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)

In this excellent book Sebastien Gandon focuses mainly on Russell's two major texts, Principa Mathematica and Principle of Mathematics, meticulously unpicking the details of these texts and bringing a new interpretation of both the mathematical and the philosophical content. Winner of The Bertrand Russell Society Book Award 2013.

Cet article est une contribution à l’étude de l’expression de la généralité et de l’abstraction en mathématique. Il propose dans cette perspective une analyse des conditions de possibilité de la réflexivité. Certaines de ces conditions sont d’abord dégagées à partir d’exemples élémentaires. Une brève analyse des Disquistiones arithmeticae de Gauss permet ensuite de proposer un critère de démarcation sémiotique pour les mathématiques ensemblistes. L’application de ce critère permet ensuite d’expliquer que la réflexivité s’exprime dans ces mathématiques par des hiérarchies.

This article argues that historical epistemology offers the history of philosophy and science more than a mere tool to write the history of concepts. It does this, first of all, by rereading historical epistemology through Michel Foucault's “techniques of the self.” Second, it turns to the work of Léon Brunschvicg and Gaston Bachelard. In their work we see a proposal for what the subjectivity of scientists and philosophers should be. The article thus argues that their work is driven by a (...) normative psychology: a set of prescriptions for which mental constitution a scholarly self has to have. In the Conclusion, it returns to existing analyses of “open‐mindedness” as a virtue and explores in what way these cases challenge these analyses, as well as to what extent Foucault's “techniques of the self” can be applied to other cases in the history of French philosophy. (shrink)

The analytical notion of ‘scientific style of reasoning’, introduced by Ian Hacking in the middle of the 1980s, has become widespread in the literature of the history and philosophy of science. However, scholars have rarely made explicit the philosophical assumptions and the research objectives underlying the notion of style: what are its philosophical roots? How does the notion of style fit into the area of research of historical epistemology? What does a comparison between Hacking’s project on styles of thinking and (...) other similar projects suggest? My aim in this paper is to answer these questions. Hacking has denied that his project of styles of thinking falls into the field of historical epistemology. I shall challenge his remark by tracing out the connections of the notion of style with historical epistemology and, more in general, with a tradition of thought born in France in the beginning of twentieth-century. (shrink)

When an English translation of Being and Event appeared in 2005, Alain Badiou took the opportunity to reminisce about the initial French publication some twenty years before: “at that moment I was quite aware of having written a ‘great’ book of philosophy.” He located that greatness in four “affirmations” and one “radical thesis.”.

Não há, para Leibniz, separação entre uma esfera ética, na qual seriam consideradas as ações voluntárias dos homens e sua busca pelo bem, e uma esfera epistemológica, pela qual se explicaria a busca pela verdade. Procuramos mostrar, primeiro, em que sentido se pode dizer que o conhecimento é uma ação voluntária; e, segundo, como a ação moral só é verdadeiramente livre se fundamentada no conhecimento do verdadeiro, por isso o modelo de homem livre é o homem sábio.

A. Kenny, Frege, an introduction to the founder of modern analytic philosophy. London:Penguin, 1995. viii-h223pp. £7.99 T. Willamson, Vagueness. London:Routledge, 1994. xiii-f-325 pp. £35.00 TOM BU...

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)

The development of philosophical ideas throughout history has sometimes been assisted by the use of handcrafted instruments. Some paradigmatic cases, such as the invention of the telescope or the microscope, show that many philosophical approaches have been the result of the intervention of such instruments. The aim of this article is to show the determining role that stringed musical instruments with frets had in the crisis and generation of philosophical paradigms. In fact, just as the observations of the moon with (...) the telescope broke more than a thousand years of Aristotelian hegemony, the fretted string instruments, predecessors of the guitar, played a central role in the collapse of one of the most influential approaches in the history of Philosophy: Pythagorism. We focus on the fundamental hallmarks of Pythagorism and on how, during the 16th century and from the fretted string instruments, the mathematical-musical notion of equal temperament emerged, which from the middle of the 19th century will be established as the prevailing philosophical-musical paradigm of the West. (shrink)

Albert Lautman (b. 1908–1944) was a philosopher of mathematics whose views on mathematical reality and on the philosophy of mathematics parted with the dominant tendencies of mathematical epistemology of the time. Lautman considered the role of philosophy, and of the philosopher, in relation to mathematics to be quite specific. He writes that: ‘in the development of mathematics, a reality is asserted that mathematical philosophy has as a function to recognize and describe’ (Lautman 2011, 87). He goes on to characterize this (...) reality as an ‘ideal reality’ that ‘governs’ the development of mathematics. The relation between mathematical problems as they arise in the historical development of mathematics and the solutions that are provided to these problems by mathematicians, in the form of new mathematical theories, definitions or axioms, are governed by a what Lautman characterises as a dialectics of mathematics. The aim of this paper is to give an account of this Lautmanian dialectic and of how it can be understood to govern the development of solutions to mathematical problems. (shrink)

