Sophie Roux

L'histoire des sciences suffit à réfuter la thèse de la mathématisation impossible, selon laquelle la mathématisation procéderait d'un formalisme abstrait manquant les choses mêmes ou la spécificité d'un domaine d'objets. Cette histoire montre en effet qu'on n'a pas cessé de mathématiser des choses dont il avait été longtemps dit qu'elles devaient, étant donné leur nature, éternellement résister à la mathématisation. À la thèse de la mathématisation impossible, il est dès lors tentant d'opposer la thèse de la mathématisation inéluctable, selon laquelle une science bien formulée est une science mathématiquement constituée. Cette dernière thèse est cependant souvent associée à l'idée que l'introduction des mathématiques dans une discipline constituerait un processus qui ne présenterait aucune difficulté et qui ne serait suivie d'aucune perte. Contre cette idée, les articles rassemblés dans ce volume ont voulu, comme le titre de ce dernier l'indique, restituer à la mathématisation son caractère problématique. La méthode adoptée dans ce volume est résolument empirique : il s'appuie non pas sur des réflexions a priori sur ce que seraient les objets mathématiques, mais sur des études de cas empruntées à différentes périodes de l'histoire et à différentes sciences. Le pari a été fait que ces différentes rubriques pourraient être articulées les uns aux autres dans trois parties qui ont respectivement pour titres "Les exigences de la mesure : la quantification", "La formalisation : la puissance des formes symboliques", et "La modélisation, une nouvelle forme de mathématisation?" Assurément, une tendance historiographique générale de ces trente dernières années a consisté à privilégier les études de cas locales et singulières, ce qui nous a permis de prendre congé du style épique, qui néglige les détails de l'histoire. Cependant, l'espèce de nominalisme qui en résulte tourne rapidement court, si l'on en reste à ces études de cas singulières. Ce volume repose donc sur un pari : essayer d'articuler différentes études de cas de manière à dégager certaines formes de la mathématisation. Que ce pari ait été gagné ou perdu, ce sera au lecteur d'en juger.

The History of scepticism from Erasmus to Spinoza is often called upon to support three theses: first, that Descartes had a dogmatic notion of systematic knowledge, and therefore of physics; second, that the hypothetical epistemology of physics which spread during the xviith century was the result of a general sceptical crisis; third, that this epistemology was more successful in England than in France. I reject these three theses: I point first to the tension in Descartes’ works between the ideal of a completely certain science and a physics replete with hypotheses; further, I argue that the use of hypotheses by mechanical philosophers cannot be separated from their conception of physics; finally I show that, at the end of the xviith century, physicists in France as well as in England spoke through hypotheses and I examine different ways of explaining this shared practice. Richard H. Popkin’s book serves therefore as a starting point for insights into the general problem: to what extent and for what reasons some propositions in physics have been presented as hypotheses in the xviith century?

Four propositions relative to the laws of nature in the classical period must be noted. 1. Certain regularities in phenomena had been discovered. 2. A concept of law had emerged. 3. Classical science is characterized by the introduction of the notion of the legality of nature. 4. New uses of the word «law» had appeared in scientific texts. This article is devoted to the analysis of only this last proposition, that is to say to a terminological problem. First we will describe the semantic uses of the word «law» that may have contributed to the constitution of its classical meaning: its specific usage, disciplinary usage, usage in physics, in mathematics, in metaphysics. Second we will analyze the various moments of the diffusion of the word in the sciences in the xviith century. The general thesis defended in the end is that if the term «law» had indeed been used in science with no reference to a law-giving God prior to the xviith century, only this reference, that is, the coming together of its physical and its metaphysical usage, allowed its generalization in the period between the xviith and xviiith centuries.

I begin by briefly recalling two facts of seventeenth century intellectual history: not only is a fourth part devoted to method added to the three parts traditionally contained in logic treatises, but in a number of texts the terms "logic" and "method" are blurred. I then give an explanation of these two facts with the following ideas: 1/ Since the criticism of Aristotelian sciences at the beginning of the seventeenth century was in particular focused on logic, the question was asked as to what could replace or supplement it. 2/ Although method was very generally supposed to fulfil this function, its objectives were not clear. I propose a distinction between four ideas of method, which I call method-disposition, method-demonstration, method-invention and method-purification. 3/ The blurring between logic and method comes partly from the fact that some of these ideas take up elements of the then existant logic, but, in a context marked by the reception of Descartes's work, we must also take into account the fact that, since Descartes had not composed a logic and since he had not really explain what his method was, his successors were looking for what both might have been.

