In this chapter we will argue, firstly, that Bacon’s engages in a pecu-liar form of mathematization of nature that develops a quali-quantitative methodology of measurement. Secondly, we will show that medicine is one of the disciplines where that dual way of measurement is practiced. In the first section of the chapter, we will expose the ontology involved in the Baconian proposal of measurement of nature. The second section will address the place that mixed mathematics occupies in Bacon’s scheme of scientific (...) branches and will suggest that a proper advancement of medicine can generate a new branch of mixed mathematics. The next section will reconstruct Bacon’s approach to measurement and expose its quali-quantitative import. In the last section, we will show some examples of medicine in which this quali-quantitative measurement is applied. (shrink)

ArgumentThe purpose of this paper is to provide an analysis of Giordano Bruno’s conception of mathematics. Specifically, it intends to highlight two aspects of this conception that have been neglected in previous studies. First, Bruno’s conception of mathematics changed over time and in parallel with another concept that was central to his thought: the concept of infinity. Specifically, Bruno undertook a reform of mathematics in order to accommodate the concept of the infinitely small or “minimum,” which was introduced at a (...) later stage. Second, contrary to what Héléne Védrine claimed, Bruno believed that mathematical objects were mind-dependent. To chart the parallel development of the conceptions of mathematics and infinity, a seven-year time span is considered, from the publication of Bruno’s first Italian dialogue (La cena de le ceneri,1584) to the publication of one of his last Latin works (De minimo, 1591). (shrink)

Ever since its foundation in 1540, the Society of Jesus had had one mission—to restore order where Luther, Calvin and the other instigators of the Reformation had brought chaos. To stop the hemorrhage of believers, the Jesuits needed to form a united front. No signs of internal disagreement could to be shown to the outside world, lest the congregation lose its credibility. But in 1570s two prominent Jesuits, Cristophorus Clavius and Benito Perera, had engaged in a bitter controversy. The issue (...) at stake had apparently nothing to do with the values on which Ignazio of Loyola had built the Society of Jesus. And yet the dispute between Clavius and Perera was matter of concern for the entire Jesuit community. They were... (shrink)

The eighteenth century was the century of self-organization, but also that of materialism, inasmuch as it was then that certain thinkers proclaimed themselves to be materialists (rather than just being labelled as such by enemies of various sorts). If one seeks to read these two features – one hesitates to call them ‘facts’ or ‘events’ – together, one arrives rather quickly at an influential metaphor, the beeswarm. But a metaphor of or for what? Irreducible organic unity, most broadly – spelled (...) out in the vocabulary of the period in terms of synergy, sympathy and sensibility, but also of cohesion, consensus and conspiration: individual bees have their characteristics, their intentions, and their own purposes, but they also ‘conspire’, ‘cohere’ and ‘consent’ in the name of a larger living unity, the swarm, although this leaves open further questions such as the exact nature of the order or organization yielded or enacted by the swarm (bottom-up? top-down? emergent? etc.). The fascination with the beeswarm is a fascination with organic unity and in some cases (notably Diderot), with the organic or organismic features of living matter itself. This is the story we wish to tell. (shrink)

Intellectual history still quite commonly distinguishes between the episode we know as the Scientific Revolution, and its successor era, the Enlightenment, in terms of the calculatory and quantifying zeal of the former—the age of mechanics—and the rather scientifically lackadaisical mood of the latter, more concerned with freedom, public space and aesthetics. It is possible to challenge this distinction in a variety of ways, but the approach I examine here, in which the focus on an emerging scientific field or cluster of (...) disciplines—the ‘life sciences’, particularly natural history, medicine, and physiology —is, not Romantically anti-scientific, but resolutely anti-mathematical. Diderot bluntly states, in his Thoughts on the interpretation of nature, that “We are on the verge of a great revolution in the sciences. Given the taste people seem to have for morals, belles-lettres, the history of nature and experimental physics, I dare say that before a hundred years, there will not be more than three great geometricians remaining in Europe. The science will stop short where the Bernoullis, the Eulers, the Maupertuis, the Clairauts, the Fontaines and the D’Alemberts will have left it.... We will not go beyond.” Similarly, Buffon in the first discourse of his Histoire naturelle speaks of the “over-reliance on mathematical sciences,” given that mathematical truths are merely “definitional” and “demonstrative,” and thereby “abstract, intellectual and arbitrary.” Earlier in the Thoughts, Diderot judges “the thing of the mathematician” to have “as little existence in nature as that of the gambler.” Significantly, this attitude—taken by great scientists who also translated Newton or wrote careful papers on probability theory, as well as by others such as Mandeville—participates in the effort to conceptualize what we might call a new ontology for the emerging life sciences, very different from both the ‘iatromechanism’ and the ‘animism’ of earlier generations, which either failed to account for specifically living, goal-directed features of organisms, or accounted for them in supernaturalistic terms by appealing to an ‘anima’ as explanatory principle. Anti-mathematicism here is then a key component of a naturalistic, open-ended project to give a successful reductionist model of explanation in ‘natural history’, a model which is no more vitalist than it is materialist—but which is fairly far removed from early modern mechanism. (shrink)

