée de Brouwer, 2001, Herméneutique et cognition, Lille, Presses universitaires du Septentri..

Pour le philosophe intéressé aux structures et aux fondements du savoir théorétique, à la constitution d'une « méta-théorétique «, θεωρíα., qui, mieux que les « Wissenschaftslehre » fichtéenne ou husserlienne et par-delà les débris de la métaphysique, veut dans une intention nouvelle faire la synthèse du « théorétique », la logique mathématique se révèle un objet privilégié.

The project of this Précis de philosophie de la logique et des mathématiques (vol. 1 under the direction of F. Poggiolesi and P. Wagner, vol. 2 under the direction of A. Arana and M. Panza) aims to offer a rich, systematic and clear introduction to the main contemporary debates in the philosophy of mathematics and logic. The two volumes bring together the contributions of thirty researchers (twelve for the philosophy of logic and eighteen for the philosophy of mathematics), (...) specialists in the history or philosophy of logic or mathematics. Each volume consists of ten chapters; each chapter is about forty pages, is independent from the others and deals with a particular philosophical question about logic or mathematics. The objective is to offer to the French-speaking reader a reference book. On many of the issues addressed, this Précis offers the first clear and thorough synthesis that is available in French. -/- This book is intended for third year / master's students, but also for teachers of philosophy in secondary or higher education, for logicians and mathematicians willing to take a philosophical look at the fundamental concepts of their discipline, and for any reader interested in the basic issues discussed in the philosophy of logic and mathematics. Each chapter is self-contained, although some basic training in logic (or in mathematics, as the case may be) is required. The book should be looked at as a comprehensive framework for contemporary discussions in the philosophy of logic and mathematics, while also giving some guidance on the disciplinary content required by these discussions. -/- The first three chapters of the present volume (on the philosophy of mathematics) are devoted to the history of the philosophy of mathematics: from antiquity to the modern era, and from this period to the foundational crisis of the nineteenth century, to the twentieth century. Next, four chapters deal with the crucial questions for the philosophy of mathematics of the twentieth century: the opposition and/or comparison of set theory and category theory as foundational frameworks for mathematics; mathematical constructivism; the analysis of calculability; and Benaceraff's problem. The two following chapters are focused on the philosophy of mathematical practice, by treating the notion of ideals of proof, in particular explanation and purity, and the notion of informal proof and the use of visual artifacts in mathematical argumentation. Finally the last chapter deals with the applicability of mathematics, including the role of probability. -/- In Chapter 1, Sébastien Maronne and David Rabouin present the history of the philosophy of mathematics from antiquity to the modern era. The goal is to show that the philosophy of mathematics has a history as ancient as mathematics and philosophy itself, a history largely continuous with the latter, often by staying out of sync with respect to the former. In Chapter 2, Sébastien Gandon discusses the question of the foundations of mathematics from Kant to the end of the nineteenth century. By tying this question to Kant, he is able to show that the discussion of the foundations of mathematics that occupied a large part of these two centuries has much older roots. Accent is placed on the differences between the framework proposed by Kant for making sense of classical mathematics and the use made by Frege and then Russell of logical tools for giving a different reading of arithmetic and real analysis. The link between between these different conceptions and the evolution of mathematics itself is also underlined. -/- Chapter 3, written by Hourya Benis-Sinaceur and Mirna Džamonja, deals with the evolution of mathematics in the twentieth century, notably the structuralist turn of Dedekind, E. Noether, and Bourbaki, and certain more recent developments. They devote themselves, among other things, to explaining how these developments can be understood as responses to and extensions of questions posed during the foundational debate of the immediately preceding era. -/- In Chapter 4, Jean-Pierre Marquis and Jean-Jacques Szczeciniarz discuss the different foundational options coming from set theory and category theory. They discuss, among other things, Lawvere's program to replace ZF with an axiomatization of the category of sets, the latter judged better adapted to the needs of contemporary mathematics. Different reactions, as much philosophical as technical, raised by this program are also taken into account. -/- Constructive mathematics are the object of Chapter 5, in which Gerhard Heinzmann and Mark van Atten examine in a critical and comparative manner a variety of constructivisms, including intuitionism, constructive type theory, predicativism, and finitism. -/- In Chapter 6, Guido Gherardi and Maël Pegny present computability theory, stressing its philosophical consequences. They discuss the basic concepts of this theory, for the computability of both the natural numbers and the real numbers. They then tackle the theory of computational complexity. -/- In Chapter 7, Andrea Sereni and Fabrice Pataut present Benacerraf's problem, opposing every possible account of mathematical knowledge to the availability of a theory of truth founded on an abstract ontology of mathematical objects. They give an overview of the original problem and its reformulation by Hartry Field, and reconstruct the debate that this problem has raised, by meeting at once the major opposing positions today of the analytic affiliation of the philosophy of mathematics, as much those on the platonist side as on that of nominalism. -/- In Chapter 8, Valeria Giardino and Yacin Hamami treat certain crucial aspects of the philosophy of mathematical practice. They deal in particular with the notion of an informal proof and on the role of artifacts in mathematical reasoning. They consider how such proofs, making use of these means, can contribute to the understanding of theorems and mathematical theories, thanks also to the role of visualisation and the aid of computer tools. -/- Why do mathematics often give several proofs of the same theorem? This is the question that Andrew Arana raises in Chapter 9, introducing the notion of an epistemic ideal and discuss two such ideals, the explanatoriness and purity of proof. -/- Chapter 10 is devoted to the applicability of mathematics in the study of empirical phenomena. After summarizing the history of this question, going back to Plato and Aristotle and passing by Mill and Kant, Daniele Molinini and Marco Panza reconstruct the contemporary debate, in relation, among other things, to questions raised in the philosophy of science. -/- Two appendices complete the volume. In the first Frédéric Patras treats the French tradition in philosophy of mathematics of the 20th century. In the second, Maria Carla Galavotti discusses the notion of probability and its use in science. (shrink)

