This monograph offers a fresh perspective on the applicability of mathematics in science. It explores what mathematics must be so that its applications to the empirical world do not constitute a mystery. In the process, readers are presented with a new version of mathematical structuralism. The author details a philosophy of mathematics in which the problem of its applicability, particularly in physics, in all its forms can be explained and justified. Chapters cover: mathematics as a formal science, mathematical ontology: what (...) does it mean to exist, mathematical structures: what are they and how do we know them, how different layers of mathematical structuring relate to each other and to perceptual structures, and how to use mathematics to find out how the world is. The book simultaneously develops along two lines, both inspired and enlightened by Edmund Husserl’s phenomenological philosophy. One line leads to the establishment of a particular version of mathematical structuralism, free of “naturalist” and empiricist bias. The other leads to a logical-epistemological explanation and justification of the applicability of mathematics carried out within a unique structuralist perspective. This second line points to the “unreasonable” effectiveness of mathematics in physics as a means of representation, a tool, and a source of not always logically justified but useful and effective heuristic strategies. (shrink)

The ArgumentAlexandre Koyré is one of the most important historians of philosophic and scientific though since the thirties. Research on the Scientific Revolution, on Galileo, Descartes, Newton, as well as on Paracelsus and Boehme has deeply changed under his influential method: it has been a model for Kuhn's methodology of paradigms and revolutions in the histroy of science. Whereas Koyré used to be considered opposed in his ideology and method to sociological approaches, he has recently been characterized by Yehuda Elkana (...) as a sociologist of knowledge. In fact, until now one of the main sources of his method had not been identified: it is only by acknowledging the influence of Lucien Lévy-Bruhl on Koyré that it is possible to explain how the latter wrote his thesis on Boehme's mystical thought just before his Etudes galiléennes. Lucien Lévy-Bruhl was teaching history of philosophy at the Sorbonne, and Koyré was strongly influenced by his idea of “prelogical thinking” as a universel phenomenon and in a general way by the sociological school of Durkheim. Conceptual analysis deriving from Husserl, collective representations and attitude mentale, came together in Alexandre Koyrè's method. (shrink)

ArgumentThis paper explores how the Belgian mathematician Paul Mansion became interested in probability theory. In comparison to many other countries at the time, probability theory had a much stronger presence in Belgium. In addition, Mansion, who was an avowed Catholic militant, had found probability theory to be a useful means of reflecting on certain problems pertaining to determinism and randomness that were arising in scientific debates at the time. Mansion’s work took place during a time of consolidation of mathematical education (...) in Belgium, as well as a new interest in probabilistic results and the foundation of the Institute for Philosophy in Louvain by his friend Désiré Mercier. The present paper addresses how these aspects intersected at the turn of the twentieth century. (shrink)

En este artículo damos a conocer la concepción de la “ciencia” del filósofo chileno Jenaro Abasolo (1833-1884) desde la indagación de algunas de sus deudas conceptuales contraídas con seis autores fuente. Fichte, Schelling, Krause, Ahrens, Darwin y Quinet son quienes le permiten a Abasolo establecer un particular intento de síntesis entre los descubrimientos de la ciencia positiva y los principios comunes a las formas de idealismo. Dichos principios sirven a Abasolo para mantener una perspectiva espiritualista que es compatible con los (...) nuevos hallazgos que las ciencias de la naturaleza aportan desde los datos sensibles. Como correlato, Abasolo considera que la filosofía tiene un rol fundamental cuando se trata de dar una visión unitaria del universo a partir de encontrar el nexo entre las relaciones cualitativas y cuantitativas de lo real. A su vez, se esboza la manera en que el análisis de la dimensión físico-natural aportaría a la interpretación del mundo humano de la libertad y cómo las ciencias permitirían hallar diversas coordenadas capaces de orientar adecuadamente al hombre en su desarrollo hacia la constitución de lo que Abasolo define como personalidad política. (shrink)

Léon Brunschvicg a été un pionnier de la philosophie des mathématiques et de l’épistémologie des sciences exactes en France. Gaston Bachelard peut être considéré comme un des héritiers les plus importants de l’épistémologie constructiviste de Brunschvicg. Les deux philosophes partagent la même conception d’une dialectique immanente de la pensée scientifique et l’idée que la philosophie consiste à construire et à analyser la logique interne du discours scientifique.