Roger Ariew's new book, Descartes and the First Cartesians, will not be a methodological surprise for those who already read his previous work, Descartes and the Last Scholastics, as well as its expanded version, Descartes Among the Scholastics. Right at the beginning of DAS, Ariew justified the title of this book in the following way: A philosophical system cannot be studied adequately apart from the intellectual context in which it is situated. Philosophers do not usually utter propositions in a vacuum, but accept, modify or reject doctrines whose meaning and significance are given in a particular culture. Thus Cartesian philosophy should...

During the seventeenth century there were different ways of opposing the new mechanical philosophy and the old Aristotelian philosophy. Remarkably enough, one of this way succeeded in becoming stable beyond the moment of its formulation, one according to which Descartes would be the benchmark by which the works of other natural philosophers of the seventeenth century fall either on the side of the old or the new. I consequently examine the French debate where this representation emerges, a debate that took place along with the development of a Cartesian propaganda in the 1660s and the ensuing official condemnations of the philosophy of Descartes, which was said to constitute a danger for the mystery of Eucharist. But these condemnations pronounced in the name of theology, as numerous and radical as they were, were not considered to be sufficient. They were assisted by numerous polemical works, the audience of which were learned companies of courteous honnêtes gens, and the object of which was to defend a certain way of proceeding in natural philosophy. I consequently concentrate on two correlated questions, the question of what kind of ontological entities are necessary for the establishment of a good physics, and the correlated question of what norms should be adopted in natural philosophy. I show quite systematically that the criticisms of Cartesian philosophers by the Oratorian Jean-Baptiste de La Grange, the bishop Pierre-Daniel Huet, and various Jesuits, Ignace Pardies, Antoine Rochon, Louis Le Valois, Gabriel Daniel, René Rapin, and Honoré Fabri respond to the mockeries of Gérauld de Cordemoy, Jacques Rohault, Louis de La Forge, Bernard Lamy, Nicolas Malebranche or Antoine Arnauld concerning the scholastic entities. Not only do I contrast their philosophical arguments concerning ontological entities and the norms to be respected in physics, but also their ways of defining the philosophical enterprise and its public.

The ontological status of forces and their causal role in Descartes’ physical world is debated among Descartes scholars. The question of forces is embedded in another more general question, namely to determine which causal activity should be attributed to God, and which causal activity should be attributed to physical bodies. Three distinct positions were attributed to Descartes: 1. he was an occasionalist and he attributed no causal power to forces, 2. he was a pure conservationist and he conceived forces as entities distinct from matter and motion, 3. he was a concurrentist who attributed causal activity both to God and to second causes, especially to forces and laws. These three interpretations seem to exhaust the possibilities. In this chapter however, I defend another interpretation of Descartes’ position, according to which God intervenes in this world only to conserve it by his ordinary concurrence (in this I agree with the conservationist interpretation), without, for all that, forces or laws being specific entities (in this I agree with the occasionalist interpretation). My interpretation leads to downplaying the strong relationship which is assumed to exist between Descartes’ physics and his metaphysics.

The mechanical (or corpuscular philosophy) has been well-established as a historiographical category for some years now. While it certainly began as an actor’s category, it has slipped into being something else, a kind of broad catch-all category that is taken to include most of those who opposed the Aristotelian philosophy of the schools throughout the entire seventeenth century, part of a broad master narrative about the demise of the scholastic Aristotelian philosophy of the schools and the rise of modern mathematical and experimental science, the titanic intellectual clash that gave birth to modernity.