RésuméExiste‐t‐il une Méthode selon laquelle les résultats de la recherche dans une discipline telle que les mathématiques peuvent être intégrés à la recherche philosophique? Cette question est traitée à l'aide de trois exemples: celui de la philosophie cartésienne, celui de l'empirisme logique et finalement celui de la philosophie ouverte.Une seule et même procédure méthodologique se dégage d'ailleurs à toute autre tentative d'intégrer les résultats d'une recherche scientifique à la philosophie. Elle est essentiellement la suivante:Une telle procédure (...) n'est valable que si l'idée de justesse fait son entrée dans la philosophie. Une philosophie qui s'établit dans une intention de justesse s'engendre et s'organise en une perspective mentale telle que le savoir assuré y trouve sa place légitime. Le savoir n'y entre d'ailleurs que sous la forme d'hypothèses à faire valoir, d'hypothèses éprouvées et non totalement certaines.La philosophie cartésienne et la doctrine de l'empirisme logique sont à rejeter parce qu'elles proposent chacune pour son compte une conception des mathématiques que la recherche réelle ne confirme pas.La philosophie ouverte cherche au contraire à en tenir compte dans la conception des mathématiques qu'elle prétend engager dans une perspective philosophique adéquate, des résultats les plus récents de la recherche sur les fondements et la méthode des mathématiques. (shrink)

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From Euclid to the second half of the 17th century, mathematicians as well as philosophers continued to raise the question of the angle of contact and, generally, of the concept of angle. This article is the first essay devoted to this subject in Arabic mathematics. It deals with Greek writings translated into Arabic on the one hand, and contributions of Arabic mathematicians on the other hand: al-Nayrīzī, Ibn al-Haytham, al-Samawʾal, al-Shīrāzī, al-Fārisī, al-Qūshjī, among others. Most of these contributions are hitherto (...) unknown.Depuis Euclide jusque tard dans le xviie siècle, mathématiciens et philosophes n'ont cessé de s'interroger sur l'angle de contingence et, plus généralement, sur la notion d'angle. Cet article est le premier essai où sont examinés les écrits grecs portant sur ce thème et transmis en arabe, ainsi que les nombreux travaux, pour la plupart jusqu'ici inconnus, des mathématiciens arabes eux-mêmes: al-Nayrīzī, Ibn al-Haytham, al-Samawʾal, al-Shīrāzī, al-Fārisī, al-Qūshjī, entre autres.Send article to KindleTo send this article to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle. Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply. Find out more about the Kindle Personal Document Service.L'ANGLE DE CONTINGENCE: UN PROBLÈME DE PHILOSOPHIE DES MATHÉMATIQUESVolume 22, Issue 1Roshdi Rashed DOI: https://doi.org/10.1017/S0957423911000087Your Kindle email address Please provide your Kindle [email protected]@kindle.com Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Dropbox To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Dropbox. L'ANGLE DE CONTINGENCE: UN PROBLÈME DE PHILOSOPHIE DES MATHÉMATIQUESVolume 22, Issue 1Roshdi Rashed DOI: https://doi.org/10.1017/S0957423911000087Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Google Drive To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Google Drive. L'ANGLE DE CONTINGENCE: UN PROBLÈME DE PHILOSOPHIE DES MATHÉMATIQUESVolume 22, Issue 1Roshdi Rashed DOI: https://doi.org/10.1017/S0957423911000087Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Export citation Request permission. (shrink)

Créativité et mathématiques ne font pas bon ménage dans l'inconscient collectif. Pourtant à l'heure d'Internet, il existe bien une manière de revisiter la géométrie, l'algèbre ou même la logique. Il suffit de prendre un peu de distance par rapport aux calculs, de se donner quelques libertés par rapport à l'histoire, de changer de point de vue... En vagabondant à travers l'histoire et l'application des mathématiques, les auteurs se livrent ici à un jeu de vulgarisation d'une saveur et d'une (...) subtilité inédites. [4e de couv.]. (shrink)