The foundational case of the parabolic motion: design of an interdisciplinary activity for the IDENTITIES project

Questa tesi si inserisce nel campo di ricerca in Didattica della Fisica. In particolare, il lavoro si colloca all’interno del progetto Erasmus+ IDENTITIES, avviato nel settembre 2019, in collaborzione con le università di Montpellier, Creta, Parma e Barcellona. IDENTITIES ha lo scopo di sviluppare moduli didattici interdisciplinari (fisica – matematica – informatica), rivolti ai futuri insegnanti. I moduli riguardano sia temi curricolari sia temi STEM come contesto in cui sviluppare competenze interdisciplinari e progettare nuovi modelli di co-teaching. I temi di IDENTITIES hanno ispirato e guidato l’attuazione di un corso rivolto a insegnanti di scuola secondaria di secondo grado, organizzato dal PLS di Fisica di Bologna assieme al PLS di Matematica e il POT di Bologna. Il corso, svoltosi tra novembre e dicembre 2019 ha rappresentato la principale fonte di materiale e di riflessioni per questo lavoro. L’obiettivo di questa tesi è contribuire al progetto tramite la creazione di un’attività didattica, rivolta ai futuri insegnanti, sul tema della parabola e del moto parabolico. L’attività è stata progettata con lo scopo di guidare attraverso i principali passaggi che hanno caratterizzato, da un punto di vista epistemologico, l’evoluzione del pensiero fisico dalla teoria sul moto del proiettile di Tartaglia fino alla dimostrazione della traiettoria parabolica del proiettile di Galileo. Nella tesi sono descritti il quadro teorico di base per il lavoro, la rielaborazione del materiale del corso PLS per costruire lenti per l’analisi dei libri di testo, l’analisi di un capitolo del libro di testo sulla cinematica bidimensionale e la conseguente progettazione dell’attività didattica. Nelle conclusioni sono discussi i principali risultati ottenuti, tra i quali la produzione delle griglie originali per l’analisi di libri di testo, l’individuazione della simmetria e dell’indipendenza dei moti come attivatori epistemologici e la produzione dell’attività

How do planets find their way? Laws of nature and the transformations of knowledge in the Scientific Revolution

Laws of nature are perceived as playing a central role in modern science. This thesis investigates the introduction of laws of nature into natural philosophy in the seventeenth century, from which modern science arguably evolved. Previous work has indicated that René Descartes was responsible for single-handedly introducing a mathematical concept of laws into physics under the form of ‘laws of nature’. However, there is less agreement on the originality, causes and aftermath of this manoeuvre. This thesis is sensitive to the circumstance that the introduction of ‘laws of nature’ in the seventeenth century is a problem for us given our hindsight perspective of the origins of modern science, not an explicit concern of the actors; ‘laws of nature’ emerged as part of a network of problems and possibilities converging in Descartes’ reform of natural philosophy. Then, the appropriation of his laws was not an assessment of isolated statements on nature, but a process bounded by critical stances towards the Cartesian enterprise involving theological and social underpinnings. Accordingly, this thesis approaches ‘laws of nature’ as by-products of the changing boundaries between mechanics, mathematics and natural philosophy in the seventeenth century and interprets them as embedded within the circumstances and interactions among the practitioners of these disciplines in which these laws were introduced, criticised and appropriated. Based on this approach, this thesis tracks the background of Descartes’s project of reform of physics from the sixteenth-century fascination for machines that led to codifications of mechanics as a mixed-mathematical science, generating quantitative ways to design and fabricate physical (artificial) objects (Chapter 1). This approach was picked up by Galileo, who transformed it to include natural motion. In so doing, Galileo developed a mathematical approach to natural philosophy—a mathematical science of motion—which ultimately relied on the physical assumption of the motion of the Earth (Chapter 2). An alternative reorganization of mathematics and natural philosophy was put forward by the Lutheran theologian Kepler, wh o considered that the natural knowledge of the world may be founded a priori by deciphering the archetypes that God followed when creating the world. His archetypal cosmology provided a link between geometry and natural philosophy, involving mechanics (Chapter 3). However, Descartes moved in a different direction. Instead of connecting mathematics to natural philosophy, he tried to anchor both mathematics and natural philosophy on certainty, claiming that matter is but extension and that a few principles codified all possible interactions among parts of this geometrical matter. These principles were three ‘laws of nature’ erected as foundations of an a priori physics (Chapter 4). These ‘laws of nature’ received considerable attention in England. Informed by local traditions, English writers rejected the causal role attributed to laws but reworked their contents in laws of motion that were moved to mechanics and extended to astronomy, in line with the local practices of the ‘elliptical astronomy’ (Chapter 5). The relocation of ‘laws of nature’ from physics to mechanics was connected with English debates concerning the role of motion in geometry. These discussions drew different consequences for the connections between mathematics and nature (Chapter 6). In line with the English appropriation of Descartes, the young Newton assumed laws of motion as mathematical explanations in mechanics. When asked by Halley about orbital motion, his answer displayed characteristics of the English disciplinary setting. However, in connection with his historical studies, Newton realised that his laws of motion were capable of accounting for the true system of the world and then they were transformed into mathematical principles of natural philosophy, redrawing the contours of mathematics, natural philosophy and mechanics. The most important outcome of this reorganization—the law of gravitation—raised suspicions for going beyond the boundaries of established practices in the Continent (Chapter 7). The thesis concludes that ‘laws of nature’ did not emerge as a generic label to denominate findings in science. On the contrary, they appeared as concrete achievements with an operative function within Descartes’ reform of natural philosophy and consequently embedded within a network of assumptions, traditions and practices that were central to the appropriation of ‘laws of nature’. English natural philosophers and mathematicians reworked these ‘laws of nature’ within different disciplinary settings and put forward alternative ‘laws of motion’ in ways not previously noticed. The picture that emerges is not that of an amalgamation of previous meanings into a more complex one that was subsequently disseminated. Instead of a unified concept of ‘laws of nature’, Descartes’ project triggered reactions framed within local traditions and therefore it is hard to claim that at the end of the seventeenth century there was any agreement on the meaning of ‘laws of nature’ or even laws of motion beyond the narrow circles that shared disciplinary commitments and values. It was during the appropriation of Newton in the eighteenth century that his achievements and those honoured as his peers were labelled with a non- Newtonian concept of ‘laws of nature’, creating a foundational myth of the origins of modern science that reached up to the twentieth century