In this article I am going to reconstruct Stephen Toulmin’s procedural theory of concepts and explanations in order to develop two overlooked ideas from his philosophy of science: methods of representations and inferential techniques. I argue that these notions, when properly articulated, could be useful for shedding some light on how scientific reasoning is related to representational structures, concepts, and explanation within scientific practices. I will explore and illustrate these ideas by studying the development of the notion of instantaneous speed (...) during the passage from Galileo’s geometrical physics to analytical mechanics. At the end, I will argue that methods of representations could be considered as constitutive of scientific inference; and I will show how these notions could connect with other similar ideas from contemporary philosophy of science like those of models and model-based reasoning. (shrink)

The aim of this article is to shed light on an understudied aspect of Giordano Bruno's intellectual biography, namely, his career as a mathematical practitioner. Early interpreters, especially, have criticized Bruno's mathematics for being “outdated” or too “concrete”. However, thanks to developments in the study of early modern mathematics and the rediscovery of Bruno's first mathematical writings (four dialogues on Fabrizio's Mordente proportional compass), we are in a position to better understand Bruno's mathematics. In particular, this article aims to reopen (...) the question of whether Bruno anticipated the concept of infinitesimal quantity. It does so by providing an analysis of the dialogues on Mordente's compass and of the historical circumstances under which those dialogues were written. (shrink)

Intellectual history still quite commonly distinguishes between the episode we know as the Scientific Revolution, and its successor era, the Enlightenment, in terms of the calculatory and quantifying zeal of the former—the age of mechanics—and the rather scientifically lackadaisical mood of the latter, more concerned with freedom, public space and aesthetics. It is possible to challenge this distinction in a variety of ways, but the approach I examine here, in which the focus on an emerging scientific field or cluster of (...) disciplines—the ‘life sciences’, particularly natural history, medicine, and physiology —is, not Romantically anti-scientific, but resolutely anti-mathematical. Diderot bluntly states, in his Thoughts on the interpretation of nature, that “We are on the verge of a great revolution in the sciences. Given the taste people seem to have for morals, belles-lettres, the history of nature and experimental physics, I dare say that before a hundred years, there will not be more than three great geometricians remaining in Europe. The science will stop short where the Bernoullis, the Eulers, the Maupertuis, the Clairauts, the Fontaines and the D’Alemberts will have left it.... We will not go beyond.” Similarly, Buffon in the first discourse of his Histoire naturelle speaks of the “over-reliance on mathematical sciences,” given that mathematical truths are merely “definitional” and “demonstrative,” and thereby “abstract, intellectual and arbitrary.” Earlier in the Thoughts, Diderot judges “the thing of the mathematician” to have “as little existence in nature as that of the gambler.” Significantly, this attitude—taken by great scientists who also translated Newton or wrote careful papers on probability theory, as well as by others such as Mandeville—participates in the effort to conceptualize what we might call a new ontology for the emerging life sciences, very different from both the ‘iatromechanism’ and the ‘animism’ of earlier generations, which either failed to account for specifically living, goal-directed features of organisms, or accounted for them in supernaturalistic terms by appealing to an ‘anima’ as explanatory principle. Anti-mathematicism here is then a key component of a naturalistic, open-ended project to give a successful reductionist model of explanation in ‘natural history’, a model which is no more vitalist than it is materialist—but which is fairly far removed from early modern mechanism. (shrink)

The Jesuit C.F. Milliet Dechales, author of one of the most famous early modern mathematical encyclopedias, Cursus seu mundus mathematicus, wrote a hundred-folio-page long treatise devoted to the “progress of mathematics,” which was published in the second, enlarged edition of his encyclopedia. His historical treatise covers the gamut of mixed mathematics—including astronomy, mechanics, optics, music, geography and navigation, ars tignaria, and architecture. The early modern historical narratives about the mathematical sciences, from Regiomontanus’s Oratio onwards, have been aptly characterized by their (...) literary form and goals rather than their historical content. Rhetoric, humanistic topoi, and philosophical filiation turned the histories of mathematics into powerful tools for different purposes. My account of Dechales’ tract on the “progress of mathematics” analyzes the ways in which it dovetails with Jesuit approaches to mathematics, provides legitimation to the mathematical sciences as well as to their authors, and contributes to define the role and boundaries of the discipline, in particular vis-à-vis natural philosophy. (shrink)