The mechanical philosophy that emerged during the Scientific Revolution can be characterised as a reductionism according to which all physical phenomena are to be explained in terms of corpuscles of different sizes, shapes, and motions. It provided early modern natural philosophers with a unified view of nature that contrasted primarily with the Aristotelian view of nature, but also with other naturalist, hermetic, mystic, occultist, Paracelsian, and chymical accounts. Indeed, early modern natural philosophers devised mechanical explanations of almost every kind of phenomena; even supranatural phenomena, such as transubstantiation, or phenomena today described as paranormal, like the action of sympathetic powder on a wound, the influence of the stars on terrestrial events, or the identification of thieves with a divining rod, were given mechanical explanations. Confronting this explanatory deluge, historians adopted two opposing narratives. The first narrative amounted to view the mechanical philosophy as a well-identified category that captured something about early modern natural philosophy. The critical narratives of the last thirty years, however, have deconstructed both the belief that it is an adequate historical category and the conviction that it made a positive contribution to the sciences. First, in an era that privileged microscopic case studies over sweeping panoramas, and emic over etic categories, ‘mechanical philosophy’ came to be thought of as lacking any real descriptive significance. Second, from an evaluative point of view, it has been claimed that the mechanical philosophy had no impact on the development of the new sciences. Considering these descriptive and evaluative challenges, taking the mechanical philosophy at face value is historically inadequate and philosophically naïve. Hence the questions addressed in this chapter: What to do with the mechanical philosophy? Does it have to be dropped completely as a category? And, if not, how is it to be used? My view is that it captures something of the transformations that we put under the umbrella of the Scientific Revolution, but that it is not possible to treat it as a naturally given category. Rather, to be employed usefully, it needs conceptual clarification and historical specification.

In November 1664, a comet appeared in the European skies; by early March 1665, it had disappeared, but, at this very moment, another comet appeared, which stayed among the stars until mid-April. Observations of these two comets were made all over Europe, and even beyond. Although most secondary literature dedicated to these two comets has been focused on England and Italy, France was not to be outdone in terms of observations, small talk and publications. In this paper, I would like to use the books that were published on these comets in France as a dispersive prism to understand the ongoing controversies about which principles should be accepted in natural philosophy in the mid-sixties. There were indeed several intellectual and institutional circumstances that gave a specific twist to the French debate: the sixties marked the beginning of the war between Jesuits and Cartesians that was to enflame learned France until the late nineties at least, they were also the period of gestation for the future Académie des sciences, and, finally, it is also during these years that French elites began to discredit systematically the practice of astrology as a popular superstition. My working hypothesis in this paper is that, in such circumstances, the two comets of 1664–1665 were an occasion for each camp to advocate publicly its positions about the proper principles for natural philosophy. By natural philosophical principles, I mean both ontological and epistemological principles.

Leaving aside here the question of the author of the Essai de logique, I show that, if Mariotte insisted on the specificity of physics, he also sought a certain inspiration in mathematics as to the way in which to lay out the propositions in a proof. To do so, I start off from the ontological distinction made in the Essai among three types of possibles; next we will show that the three types of propositions correspond to three types of knowledge, and, correlatively, that the main problem of physics is that it is impossible to establish in a certain fashion universal sensory propositions; I next characterize the notion of principles of experience; finally, I examine a bit systematically the relationship between mathematics and physics.

Early modern natural philosophers put forward the ontological program that was called "mechanical philosophy" and they gave mechanical explanations for all kinds of phenomena, such as gravity, magnetism, the colors of the rainbow, the circulation of the blood, the motion of the heart and the development of animals. For a generation of historians, the mechanical philosophy was regarded as the main alternative to Aristotelian orthodoxy during the so-called Scientific Revolution and mechanical explanations were presented as paving the way for the use of experiments and mathematics in the understanding of natural phenomena. However, the historical category of mechanical philosophy was later criticized as being too broad, while early modern mechanical explanations were condemned by more epistemologically oriented minds for being incompatible with, or at least not necessarily connected to, the use of experiments and mathematics. In the last ten years, just as the new mechanistic literature emerged in philosophy of science, there has been a reevaluation of early modern mechanical explanations in a domain that had been until then considered as peripheral to the so-called Scientific Revolution, namely the domain of biology, anatomy, physiology and medicine. Although they were neither confirmed nor predictive, some early modern explanations in these domains appear to have a cognitive value similar to the value of contemporary mechanistic explanations.

This volume explores the relationship between physics and metaphysics in Descartes’ philosophy. According to the standard account, Descartes modified the objects of metaphysics and physics and inverted the order in which these two disciplines were traditionally studied. This book challenges the standard account in which Descartes prioritizes metaphysics over physics. It does so by taking into consideration the historical reception of Descartes and the ways in which Descartes himself reacted to these receptions in his own lifetime. The book stresses the diversity of these receptions by taking into account not only Cartesianisms but also anti-Cartesianisms, and by showing how they retroactively highlighted different aspects of Descartes’ works and theoretical choices. The historical aspect of the volume is unique in that it not only analyzes different constructions of Descartes that emerged in the 18th, 19th and 20th centuries, but also reflects on how his work was first read by philosophers across Europe. Taken together, the essays in this volume offer a fresh and up-to-date contribution to this important debate in early modern philosophy.