Proclus on the elements and the celestial bodies: physical thought in late Neoplatonism

Until recently, the period of Late Antiquity had been largely regarded as a sterile age of irrationality and of decline in science. This pioneering work, supported by first-hand study of primary sources, argues that this opinion is profoundly mistaken. It focuses in particular on Proclus, the head of the Platonic School at Athens in the 5th c. AD, and the chief spokesman for the ideas of the dominant school of thought of that time, Neoplatonism. Part I, divided into two Sections, is an introductory guide to Proclus' philosophical and cosmological system, its general principles and its graded ordering of the states of existence. Part II concentrates on his physical theories on the Elements and the celestial bodies, in Sections A and B respectively, with chapters (or sub-sections) on topics including the structure, properties and motion of the Elements; light; space and matter; the composition and motion of the celestial bodies; and the order of planets. The picture that emerges from the study is that much of the Aristotelian physics, so prevalent in Classical Antiquity, was rejected. The concepts which were developed instead included the geometrization of matter, the four-Element composition of the universe, that of self-generated, free motion in space for the heavenly bodies, and that of immanent force or power. Furthermore, the desire to provide for a systematic unity in explanation, in science and philosophy, capable of comprehending the diversity of entities and phenomena, yielded the Neoplatonic notion that things are essentially modes or states of existence, which can be arranged in terms of a causal gradation and described accordingly. Proclus, above anyone else, applied it as a scientific method systematically. Consequently, that Proclus' physical thought is embedded in his Neoplatonic philosophy is not viewed as something regrettable, but as proof of his consistent adherence to the belief, that there must be unity in explanation, just as there is one in the universe, since only the existence of such unity renders the cosmos rational and makes certainty in science attainable

Natural knowledge and Aristotelianism at early modern protestant universities

This volume aims to shed new light on the ways in which science was institutionalized and the central role played by university culture at reformed universities in the early modern period. It particularly explores the relationship between the Aristotelian legacy in Protestant centers of learning and the new natural knowledge which emerged from the mid-sixteenth to the mid-seventeenth century. Within the university context, Aristotelianism proved to be a dynamic tradition which we would term a ‘mobile episteme’ in line with the research program of the Collaborative Research Centre Episteme in Motion and the ERC endeavor EarlyModernCosmology (Horizon 2020, GA 725883). The transformation of academic science depended on its circulation in institutional and intellectual networks. The transfer and exchange of knowledge always implied its reformulation and often its deep alteration as well, even in those cases in which the explicit intention of the historical actors was to preserve and secure a received canon of knowledge, such as the corpus Aristotelicum or the Scholastic style of thought. As a matter of fact, the cross-pollination between ‘early’ forms of knowledge and ‘modern’ perspectives produced changes of content, theory, and experience. The fields that underwent major hybridizations and shifts range from astronomy to astrology, medicine, theories of the soul, alchemy, physics, and biology. Because methodologies were revised throughout this process, later instantiations of method, including rhetoric, epistemology, and theories of argumentation must be reevaluated within the terms of this transformative episteme

Space, spatiality, and epistemology in Hooke, Boyle, Newton, and Milton

In this thesis I trace the relations between thinking about space and the spatiality of thought as it relates to epistemology in the eponymous authors. I argue that the verbal,visual, and mental tools used to negotiate the ideas and objects under consideration are not merely representative or rhetorical, but are part of the process of knowledge-making itself. I contend that the spatialities of language, visual presentation, and mental image facilitate new ways of seeing and the exploring of previously invisible relationships. I show how the dynamic spatiality of the imagination is used for testing hypothesis, considering multiple points of view, accommodating uncertainties, and thinking about expansive ideas that push at (or exist beyond) the boundaries of the known or possible. In this way I offer new readings of key texts that foreground the inherent relativity of human experience, which I contend is at the heart of a scientific uncertainty found even in the new science that strove for objectivity. In four case studies I explore the elationship between external and internal space in the thinking and perceiving subject, building on Steven Connor’s assertion that ‘thinking about things is unavoidably a kind of thinking about the kind of thing that thinking is’ (‘Thinking Things’, 2010). In addition to this unidirectional relation between thinking and things, I demonstrate a complex dialogue between interior (thought) and exterior (thing) that occurs in the ways processes of thought and perception are externalized on the page and with instruments of viewing; in the way objects are brought into the mind; and in the way the mind creates infinities within by tracing expansive external spatialities

Mathematics and Its Applications, A Transcendental-Idealist Perspective

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

Enacting Inquiry Learning in Mathematics through History

International audienceWe explain how history of mathematics can function as a means for enacting inquiry learning activities in mathematics as a scientific subject. It will be discussed how students develop informed conception about i) the epistemology of mathematics, ii) of how mathematicians produce mathematical knowledge, and iii) what kind of questions that drive mathematical research. We give examples from the mathematics education at Roskilde University and we show how (teacher) students from this program are themselves capable of using history to establish inquiry learning environments in mathematics in high school. The realization is argued for in the context of an explicit-reflective framework in the sense of Abd-El-Khalick (2013) and his work in science education