This thesis studies the role of conceptual content in inference and reasoning. The first two chapters offer a theoretical and historical overview of the relation between inference and meaning in philosophy and psychology. In particular, a critical analysis of the formality thesis, i.e., the idea that rational inference is a rule-based and topic-neutral mechanism, is advanced. The origins of this idea in logic and its influence in philosophy and cognitive psychology are discussed. Chapter 3 consists of an analysis of the (...) relationship between inference and representation. It is argued that inference has to be studied from a pluralistic per- spective due to its dependence on different formats of representing information. The following four chapters apply conceptual spaces, a formal theory of concepts within cognitive semantics, to three concept-based inference-types. First, an explication of Sellars notion of material inference is advanced. Later, the model is extended to account for nonmonotonic inference by studying the role expectations in reasoning. Finally, a conceptual space-model of category-based induction is presented. This model predicts most of the empirical properties of this psychological phenomenon and subsumes some of the previous theories in psychology. It is stated that the explanatory fruitfulness of this new approach is evidence for the failure of the formality thesis and calls for a unified model of rational inference that puts semantics at center stage. The last chapter of the thesis discusses how inference and concepts interact in scientific reasoning, which makes constant use of hybrid symbolic structures for representing conceptual information. Stephen Toulmin’s notions of method of representation and inferential technique are developed and applied in a case study about the emergence of the notion of instantaneous speed during the passage from geometrical physics to analytical mechanics. It is claimed that this analysis provides support to the pluralistic perspective for theorizing about reasoning. (shrink)

Like many of their contemporaries Bernard Nieuwentijt and Pieter van Musschenbroek were baffled by the heterodox conclusions which Baruch Spinoza drew in the Ethics. As the full title of the Ethics—Ethica ordine geometrico demonstrata—indicates, these conclusions were purportedly demonstrated in a geometrical order, i.e. by means of pure mathematics. First, I highlight how Nieuwentijt tried to immunize Spinoza’s worrisome conclusions by insisting on the distinction between pure and mixed mathematics. Next, I argue that the anti-Spinozist underpinnings of Nieuwentijt’s distinction between (...) pure and mixed mathematics resurfaced in the work of van Musschenbroek. By insisting on the distinction between pure and mixed mathematics, Nieuwentijt and van Musschenbroek argued that Spinoza abused mathematics by making claims about things that exist in rerum natura by relying on a pure mathematical approach. In addition, by insisting that mixed mathematics should be painstakingly based on mathematical ideas that correspond to nature, van Musschenbroek argued that René Descartes’ natural-philosophical project abused mathematics by introducing hypotheses, i.e. ideas, that do not correspond to nature. (shrink)

Reading Galileo: Scribal Technologies and The Two New Sciences, by Renée Raphael, Baltimore, Johns Hopkins University Press, 2017, ix + 265 pp., figs., bibl., index, £40.50 (cloth), ISBN 9781421421...

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... (shrink)

In On Local Motion in the Two New Sciences, Galileo distinguishes between ‘time’ and ‘quanto time’ to justify why a variation in speed has the same properties as an interval of time. In this essay, I trace the occurrences of the word quanto to define its role and specific meaning. The analysis shows that quanto is essential to Galileo’s mathematical study of infinitesimal quantities and that it is technically defined. In the light of this interpretation of the word quanto, Evangelista (...) Torricelli’s theory of indivisibles can be regarded as a natural development of Galileo’s insights about infinitesimal magnitudes, transformed into a geometrical method for calculating the area of unlimited plane figures. (shrink)

The article deals with the problem concerning the possibility of qualitative physics paradigm development and its close connection with metaphysics. The idea of qualitative physics is based on the principles of Aristotelian physics and is opposed to quantitative modern physics. It is stated that the essential difference between the two physical paradigms lies in the ways of describing physical objects. Qualitative physics presuppose the qualitative description of physical objects independent of their quantitative description. In normal nowadays physics, on the contrary, (...) physical objects are regarded to be fully determined through quantitative relationships with other objects. The statements of modern physics are considered reasonable if they can be self-consistently expressed by the apparatus of mathematics. The article shows that this way of describing and explaining physical reality is incomplete. There is ground to assert that the quantitative relations of physical objects do not encompass everything that exists in the relations of physical objects. It is argued that there are qualitative aspects of physical reality that are not defined quantitatively and may become the content of special qualitative physics. The conclusion is made that such qualitative physics in its principles and language must be close to traditional metaphysics and can appear to be an application of metaphysics to the field of physical reality. (shrink)