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Readings and Issues of Jules Vuillemin’s Philosophie de l’algèbre

Thematic issue of Philosophia Scientiæ 24/3 (November 2020)

Guest editors : Sébastien Maronne et Baptiste Mélès

Submission Deadline: 1 July 2019

Acceptance Notification: 1 October 2019

Final version due: 1 May 2020

Please send submissions to: smaronne@math.univ-toulouse.fr, baptiste.meles@gmail.com

In the first volume, the only one published, of his Philosophie de l’algèbre (Paris, Presses Universitaires de France, 1962, rééd. 1993), Jules Vuillemin aimed to examine “how pure knowledge is possible [by using] the analogies of mathematical knowledge”. In order to emphasize this “close relationship between pure Mathematics and theoretical Philosophy”, he described the intrusion of Abel’s and Galois’ general methods into the theory of algebraic equations. They would supersede genetic methods at work in Lagrange’s works and allow a “glimpse” of the new idea of structure. By transferring his analysis to philosophy, he brought out some connections, sometimes surprising, between mathematical and philosophical methods, for instance between Lagrange’s and Fichte’s methods. He then concluded by a long study of Mathesis Universalis in which he presented and compared Klein’s and Lie’s mathematical theories with Husserlian phenomenology. It is in volume 2 of the Philosophie de l’algèbre, which he chose not to publish, that Vuillemin finally studied the developments of the new idea of structure, from Dedekind’s theory of ideals to Garett Birkhoff’s universal algebra, which he described as an “algebra of algebra” ; he therefore proposed to derive from that a research program of a general criticism of pure reason, namely a structural analysis per se of philosophical works. By focussing on the very notion of structure, Vuillemin himself carried away by the impetus he had put forward and then described, being contemporary to Bourbaki and to his theory of algebraic or topological “mother structures”.

More than sixty years after the publication of the first volume of the Philosophie de l’algèbre, contemporary mathematics have gone through major changes which Vuillemin could of course not foresee, such as the development of category theory (which anyway he had not studied in his works), algebraic geometry and model theory. Notwithstanding, convergences between geometry, logic, topology and algebra that had been noticed by Vuillemin, through the example of boolean algebra and topology, have been ceaselessly emphasized and deepened in contemporary mathematics : quite recently and still today, with topos theory, these very last years, with homotopy type theory. In the light of these recent developments, which lessons can be drawn from Jules Vuillemin’s Philosophie de l’algèbre, either in history and philosophy of mathematics, or in “theoretical philosophy”?

We will publish in this volume two kinds of contributions : on the one hand, historical analysis wholly or partly dealing with Jules Vuillemin’s book, on the other, reflections on the extension or the limits of Vuillemin’s project in the light of contemporary mathematics.

One could deal with the following issues :

  • How does Vuillemin’s project fit into his previous and following works? Into the tradition of historical epistemology of mathematics?

  • How can be characterised the forms of mathematical reflexivity today? Do they stand comparison with these of philosophical reflexivity like that of Cartesian or Husserlian cogito? If so, do they assume a conscience or, on the contrary, do they pertain to a philosophy of the mere concept? In the first case, to which figures of conscience would they pertain? One could particularly draw comparisons with Hegel, Husserl, Cavaillès, Lautman, Granger and Desanti.

  • How does Vuillemin’s project fit into the mathematics of his period? into our own mathematics?

  • How can be read anew today the parallelism proposed by Vuillemin between history of mathematics and history of theoretical philosophy?

  • How, finally, can be understood the mathematical and the philosophical projects gathered by the plurivocal and many times reinterpreted expression of mathesis universalis?

  • Manuscripts should be submitted in French, English, or German, and prepared for anonymous peer review.

  • Abstracts in French and English of 200-300 words in length should be included.

  • Articles should not exceed 50,000 characters (spaces, list of references and footnotes included).

  • Please send submissions to: smaronne@math.univ-toulouse.fr, baptiste.meles@gmail.com.

Guidelines for authors are to be found on the journal’s website: http://philosophiascientiae.revues.org/633.

Bibliographie

[Marquis(2009)] Jean-Pierre Marquis : From a geometrical point of view. A study of the History and Philosophy of Category Theory. Springer, New York, 2009.

[Rabouin(2009)] David Rabouin : Mathesis Universalis. L’idée de “mathématique universelle” d’Aristote à Descartes. Épiméthée. Presses Universitaires de France, 2009.

[Timmermans(2012)] Benoît Timmermans : Histoire philosophique de l’algèbre moderne : les origines romantiques de la pensée abstraite. Classiques Garnier, Paris, 2012.

[Vuillemin(1962)] Jules Vuillemin : La Philosophie de l’algèbre. Tome premier, Recherches sur quelques concepts et méthodes de l’algèbre moderne. Épiméthée. Presses Universitaires de France, Paris, 1962.

[Zalamea(2018)] Fernando Zalamea : Philosophie synthétique de la mathématique contemporaine. Hermann, Paris, 2018.

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