In inter-war France, history of philosophy was a very important academic discipline, but nevertheless its practitioners thought it necessary to defend its identity, which was threatened by its vicinity to many other disciplines, and especially by the emergent social sciences and history of science. I shall focus on two particular issues that divided traditional historians of philosophy from historians of science, ethnologists and sociologists, and that became crucial in the definition of the identity of their disciplines: the conception of history (...) and the interpretation of texts. By analysing representative discussions and positions, I shall show that traditional historians of philosophy needed to reassert their own approach to history, which, borrowing the term from Bergson, I define as ‘snapshot’. This approach is focused on a particular idea or text rather than a narrative. I shall also show that history of philosophy, in its traditional form, would have been undermined both intellectually and institutionally by the opposite ‘narrative’ approach of history of science and of the social sciences. Social scientists openly attacked history of philosophy’s methods and, in the eyes of traditional philosophers, its existence as an academic discipline. The same opposition is to be found in evaluation of past texts, which for traditional historians of philosophy were to be read as timeless documents, while for historians of science, ethnologists and sociologists were to be considered as documents exhibiting a particular mentality. However, between these alternatives there were intermediate positions. I shall in particular consider that of Léon Brunschvicg: he embraced a narrative approach and considered texts as documents of different ways of thinking, but at the same time carried on employing philosophical methods and defending the institutional position of philosophy. I shall argue that this was possible partly because of the considerable amount of power he enjoyed at the Sorbonne. (shrink)

Au début des années 1930, l'épistémologie, l'histoire et l'histoire des sciences ont eu une influence réciproque importante mais indirecte. Tant les philosophes que les historiens mettaient en jeu des concepts épistémologiques issus de la crise des sciences exactes et rejetaient l'empirisme et l'idéalisme au nom d'un rationalisme expérimental qui mettait en relation théorie et expérience. Il en résultait une conception dialectique de la science, essentiellement historique mais à propos de laquelle les historiens étaient encore timides du point de vue théorique. (...) Désormais l'histoire était pourtant au coeur des réflexions philosophiques et épistémologiques. (shrink)

La question du rapport de la pensée bachelardienne à la mathématique contemporaine a longtemps été éludée au profit exclusif d'une interprétation fautive. Un Bachelard formé à la physique et à la chimie qui n'auraitjamais donné sa véritable place à l'étude mathématique est l'interprétation qui prédomine depuis le colloque de Cerisy (1974). Le logicien Roger Martin affinne qu'il existe un silence coupable autour du problème des fondements (problématique ensembliste, axiomatique et logicisme). À partir d'une analyse serrée des textes, cette étude renoue (...) avec une image beaucoup plus juste d'un Bachelard pour qui la mathématique n'est rien de moins que « le sur-moi de la science ». (shrink)

Arthur Pap’s work played an important role in the development of the analytic tradition. This role goes beyond the merely historical fact that Pap’s views of dispositional and modal concepts were influential. As a sympathetic critic of logical empiricism, Pap, like Quine, saw a deep tension in logical empiricism at its very best in the work of Carnap. But Pap’s critique of Carnap is quite different from Quine’s, and represents the discovery of limits beyond which empiricism cannot go, where there (...) lies nothing other than intuitive knowledge of logic itself. Pap’s arguments for this intuitive knowledge anticipate Etchemendy’s recent critique of the model-theoretic account of logical consequence. Pap’s work also anticipates prominent developments in the contemporary neo-Fregean philosophy of mathematics championed by Wright and Hale. Finally, Pap’s major philosophical preoccupation, the concepts of necessity and possibility, provides distinctive solutions and perspectives on issues of contemporary concern in the metaphysics of modality. In particular, Pap’s account of modality allows us to see the significance of Kripke’s well-known arguments on necessity and apriority in a new light. (shrink)

In this contribution, I comment on Raffaella Campaner’s defense of explanatory pluralism in psychiatry (in this volume). In her paper, Campaner focuses primarily on explanatory pluralism in contrast to explanatory reductionism. Furthermore, she distinguishes between pluralists who consider pluralism to be a temporary state on the one hand and pluralists who consider it to be a persisting state on the other hand. I suggest that it would be helpful to distinguish more than those two versions of pluralism – different understandings (...) of explanatory pluralism both within philosophy of science and psychiatry – namely moderate/temporary pluralism, anything goes pluralism, isolationist pluralism, integrative pluralism and interactive pluralism. Next, I discuss the pros and cons of these different understandings of explanatory pluralism. Finally, I raise the question of how to implement or operationalize explanatory pluralism in scientific practice; how to structure the “genuine dialogue” or shape “the pluralistic attitude” Campaner is referring to. As tentative answers, I explore a question-based framework for explanatory pluralism as well as social-epistemological procedures for interaction among competing approaches and explanations. (shrink)

The collection Virtual Mathematics: the logic of difference brings together a range of new philosophical engagements with mathematics, using the work of French philosopher Gilles Deleuze as its focus. Deleuze’s engagements with mathematics rely upon the construction of alternative lineages in the history of mathematics in order to reconfigure particular philosophical problems and to develop new concepts. These alternative conceptual histories also challenge some of the self-imposed limits of the discipline of mathematics, and suggest the possibility of forging new connections (...) between philosophy and more recent developments in mathematics. This component of Deleuze’s work has, to date, been rather neglected by commentators working in the field of Deleuze studies. One of the aims of this collection is to address this critical deficit by providing a philosophical presentation of Deleuze’s relation to mathematics; one that is adequate to his project of constructing a philosophy of difference and to the exploration of some of its applications. This project developed as a result of encounters with an increasing number of researchers who have been working on the mathematical aspects of Deleuze’s work and the key role that these play in his philosophy. By bringing this work together for the first time, this collection makes an important contribution to the emerging body of work that endeavours to explore the broad range of Deleuze’s philosophy. (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)