Résumé Cet article esquisse une généalogie du privilège que le terme concept a acquis en français par rapport à notion et à idée en se fondant non seulement sur les ouvrages des philosophes, mais sur des dictionnaires de langue philosophique. Il comprend quatre parties chronologiques. Après avoir étudié l’introduction des termes concept, notion, idée dans la langue philosophique, la première partie répertorie leurs usages dans les dictionnaires scolastiques du SVIIe siècle. La deuxième montre que Descartes a imposé idée en donnant de ce terme une acception large et que les dictionnaires post-cartésiens la reprenne. La troisième établit que Kant a, contre cette acception large, introduit une distinction entre Begriff et Idee, mais que, initialement, les dictionnaires français refusèrent cette distinction. La quatrième et dernière partie est consacrée à un XXe siècle marqué par l’idéalisme allemand et l’analyse logique : aussi différentes qu’aient été ces deux traditions, elles ont été d’accord pour consacrer la prééminence du terme concept.

This volume explores the relationship between physics and metaphysics in Descartes' philosophy. According to the standard account, Descartes modified the objects of metaphysics and physics and inverted the order in which these two disciplines were traditionally studied. This book challenges the standard account in which Descartes prioritizes metaphysics over physics. It does so by taking into consideration the historical reception of Descartes and the ways in which Descartes himself reacted to these receptions in his own lifetime. The book stresses the diversity of these receptions by taking into account not only Cartesianisms but also anti-Cartesianisms, and by showing how they retroactively highlighted different aspects of Descartes' works and theoretical choices. The historical aspect of the volume is unique in that it not only analyzes different constructions of Descartes that emerged in the 18th, 19th and 20th centuries, but also reflects on how his work was first read by philosophers across Europe. Taken together, the essays in this volume offer a fresh and up-to-date contribution to this important debate in early modern philosophy.

Mes réflexions sur Meyerson et les mathématiques ont pour origine trois questions : 1) Une idée reçue est que, des trois synthèses de Meyerson -- Identité et réalité, De l'explication dans les sciences et Du cheminement de la pensée -- , seule la dernière analyse les mathématiques, en elles-mêmes aussi bien que dans leurs rapports avec la pensée. La première question est donc de déterminer si cette idée reçue est correcte ou bien si l'on peut trouver dans les deux autres ouvrages des indications significatives sur le statut des mathématiques et sur leur fonction dans une théorie de la connaissance. 2) La deuxième question consiste à examiner une autre idée reçue, cautionnée cette fois par Meyerson. À plusieurs reprises, ce dernier a déclaré qu'il avait répété la même chose dans tous ses ouvrages : les outils conceptuels seraient restés identiques, mais appliqués à des objets ou dans des domaines divers, de la physique quantique à la pensée primitive. Il s'agit dans ce cas de savoir si cette déclaration est fondée, ou bien plutôt si l'on doit, par-delà l'identité des termes, souligner l'existence de différences de signification ou d'application telles que l'identité des outils conceptuels en devient problématique. 3) Meyerson s'est trouvé confronté à un ensemble d'épistémologues que, reprenant une expression de Léon Brunschvicg, il qualifie d'idéalistes mathématiques, d'un côté l'école de Marbourg avec Ernst Cassirer, Paul Natorp et Hermann Cohen, de l'autre, Gaston Milhaud, mais surtout Brunschvicg lui-même . S'il s'avère qu'il y a des thèses sur les mathématiques ailleurs que dans Du cheminement de la pensée, et qu'il y a certains infléchissements d'un lieu à l'autre de son œuvre, on peut se demander dans quelle mesure les critiques adressées à l'idéalisme mathématique ont évolué. Dans cet article, j'analyse les trois synthèses de Meyerson avec ces trois questions en arrière-plan : je ne les examine pas explicitement l'une après l'autre, ni a fortiori dans un ordre prédéterminé, mais plutôt de biais et au fur et à mesure qu'elles interviennent