The paper is devoted to an overview of Cassirer’s work both as historian of philosophy and historian of science. Indeed, the “intelletcual cooperation” between history of philosophy and history of science represents an essential feature of Cassirer’s style of philosophizing: while the roots of a wide exploration stretching from Renaissance thought to modern physics go back to the Neo-Kantianism of the Marburg School, the results of a similar cross-fertilization of research fields have deeply contributed to shaping new standards of inquiry. (...) Cassirer’s relationship with the Warburg milieu in Hamburg and late in his life with the American intellectual scenario are surely worthy of closer investigation. Distinguished scholars such as Meyerson, Brunschvicg, Burtt, Koyré, Metzger, Lovejoy, Kristeller, have disussed, appreciated, critizised Cassirer’s still today fascinating studies devoted to Pico della Mirandola, Galileo, Newton, Leibniz, to mention but a few. To explore some of these aspects focusing both on affinities and differences within a cosmpolitian intellectual community can provide a better understanding of philosophy and history of science in the first half of 20th century. Cassirer’s legacy requires, therefore, a new assessment. (shrink)

On the basis of recent work concerning the meaning of the term stoicheion in Aristotle’s Analytics, we strengthen the view that this treatise can be viewed as a precursor of Euclid’s Elements.

Le développement et la complexité, qui ont caractérisé les débats sur la structure conceptuelle des sciences au xxe siècle, ont déterminé la naissance d'une nouvelle discipline, l'histoire de l'épistémologie, avec l'objectif de tracer une histoire critique de notre « patrimoine épistémologique » avec des méthodologies plus appropriées de nature historique et théorique. Une histoire critique de la philosophie des sciences nous montre beaucoup de « Wenden » au-delà de celles plus bien connues, néopositiviste et post-néopositiviste ; en Italie et en (...) France, dans les dix premières années du xxe siècle, a pris naissance une tradition de recherche épistémologique de nature néorationaliste, avec des noyaux théoriques spécifiques pour l'importance accordée à l'historicité de la science. L'épistémologie néorationaliste italo-française a privilégié l'analyse des rapports entre la pensée mathématique et la pensée physique, telle que l'on peut la caractériser comme une véritable épistémologie de la physique mathématique; de plus, cette approche différente, dessinée d'abord par Federigo Enriques et ensuite par Gaston Bachelard, Albert Lautman et Ferdinand Gonseth, a permis de comprendre déjà dès les années trente la « philosophie implicite » dans les travaux de Kurt Godel et d'Hermann Weyl. (shrink)

In my dissertation, I present Hermann Cohen's foundation for the history and philosophy of science. My investigation begins with Cohen's formulation of a neo-Kantian epistemology. I analyze Cohen's early work, especially his contributions to 19th century debates about the theory of knowledge. I conclude by examining Cohen's mature theory of science in two works, The Principle of the Infinitesimal Method and its History of 1883, and Cohen's extensive 1914 Introduction to Friedrich Lange's History of Materialism. In the former, Cohen gives (...) an historical and philosophical analysis of the foundations of the infinitesimal method in mathematics. In the latter, Cohen presents a detailed account of Heinrich Hertz's Principles of Mechanics of 1894. Hertz considers a series of possible foundations for mechanics, in the interest of finding a secure conceptual basis for mechanical theories. Cohen argues that Hertz's analysis can be completed, and his goal achieved, by means of a philosophical examination of the role of mathematical principles and fundamental concepts in scientific theories. (shrink)