According to a grand narrative that long ago ceased to be told, there was a seventeenth century Scientific Revolution, during which a few heroes conquered nature thanks to mathematics. This grand narrative began with the exhibition of quantitative laws that these heroes, Galileo and Newton for example, had disclosed: the law of falling bodies, according to which the speed of a falling body is proportional to the square of the time that has elapsed since the beginning of its fall; the law of gravitation, according to which two bodies are attracted to one another in proportion to the sum of their masses and in inverse proportion to the square of the distance separating them -- according to his own preferences, each narrator added one or two quantitative laws of this kind. The essential feature was not so much the examples that were chosen, but, rather, the more or less explicit theses that accompanied them. First, mathematization would be taken as the criterion for distinguishing between a qualitative Aristotelian philosophy and the new quantitative physics. Secondly, mathematization was founded on the metaphysical conviction that the world was created pondere, numero et mensura, or that the ultimate components of natural things are triangles, circles, and other geometrical objects. This metaphysical conviction had two immediate consequences: that all the phenomena of nature can be in principle submitted to mathematics and that mathematical language is transparent; it is the language of nature itself and has simply to be picked up at the surface of phenomena. Finally, it goes without saying that, from a social point of view, the evolution of the sciences was apprehended through what has been aptly called the "relay runner model," according to which science progresses as a result of individual discoveries. Grand narratives such as this are perhaps simply fictions doomed to ruin as soon as they are clearly expressed. In any case, the very assumption on which this grand narrative relies can be brought into question: even in the canonical domain of mechanics, the relevant epistemological units crucial to understanding the dynamics of the Scientific Revolution are perhaps not a few laws of motion, but a complex set of problems embodied in mundane objects. Moreover, each of the theses just mentioned was actually challenged during the long period of historiographical reappraisal, out of which we have probably not yet stepped. Against the sharp distinction between a qualitative Aristotelian philosophy and the new quantitative physics, numerous studies insist that Rome wasn't built in a day, so to speak. Since Antiquity, there have always been mixed sciences; the emergence of pre-classical mechanics depends on both medieval treatises and the practical challenges met by Renaissance engineers. It is indeed true that, for Aristotle, mathematics merely captures the superficial properties of things, but the Aristotelianisms were many during the Renaissance and the Early Modern period, with some of them being compatible with the introduction of mathematics in natural philosophy. In addition, the gap between the alleged program of mathematizing nature and its effective realization was underlined as most natural phenomena actually escaped mathematization; at best they were enrolled in what Thomas Kuhn began to rehabilitate under the appellation of the "Baconian sciences," i.e., empirical investigations aiming at establishing isolated facts, without relating them to any overarching theory. Hence, mathematization of nature cannot pretend to capture a historical fact: at most, it expresses an indeterminate task for generations to come. On top of these first two considerations, and against the thesis of the neutrality of the mathematical language, it was urged that mathematics is not "only a language" and that, exactly as other symbolic means or cognitive tools, it has its own constraints. For example, it has been thoroughly explained that the Euclidean theory of proportions both guides and frustrates the Galilean analysis of motion; its shortages were particularly clear with respect to the expression of continuity, which is crucial in the case of motion. Consequently, when calculus was invented and applied to the analysis of motion, it was not a transposition that left things as they stood. Even more clearly than in the case of a translation from one natural language to another, the shift from one symbolic language to another entails that certain possibilities are opened while others are closed. The cognitive constraints imposed by established mathematical theories, as seen in the theory of proportions or calculus, were not the only ones to be studied in relation to mathematization. Certain schemes dependent on the grammar of natural languages, e.g., the scheme of contrariety, or certain symbolic means of representation, e.g. geometrical diagrams and numerical tables, were also subject to such scrutiny. Lastly, it was insisted that, even if we concede the existence of scientific geniuses, mathematics is largely produced by intellectual communities and embedded within social practices. More attention was consequently paid to the forms of communication in given mathematical networks, or to the teaching of the discipline in, for example, Jesuit colleges and universities. The set of mathematical practices specific to specialized craftsmen, highly-qualified experts and engineers began to be studied in its own right. All these reflections may have helped us change our perspectives on the question of mathematization. It seems, however, that they were instead set aside, both because of a general distrust towards sweeping narratives that are always subject to the suspicion that they overlook the unyielding complexity of real history, and because of a shift in our interests. The more obscure and idiosyncratic they are, an alchemist, a patron of the sciences or a lunatic collector is nowadays honored in journals of the history of sciences. As for the general issues involved in the question of mathematization, they are rejected as obsolete, or reserved for specialized journals in the history of mathematics. Consequently, before presenting the essays of this fascicle, I would like to say a few words in favor of a renewed study of the forms of mathematization in the history of the early sciences.